Modern Classics

An Investigation of the Laws of Thought

The Mathematical Theory of Logic

by George Boole

1854 2026

This is an AI modernization of An Investigation of the Laws of Thought into contemporary English. The original is available from Project Gutenberg. Also available as an epub.

Editor's Introduction

George Boole published this book in 1854, when he was thirty-nine and Professor of Mathematics at Queen's College, Cork. He had not been to university. He was the son of a shoemaker in Lincoln, taught himself Latin, Greek and then mathematics, and was supporting his family as a schoolmaster from the age of sixteen. By his late twenties he was publishing original mathematics in the journals and had won a medal from the Royal Society, and in 1849 the new college in Cork made him its first professor of mathematics without his ever having taken a degree.

His idea was that reasoning has laws as definite as the laws of algebra, and that they can be written down as an algebra. Let a letter stand for a class of things: x for sheep, y for white things. Then xy is the class of things that are both, the white sheep; 1 stands for everything under discussion and 0 for nothing; 1 − x is everything that is not a sheep. "All x are y" becomes an equation, and reasoning becomes the solving of equations. Almost every rule of ordinary algebra carries over. The one that does not is the law Boole put at the centre of the book, x2 = x: a class combined with itself is the same class. Of all numbers, only 0 and 1 obey it, and so the algebra of logic turns out to be the algebra of 0 and 1. He had sketched the system in a pamphlet of 1847, The Mathematical Analysis of Logic; this is the full statement.

It was admired and little used in his lifetime. Logicians found the algebra forbidding, and mathematicians found parts of it strange, above all his habit of working through steps that have no logical meaning, such as dividing by zero, and trusting the answer at the end. Jevons, Venn and Peirce rebuilt it in the following decades into a simpler form, and in that form it became the foundation of modern logic. In 1937 Claude Shannon showed that it also describes electrical switching circuits, which is why every digital computer now calculates in Boole's algebra.

This edition retells the first fifteen chapters, the logic. Boole's title promises a second subject as well, probability, and Chapters XVI to XXII apply the method to it; they are not included here, and the places where the first half points ahead to them say so. His prose has been put into plain modern English. His symbols, equations and equation numbers are exactly his, except for a handful of printer's errors in the 1854 edition and the transcription it was set from, corrected against the original and listed in the source files. His numbered articles are kept, so that his own cross-references, such as "(II. 9)", meaning Chapter II, article 9, still work. Where an equation would stop a reader, a few words in the sentence say what it means. And seven short notes marked "Editor's note" say where modern Boolean algebra does things differently, or where an idea of his went next; those are ours, not his.

On the cover. Drawn for this edition, in the manner of the Dover science paperbacks of the 1950s and 60s. It is Boole's "development" from Chapter V: the universe, 1, cut by a class x into x and not-x, each part cut again by y and again by z, until eight constituents stand in the bottom row. A premise says which of them are empty, and those are struck out; reading off what survives is how every problem in the book is solved, and it is the ancestor of the truth table.

Dedication

To
	John Ryall, LL.D.
	Vice-President and Professor of Greek
	in Queen’s College, Cork,
	this work is inscribed
	in testimony of friendship and esteem

Preface

This book is not a reissue of an earlier treatise of mine, “The Mathematical Analysis of Logic.” Its first part does pursue the same object, and it begins by laying down the same system of fundamental laws. But its methods are more general, and it applies them far more widely. It gives the results of some years of study and reflection on a way of investigating the operations of the intellect. My earlier account of that idea was written within a few weeks of first conceiving it.

The part of this book that deals with logic assumes that the reader knows the most important terms of the subject as it is usually taught, and what the subject is for. On these points there is no better guide than Archbishop Whately’s “Elements of Logic” or Mr. Thomson’s “Outlines of the Laws of Thought.” The present revival of interest in studies of this kind seems largely due to the first of these. The reader also needs some algebra, but no more than how to solve simple equations. The chapters on the theory of probability (not included in this edition) call for rather more algebra, especially the method of elimination and the solution of equations with more than one unknown. For background on that subject, see the treatises on probability in “Lardner’s Cabinet Cyclopædia,” by Professor De Morgan, and in the “Library of Useful Knowledge,” by Sir John Lubbock, and an interesting series of letters translated from the French of M. Quetelet. Other references are given in the book itself. On a first reading the reader may skip, as he sees fit, Chapters X., XIV., and XIX. (the last of these is in the chapters on probability, not included in this edition), along with any of the applications he finds uninviting or beside his purpose.

In various places, and especially in the notes to the last chapter (which, like the chapters on probability, is not included in this edition), the reader will find references to writers ancient and modern. They are meant mainly to illustrate a certain view of the history of philosophy. On these I should add one thing. I have never given a citation that I did not believe, after careful examination, to be supported either by other authorities or by the general drift of the work it came from. I would gladly have avoided anything that might be taken for a parade of learning. But I felt I owed it both to my subject and to the truth to give the reader the means of checking the statements in the text.

I now bring to a close a labour whose extent few people will be able to judge from its visible fruits. If I may speak for a single moment of the feelings with which I have pursued this task, and with which I now lay it aside, I would say this. I never doubted that it was worthy of my best efforts. I felt that whatever truth it might bring to light was not a private or arbitrary thing, and did not depend, in its essence, on anyone’s opinion. I knew well that learned and able men held views on logic directly opposed to those on which the whole argument and method of this book rest. I believed their views to be mistaken, but I knew that my own might be bent, without my noticing, by an influence of another kind. I felt especially the danger of the bias that comes from long attention to one particular aspect of the truth. But I trust that out of this conflict of opinions the truth will only emerge the freer from anything merely personal. I trust that its different parts will be seen in their proper proportion, and that none of them will in the end be valued too highly or too lightly because of prejudices attached to the mere form in which it was presented.

To my valued friend the Rev. George Stephens Dickson, of Lincoln, I wish to record my thanks for much kind help in revising this work, and for some important suggestions.

5, Grenville-place, Cork,

Nov. 30th. 1853.

Chapter 1: Nature and Design of This Work

1. This book has four aims. The first is to investigate the fundamental laws of those operations of the mind by which we reason. The second is to express those laws in the symbolic language of a calculus (a system of symbols with fixed rules for working with them, as algebra is), and on that foundation to build the science of logic and its method. The third is to make that method the basis of a general method for applying the mathematical theory of probability (in the chapters on probability, not included in this edition). The last is to gather, from the pieces of truth these inquiries bring to light, some likely hints about the nature and make-up of the human mind.

2. That this plan is not entirely new hardly needs saying. Its two main practical branches, logic and probability, have drawn a great deal of attention from philosophers. In its ancient and medieval form, logic is tied almost entirely to one great name, Aristotle. It reached the ancient Greeks in the treatises of his Organon, which are partly technical and partly metaphysical, and it has come down to our own day with hardly any essential change. Original inquiry has flowed instead toward questions of general philosophy. These questions began in the quarrels of the logicians, but they outgrew their origin and gave each later age of speculation its own particular bent. Think of the eras of Porphyry and Proclus, of Anselm and Abelard, of Ramus, and of Descartes, and of the final protests of Bacon and Locke. Each shows the study of logic reaching out to shape the course of human thought, sometimes by raising topics rich in debate, sometimes by provoking protest against logic's own inflated claims.

The history of the theory of probability looks quite different. It has grown steadily, the way a science grows. At its birth it had the young genius of Pascal; in its maturity, the most difficult of all the mathematical researches of Laplace; and other names hardly less distinguished could be added. Logic has been remarkable for the related questions of metaphysics it gave rise to; probability has been remarkable for the push it gave to the higher branches of mathematics. Both subjects, moreover, have rightly been seen as serving a theoretical end as well as a practical one. Logic is not only for drawing correct conclusions from given premises. And probability does more than teach us how to put life insurance on a secure footing, or how to boil down whatever is valuable in countless observations in astronomy, in physics, or in that field of social inquiry which is fast becoming very important. Both studies matter in another way too: they shed light on the powers of the mind. They show how language and number serve as tools for reasoning. They reveal something of how the different powers of the intellect we all share are connected. They set out the essential standards of truth and correctness in the two realms of knowledge, the demonstrative (what can be proved) and the probable. And those standards are not imposed from outside; they are rooted deep in the way the human faculties are made. These theoretical goals are no less interesting, no less dignified, and, I would add, no less important than the practical aims that history has tied them to. Our higher powers of thought give us all our knowledge of the world and of ourselves that goes beyond what the senses simply perceive. To uncover their hidden laws and relations is a goal that no rational mind needs to be persuaded of.

3. Some parts of this plan have occurred to others before me. But its overall conception, its method, and to a large extent its results are, I believe, original. So in this chapter I will offer some preliminary statements and explanations, to make the real aim of the book clear and its subject easier to follow.

The first aim, then, is to investigate the fundamental laws of those operations of the mind by which reasoning is carried out. There is no need to argue here that the operations of the mind really do, in some real sense, obey laws, and therefore that a science of the mind is possible. If anyone doubts this, the way to meet the doubt is not to try to settle it à priori, by argument in advance of the evidence. The way is to point the doubter to the evidence of actual laws, which means pointing him to an actual science. So the answer to that doubt belongs not in the introduction to this book but in the book itself. Let us grant, then, that a science of the powers of the intellect is possible, and consider for a moment how we are to obtain it.

4. Like every other science, the science of the mind's operations must rest first of all on observation. Here, what we observe are the very operations and processes whose laws we want to find. Every science needs this foundation in experience. But there are some special differences in how experience yields general truths when the subject is the mind and when the subject is external nature. We need to look at these differences.

The general laws of nature are not, for the most part, things we directly perceive. Either they are inductive inferences from a large body of facts, stating the truth those facts have in common; or, at least to begin with, they are physical hypotheses about causes, which explain the phenomena with unfailing precision and let us predict new combinations of them. In every case, and in the strictest sense of the word, they are probable conclusions. They come ever closer to certainty as experience confirms them more and more. But they never entirely lose the character of probability, in the strict and proper sense of that term. Knowledge of the laws of the mind is different. It does not need a large collection of observations as its basis. We see the general truth in the particular instance, and repeating the instance does not confirm it any further. An obvious example will show this. There is a rule of reasoning called the dictum of Aristotle, de omni et nullo ("about all and about none": roughly, whatever is affirmed or denied of a whole class is affirmed or denied of everything in it). We may ask whether this rule expresses a primary law of human reasoning. But there is no question that it expresses a general truth of logic. And reflecting on a single instance of it is enough to show that truth in all its generality. This is evidence that the rule rests on some general law or laws of the mind. It also illustrates the point that we do not reach such general truths by induction from many instances: they are contained in seeing one instance clearly. Alongside this comes another truth that is just as important. Our knowledge of the laws on which the science of the intellect rests, however much or little of it we have, is not probable knowledge. For in the particular example we see not only the general truth but also that it is a certain truth, a truth in which our confidence will not keep growing as we see it confirmed again and again in practice.

5. But if the general truths of logic win our agreement the moment the mind sees them, what makes it hard to build a science of logic? Not, we may answer, gathering the material, but sorting it out: telling what kind of truth each one is, and settling its place and its relation to the rest. Every science consists of general truths, but only some of those truths are primary and fundamental; the rest are secondary and derived. Kepler's laws of elliptical motion are general truths of astronomy, but they are not its fundamental truths. The same holds in the purely mathematical sciences. An almost endless variety of known theorems, and an infinite number of others still unknown, all rest together on the foundation of a few simple axioms; and yet all of them are general truths. I would add that to a mind refined enough, these truths would shine out by their own light. It would not need the connecting links of thought, the steps of tiring and often painful deduction, by which we actually come to know them. Let us call fundamental those laws and principles from which all the other general truths of a science can be deduced, and into which they can all be broken down again. Are we wrong, then, to count this as the true science of logic: one that lays down certain elementary laws, confirmed by the mind's own testimony; that lets us deduce from them, by uniform processes, the whole chain of their secondary consequences; and that supplies, for practical use, methods of perfect generality? Consider whether any science, taken either as a system of truth or as the basis of a practical art, can properly have any other test of whether its laws are complete and fundamental. The test is how complete its system of derived truths is, and how general the methods it supports. Other questions may certainly come up. Convenience, custom and personal preference may press their claims, and they deserve attention. But on the question of what makes a science in its pure, abstract integrity, I believe nothing besides the test above really counts.

6. The second aim of this book is to express the fundamental laws of reasoning in the symbolic language of a calculus. Here it is enough to say that the laws themselves suggest this way of expressing them, and make it uniquely fit for our purposes. There is a close analogy between the operations of the mind in reasoning generally and its operations in the particular science of algebra. More than that: to a large extent the two kinds of operation follow exactly the same laws. Of course, the laws must be worked out separately in each case. Any agreement in form between them can only be established à posteriori, after the fact, by actually comparing them. To borrow the notation of the science of number and simply assume that the rules for using it will stay the same in a new use would be mere hypothesis. There are, it is true, certain general principles, grounded in the very nature of language, that govern how symbols are used; and symbols are just the elements of a scientific language. Up to a point these elements are arbitrary. What they mean is purely a matter of convention: we may use them in any sense we like. But that freedom has two essential limits. First, once we have fixed a symbol's meaning, we must never depart from it within the same piece of reasoning. Second, the laws by which we reason with the symbols must be based only on that fixed meaning. Under these principles, if the laws of the symbols of logic turn out to agree with those of algebra, the result can only be an agreement of processes. The two fields of meaning stay separate and independent, each with its own laws and conditions.

Now, the investigations in the following pages present logic, on its practical side, as a system of processes carried out with symbols that have a definite meaning and that obey laws based on that meaning alone. But at the same time they show those laws to be identical in form with the laws of the general symbols of algebra, with one single addition: the symbols of logic obey a further special law (Chap. II.), which the symbols of quantity, as such, do not obey. I will not dwell here on what this law is or on the evidence for it; those questions are fully discussed later. But this law is the essential ground of difference between the kinds of inference logic deals with and those that arise in the particular science of number, and so it deserves more than a passing mention. It may be said to lie at the very foundation of general reasoning. It governs those acts of conception or imagination that come before the processes of logical deduction, and it gives those processes much of their actual form and expression. So we may say that this law is the seed, the first principle, and that every approach to a general method in logic is a more or less complete growth from it.

7. We have already laid down the principle (5) that, in the science of logic, whether an assumed system of laws is sufficient and truly fundamental must be seen partly in how perfect the methods are that it leads to. It remains, then, to ask what a general method in logic requires, and how far the system of this book meets those requirements.

Logic deals with two kinds of relation: relations among things, and relations among facts. But facts are expressed by propositions, so the second kind can, at least for the purposes of logic, be turned into relations among propositions. Suppose we say that the fact or event A always follows the fact or event B. To this extent, at least, that is the same as saying that the truth of the proposition asserting that the event B happens always implies the truth of the proposition asserting that the event A happens. So instead of saying that logic deals with relations among things and relations among facts, we may say that it deals with relations among things and relations among propositions. An example of the first kind is the proposition “All men are mortal”; of the second, the proposition “If the sun is totally eclipsed, the stars will become visible.” The first expresses a relation between “men” and “mortal beings”; the second, a relation between the simple propositions “The sun is totally eclipsed” and “The stars will become visible.” I count among such relations those that affirm or deny that things exist, and those that affirm or deny that propositions are true. Now let us call the things or propositions among which a relation is expressed the elements of the propositions that express it. With this definition we can say that the premises of any logical argument express given relations among certain elements, and that the conclusion must express an implied relation among those elements, or among some of them: that is, a relation that the premises imply, or contain by way of inference.

8. With this settled, a general method in logic seems to need the following.

First, since the conclusion must express a relation among all or some of the elements in the premises, we need a way of eliminating the elements we do not want to appear in the conclusion, and of finding the full relation the premises imply among the elements we want to keep. In the language of ordinary logic, the elements that do not appear in the conclusion are called middle terms. The kind of elimination shown in logic textbooks takes two propositions that share an element, the middle term, and deduces a conclusion connecting the two terms that remain. But the problem of elimination as this book sees it is much wider. It aims not just to eliminate one middle term from two propositions, but to eliminate middle terms from propositions in general, however many terms and propositions there are and however they are connected. Neither logic nor algebra, as they now stand, offers any exact parallel to this. In algebra, as is well known, elimination (combining equations so that a symbol drops out) is limited in this way: from two equations we can eliminate one symbol of quantity; from three equations, two symbols; and in general, from n equations, n−1 symbols. This condition, necessary in algebra, seems to hold in the existing logic too, but it has no essential place in logic as a science. There, no relationship at all can be proved between the number of terms to be eliminated and the number of propositions they are to be eliminated from. From the equation representing a single proposition we can eliminate any number of the symbols that stand for terms or elements in logic; and from any number of equations representing propositions, we can eliminate one such symbol, or any other number of them, in the same way. One general process does this in every case. This is one of the many remarkable consequences of that distinctive law of the symbols of logic which I have already pointed out.

Second, a general method in logic should be able to express the final relation among the elements of the conclusion as any admissible kind of proposition, or with the terms in any chosen order. Among the kinds we may count those that logicians call categorical, hypothetical, disjunctive, and so on (plain statements, “if” statements, “either-or” statements). Under the choice of order we may put everything that depends on which elements appear in the subject or in the predicate, or in the antecedent or in the consequent (the “if” part or the “then” part), of the proposition that forms the “conclusion.” But setting aside the technical language of the old schools, let us ask what genuinely different kinds of problem may come up. We have seen that the elements of the final, inferred relation may be either things or propositions. Take the first case. We might be asked to deduce from the premises a definition or description of some one thing, or class of things, among the elements of the conclusion, in terms of the other things it involves. Or we might form the idea of some thing or class of things that involves more than one of the elements of the conclusion, and ask for its expression in terms of the other elements. Now take the second case, where the elements kept in the conclusion are propositions. Then we might want to settle questions like these. Given the premises, is any of these propositions, taken by itself, true or false? Are particular combinations of them true or false? If we assume a particular proposition to be true, does anything follow about the other propositions, and if so, what? If we assume some particular condition about certain of the propositions, does anything follow about the others, and what? And so on. These, I say, are general questions, and a general method in logic ought to be able to answer them. Perhaps we can gather them all into one statement of the final problem of practical logic. Given a set of premises expressing relations among certain elements, whether things or propositions: required, explicitly, the whole relation that follows among any of those elements, under any proposed conditions, and in any proposed form. That this problem can be solved in all its aspects will appear later. But I have not gone into the nature and functions of a general method in logic just to point that out. The reader needs to understand the specific goals of the investigation we are now beginning, and the principles that are to guide us toward them.

9. It might be said at this point that Aristotle's logic, in its rules of syllogism and conversion, already sets out the elementary processes of which all reasoning consists, and that beyond these there is neither room nor need for a general method. (A syllogism draws a conclusion from two premises that share a middle term; conversion swaps the subject and predicate of a proposition.) I have no wish to point out the defects of the common logic, and I will refer to it only as far as I must to show the real nature of this book in its true light. With that purpose alone, I would make two remarks. First, syllogism, conversion and the rest are not the ultimate processes of logic. This book will show that they rest on, and can be broken down into, deeper and simpler processes, and that these are the real elements of method in logic. Nor is it in fact true that all inference can be reduced to the particular forms of syllogism and conversion (see Chap. xv.). Second, even if all inference could be reduced to these two processes (and some have held that it can be reduced to syllogism alone), a general method would be just as necessary. We would still have to decide in what order the processes should follow one another, as well as what each one is, in order to reach the relation we want. By the relation we want, I mean the full relation that, given the premises, connects any elements we choose from them, and that expresses this relation in any form and order we choose. If we may judge by the mathematical sciences, which are the most perfect examples of method we know, this directive function, deciding what to do and in what order, is method's chief task and its mark of distinction. The fundamental processes of arithmetic, for example, are in themselves only the elements of a possible science. Saying what they are is the first business of its method; arranging their order is its later and higher task. In the more complex examples of logical deduction, and especially in those that underlie the solution of hard questions in the theory of probability (in the chapters on probability, not included in this edition), we cannot do without the help of a directive method, and only a calculus can supply one.

10. Why the ultimate laws of logic are mathematical in form; why they are identical, except at a single point, with the general laws of number; and why they differ at that one point: on these questions it would come close to presumption to try to give a firm answer. They probably lie beyond the reach of our limited faculties. The mind may perhaps be allowed to know the laws it is itself subject to without being given any understanding of their ground and origin, or even, except to a very limited degree, of how well they suit their purpose compared with other systems of law we can imagine. Science does not need that knowledge anyway. Science properly concerns itself with what is; it does not look for reasons why one thing was preferred or appointed over another. These considerations are a sufficient answer to every protest against presenting logic in the form of a calculus. We do not adopt this form because we choose to give logic this way of showing itself. We adopt it because the ultimate laws of thought make it possible, dictate its character, and seem to rule out presenting the science perfectly in any other form. We should remember that the business of science is not to create laws but to discover them. We do not make the constitution of our own minds, however much it may be in our power to shape their character. And just as the laws of the human intellect do not depend on our will, so the forms of the science built on them are, in everything essential, independent of personal choice.

11. Besides setting out the principles of this method in general, the book will apply it to the analysis of a wide variety of propositions, and of chains of propositions that form the premises of demonstrative arguments. These examples are taken from various writers; they differ greatly in complexity, and they cover a wide range of subjects. Some may feel that I have allowed myself too much freedom in choosing them, but I do not think I need to apologize for that. Everyone admits that logic, as a science, can be applied very widely; and it is just as certain that its ultimate forms and processes are mathematical. So any objection à priori to using such forms and processes on a problem of morals or of general philosophy must rest on a misunderstanding or a false analogy. It is not of the essence of mathematics to deal with the ideas of number and quantity. Whether it would be good, as a general habit of mind, to apply symbolic processes to moral argument is another question. Perhaps, as I have said elsewhere, the perfection of the method of logic may be valuable chiefly as evidence that its principles are true in theory. The last thing that anyone who knows the value of mental toil and struggle would want is to replace ordinary reasoning, or to bind it to the rigor of technical forms. That toil and struggle give the mind an athlete's strength; they teach it to wrestle with difficulties and to rely on itself in an emergency. Still, there may be cases, even among matters that everyone agrees belong to ordinary reasoning, where the value of a scientific procedure comes to be felt and recognized. Some examples of this kind will be found in this book.

Footnote: Mathematical Analysis of Logic. London: G. Bell. 1847.

12. The general doctrine and method of logic explained above also form the basis of a theory of probability, and of a method to go with it. So building that theory and method on these principles will be a separate aim of this book (in the chapters on probability, not included in this edition). It may help to say something here about the nature of this application, and especially about the kind of solutions it leads to. For that purpose I will need to say a little more about the forms in which the results of the logical analysis are presented.

Why must a method in logic come first, as the basis of a theory of probability? The reason can be put in a few words. Suppose we want to know how the expected frequency of some event depends on the known frequencies of other events. Before we can find that out, we must know how the events themselves depend on one another. To put it technically, we must be able to express the event whose probability we are looking for as a function of the events whose probabilities are given. And working out that expression always belongs to logic. Probability, however, in its mathematical sense, can be measured in numbers. So the subject of probability belongs equally to the science of number and to logic. This book differs from all earlier ones in recognizing that both these elements stand side by side. That difference decides, in a great many cases, whether a problem can be solved at all, and it also brings new and important elements into the solutions. So I think it necessary to set out here, at some length, the particular consequences of the theory developed in the following pages.

13. The measure of the probability of an event is usually defined as a fraction. Its numerator is the number of cases favourable to the event, and its denominator is the whole number of cases, favourable and unfavourable, all of them supposed equally likely to happen. This book adopts that definition. At the same time, it shows that there is another way of looking at the subject (I will come to it shortly) that could just as well be taken as fundamental, and that would lead to exactly the same methods and conclusions. I may add this. As far as the accepted conclusions of the theory of probability go, and as far as they follow from its basic definitions, they do not differ from the results of the method of this book (which I take to be equally sound as inference).

Questions in the theory of probability come in many shapes, and algebraic and other conditions can change them in many ways. Even so, there seems to be one general type to which every such question can be brought, or at least as much of each question as truly belongs to the theory of probability. In terms of the data (what is given) and the quæsitum (what is sought), that type can be described as follows. First, the data are the probabilities of one or more given events. Each is either the probability that its event simply happens, or the probability that it happens under certain stated conditions. Second, the quæsitum, the thing sought, is the probability that some other event happens, either simply or under conditions. This event is expressed differently from those in the data, but it involves, more or less, the same elements. As for the data, they are either causally given, as when we work out the probability of a particular throw of a die from what we know of how the die is made, or they come from watching repeated cases of events succeeding or failing. In the second case, the probability of an event can be defined as the limit that the ratio of favourable cases to all observed cases approaches as the observations go on without end (assuming that nature is uniform). Last, one important distinction remains about the nature of the events, or how they relate. Events are either simple or compound. A compound event is one whose expression in language, or whose conception in thought, depends on the expression or conception of other events, which may be called simple in relation to it. To say “it rains” or “it thunders” is to state that a simple event happens. But to say “it rains and thunders,” or “it either rains or thunders,” is to state that a compound event happens, because the expression of that event depends on the elementary expressions “it rains” and “it thunders.” So the test of a simple event is not that its nature is somehow simple. It rests entirely on the way the event is expressed in language or conceived in thought.

14. Now one general problem that the existing theory of probability can solve is this: given the probabilities of any simple events, find the probability of a given compound event, that is, of an event put together in a given way out of those simple events. The problem can also be solved when the compound event whose probability is sought is subject to given conditions, that is, to conditions that also depend in a given way on the given simple events. Besides this general problem, there are some particular problems whose principle of solution is known. Various questions about causes and effects can be solved by known methods on the particular assumption that the causes exclude one another, but apparently not otherwise. Beyond this, it is not clear that anyone has made progress toward what may be called the general problem of the science, namely: given the probabilities of any events, simple or compound, conditional or unconditional, find the probability of any other event, just as freely chosen in its expression and conception. In stating this problem I do not even require that the events whose probabilities are given and the event whose probability is sought should share some elements. It is the job of a method to decide whether the data of a problem are enough for the purpose, and, when they are not, to show where they fall short.

The method of this book can solve this problem in its most unrestricted form. To be more precise, its theoretical solution is given completely, and its practical solution comes down to purely mathematical processes, such as solving and analysing equations. The order and character of the general solution can be described as follows.

15. First, it is always possible, by the preliminary method of the calculus of logic, to express the event whose probability is sought as a logical function of the events whose probabilities are given. The result looks like this. Suppose that X stands for the event whose probability is sought, and A, B, C, &c. for the events whose probabilities are given, simple or compound. Then the whole relation of the event X to the events A, B, C, &c. comes out in the form of what mathematicians call a development. In the most general case it has four distinct classes of terms. The first class expresses those combinations of the events A, B, C that both necessarily go with the event X and necessarily signal that it has happened. The second class expresses the combinations that necessarily go with the event X but do not necessarily imply it. The third class expresses the combinations that cannot happen together with the event X, though they are not otherwise impossible. The fourth class expresses the combinations that cannot happen under any circumstances. I will not dwell on this account of what the logical analysis of the problem yields, except to remark that its elements are exactly the ones by which our expectation of the event X, as it depends on what we know of the events A, B, C, is affected, and the only ones by which it can be. General reasoning would confirm this conclusion. But general reasoning would not usually manage to untangle the complicated web of events and circumstances out of which the solution just described has to be drawn. Reaching that result is the first step toward the complete solution of the question I set. Note that up to this point the process is logical, that is, it is carried out with symbols that have a logical meaning, and it ends in an equation that can be interpreted as a proposition. Let us call this result the final logical equation.

The second step of the process deserves close attention. From the final logical equation that the first step has led us to, we read off, by inspection, a series of algebraic equations that contain the complete solution of the problem implicitly. Of how this passage is made, it is enough to say this. There is a definite relation between two sets of laws: the laws by which the probabilities of events are expressed as algebraic functions of the probabilities of the other events they depend on, and the laws by which the logical connection of the events is itself expressed. Like the other agreements of formal law mentioned already, this relation is not founded on a hypothesis. We learn it by observation (I.4) and reflection. If, however, we assumed it à priori, in advance of the evidence, as the very basis of the definition of probability, strict deduction would lead from it to the accepted numerical definition as a necessary consequence. As I have already remarked (I.12), the theory of probability stands in an equally close relation to logic and to arithmetic. As far as results go, it makes no difference whether we regard it as springing from arithmetic, or as founded on the relations that connect the two together.

16. The general solutions this method yields have some features, of interest perhaps to the mathematician, that are worth pointing out.

First, since the method does not depend on how many data there are or what kind, it still works when the data are not enough to fix the value sought. When that happens, the final expression of the solution will contain terms with arbitrary constant coefficients. Matching these, there will be terms in the final logical equation (I. 15). Interpreting those terms tells us what new data we need to fix the values of the constants, and so to make the numerical solution complete. If such data cannot be had, we can still give the constants their limiting values 0 and 1, and so find the limits within which the probability sought must lie, whatever further experience may show. When the event whose probability is sought is quite independent of those whose probabilities are given, the limits found this way will be 0 and 1, just as they plainly ought to be, and interpreting the constants will only lead us back to a restatement of the original problem.

Second, the expression of the final solution will always involve one particular quantity, found by solving an algebraic equation. When that equation is of high degree, there may seem to be a difficulty about which root to choose. In some cases, it is true, both the given elements and the element sought are so obviously restricted by necessary conditions that no choice is left. But in complicated cases it would be hopeless to find such conditions by unaided reasoning. A separate method is needed for this purpose, one that might fairly be called the Calculus of Statistical Conditions. I will not go further into it here than to say two things. Like the first method, it is based on the use of the “final logical equation.” And it definitely fixes, first, the conditions that the numbers in the data must satisfy for the problem to be real, that is, drawn from a possible experience; and second, the numerical limits within which the probability sought would have had to fall if, instead of being found by theory, it had been found directly by observation of the same system of phenomena the data came from. Clearly these limits will be actual limits of the probability sought. Now, if the data are taken to satisfy the conditions just assigned to them, then in every case I have examined there turns out to be one root, and only one, of the final algebraic equation that meets the required limits. Every source of ambiguity is thus removed. It even seems that new truths about the theory of algebraic equations come to light along the way. It is remarkable that this special quantity depends only on the data of the problem, and not at all on its quæsitum. So solving one particular problem unties the knot for a whole system of problems, namely all the problems that share the same data, whatever their quæsita may be. This matters whenever we need to draw a series of connected conclusions from one particular set of data. It also gives the solutions of particular problems the family likeness, drawn from their dependence on one central and fundamental unity, that often marks the use of general methods.

17. These considerations, and others like them, justify calling the method of this book a general method for solving questions in the theory of probability. But it does not follow that we can always do without hypotheses. As I have noted, a solution may consist entirely of terms with arbitrary constant coefficients; it may, in fact, be wholly indefinite. If we used only the data of experience, the method of this book would give results of that kind for some of the most important questions it covers. To get a definite solution in such cases, we must fall back on hypotheses that have more or less probability of their own but cannot be exactly verified. Generally such hypotheses differ from the direct results of experience in being logical rather than numerical in character: they lay down the conditions under which phenomena occur, rather than how often they occur. That difference, however, does not matter. Whatever their nature, once assumed, the hypotheses must be counted among the actual data, though they will obviously give the solution itself something of a hypothetical character. Understood this way, with these possible sources for the data actually used, the method is perfectly general. But it is no more answerable for the correctness of the hypotheses brought in than for the correctness of the numerical data taken from experience.

To illustrate, the theory of reducing astronomical observations rests partly on hypotheses. It makes certain assumptions about the nature of error, such as that an error is equally likely to fall on the side of excess or of defect, and so on. Without them it would be impossible to draw any definite conclusions from a set of observations that disagree. But once such assumptions are granted, the rest of the investigation falls strictly within the theory of probability. The same holds for the important problem of working out, from the records of the majorities of a deliberative assembly, the average probability that any one of its members judges correctly. Applied to the bare numerical data, the method of this book gives a solution of the wholly indefinite kind described above. And, to show still more strikingly how insufficient those data are on their own, the interpretation of the arbitrary constants (I. 16) in the solution merely gives back a restatement of the original problem. But if we admit the hypothesis that each mind forms its opinion independently, either absolutely, as in the speculations of Laplace and Poisson, or within limits set by the actual data, as will be seen in this treatise, Chap. XXI. (in the chapters on probability, not included in this edition), the problem takes on a far more definite character. Clearly, then, the further value of the theory of probability must depend a great deal on forming such intermediate hypotheses correctly, wherever the purely experimental data are not enough for a definite solution and the further experience that the interpretation of the final logical equation points to cannot be had. On the other hand, too ready a habit of forming hypotheses on subjects that by their very nature lie beyond human knowledge is bound to damage the credit of the theory of probability, and to throw doubt, in the public mind, on even its most legitimate conclusions.

Footnote: The author designs to treat this subject either in a separate work or in a future Appendix. In the present treatise he avoids the use of the integral calculus.

18. It would perhaps be premature to speculate here on whether the methods of abstract science are ever likely to serve the study of social problems as well as they have served the various branches of physical inquiry. Trying to settle that question on purely à priori grounds, by argument in advance of the evidence, would very likely mislead us. For example, the fact of human free will would seem at first sight to rule out the idea that the movements of society should ever show the kind of orderly development we expect under the rule of physical necessity. Yet the researches of the statistician already reveal facts that contradict that expectation. The records of crime and poverty, for instance, show a degree of regularity unknown in regions untouched by the disturbing influence of human wants and passions. On the other hand, bad seasons, volcanic eruptions, the spread of blight among plants or of epidemic disease among animals, things apparently or mainly the product of natural causes, refuse to be brought under regular laws that we can grasp. “Fickle as the wind” is a proverb. Reflecting on these points teaches us to correct our first judgments somewhat. We learn not to expect, under the rule of necessity, an order that human observation can perceive, unless the interplay of the causes producing it is simple enough. Nor should we think, on the other side, that free agency in the individual cannot coexist with regularity in the movements of the system he is one unit of. Human freedom stands out as an apparent fact of our consciousness. It is also, I believe, a highly probable conclusion by analogy (Chap. XXII., not included in this edition) from the nature of that part of the mind whose scientific constitution we are able to investigate. But whether we accept it as a fact resting on consciousness or as a conclusion approved by reason, it must be understood in a way that does not conflict with an established result of observation. That result is this: phenomena that involve large masses of people do actually show a very remarkable degree of regularity. This lets us collect, in each age, the materials on which any estimate of its state and progress, as far as these show in outward results, must depend. So there is no sound à priori objection to the possibility of the kind of data needed as the experimental foundation of a science of social statistics. Further, whatever else this book may accomplish, I expect it to leave no doubt that there exists a system of abstract principles, and of methods founded on them, by which any collection of social data can be made to yield, explicitly, whatever information it implicitly contains. Where the data are extremely complex, getting this information out may be very hard. The difficulty would not come from any flaw in the theory, but from the sheer labour of the calculations it calls for. It is quite possible that in many cases the difficulty would be one that only combined effort could overcome. But this much, I believe, can be affirmed with perfect safety: we have in theory in every case, and in practice as far as the necessary labour of calculation can be supplied, the means of drawing out from statistical records the seeds of general truths that lie buried in the mass of figures.

19. Beyond these general positions, though, I do not venture to speak with confidence. Applying scientific methods to statistical records might produce results beyond those that can be found without such help. Whether those extra results would repay the labour enough to make it worth carrying out such investigations on a proper scale is a question that perhaps only experience could settle. It is to be hoped, and it might be expected without much presumption, that here as elsewhere the abstract doctrines of science will serve more than intellectual pleasure. Consider the apparent order in which the sciences have developed and, one after another, lent their help to the service of mankind. In that light it does not seem very unlikely that a day may come when similar help will come from branches of knowledge still more closely bound up with human welfare. For the present, though, let the speculations of this book rest simply on their claim to be true.

20. Last, I intend to try to draw from the scientific results of the earlier inquiries some general hints about the nature and constitution of the human mind (in the final chapter, not included in this edition). There is no need here to go into the grounds on which such an inference is possible. One or two general remarks may show the path I will try to follow. It must be admitted that our views of the science of logic are bound to influence greatly, perhaps mainly to decide, our opinions about the nature of the intellectual faculties. Take, for example, this question. Does reasoning consist merely in applying certain first or necessary truths that were stamped on the mind from the start? Or is the mind itself a seat of law, whose working is as plain and as conclusive in the particular case as in the general formula? Or, as some writers of note seem to hold, is all reasoning about particulars? This question, I say, does not merely affect the science of logic. It also bears on forming sound views of how the intellectual faculties are constituted. Again, suppose we conclude that the mind is by its original constitution a seat of law. Then the question of what kind of subjection to law this is also has deep speculative interest. Is it, for instance, an obedience founded on necessity, like the obedience that keeps the heavens turning and preserves the order of nature? Or is it a subjection of some quite different kind? Further, if the mind really is shown to be subject to law, and if its laws are correctly stated, then the question of how those laws are likely, or bound, to influence the course of human thought in different ages is one of great importance. It well deserves patient investigation, as a matter both of philosophy and of history. These and other questions I propose to discuss, however imperfectly, in the concluding part of this book. They belong, perhaps, to the domain of probable or conjectural knowledge rather than to that of positive knowledge. But it may happen that where there is not enough warrant for the certainties of science, there are grounds of analogy good enough to suggest highly probable opinions. It has seemed better to me to keep this discussion entirely for after the main business of this book, which is the investigation of scientific truths and laws. Experience teaches us clearly enough that the right order of progress in every search for truth is from the known to the unknown. There are parts, even of the philosophy and constitution of the human mind, that have been put fully within reach of our investigation. To make a proper knowledge of those parts of our nature the basis of every attempt to see into the shadows and uncertainties of the conjectural realm that lies beyond and above them: that is the course best suited to the limits of our present condition.

Chapter 2: Signs and Their Laws

1. Language is a tool of human reason, not just a way of expressing thought. That much is generally agreed. In this chapter I want to ask what it is that makes language able to serve the most important of our intellectual powers in this way. Along the way we will look at how language is built, treating it as a system designed for a purpose. We will pick out its elements, try to settle how they relate to and depend on one another, and ask how each of them, as a working part of the system, helps it reach the end it serves.

We need not, for these inquiries, enter the famous old debate of the schools: is language an essential tool of reasoning, or could we reason without it? I take that question to lie outside the plan of this book, and here is why. The business of science is to investigate laws. We may regard signs as standing for things and their relations, or as standing for the conceptions and operations of the human intellect. Either way, in studying the laws of signs we are in effect studying the laws of reasoning as they show themselves. If the two inquiries differ at all, the difference does not touch the scientific statement of formal laws, which is what this part of the book is after. It touches only the way the results are presented to the mind. When we investigate the laws of signs à posteriori, from the outside, what we examine directly is language and the rules that govern its use. When we take the inner processes of thought as our direct subject, we appeal more immediately to our own consciousness. But in both cases, as we will find, the results come out formally the same. Nor could we easily believe that the countless languages and dialects of the earth would have kept so much in common, over so many ages, unless their agreement had some deep foundation in the laws of the mind itself.

2. All language is made of signs, or symbols. Words are signs. Sometimes words are said to stand for things. Sometimes they stand for the operations by which the mind combines the simple notions of things into complex conceptions. Sometimes they express relations of action, of being acted on, or of mere quality, which we see among the objects of our experience. Sometimes they express the feelings of the mind that perceives. But words, though they serve as signs in these ways and others, are not the only signs we can use. Arbitrary marks that speak only to the eye, and arbitrary sounds or gestures that speak to some other sense, are signs just as much, provided that what they stand for is defined and understood. In the mathematical sciences, letters and the symbols +, −, =, and so on, are used as signs. There the word “sign” is usually kept for the second kind, the symbols that stand for operations or relations, rather than for the letters, which stand for the elements of number and quantity. What a sign really means does not depend on the particular shape it takes, and neither do the laws that govern its use. In this book, however, we are dealing with written signs, and the word “sign” will be used for these alone. The essential properties of signs are listed in the following definition.

Definition.—A sign is an arbitrary mark, having a fixed interpretation, and susceptible of combination with other signs in subjection to fixed laws dependent upon their mutual interpretation.

In plainer words: a sign is a mark chosen at will, which has a fixed meaning, and which can be combined with other signs according to fixed laws, laws that depend on what the signs mean.

3. Let us take the parts of this definition one at a time.

(1.) First, a sign is an arbitrary mark. It clearly makes no difference which word or token we attach to a given idea, as long as the attachment, once made, is kept. The Romans said “civitas” for what we call a “state.” But they and we could just as well have used any other word for the same conception. In fact nothing in the nature of language would stop us from using a single letter in that sense. If we did, the laws governing the letter would be essentially the same as the laws governing “civitas” in Latin and “state” in English, at least so far as those words are governed by general principles common to all languages.

(2.) Second, each sign must have a fixed meaning throughout any one discussion or chain of reasoning. The need for this is obvious; it seems to come from the very nature of the subject. There is, however, a dispute about exactly what words or symbols stand for when they are used as names in reasoning. Some hold that they stand only for the conceptions of the mind; others, that they stand for things. The question matters little here, since its answer cannot change the laws by which signs are used. My own view is this. In reasoning, signs take the place of the mind's conceptions and operations and do their work. Those conceptions and operations represent things and the connections and relations of things, and so the signs represent things with their connections and relations too. And because signs take the place of the mind's conceptions and operations, they are subject to the laws of those conceptions and operations. This view will be explained more fully in the next chapter. Here it serves to explain the third part of the definition of a sign, namely, that its combinations are subject to fixed laws, and that these laws depend on what it means.

4. The following Proposition analyses and classifies the signs by which reasoning is carried on.

Proposition I.

Everything language does as a tool of reasoning can be done by a system of signs made of the following elements:

First, literal symbols, such as x, y, and so on, standing for things as objects of our conceptions.

Second, signs of operation, such as +, −, ×, standing for those operations of the mind by which conceptions of things are combined or taken apart to form new conceptions out of the same elements.

Third, the sign of identity, =.

These symbols of logic obey definite laws in their use. Those laws partly agree with, and partly differ from, the laws of the matching symbols in algebra.

Let us take this as our test of the true elements of rational discourse: they should combine in the simplest forms and by the simplest laws, and by combining in this way they should produce every other form of language, known or conceivable. With that principle in hand, consider the following classification.

Class I.

5. Naming or describing signs: signs that give the name of a thing, or some quality or circumstance belonging to it.

To this class we can plainly assign the noun, proper or common, and the adjective. The two really differ in only one respect. The noun states outright that the individual thing or things it refers to exist; the adjective takes that existence for granted. If we attach to an adjective the subject everyone understands anyway, “being” or “thing,” it becomes in effect a noun, and for every essential purpose of reasoning the noun can replace it. Perhaps, in every shade of the mind's view, it is not quite the same to say “Water is a fluid thing” as to say “Water is fluid.” But for expressing the processes of reasoning the two are at least equivalent.

It is also clear that this class must include any sign used by convention for some circumstance or relation that would otherwise take many signs to spell out. The epithets of poetry are very often of this kind. They are usually compound adjectives, each doing the work of a description of many words. Homer's “deep-eddying ocean” packs a whole description into the single word βαθυδίνης. In the same way, any other description, whether aimed at the imagination or at the intellect, could by agreement be represented by a single sign. That sign would obey, in every essential point, the same laws as the adjective “good” or “great.” Joined to the subject “thing,” it would become in effect a noun, and a single noun could then carry the combined meaning of the thing and the quality.

6. Now, since a sign has been defined as an arbitrary mark, we are free to replace every sign of the kind just described by a letter. So let us agree to represent by a single letter, such as x, the class of individuals to which a particular name or description applies. If the name is “men,” for instance, let x stand for “all men,” or the class “men.” By a class we usually mean a collection of individuals, each of which the particular name or description fits. In this book the term is stretched a little. It includes the case where just one individual answers to the name or description. It also includes the cases named by the words “nothing” and “universe,” which as classes must be understood to contain “no beings” and “all beings” respectively. Next, if an adjective such as “good” is used to describe, let a letter, say y, stand for all the things the description “good” fits, that is, “all good things,” or the class “good things.” Let us further agree that the combination xy shall stand for the class of things to which the names or descriptions represented by x and y both apply at once. So if x alone stands for “white things” and y for “sheep,” let xy stand for “white sheep.” Likewise, if z stands for “horned things,” while x and y keep their meanings, let zxy stand for “horned white sheep,” that is, the collection of things to which the name “sheep” and the descriptions “white” and “horned” all apply together.

Let us now look at the laws that the symbols x, y, and so on, used in this sense, obey.

7. First, it is plain that in combinations like these, the order in which two symbols are written makes no difference. The expressions xy and yx both stand for the class of things to every member of which the names or descriptions x and y both apply. So we have xy=yx.(1)

If x stands for white things and y for sheep, either side of this equation stands for the class of “white sheep.” We may form the conception in a different order, but the individual things it covers are the same. In the same way, if x stands for “estuaries” and y for “rivers,” the expressions xy and yx both stand, without distinction, for “rivers that are estuaries” or “estuaries that are rivers.” In ordinary language this combination joins two nouns, rather than a noun and an adjective as in the first example. Let a third symbol, z, stand for the class of things to which the word “navigable” applies. Then any one of the expressions zxy,zyx,xyz,&c., will stand for the class of “navigable rivers that are estuaries.”

Sometimes one descriptive term carries a hidden reference to another. Then all we need do is write that reference openly into its stated meaning, and the remarks above apply again. For instance, if x stands for “wise” and y for “counsellor,” we must say whether x means wisdom in the absolute sense or only wisdom in giving counsel. Once that is settled, the law xy=yx holds as before.

So we may use the symbols x, y, z, and so on, in place of nouns, adjectives, and descriptive phrases, subject to two conditions. The first is a rule of interpretation: any expression in which several of these symbols are written together stands for all the objects or individuals to which their meanings all apply at once. The second is a law: the order in which the symbols follow each other makes no difference.

The rule of interpretation has now been illustrated enough. So I will not always bother to state the subject “things” when I give the meaning of a symbol that stands for an adjective. When I say, let x represent “good,” it will be understood that x means just “good” only when some other symbol supplies a subject for that quality; used alone, it means “good things.”

8. A few remarks can be added about the law just found. They will also apply, more or less, to some other laws to be derived later.

First, this law is a law of thought, not, strictly speaking, a law of things. A difference in the order of the qualities or attributes of an object, setting aside all questions of cause, is merely a difference in how we conceive it. The law (1) states as a general truth that the same thing can be conceived in different ways, and it says what that difference amounts to. It does nothing more.

Second, as a law of thought, it is actually worked out in a law of language, which is both the product and the tool of thought. Prose tends toward a uniform order. Yet even in prose, when several adjectives with a fixed meaning are applied to the same subject, the order in which they come makes no difference. And poetry owes much of its rich variety to extending this same lawful freedom to the noun as well. Milton's language is especially marked by this kind of variety. The noun often comes before the adjectives that qualify it, and it is often placed right in the middle of them. In the first few lines of his invocation to Light we meet examples like these:

	“Offspring of heaven first-born.”
	“The rising world of waters dark and deep.”
	“Bright effluence of bright essence increate.”

These inverted forms are not just the fruit of poetic license. They are the natural expression of a freedom that the deep laws of thought allow, but that ordinary speech, for convenience, does not use.

Third, the law expressed by (1) can be summed up by saying that the literal symbols x, y, z, are commutative, like the symbols of algebra: their order can be swapped. In saying this, I do not claim that multiplication in algebra, whose fundamental law is expressed by the equation xy=yx, is in itself anything like the process of logical combination that xy has been made to stand for above. I claim only that if the arithmetical process and the logical process are written in the same way, their symbolic expressions will obey the same formal law. The evidence that they obey it is quite separate in the two cases.

9. The combination of two literal symbols in the form xy expresses the whole class of objects to which the names or qualities represented by x and y both apply. It follows that if the two symbols mean exactly the same thing, their combination expresses no more than either symbol alone. In that case we should have xy=x. But since y is supposed to mean the same as x, we may put x in its place in this equation, and we get xx=x. that is, a class combined with itself gives that same class. Now in ordinary algebra the combination xx is written more briefly as x2. Let us adopt the same notation here. The way we write down a particular sequence of mental operations is just as arbitrary as the way we write down a single idea or operation (II. 3). In this notation, then, the equation above takes the form x2=x,(2) and it is, in fact, a second general law of the symbols by which names, qualities, or descriptions are represented.

The reader should keep in mind that, although in the earlier examples the symbols x and y were given meanings different from each other, nothing stops us from giving them exactly the same meaning. Clearly, the closer their actual meanings come to each other, the closer the class named by the combination xy comes to being the same as the class named by x, and the same as the class named by y. The case assumed in proving equation (2) is that of absolute sameness of meaning. Language shows this law at work in practice. To say “good, good” about anything, though a clumsy and useless repetition, is the same as saying “good.” So “good, good” men are just “good” men. People do sometimes repeat words in this way to heighten a quality or strengthen a statement. But that effect is secondary and a matter of convention; it is not rooted in the inner relations of language and thought. Most of the operations we see in nature, or carry out ourselves, have a greater effect when repeated. This prepares us to expect the same thing in language, and even to repeat ourselves when we want to speak with emphasis. But neither in strict reasoning nor in exact speech is there any real justification for the practice.

10. We now turn to another class of the signs of speech, and to the laws connected with their use.

Class II.

11. Signs of the mental operations by which we gather parts into a whole, or separate a whole into its parts.

We can do more than form conceptions of objects marked out by names, qualities, or circumstances that apply to each individual in the group we are thinking about. We can also form the combined conception of a group of objects made up of smaller groups, each named or described on its own. For this we use the conjunctions “and,” “or,” and so on. “Trees and minerals” and “barren mountains, or fertile vales” are examples of this kind. Strictly speaking, when the words “and” or “or” are placed between the terms describing two or more classes of objects, they imply that those classes are entirely separate, so that no member of one is found in another. In this respect, and in every other, the words “and” and “or” are like the sign + in algebra, and their laws are the same. So the expression “men and women,” setting aside conventional meanings, is equivalent to “women and men.” Let x stand for “men” and y for “women,” and let + stand for “and” and “or.” Then we have x+y=y+x,(3) an equation that would hold just as well if x and y stood for numbers and + were the sign of arithmetical addition.

Let the symbol z stand for the adjective “European.” Since it comes to the same thing to say “European men and women” as to say “European men and European women,” we have z(x+y)=zx+zy.(4) This equation, too, would hold just as well if x, y, and z were symbols of number and writing two letters side by side meant their algebraic product, just as, in the logical meaning given earlier, it means the class of objects to which both of the joined descriptions apply.

Those are the laws governing the sign +, used here for the positive operation of gathering parts into a whole. But the very idea of an operation that makes some positive change seems to suggest the idea of an opposite, negative operation, one that undoes what the first has done. We cannot think it possible to gather parts into a whole without also thinking it possible to separate a part from a whole. In ordinary speech we express this operation by the word “except,” as in “All men except Asiatics,” or “All states except those which are monarchical.” Here it is understood that the things excepted are part of the things they are excepted from. Just as we wrote the operation of gathering with the sign +, so we may write this negative operation with −, the minus sign. So if x stands for men and y for Asiatics, that is, Asiatic men, then the conception “All men except Asiatics” will be written x−y, the class of men with the Asiatic men taken out. And if x stands for “states” and y for the descriptive property “having a monarchical form,” then the conception “All states except those which are monarchical” will be written x−xy: from the states, take away the states that are monarchical.

Editor's note: Boole's plus sign is not the modern "or". He uses it only to join classes that have no members in common, so that "x plus y" means things that are either x or y but never both, and he allows "x minus y" only when all the y's are already among the x's. Modern Boolean algebra, following Jevons in 1864 and later Peirce, reads plus as an inclusive "or" (either one, or both) and replaces subtraction with the complement, "not-y". Boole's results still come out right in his own system, but a reader who knows modern notation should not read his plus as "or".

For every essential purpose of reasoning it makes no difference whether we mention the excepted cases first or last. In the same way, it makes no difference in what order we write a series of terms, some of which carry the sign −. So we have, as in ordinary algebra, x−y=−y+x.(5) Still letting x stand for the class “men” and y for “Asiatics,” let z stand for the adjective “white.” Now to apply the adjective “white” to the group of men described by the phrase “Men except Asiatics” is the same as to say “White men, except white Asiatics.” So we have z(x−y)=zx−zy.(6) This too agrees with the laws of ordinary algebra.

Equations (4) and (6) can be seen as two cases of a single general law: the literal symbols x, y, z, and so on, are distributive in their operation. The general fact this law expresses is this. Suppose some quality or circumstance is ascribed to all the members of a group, where the group was formed by gathering smaller groups together or by excluding some of them. The resulting conception is the same as if the quality or circumstance had first been ascribed to the members of each smaller group, and the gathering or excluding done afterwards. Whatever is said of the members of the whole is said of the members of all its parts, however those parts are put together.

Class III.

12. Signs that express relation, and by which we form propositions.

All verbs could properly be put in this class. But for the purposes of logic it is enough to include only the verb “to be,” in the form is or are. Every other verb can be broken down into this element plus one of the signs of Class I. Those signs express quality or circumstance of every kind, so they can express the subject's relation of acting or being acted on, whether in the past, present, or future. Thus the proposition “Cæsar conquered the Gauls” can be broken down into “Cæsar is he who conquered the Gauls.” My reason for this analysis is as follows. Unless we understand what it means to have conquered the Gauls, that is, what the expression “One who conquered the Gauls” means, we cannot understand the sentence. So that expression is truly an element of the sentence. “Cæsar” is another element. And one more is needed: the copula is, to show the connection between the two. I do not claim that this is the only way to view the relation expressed by the proposition “Cæsar conquered the Gauls.” I claim only that this analysis is correct from the particular point of view we have taken, and that it is enough for the purposes of logical deduction. It may be noted that the passive and future participles of the Greek language imply the principle asserted here, namely, that the sign is or are can be treated as an element of every personal verb.

13. This sign, is or are, can be represented by the symbol =. Next we consider the laws, or as they would usually be called, the axioms, that this symbol brings with it.

Take the proposition “The stars are the suns and the planets,” and let stars be represented by x, suns by y, and planets by z. Then we have x=y+z.(7) Now if it is true that the stars are the suns and the planets, it follows that the stars, except the planets, are suns. That gives the equation x−z=y,(8) which must therefore follow from (7). So a term z has been moved from one side of an equation to the other by changing its sign. This agrees with the algebraic rule of transposition.

But rather than dwell on particular cases, we can state the general axioms at once:

First, if equal things are added to equal things, the wholes are equal.

Second, if equal things are taken from equal things, the remainders are equal.

It follows that we may add or subtract equations, and use the rule of transposition just given, exactly as in ordinary algebra.

Again: suppose two classes of things, x and y, are identical; that is, every member of the one is a member of the other. Then the members of the one class that have a given property z will be identical with the members of the other that have the same property z. So if we have the equation x=y; then whatever class or property z may stand for, we also have zx=zy. This is the same in form as the algebraic law: if both sides of an equation are multiplied by the same quantity, the products are equal.

In the same way it can be shown that if the matching sides of two equations are multiplied together, the resulting equation is true.

14. Here, however, the likeness between this system and algebra, as algebra is usually stated, seems to stop. Suppose it is true that the members of a class x that have a certain property z are identical with the members of a class y that have the same property z. It does not follow that the members of the class x, all of them, are identical with the members of the class y. So we cannot infer from the equation zx=zy, that is, “the x's that are z are the same as the y's that are z,” that the equation x=y is also true. In other words, the algebraists' axiom that both sides of an equation may be divided by the same quantity has no formal equivalent here. I say no formal equivalent because, in keeping with the general spirit of these inquiries, we do not even try to decide whether the mental operation represented by removing a logical symbol z from a combination zx is in itself like division in arithmetic. That mental operation is in fact what is usually called abstraction, and it will appear later that its laws depend on the laws already derived in this chapter. What has been shown now is that nothing among those laws matches, in form, a commonly accepted axiom of algebra.

But a little thought will show that even in ordinary algebra this axiom is not as general as the other axioms we have looked at. Deducing the equation x=y from the equation zx=zy is valid only when we know that z is not equal to 0. So if the value z=0 is allowed in the algebraic system, the axiom stops applying, and the analogy shown before remains, at least, unbroken.

15. However, there is little point, except as a matter of speculation, in tracing such likenesses with the symbols of quantity in general. We have seen (II. 9) that the symbols of logic obey the special law x2=x. Now among the symbols of number there are only two, namely 0 and 1, that obey the same formal law. We know that 02=0, and that 12=1; and the equation x2=x, taken as an equation of algebra, has no roots other than 0 and 1. So instead of measuring how far the symbols of logic agree in form with the symbols of number in general, the more natural step is to compare them with symbols of quantity admitting only of the values 0 and 1. Let us imagine, then, an algebra in which the symbols x, y, z, etc. can take the values 0 and 1, either one, and these values only. The laws, the axioms, and the processes of such an algebra will be identical, through their whole range, with the laws, the axioms, and the processes of an algebra of logic. Only a difference of interpretation will separate them. This is the principle on which the method of this book is built.

16. It remains to show that the parts of ordinary language not yet considered in this chapter either break down into the same elements we have considered, or serve those elements by helping to define them more precisely.

We have already dealt with the noun, the adjective, and the verb, along with the particles and and except. The pronoun can be treated as a particular form of the noun or the adjective. The adverb modifies the meaning of the verb but does not change its nature. Prepositions help to express circumstance or relation, and so they add precision and detail to the meaning of the literal symbols. The conjunctions if, either, or, are used mainly to express relations among propositions. It will be shown later that those same relations can be fully expressed by elementary symbols that are similar in meaning, and identical in form and law, to the symbols whose use and meaning have been explained in this chapter. As for any remaining elements of speech, examination will show that they do one of two things. Either they give a more definite meaning to the terms of discourse, and so enter into the interpretation of the literal symbols already considered; or they express some emotion or feeling that goes with the utterance of a proposition, and so do not belong to the province of the understanding, which is our only concern here. Experience in using this classification will show that it is sufficient.

Chapter 3: Derivation of the Laws

1. The object of science, in the proper sense of the word, is the knowledge of laws and relations. One of the most important conditions of scientific progress is the ability to tell what is essential to that end from what is only accidentally attached to it. I say tell apart, because a steady devotion to science does not require us to turn our attention away completely from other speculations, often metaphysical ones, with which science is frequently bound up. Take questions like these: whether there is some underlying ground that sustains the phenomena we observe; whether causes are real; whether it is proper to speak as if one state of things leads to the next by some operation. Questions of this kind may have deep interest and meaning in relation to science without being, in their essence, scientific. In fact it is hardly possible to state the conclusions of natural science without borrowing the language of these ideas. Nor does this necessarily cause any practical trouble. Some people believe that there is more to the relation of cause and effect than a fixed order in which one thing follows another; some refuse to believe it. Both sides read the conclusions of physical astronomy in exactly the same way. But they agree only because they recognise a shared element of scientific truth, one that does not depend on their particular views about what causation is.

2. If this distinction matters in physical science, it deserves even more attention in the science of the powers of the intellect. For the questions this science raises are almost bound to get mixed up, at least in how they are expressed, with ways of thinking and speaking that betray a metaphysical origin. The idealist would state the laws of reasoning in one form; the sceptic, if he were true to his principles, in another. Some people regard the phenomena we are studying here as nothing more than a series of states of the thinking subject, with no causal link between them. Others trace them to the operations of an active intelligence. If each side were consistent, they would state those laws differently too. A different way of classifying the faculties of the mind would produce differences of the same kind. Amid so much variety, some of it only apparent and some of it real, I want to assert one principle as our only firm and stable ground. It is this: if the laws in question really are drawn from observation, then they have a real existence as laws of the human mind, whatever metaphysical theory their wording may seem to involve. They contain an element of truth that no later criticism of the nature of the mind's operations, or even of whether those operations are real, can essentially touch. Grant, if you like, that the mind is nothing but a succession of states of consciousness: a series of passing impressions, caused neither from outside nor from within, rising out of nothing and sinking back into nothing again. That is the last refinement of the sceptical intellect. Even so, the results that observation had led us to would remain true as laws of succession, or at least of a succession that has already happened. They would have to be translated into a language whose vocabulary had been stripped of every term like cause and effect, operation and subject, substance and attribute. But they would still be valid as scientific truths.

There is more. Any statement of the laws of thought that rests on actual observation must contain scientific elements that do not depend on metaphysical theories about the nature of the mind. So the practical use of those elements, in building a system or method of reasoning, must also be independent of metaphysical distinctions. For any practical application will rest on the scientific elements contained in the statement of the laws. In the same way, the practical conclusions of physical astronomy do not depend on any theory of what causes gravitation; they rest only on knowing its observed effects. And so, both in finding the laws of thought and in putting them to use once found, we are, for every truly scientific purpose, not concerned with whether any metaphysical speculation whatever is true or false.

3. Under these circumstances, the course I think best is to use the language of everyday speech as far as I can, without regard to any theory of the nature and powers of the mind that it might be thought to carry with it. For instance, it is ordinary usage to say that we talk with each other by communicating ideas, or conceptions, and that communicating them is the job of words. It is also ordinary usage to say that, when particular ideas or conceptions are presented to the mind, it has certain powers or faculties for dealing with them. By these it can fix its regard on some ideas and shut out others, or combine the given conceptions or ideas in various ways. These faculties or powers have been given different names, such as Attention, Simple Apprehension, Conception or Imagination, Abstraction, and so on. These names have not only supplied the titles of separate branches of the philosophy of the human mind; they have also passed into the common language of people. So whenever I have occasion to use these terms, I shall use them without implying that I accept the theory that the mind has such and such powers and faculties as distinct parts of its activity. Nor is it really necessary to ask whether these powers of the understanding exist separately or not. We may gather all these different titles under one general name, the Operations of the human mind. We can define these operations as far as this book needs, and then try to express their ultimate laws. That is the general plan I shall follow, though now and then I will refer to the names that common agreement has given to the particular states or operations of the mind that come up along the way.

It will be most convenient to divide the more definite results of the investigation that follows into separate Propositions.

Proposition I.

4. To deduce the laws of the symbols of Logic from a consideration of those operations of the mind which are implied in the strict use of language as an instrument of reasoning.

In every discourse, whether it is the mind talking with its own thoughts or a person talking with others, there is a limit, assumed or stated, within which the subjects being talked about are confined. The most unrestricted discourse is the one in which our words are taken in the widest possible sense; for such words, the limits of discourse are as wide as those of the universe itself. But more often we confine ourselves to a smaller field. Sometimes, when we speak of men, we mean (without saying so) men only under certain circumstances and conditions: civilized men, for example, or men in the prime of life, or men under some other condition or relation. Now, whatever the extent of the field within which all the objects of our discourse are found, that field may properly be called the universe of discourse.

5. Furthermore, this universe of discourse is, in the strictest sense, the ultimate subject of the discourse. Under the limits we have just described, the job of any name or descriptive term is not to call up in the mind the conception of every being or object it could apply to. It is to call up only those that exist within the assumed universe of discourse. If that universe of discourse is the actual universe of things, as it always is when our words are taken in their real and literal sense, then by men we mean all men that exist. But if the universe of discourse is limited by some understanding implied in advance, then we are speaking of men under the limit so introduced. In both cases the business of the word men is to direct a certain operation of the mind: from the proper universe of discourse, we pick out, or fix upon, the individuals the word signifies.

6. The mental operation implied by using an adjective is of exactly the same kind. Suppose, for instance, that the universe of discourse is the actual Universe. The word men tells us to select in our minds, from that Universe, all the beings the term "men" applies to. In just the same way the adjective "good," in the combination "good men," tells us to go further and select from the class of men all those who have the further quality "good." And if another adjective were put in front of "good men," it would direct a further operation of the same kind, concerning whatever further quality it was chosen to express.

It is important to notice carefully what the operation described here really is, because it is conceivable that it might have been different. If the adjective were simply attributive, that is, if it simply attached a quality, then once a particular set of beings had been marked out by men, putting the adjective good in front would tell us to attach the quality of goodness, in our minds, to every one of those beings. But that is not what the adjective really does. The operation we actually perform is one of selection according to a prescribed principle or idea. It does not matter here which faculties of the mind such an operation would be assigned to under the accepted classification of its powers. I suppose it would be thought to depend on two of them, Conception or Imagination, and Attention. To the first we might assign the forming of the general conception; to the second, the fixing of the mind's regard on those individuals, within the prescribed universe of discourse, that answer to the conception. But it seems quite possible that the power of Attention is nothing more than the power of keeping up the exercise of some other faculty of the mind. If so, we might properly assign the whole mental process just described to the single faculty of Imagination or Conception. The first step of the process would be the conception of the Universe itself, and each step after that would narrow, in a definite way, the conception so formed. Taking this view, I shall call each such step, or any definite combination of such steps, a definite act of conception. And I shall stretch this term to cover not only the conception of classes of objects named by particular words or by simple qualities, but also the combining of such conceptions in any way that the powers and limits of the human mind allow. In fact it covers any intellectual operation short of the one involved in building a sentence or proposition. We must now consider the general laws that such operations of the mind obey.

7. I shall now show that the laws found in the preceding chapter for the use of the literal symbols of Logic, which there were reached à posteriori, after the fact, from the structure of language, are really the laws of the definite mental operation just described. We begin our discourse with a certain understanding about the limits of its subject, that is, about the limits of its Universe. Every name, every descriptive term we use, directs the person we are speaking to to carry out a certain mental operation on that subject. That is how thought is communicated. But on this view each name or descriptive term is only the representative of an intellectual operation, and that operation comes first in the order of nature. So it is clear that the laws of the name or symbol must be derived ones. They must, in fact, come from the laws of the operation they represent. I shall now show that the laws of the symbol and the laws of the mental process are identical in expression.

8. Let us suppose, then, that the universe of our discourse is the actual universe, so that words are used in the full extent of their meaning. And let us consider the two mental operations implied by the words "white" and "men." The word "men" implies the operation of selecting in thought, from its subject, the universe, all men; and the resulting conception, men, becomes the subject of the next operation. The operation implied by the word "white" is that of selecting, from its subject, "men," all members of that class who are white. The final resulting conception is that of "white men." Now it is perfectly plain that if these operations had been performed in the reverse order, the result would have been the same. We may begin by forming the conception "men" and then, by a second act of the intellect, narrow it to "white men." Or we may begin by forming the conception "white objects" and then narrow it to those members of that class who are "men." As far as the result goes, it makes no difference at all. Clearly, the order of the mental processes would make no difference either if we put any other descriptive or naming terms in place of "white" and "men," provided only that their meaning was fixed and absolute. So the fact that the order of two successive acts of the faculty of Conception does not matter, where one act supplies the subject that the other works on, is a general condition of the use of that faculty. It is a law of the mind. And it is the real origin of the law of the literal symbols of Logic that is its formal expression, law (1) of Chap. II, the law that xy is the same as yx.

9. It is just as clear that this mental operation is of such a kind that repeating it does not change its effect. Suppose that by one definite act of conception the attention has been fixed on men, and that by another exercise of the same faculty we narrow it to those of the race who are white. Then repeating the second mental act any further, narrowing the attention to white objects again, does not change in any way the conception we have reached, namely, that of white men. This too is an example of a general law of the mind, and its formal expression is the law ((2) Chap, II.) of the literal symbols, the law that xx is the same as x.

10. Again, it is plain that from the conceptions of two distinct classes of things we can form the conception of the collection of things that the two classes make up together. And it obviously makes no difference in what order of position or of priority the two classes are presented to the mind. This is another general law of the mind, and its expression is found in (3) Chap. II, the law that x + y is the same as y + x.

11. There is no need to carry this line of inquiry and comparison further. Enough illustration has been given to make the following two positions clear.

First, the operations by which the mind, using its power of imagination or conception, combines and modifies the simple ideas of things or qualities are subject to general laws, no less than those operations of the reason that work on truths and propositions.

Second, those laws are mathematical in their form, and they are actually worked out in the essential laws of human language. So the laws of the symbols of Logic can be deduced from a consideration of the operations of the mind in reasoning.

12. The rest of this chapter deals with questions about the law of thought whose expression is x2=x (II. 9). As has been implied (II. 15), this law is what marks off the operations of the mind in its ordinary discourse and reasoning from its operations when it is busy with the general algebra of quantity. An important part of the inquiry that follows will be to prove that the symbols 0 and 1 have a place among the symbols of Logic, and can be given a meaning there. But first it may be necessary to show how particular symbols like these can properly, and usefully, be used to represent quite separate systems of thought.

This propriety cannot rest on any shared meaning. Logic and Arithmetic (I use the second term in its widest sense, as the science of Number) are truly separate systems of thought, and properly speaking they have no subject in common. The one deals with the very conceptions of things; the other takes account only of their numerical relations. But the forms and methods of any system of reasoning depend directly on the laws the symbols obey, and only indirectly, through that link, on what the symbols mean. So it can be both proper and useful to use the same symbols in different systems of thought, provided they can be given meanings that make their formal laws identical and their use consistent. The ground for using them will then not be a shared meaning but the shared formal laws they obey in their respective systems. And that sharing of formal laws must be established on no other ground than careful observation and comparison of the results that are seen to follow, separately, from the meanings within each system under consideration.

These remarks explain the line of inquiry taken in the next Proposition. The literal symbols of Logic all obey, without exception, the law whose expression is x2=x. Among the symbols of Number there are only two, 0 and 1, that satisfy this law. But each of these two symbols also obeys a law of its own in the system of numerical quantity. This suggests the question: what meanings must we give to the literal symbols of Logic so that these same special formal laws hold in the logical system too?

Proposition II.

13. To determine the logical value and significance of the symbols 0 and 1.

The symbol 0, as used in Algebra, satisfies the following formal law, 0×y=0, or 0y=0,(1) whatever number y may stand for. For this formal law to hold in the system of Logic, we must give the symbol 0 a meaning such that the class represented by 0y is identical with the class represented by 0, whatever the class y may be. A little thought will show that this condition is met if the symbol 0 stands for Nothing. Following an earlier definition, we may call Nothing a class. In fact, Nothing and Universe are the two limits of how far a class can extend. They are the limits of the possible meanings of general names, since no general name can refer to fewer individuals than Nothing contains, or to more than the Universe contains. Now, whatever the class y may be, the individuals it shares with the class "Nothing" are identical with those contained in the class "Nothing," for there are none. So by giving 0 the meaning Nothing, we satisfy law (1); and no other meaning satisfies it consistently with the perfectly general character of the class y.

Second, in the system of Number the symbol 1 satisfies the following law, namely, 1×y=y, or 1y=y, whatever number y may stand for. Suppose this formal equation holds equally in the system of this book, in which 1 and y stand for classes. Then the symbol 1 must stand for a class such that all the individuals found in any proposed class y are also all the individuals 1y that the class y shares with the class represented by 1. A little thought will show here that the class represented by 1 must be "the Universe," since this is the only class in which all the individuals that exist in any class are found. So the meanings of the symbols 0 and 1 in the system of Logic are, respectively, Nothing and Universe.

14. The idea of any class of objects, such as "men," brings to mind the idea of the opposite class, the beings that are not men. And the whole Universe is made up of these two classes together, since of every individual in it we may say either that it is a man or that it is not a man. So it becomes important to ask how such opposite names are to be expressed. That is the object of the following Proposition.

Proposition III.

If x represent any class of objects, then will 1−x represent the contrary or supplementary class of objects, that is, the class including all objects which are not comprehended in the class x.

To make the idea clearer, let x stand for the class men, and, following the last Proposition, let us express the Universe by 1. Now if from the conception of the Universe, as made up of "men" and "not-men," we take away the conception of "men," what remains is the conception of the opposite class, "not-men." So the class "not-men" will be represented by 1−x. And in general, whatever class of objects the symbol x stands for, the opposite class will be expressed by 1−x, which reads as "the Universe, leaving out the x's."

15. Strictly speaking, the following Proposition belongs to a later chapter of this book, the one devoted to maxims or necessary truths. But the law of thought it concerns is so important that it seemed right to introduce it here.

Proposition IV.

That axiom of metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for any being to possess a quality, and at the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x2=x.

Let us write this equation in the form x−x2=0, from which we get x(1−x)=0;(1) both of these steps being justified by the axiomatic laws of combination and transposition (II. 13). In words, the equation (1) says that the class of things that are both x and not-x is empty. To make the idea simpler, let us give the symbol x the particular meaning men. Then 1−x will represent the class of "not-men" (Prop. III.). Now the formal product of the expressions for two classes represents the class of individuals common to both (II. 6). So x(1−x) will represent the class whose members are at once "men" and "not men," and the equation (1) thus expresses the principle that a class whose members are at the same time men and not men does not exist. In other words, it is impossible for the same individual to be at the same time a man and not a man. Now let us widen the meaning of the symbol x from "men" to any class of beings marked out by having any quality whatever. The equation (1) will then express that it is impossible for a being to have a quality and not to have that quality at the same time. But this is exactly the "principle of contradiction" that Aristotle described as the fundamental axiom of all philosophy. "It is impossible that the same quality should both belong and not belong to the same thing.…This is the most certain of all principles.…Wherefore they who demonstrate refer to this as an ultimate opinion. For it is by nature the source of all the other axioms."

Footnote: Τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό… Αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν… Διὸ πάντες οἱ ἀποδεικνύντες εἰς ταύτην ἀνάγουσιν ἐσχάτην δόξαν· φύσει γὰρ ἀρχὴ καὶ τῶν ἄλλων ἀξιωμάτων αὕτη πάντων.—Metaphysica, III, 3.

I have brought in this interpretation not for its immediate value in the present system, but to illustrate a significant fact in the philosophy of the powers of the intellect. What has commonly been regarded as the fundamental axiom of metaphysics is only the consequence of a law of thought, and that law is mathematical in form. I also want to draw attention to the fact that the equation (1), which expresses this fundamental law of thought, is an equation of the second degree: x appears in it squared. I will not speculate in this chapter about whether this has to be so by its very nature. But we may venture to say that if it were not so, the whole working of the understanding would be different from what it is. It is because the fundamental equation of thought is of the second degree that we analyse and classify by dividing things into pairs of opposites, or, as the technical term has it, by dichotomy (cutting in two). Now if the equation in question had been of the third degree, and could still be interpreted as such, the mind's division of things would have had to be threefold. We would have had to proceed by a kind of trichotomy (cutting in three). With the faculties we actually have, we cannot properly conceive what that would really be like; but we could still investigate its laws as an object of intellectual speculation.

Footnote: Someone might object here that if the equation x2=x exists, then so must the equation x3=x, which is of the third degree, and ask whether that equation does not point to a process of trichotomy. The answer is that the equation x3=x cannot be interpreted in the system of logic. For if we write it in either of the forms x(1−x)(1+x)=0,(2)x(1−x)(−1−x)=0,(3) we see that its interpretation, if it has one at all, must involve the meaning of the factor 1+x, or of the factor −1−x. The first cannot be interpreted, because we cannot conceive of adding any class x to the universe 1. The second cannot be interpreted, because the symbol −1 does not obey the law x(1−x)=0, which all class symbols obey. So the equation x3=x admits of no interpretation like that of the equation x2=x. Suppose, however, that the former equation were true independently of the latter. That is, suppose the act of the mind denoted by the symbol x were such that doing it a second time over, three times in all, reproduced the result of doing it once, while doing it just twice did not. Then presumably we would be able to interpret one of the forms (1), (2), which under the actual conditions of thought we cannot do. Mathematicians know of operations whose law can be fully expressed by the equation x3=x. But they are of a nature entirely foreign to the field of general reasoning. When I say it is conceivable that the law of thought might have been different from what it is, I mean only that we can frame such a hypothesis and study its consequences. Being able to do this implies no such doctrine as that the actual law of human reason is the product either of chance or of arbitrary will.

Editor's note: This is the step that made Boolean algebra the algebra of two values. Once every class symbol must obey the law that x times x is x, and the only numbers that obey it are 0 and 1, the whole system can be worked as if each symbol were either 0 or 1, nothing or everything, false or true. In 1937 Claude Shannon showed that electrical switching circuits, with each switch either open or closed, obey these same laws. That is why every digital computer calculates in Boole's algebra.

16. For reasons that the discussion above makes plain, the law of thought expressed by the equation (1) will sometimes be referred to as the "law of duality."

Chapter 4: Division of Propositions

1. We have now investigated the laws of the mental operations involved in conception, or imagination, and explained the matching laws of the symbols that represent them. The next step is to put these results to practical use. There are three uses: first, expressing the complex terms of propositions; second, expressing propositions themselves; and last, building a general method of deductive analysis. This chapter deals mainly with the first of these. As an introduction to it, we need to establish the following Proposition:

Proposition I.

All logical propositions may be considered as belonging to one or the other of two great classes, which may be called "Primary" or "Concrete Propositions," and "Secondary" or "Abstract Propositions."

Every assertion we make belongs to one of two kinds. Either it states a relation among things, or it states (or amounts to stating) a relation among propositions. An assertion about the properties of things, or the phenomena they show, or the circumstances they are in, is strictly the assertion of a relation among things. For the purposes of logic, to say that "snow is white" is the same as to say that "snow is a white thing." An assertion about facts or events, and how they connect with and depend on one another, usually amounts, for the same purposes, to saying that certain propositions about those events are related in their truth or falsehood. The first class of propositions, about things, I call "Primary." The second class, about propositions, I call "Secondary." In practice this distinction nearly, but not quite, lines up with the usual logical division of propositions into categorical and hypothetical.

For instance, the propositions "The sun shines" and "The earth is warmed" are primary. The proposition "If the sun shines the earth is warmed" is secondary. To say "The sun shines" is to say "The sun is that which shines." It states a relation between two classes of things, namely "the sun" and "things which shine." The secondary proposition, on the other hand, states a relation of dependence between the two primary propositions "The sun shines" and "The earth is warmed." I do not claim that the relation between these propositions is a relation of cause and effect, as the relation between the facts they express is. I claim only that the relation among the propositions implies the relation among the facts, and is implied by it, so closely that for the purposes of logic it can stand in for it.

2. Suppose that instead of the proposition "The sun shines" we say "It is true that the sun shines." Then we are no longer speaking directly about things. We are speaking about a proposition concerning things, namely the proposition "The sun shines." So the proposition in which we say this is a secondary one. In this way every primary proposition can give rise to a secondary proposition: the one that asserts its truth, or declares its falsehood.

The little words if, either, and or will usually signal that a proposition is secondary, but they do not have to. The proposition "Animals are either rational or irrational" is primary. It cannot be broken into "Either animals are rational or animals are irrational," and so it does not state a relation of dependence between the two propositions joined in that second, disjunctive sentence. The words either and or are in fact no test of what kind a proposition is, even though they turn up more often in secondary propositions. Even the word if can appear in a primary proposition. "Men are, if wise, then temperate," is an example. It cannot be broken into "If all men are wise, then all men are temperate."

3. It is not my purpose to discuss the strengths or weaknesses of the usual division of propositions. I will only say this about the classification used here. Its principle is clear and definite in use; it marks a real and fundamental difference among propositions; and it is essential to developing a general method of reasoning. Nor is it any objection that a primary proposition can be put into a form in which it becomes secondary. In that case we take no direct account of the things connected in the primary proposition. We consider only the proposition itself, as true or as false.

4. In this book the same symbols will be used to express both primary and secondary propositions, and, as we shall see, they obey the same laws. The difference between the two cases is not a difference of form but of interpretation. In both, the relation the proposition sets out to express will be written with the sign =. In primary propositions, the two sides joined by this sign will usually stand for the "terms" of the proposition, or, to give them their more specific names, its subject and predicate.

Proposition II.

5. To work out a general method, based on listing the possible cases, for expressing any class or collection of things that can be a "term" of a Primary Proposition.

First, suppose the class or collection is defined only by names or qualities that all its members share. Then its expression is a single term, in which the symbols for those names or qualities are joined with no sign between them, as in algebraic multiplication. So if x stands for opaque substances, y for polished substances, and z for stones, we have xyz= opaque polished stones; xy(1−z)= opaque polished substances which are not stones; x(1−y)(1−z)= opaque substances which are not polished, and are not stones; and so on for any other combination. Notice that each of these expressions obeys the same law of duality as the single symbols it contains. So xyz×xyz=xyz;xy(1−z)×xy(1−z)=xy(1−z); and so on: taking the opaque polished stones together with the opaque polished stones gives nothing new. Any term like these we shall call a "class term," because it expresses a class of things through the properties or names its members have in common.

Second, suppose we are speaking of a collection whose different portions are defined by different properties, names, or attributes. Then the expressions for those portions must be formed separately and joined by the sign +. But if the collection was formed by removing a defined portion from some wider collection, the sign − must be placed before the symbolic expression of the portion removed. A few more remarks on these symbols are in order.

6. Broadly speaking, the symbol + is the equivalent of the words "and" and "or," and the symbol − the equivalent of the word "except." Of "and" and "or," the first is usually used when the collection described is the subject of a proposition, the second when it is the predicate. "The scholar and the man of the world desire happiness" illustrates the first case. "Things possessing utility are either productive of pleasure or preventive of pain" illustrates the second. Now whenever an expression with these words appears in a primary proposition, it matters a great deal whether the groups they separate in thought are meant to be completely distinct and mutually exclusive, or not. Does "Scholars and men of the world" include or exclude those who are both? Does "Either productive of pleasure or preventive of pain" include or exclude things that are both? My view is that, strictly speaking, "and" and "or" do have this power of separating, or excluding. On that view the statement "All x’s are either y’s or z’s," taken rigorously, means "All x’s are either y’s, but not z’s," or "z’s but not y’s." But it must also be admitted that the "jus et norma loquendi" (the law and rule of speech, that is, ordinary usage) seems rather to favour the opposite reading. "Either y’s or z’s" would generally be understood to include things that are y’s and z’s at once, along with things that are one but not the other. Remember, though, that the symbol + does have this separating power. So we must break any disjunctive expression we meet into parts that really are separate in thought, and then join their expressions with the symbol +.

So, depending on the meaning intended, the expression "Things which are either x’s or y’s" will have two different symbolic equivalents. If we mean "Things which are x’s but not y’s, or y’s but not x’s," the expression will be x(1−y)+y(1−x); the symbol x standing for x’s, and y for y’s. If, however, we mean "Things which are either x’s, or, if not x’s, then y’s," the expression will be x+y(1−x). This expression allows for things that are both x’s and y’s at once. It could be written out more fully as xy+x(1−y)+y(1−x); but adding the first two terms of this, the x’s that are y’s and the x’s that are not, simply gives back all the x’s, and so the previous expression.

Notice that the expressions just given obey the fundamental law of duality (III. 16). So we have {x(1−y)+y(1−x)}2=x(1−y)+y(1−x),{x+y(1−x)}2=x+y(1−x). We shall see later that this is only one case of a general law of expressions that stand for "classes or collections of things."

7. The results of these inquiries can be summed up in the following rule of expression.

Rule.—Express simple names or qualities by the symbols x, y, z, etc., and their contraries by 1−x, 1−y, 1−z, etc. Express classes of things defined by common names or qualities by joining the corresponding symbols as in multiplication. Express collections of things made up of portions that differ from one another by joining the expressions for those portions with the sign +. In particular, express "Either x’s or y’s" by x(1−y)+y(1−x) when the classes denoted by x and y are exclusive, and by x+y(1−x) when they are not exclusive. In the same way, express "Either x’s, or y’s, or z’s" by x(1−y)(1−z)+y(1−x)(1−z)+z(1−x)(1−y) when the classes denoted by x, y, and z are meant to be mutually exclusive, and by x+y(1−x)+z(1−x)(1−y) when they are not meant to be exclusive; and so on.

8. On this rule of expression rests the reverse rule, the rule of interpretation. The following examples should illustrate both well enough. For brevity, leaving out the universal subject "things" or "beings," let us take x=hard,y=elastic,z=metals; and we get the following results:

"Non-elastic metals" will be expressed by z(1−y), the metals that are not elastic;

"Elastic substances with non-elastic metals," by y+z(1−y);

"Hard substances, except metals," by x−z;

"Metallic substances, except those which are neither hard nor elastic," by z−z(1−x)(1−y), or by z{1−(1−x)(1−y)} (see (6), Chap. II.).

In the last example, what we really had to express was "Metals, except not hard, not elastic, metals." Words like "and" placed between adjectives usually add nothing, and so they must not be expressed in symbols.

So "Metals hard and elastic" is the same as "Hard elastic metals," and is expressed by xyz.

Take next the expression "Hard substances, except those which are metallic and non-elastic, and those which are elastic and non-metallic." Here the word those means hard substances. So the expression really means hard substances, except hard substances that are metallic and non-elastic, and hard substances that are non-metallic and elastic; the word except covers both of the classes that follow it. The complete expression is x−{xz(1−y)+xy(1−z)};or, x−xz(1−y)−xy(1−z). that is, the hard things, with the hard non-elastic metals and the hard elastic non-metals taken away.

9. The Proposition just given, with its illustrations, is a necessary first step toward the next one, which will complete the plan of this chapter.

Proposition III.

To work out, by examining their possible kinds, a general method for expressing Primary or Concrete Propositions.

A primary proposition, in the most general sense, consists of two terms and asserts that a relation holds between them. These terms need not be names of one word. They may stand for any collection of objects, of the sort we have just been considering. How to express the terms is therefore already covered by the general rules above. What remains is to find out how to express the relations between them. This will plainly depend on the kind of relation, and above all on whether, in that relation, the terms are understood to be universal or particular. That is, it depends on whether we speak of the whole of the collection a term refers to, or speak indefinitely of the whole or of a part of it, which is what the word "some" usually means.

Suppose we want to express a relation of identity between the two classes "Fixed Stars" and "Suns," that is, to express that "All fixed stars are suns" and "All suns are fixed stars." Here, if x stands for fixed stars and y for suns, we have x=y as the equation required. It reads: the fixed stars and the suns are one and the same class.

In the proposition "All fixed stars are suns," the term "all fixed stars" would be called the subject, and "suns" the predicate. Let us stretch the meaning of subject and predicate as follows. By subject we will mean the first term of any affirmative proposition, the term that comes before the copula is or are. By predicate we will mean the second term, the one that comes after the copula. And let us allow that either of them may be universal or particular, so that either the whole class is meant, or only a part of it. Then we have the following Rule for cases like the last example:—

10. Rule.—When both Subject and Predicate of a Proposition are universal, form the separate expressions for them, and connect them by the sign =.

This case usually comes up in the definitions of a science, or of subjects handled in the manner of pure science. Mr. Senior’s definition of wealth is a good example:

"Wealth consists of things transferable, limited in supply, and either productive of pleasure or preventive of pain."

Before putting this definition into symbols, notice that the word and here adds nothing. Wealth is defined by having three properties or qualities, not by being made up of three classes or collections of objects. So, leaving out the and, let us set w=wealth.t=things transferable.s=limited in supply.p=productive of pleasure.r=preventive of pain.

Now it is clear from the subject that the phrase "Either productive of pleasure or preventive of pain," in this definition, is meant to say "Either productive of pleasure; or, if not productive of pleasure, preventive of pain." So the class of things this phrase would define on its own would consist of all things productive of pleasure, together with all things that are not productive of pleasure but are preventive of pain. Its symbolic expression would be p+(1−p)r. that is, the pleasure-giving things, plus the things that give no pleasure but prevent pain.

Now we put this expression in brackets, to show that both its terms are meant, and attach to it the symbols s and t, which limit it to things "transferable" and "limited in supply." That gives the following symbolic equivalent of the original definition: w=st{p+r(1−p)}.(1) In words: wealth is exactly the class of things that are transferable, limited in supply, and either pleasure-giving or, failing that, pain-preventing.

Suppose instead that "Either productive of pleasure or preventive of pain" were meant to pick out only things productive of pleasure without being preventive of pain, p(1−r), or preventive of pain without being productive of pleasure, r(1−p), leaving out the things that are both. Then the definition in symbols would be w=st{p(1−r)+r(1−p)}.(2)

All this agrees with what was said earlier in more general terms. The reader may wonder what would happen if we translated the phrase "Things productive of pleasure or preventive of pain" word for word, as p+r, making the equation of the definition w=st(p+r).(3)

The answer is that this would be equivalent to (2), with one extra implication: that the classes of things denoted by stp and str are quite distinct. In other words, it would imply that among things transferable and limited in supply there exist none in the universe that are at once productive of pleasure and preventive of pain. How to find the full meaning of any equation will be explained later. What has been said may show that before we try to translate our data into the strict language of symbols, it is above all necessary to find out what our words are intended to mean. But no one who values correct thinking, and who regards the right use of language as both its tool and its safeguard, will count this necessity as a burden.

11. Let us next consider the case in which the predicate of the proposition is particular, for example "All men are mortal."

Here our meaning is clearly "All men are some mortal beings," and we must find the expression for the predicate, "some mortal beings." So let v stand for a class indefinite in every respect but one: some of its members are mortal beings. And let x stand for "mortal beings." Then vx will stand for "some mortal beings." So if y stands for men, the equation we want is y=vx. It reads: the men are some of the mortal beings, that is, the class of men is some part, we do not say which, of the class of mortals.

Editor's note: The indefinite class v is Boole's way of saying "some." Later logic stopped using it. Today "Some X are Y" is read as saying that the class of things that are both X and Y is not empty, and the idea of "some" became what logicians call the existential quantifier, "there exists at least one."

From these considerations we get the following Rule for expressing an affirmative universal proposition whose predicate is particular:

Rule.—Express as before the subject and the predicate, attach to the latter the indefinite symbol v, and equate the expressions.

Clearly v is a symbol of the same kind as x, y, and the rest, and it obeys the general law v2=v,orv(1−v)=0. that is, the indefinite class, like every class, taken together with itself is just itself.

So to express the proposition "The planets are either primary or secondary," we would follow the rule like this. Let x represent planets (the subject); y=primary bodies;z=secondary bodies; then, assuming that the word "or" separates the class of "primary" bodies completely from the class of "secondary" bodies, as far as they come into the proposition, we find as the equation of the proposition x=v{y(1−z)+z(1−y)}.(4) that is, the planets are some of the things that are either primary and not secondary, or secondary and not primary.

It is worth noticing that in this case the literal translation of the premises into the form x=v(y+z)(5) would be exactly equivalent, because v is an indefinite class symbol. Still, form (4) is the better one, because the expression y(1−z)+z(1−y) consists of terms that stand for classes quite distinct from each other, and it obeys the fundamental law of duality.

Take the proposition "The heavenly bodies are either suns, or planets, or comets," and represent these classes of things by w, x, y, z, respectively. If we suppose that none of the heavenly bodies belongs to two of these divisions at once, its expression will be w=v{x(1−y)(1−z)+y(1−x)(1−z)+z(1−x)(1−y)}

But suppose it were meant that the heavenly bodies were either suns, or, if not suns, planets, or, if neither suns nor planets, fixed stars. That meaning does not rule out some of them belonging at once to two, or to all three, of the divisions of suns, planets, and fixed stars. Then the expression required would be w=v{x+y(1−x)+z(1−x)(1−y)}.(6)

The examples above are descriptions, not definitions. In fact the predicates of propositions are usually particular. When they are not, either the predicate is a singular term (the name of one individual), or instead of the copula "is" or "are" we use some other form of connection that tells us the predicate is to be taken universally.

12. Consider next the case of universal negative propositions, for example "No men are perfect beings."

Plainly we are not speaking here of a class called "no men" and asserting that all its members are "perfect beings." What we are really doing is making an assertion about "all men": that they are "not perfect beings." So the true meaning of the proposition is this:

"All men (subject) are (copula) not perfect (predicate);" so if y stands for "men," and x for "perfect beings," we have y=v(1−x), that is, the men are some of the beings that are not perfect; and the same goes for any other case. So we have the following Rule:

Rule.—To express any proposition of the form "No x’s are y’s," convert it into the form "All x’s are not y’s," and then proceed as in the previous case.

13. Consider, last, the case in which the subject of the proposition is particular, for example "Some men are not wise." Here, as we have noted, the negative not may properly be attached, at least for the purposes of logic, to the predicate wise. We do not mean to say that it is untrue that "Some men are wise." We mean to say of "some men" that they lack wisdom. So the proper form of the proposition is "Some men are not-wise." Now put y for "men," and x for "wise," that is, "wise beings." Bring in v as the symbol of a class indefinite in every respect but this one: it contains some individuals of the class it is placed before. Then we have vy=v(1−x). It reads: some men are the same as some not-wise beings, that is, some part of the class of men lies inside the class of the not-wise.

14. We can gather everything we have settled into the following general Rule:

General Rule for the Symbolic Expression of Primary Propositions.

First, if the proposition is affirmative, form the expression of the subject and that of the predicate. If either of them is particular, attach to it the indefinite symbol v, and then equate the resulting expressions.

Second, if the proposition is negative, first express its true meaning by attaching the negative word to the predicate, then proceed as above.

One or two more examples should be enough to illustrate it.

Ex.—"No men are placed in exalted stations, and free from envious regards."

Let y stand for "men," x for "placed in exalted stations," and z for "free from envious regards."

Now the class described as "placed in exalted station" and "free from envious regards" is expressed by xz. So the opposite class, those to whom this description does not apply, will be represented by 1−xz, and all men are referred to this class. So we have y=v(1−xz). that is, the men are some of the beings who are not both highly placed and free from envy. If the proposition had been put in the equivalent form "Men in exalted stations are not free from envious regards," its expression would have been yx=v(1−z). We shall see later that this expression really is equivalent to the previous one, given the assumption involved, namely that v is an indefinite class symbol.

Ex.—"No men are heroes but those who unite self-denial to courage."

Let x = "men," y = "heroes," z = "those who practise self-denial," and w = "those who possess courage."

What is really asserted is that "men who do not possess courage and practise self-denial are not heroes."

So we have x(1−zw)=v(1−y) as the equation required: the men who do not have both self-denial and courage are some of the beings who are not heroes.

15. To close this chapter, it may be interesting to set side by side the great leading types of propositions as expressed in symbols. If we agree to let X and Y stand for the symbolic expressions of the "terms," the things related, these types will be X=vY,X=Y,vX=vY. In the first, only the predicate is particular; in the second, both terms are universal; in the third, both are particular. Some lesser forms are really included under these. For instance, if Y=0, the second form becomes X=0; which says that there is no such thing as an X; and if Y=1 it becomes X=1; which says that X is the whole universe of discourse. Both these forms can be interpreted. Notice also that the expressions X and Y, if they rest on a careful enough analysis of what the "terms" of the proposition mean, will obey the fundamental law of duality, which requires that X2=XorX(1−X)=0,Y2=YorY(1−Y)=0.

Chapter 5: Principles of Symbolic Reasoning

1. The chapters so far have done three things. They have investigated the fundamental laws of the operations of the mind in reasoning. They have shown how those laws become the laws of the symbols of logic. And they have set out the principles of expression by which one kind of proposition, the kind called primary, can be written in the language of symbols. All of this was, in the strictest sense, preparation. But it is preparation we cannot do without, because it leads to one of the chief aims of this book: to build a system, or method, of logic on an exact summary of the fundamental laws of thought. Before we start, I think it necessary to point out certain things about the nature of this aim and about the means of reaching it.

2. My first point is that a method in logic can be general only so far as its elementary processes and laws are general. Take one example. In the earlier sections we investigated, among other things, the laws of the logical process of addition, the process symbolized by the sign +. We found those laws by studying instances. In every one of those instances it was a necessary condition that the classes or things added together in thought should be mutually exclusive. Indeed the expression x+y seems to make no sense unless we assume that the things represented by x and the things represented by y are entirely separate, with no individuals in common. Conditions much like this one were also present in the acts of conception from which we learned the laws of the other symbolic operations. So a question arises. Must we restrict our use of these symbolic laws and processes by the same conditions of interpretability under which we first learned them? If we must, then plainly no general method in logic is possible. If we need not, then how are we to think about processes that seem to have no meaning in the very field of thought they are meant to serve? These questions do not belong to logic alone. They apply equally to every developed form of human reasoning that works through a symbolic language.

3. My second point is that this apparent failure of process and interpretation to match up never shows itself in the ordinary uses of human reason. There, no operation is performed whose meaning and use are not seen. Most minds are not satisfied when merely formal reasoning links their premises to their conclusions. Every step of the chain, every intermediate result established along the way, must make sense too. And no doubt this is both how ordinary reasoning actually works and an important safeguard in the reasoning and talk of everyday life.

Perhaps many people would want to extend the same principle to the general use of symbolic language as a tool of reasoning. They might argue like this. The laws or axioms that govern our use of symbols were established by studying only the cases in which interpretation is possible. So we have no right to apply them to other cases, where interpretation is impossible or doubtful, even if (as should be admitted) we apply them only in the intermediate steps of a proof. If this objection were conclusive, we would have to admit that a symbolic method in logic offers little advantage. Perhaps the advantage would shrink to the mechanical gain of using short, handy symbols in place of clumsier ones. But the objection itself is mistaken. Whatever we might expect à priori, it is an unquestionable fact that a conclusion reached by any symbolic process of reasoning is valid whether or not we can interpret the formal results that turned up at the different stages along the way. There are, in fact, certain general principles about the use of symbolic methods. I shall first state them as they apply to logic in particular, and then offer some remarks on their nature and on why they deserve to be accepted.

4. The conditions of valid reasoning with the help of symbols are these.

First, a fixed interpretation must be given to the symbols used to express the data, and the laws by which those symbols combine must be correctly worked out from that interpretation.

Second, the formal processes of solution or proof must be carried out from start to finish in obedience to all the laws found in this way, without asking whether the particular results obtained along the way can be interpreted.

Third, the final result must be interpretable in form, and it must actually be interpreted according to the same system of interpretation that was used to express the data. On these principles the following observations may be made.

5. The need for a fixed interpretation of the symbols has already been discussed enough (II. 3). The need for the final result to be in a form to which that interpretation can be applied rests on an obvious principle. Symbols are a means to an end, and the end is knowledge of some fact or truth we can understand. For that end to be reached, the final result, which expresses the symbolic conclusion, must be in an interpretable form. The greatest difficulty, however, is likely to be felt over the second of the general principles or conditions above (V. 4), and on this point a few more words are needed.

I would say, then, that this principle may be seen as resting on a general law of the mind. We do not know that law à priori, that is, before experience. We learn it the way we learn the other laws of the mind: by seeing the general principle clearly displayed in a particular instance. Take a single example of reasoning in which symbols are used according to laws founded on their interpretation, but without keeping that interpretation constantly in view, so that the chain of proof leads us through intermediate steps that cannot be interpreted to a final result that can. Such an example seems not only to establish that this particular application is valid, but also to reveal the general law at work in it. No pile of further instances can properly add weight to that evidence. More instances may give us a clearer idea of the common element of truth on which the principle rests, and so prepare us to accept it. Where someone does not feel the direct force of the evidence, they may serve as a check, à posteriori, that the principle works in practice. But none of this touches the point I am making: the general principle must be seen in the particular instance, seen to be general in its reach as well as true in the special example. The use of the uninterpretable symbol −1 in the intermediate steps of trigonometry illustrates what I mean. I believe there is no way of explaining that use which does not quietly assume the very principle in question. That principle is not, I think, justified by formal reasoning from any other grounds. But it seems to deserve a place among those self-evident truths which in some sense make general knowledge possible at all, and which may rightly be regarded as expressions of the mind's own laws and make-up.

6. Here is how this principle will be applied in the present work. We have seen that any system of propositions can be expressed by equations in symbols x, y, z. Wherever interpretation is possible, these symbols obey laws identical in form with the laws of a system of number symbols that can take only the values 0 and 1 (II. 15). But the formal processes of reasoning depend only on the laws of the symbols, not on what the symbols are taken to mean. So we are allowed to treat the symbols x, y, z as if they were number symbols of exactly that kind. In practice, we may set aside the logical meaning of the symbols in a given equation; turn them into number symbols that can take only the values 0 and 1; carry out on them, as numbers, every process the solution requires; and at the end give them back their logical meaning. This is the procedure actually used in this book, though I shall not stop each time to restate what kind of change has been made. When the symbols x, y, z are treated as numbers of this kind, the processes we apply to them are not bound by the conditions of thought that would bind them if they were applied to purely logical symbols. That gives us a freedom of operation without which the search for a general method in logic would be hopeless.

Now this system of processes would lead us to nothing intelligible unless the final equations it produced were in a form that could be interpreted once the symbols had their logical meaning back. There is, however, a general method of bringing equations into such a form, and the rest of this chapter is devoted to it. I shall say little here about how the method makes interpretation possible, since that question is kept for the next chapter. Instead I shall confine myself mostly to the process itself, which may be called a process of "development." Before describing it, a few observations will help.

7. Suppose we are looking at some class of things with one question in mind: which of its members have a certain property x, and which lack it? Every individual in the class either has the property or does not. So we can divide the class into two portions: those individuals that have the property, and those that do not. We can make this division in thought before we know anything else about how the class is made up. Whatever else we later learn can only tell us, more or less precisely, what further conditions hold for the portion that has the property and for the portion that lacks it. Suppose, then, we learn this much: the members that have the property x also have a certain property u, and these two conditions together are enough to define them. Then we can represent that portion of the original class by the expression ux (II. 6), that is, the things that are both u and x. Suppose we also learn that the members of the original class that do not have the property x satisfy a condition v, and are defined by it. Then clearly those members will be represented by the expression v(1−x), the things that are v and not x. So the class as a whole will be represented by ux+v(1−x); in words, the x's that are u, together with the not-x's that are v. This may be regarded as a general developed form for expressing any class of objects considered with respect to whether they have or lack a given property x.

The general form we have just reached on purely logical grounds can also be derived by a separate route, from the formal laws that the symbols x, y, z obey equally under their logical and under their numerical interpretation, as mentioned above (V. 6).

8. Definition.—Any algebraic expression containing a symbol x is called a function of x, and may be written in the short general form f(x). Any expression containing two symbols, x and y, is likewise called a function of x and y, and may be written in the general form f(x,y); and so on for any number of symbols.

So the form f(x) could stand for any of the following functions, for instance x, 1−x, 1+x1−x, and so on; and f(x,y) could stand equally for any of the forms x+y, x−2y, x+yx−2y, and so on.

By the same notation, if in any function f(x) we replace x by 1, the result is written f(1); if in the same function we replace x by 0, the result is written f(0). For example, if f(x) stands for the function a+xa−2x, then f(1) will stand for a+1a−2, and f(0) will stand for aa.

9. Definition.—Take any function f(x) in which x is a logical symbol, or a number symbol that can take only the values 0 and 1. The function is said to be developed when it has been brought into the form ax+b(1−x), with a and b chosen so that the result is equivalent to the function we started from. In words, a developed function says how much of it goes with the x's (the coefficient a) and how much with the not-x's (the coefficient b).

This definition assumes that any function f(x) can be written in this form. The following Proposition shows that the assumption is justified.

Proposition I.

10. To develop any function f(x) in which x is a logical symbol.

By the principle laid down in this chapter, we may lawfully treat x as a number symbol that can take only the values 0 and 1.

Let us assume, then, that f(x)=ax+b(1−x), and put x=1. The second term vanishes, because one minus one is zero, and we have f(1)=a. Next, in the same equation, put x=0. Now the first term vanishes, and we have f(0)=b. So the values of a and b are found. Substituting them in the first equation, we get f(x)=f(1)x+f(0)(1−x);(1) which is the development we were looking for. Read aloud, it says: the function equals what it becomes when x is 1, multiplied by x, plus what it becomes when x is 0, multiplied by one minus x. The right-hand side of this equation correctly represents the function f(x), whatever form that function may have. The reason is simple. Regarded as a number symbol, x can take only the values 0 and 1, and for each of these values the development f(1)x+f(0)(1−x), takes the same value as the function f(x). When x is 1, it reduces to the first coefficient; when x is 0, to the second; and those are exactly the values of the function itself.

Footnote: Some readers may be interested to notice that the development of f(x) found in this chapter holds, in the logical system, exactly the place that the expansion of f(x) in ascending powers of x holds in ordinary algebra. In fact it can be obtained by taking the well-known theorem of Taylor, namely f(x)=f(0)+f′(0)x+f′′(0)x21⋅2+f′′′(0)x31⋅2⋅3, &c.(1) and bringing in the condition x(1−x)=0. From that condition we find x2=x, x3=x, and so on, and so f(x)=f(0)+{f′(0)+f″(0)1⋅2+f‴(0)1⋅2⋅3+ &c.}x.(2) But putting x=1 in (1), we get f(1)=f(0)+f′(0)+f″(0)1⋅2+f‴(0)1⋅2⋅3+ &c. so that f′(0)+f″(0)1⋅2+ &c.=f(1)−f(0), and (2) becomes, on substitution, f(x)=f(0)+{f(1)−f(0)}x,=f(1)x+f(0)(1−x), which is the form in question. This proof assumes that f(x) can be expanded in a series of ascending powers of x, so it is less general than the one in the text.

As an illustration, suppose we are asked to develop the function 1+x1+2x. When x=1, we find f(1)=23, and when x=0, we find f(0)=11, or 1. So the required expression is 1+x1+2x=23x+1−x; and this equation holds for each of the values the symbol x can take. (Try it: with x equal to 1 both sides come to two-thirds, and with x equal to 0 both sides come to 1.)

Proposition II.

To expand or develop a function of any number of logical symbols.

Let us begin with the case of two symbols, x and y, and let us write the function to be developed as f(x,y).

First, we treat f(x,y) as a function of x alone and expand it by the general theorem (1). This gives f(x,y)=f(1,y)x+f(0,y)(1−x);(2) where f(1,y) stands for what the given function becomes when, in place of x, we write 1 in it; and f(0,y) stands for what the same function becomes when, in place of x, we write 0 in it.

Now we take the coefficient f(1,y), treat it as a function of y, and expand it in the same way. We get f(1,y)=f(1,1)y+f(1,0)(1−y),(3) where f(1,1) stands for what f(1,y) becomes when y is set equal to 1, and f(1,0) for what f(1,y) becomes when y is set equal to 0.

In the same way, the coefficient f(0,y) expands to f(0,y)=f(0,1)y+f(0,0)(1−y).(4) Now put into (2), in place of f(1,y) and f(0,y), their values from (3) and (4), and we have f(x,y)=f(1,1)xy+f(1,0)x(1−y)+f(0,1)(1−x)y(5)+f(0,0)(1−x)(1−y),(6) which is the expansion required. Here f(1,1) stands for what f(x,y) becomes when we set x=1, y=1 in it; f(1,0) for what f(x,y) becomes when we set x=1, y=0; and so on for the rest. In words, the four terms sort everything into four compartments: things that are x and y; things that are x but not y; things that are y but not x; and things that are neither. Each compartment is multiplied by the value the function takes when x and y are set to 1 or 0 to match it.

So, for example, if f(x,y) stands for the function 1−x1−y, we find f(1,1)=00,f(1,0)=01,f(0,1)=10,f(0,0)=1 and so the expansion of the given function is 00xy+0x(1−y)+10(1−x)y+(1−x)(1−y). We shall see in the next chapter that the forms 00 and 10 have a very important logical interpretation in expressions like this one. The first of them, zero over zero, is known to mathematicians as the symbol of an indeterminate quantity.

Next, suppose the function to be expanded contains three symbols, so that we can write it in the general form f(x,y,z). Proceeding as before, we get f(x,y,z)=f(1,1,1)xyz+f(1,1,0)xy(1−z)+f(1,0,1)x(1−y)z+f(1,0,0)x(1−y)(1−z)+f(0,1,1)(1−x)yz+f(0,1,0)(1−x)y(1−z)+f(0,0,1)(1−x)(1−y)z+f(0,0,0)(1−x)(1−y)(1−z) where f(1,1,1) stands for what the function f(x,y,z) becomes when we set x=1, y=1, z=1 in it, and so on for the rest. With three symbols there are eight compartments, one for each way of having or lacking each of the three properties.

11. It is now easy to see the general law that governs the expansion of any given function, and to turn the method of expanding into a rule. But before stating that rule, it will help to make the following observations.

Each expansion we have obtained is made up of certain terms that contain the symbols x, y, and so on, multiplied by coefficients that do not contain those symbols. So the expansion of f(x) consists of the two terms x and 1−x, multiplied by the coefficients f(1) and f(0) respectively. And the expansion of f(x,y) consists of the four terms xy, x(1−y), (1−x)y, and (1−x)(1−y), multiplied by the coefficients f(1,1), f(1,0), f(0,1), f(0,0) respectively. The terms x, 1−x in the first case, and the terms xy, x(1−y), and so on in the second, we shall call the constituents of the expansion. Each constituent names one of the compartments described above: xy the things that are both x and y, x times one minus y the things that are x but not y, and so on down to the things that are neither. Clearly their form does not depend on the form of the function being expanded. In the constituent xy, the symbols x and y are called its factors.

The general rule of development will therefore have two parts. The first tells us how to form the constituents of the expansion; the second, how to find their coefficients. It runs as follows.

First. To expand any function of the symbols x, y, z.—Form a series of constituents in this way. Let the first constituent be the product of all the symbols. In this product change any one symbol, say z, into 1−z, to get the second constituent. Then in both of these change another symbol, say y, into 1−y, to get two more constituents. Then in the four constituents now obtained change another symbol, say x, into 1−x, to get four new ones; and so on, until there are no more symbols left to change. Each change doubles the list, so two symbols give four constituents, three give eight, and so on.

Second. To find the coefficient of any constituent.—If that constituent has x as a factor, change x into 1 in the original function; but if it has 1−x as a factor, change x into 0 in the original function. Do the same for the symbols y, z, and so on. The value of the function once all these changes are made, worked out in full, is the coefficient we want.

The sum of the constituents, each multiplied by its own coefficient, is the expansion required.

Editor's note: What Boole calls the development of a function into its constituents is what is now called its disjunctive normal form, and his constituents are what computer scientists call minterms. Listing all the constituents (two of them, then four, then eight, and so on) and keeping only the ones a premise allows is the very method Jevons teaches by hand in Lesson 23 of his Elementary Lessons in Logic, which is in this collection. It is also the ancestor of the truth table.

12. It is worth noticing that a function can be developed with respect to symbols it does not actually contain. Suppose, for example, that we follow the rule to develop the function 1−x with respect to the symbols x and y. We have,

Whenx=1andy=1the given function=0.
x=1"y=0" "=0.
x=0"y=1" "=1.
x=0"y=0" "=1.

So the development is 1−x=0xy+0x(1−y)+(1−x)y+(1−x)(1−y); and this is a true development. Adding the terms (1−x)y and (1−x)(1−y) gives back the function 1−x. In words: the not-x's are just the not-x's that are y together with the not-x's that are not y.

The symbol 1, developed by the rule with respect to the symbol x, gives x+1−x. Developed with respect to x and y, it gives xy+x(1−y)+(1−x)y+(1−x)(1−y). In the same way, developed with respect to any set of symbols, it produces a series made up of all the possible constituents of those symbols. Since 1 stands for the whole universe of discourse, this says that the compartments together fill the universe.

13. A few more remarks on the nature of these general expansions are in order. Let us take as our example the general theorem (5), the pattern of development for functions of two logical symbols.

First, that theorem is entirely true and meaningful when x and y are number symbols of the kind considered in this chapter, whatever algebraic form the function f(x,y) is given. So it may be used, with full meaning, at any stage of the work between the moment we change the symbols from their logical to their numerical interpretation, as described above, and the moment we finally give them back their logical interpretation.

Second, the theorem is entirely true and meaningful when x and y are logical symbols, provided the function f(x,y) has a form that represents a class or collection of things. In that case the right-hand side can always be interpreted logically. For instance, if f(x,y) stands for the function 1−x+xy, applying the theorem gives 1−x+xy=xy+0x(1−y)+(1−x)y+(1−x)(1−y),=xy+(1−x)y+(1−x)(1−y), and this result is both meaningful and true. Read in words, the function names the not-x's together with the x's that are y; the expansion names the same things compartment by compartment: the things that are x and y, the y's that are not x, and the things that are neither. Only the compartment of x's that are not y is left out.

So we may regard the theorem as true and meaningful for number symbols of the kind described always, and for logical symbols always when interpretable. Whenever it is used in this book, then, it must be understood that if the expansion obtained cannot be interpreted, the symbols x, y are being taken as number symbols of that particular kind.

But although the expansion cannot always be interpreted directly, it always leads us straight to results that can. For example, the expression x−y gives, on development, the form x(1−y)−y(1−x), which in general cannot be interpreted. We cannot, in thought, take away from the class of things that are x's and not y's the class of things that are y's and not x's, because the second class is not contained in the first. But suppose the form x−y appeared as the left-hand side of an equation whose right-hand side was 0. Then on development we should have x(1−y)−y(1−x)=0. Now it will be shown in the next chapter that this equation, with x and y regarded as number symbols of the kind described, splits at once into the two equations x(1−y)=0,y(1−x)=0, and these equations can be read directly in logic once the symbols x and y are given logical meanings: there is nothing that is x and not y, and nothing that is y and not x. In other words, the x's and the y's are the same things. It should also be remarked that although functions do not necessarily become interpretable when developed, equations can always be brought by this process into interpretable forms.

14. The following Proposition establishes some important properties of constituents. In stating it, the symbol t is used to stand for any constituent of an expansion, whichever it may be. So if the expansion is of a function of two symbols x and y, t stands for any of the four forms xy, x(1−y), (1−x)y, and (1−x)(1−y). Where we need to represent the constituents of an expansion by single symbols and still tell them apart, we shall mark the difference with subscripts. So t1 might be used for xy, t2 for x(1−y), and so on.

Proposition III.

Any single constituent t of an expansion satisfies the law of duality, whose expression is t(1−t)=0. The product of any two different constituents of an expansion is equal to 0, and the sum of all the constituents is equal to 1.

In words: nothing both belongs to a compartment and lies outside it; no two compartments share any member; and all the compartments together make up the universe of discourse.

First, take the particular constituent xy. We have xy×xy=x2y2. But x2=x, y2=y, by the fundamental law of class symbols; so xy×xy=xy. Or, representing xy by t, t×t=t, or t(1−t)=0. In the same way the constituent x(1−y) satisfies the same law. For we have x2=x,(1−y)2=1−y,∴{x(1−y)}2=x(1−y), or t(1−t)=0. Now every factor of every constituent has either the form x or the form 1−x. So the square of each factor equals that factor, and therefore the square of the product of the factors, that is, of the constituent, equals the constituent. So, with t standing for any constituent, we have t2=t, or t(1−t)=0.

Second, the product of any two constituents is 0. This follows plainly from the general law of the symbols expressed by the equation x(1−x)=0, which says that nothing is both x and not x. For whichever two constituents of the same expansion we take, there will be at least one factor x in the one matched by a factor 1−x in the other.

Third, the sum of all the constituents of an expansion is unity, that is, 1. This is plain when we add the two constituents x and 1−x, or the four constituents xy, x(1−y), (1−x)y, (1−x)(1−y). But it is also proved, and more generally, by expanding 1 in terms of any set of symbols (V. 12). The constituents in that case are formed as usual, and all the coefficients are unity.

15. With the Proposition above we may connect the following one.

Proposition IV.

If V stands for the sum of any series of constituents, each with coefficient 1, then the condition V(1−V)=0 is satisfied. In words, any collection of whole compartments, taken together, obeys the law of duality just as a single class symbol does.

Let t1, t2 …tn be the constituents in question. Then V=t1+t2…+tn. Squaring both sides, and noting that t12=t1, t1t2=0, and so on, we have V2=t1+t2…+tn; so that V=V2. Therefore V(1−V)=0.

Chapter 6: Of Interpretation

1. We have seen that when any function is fully expanded by the general rule proved in the last chapter, the expansion is made of two distinct sets of elements: the constituents of the expansion, and their coefficients (the numbers standing in front of them). In this chapter I will ask, first, how the constituents are to be interpreted, and then how that interpretation is changed by the coefficients attached to them.

I will use the terms "logical equation," "logical function," and the like in a general sense. They mean any equation or function involving the symbols x, y, and so on, that turns up either in expressing a set of premises or in the chain of symbolic results that runs between the premises and the conclusion. If such a function or equation is in a form that cannot be read straight off in logic, then the symbols x, y, and so on, must be treated as symbols of quantity of the special kind described in earlier chapters (II. 15), (V. 6): quantities that obey the law x(1−x)=0. that is, quantities that can only be 0 or 1.

So by the problem of the interpretation of a logical function or equation, I mean two things. The first is to reduce it to a form which becomes readable once the symbols x, y, and so on, are given their logical meanings. The second is the reading that results. These agreed definitions follow the general principles for conducting the method of this book, laid down in the last chapter.

Proposition I.

2. The constituents of the expansion of any function of the logical symbols x, y, and so on, can be interpreted. They stand for the separate, non-overlapping divisions of the universe of discourse that we get by affirming and denying, in every possible combination, the qualities the symbols x, y, and so on, denote.

To make this concrete, suppose the function involves two symbols, x and y, and has been expanded with respect to both. The constituents are then: xy,x(1−y),(1−x)y,(1−x)(1−y). Clearly the first, xy, stands for the class of things that have both of the qualities named by x and y. The second, x(1−y), stands for the class that has the property x but not the property y. In the same way the third constituent stands for things that have the property represented by y but not the one represented by x. And the fourth, (1−x)(1−y), stands for things that have neither quality.

So in this case the constituents stand for all four classes of things that can be described by affirming or denying the properties expressed by x and y. These classes are distinct. No member of one is a member of another, because each class has some quality that contradicts a quality of every other class. And together they make up the universe. Any object at all can be described by whether it has or lacks a given quality, so every single thing in the universe falls into one or other of the four classes formed by combining the two given classes x and y and their contraries (their opposites, "not-x" and "not-y").

What has been said here about the constituents of f(x,y) holds in general. The constituents of any expansion stand for classes; those classes are kept distinct by having contrary qualities; and together they make up the universe of discourse.

3. These properties of constituents are expressed in the theorems proved at the end of the last chapter, and could have been deduced from them. Every constituent obeys the fundamental law that the single symbols obey; from that, one might have guessed that each constituent stands for a class. The product of any two constituents of an expansion is zero; from that, one might have concluded that the classes they stand for exclude one another. And the sum of all the constituents of an expansion is 1; from that, one might have inferred that their classes together make up the universe.

4. The analysis and interpretation of logical equations rest on these laws of constituents and on the way of interpreting them just set out. We have already said that every such equation can be interpreted by means of the theorem of development. Here I want to examine the forms of solution that come out at the end of a train of reasoning, and to show how they arise. Strictly speaking they are all versions of one basic type of expression. Still, it will be clearer to present their minor varieties one at a time.

There are three such forms, as follows:

Form I.

5. The first form arises when a logical equation V=0 is developed, broken up into its component equations, and the result is to be interpreted. We assume the function involves the logical symbols x, y, and so on, in combinations that are not fractions. Fractions in fact arise only in the kind of problem we will meet under the third form of solution.

Proposition II.

To interpret the logical equation V=0.

For simplicity, suppose V involves just two symbols, x and y, and write the development of the given equation as axy+bx(1−y)+c(1−x)y+d(1−x)(1−y)=0;(1) where a, b, c, and d are definite numbers.

Now suppose some coefficient, say a, is not zero. Multiply both sides of the equation by the constituent xy that this coefficient belongs to. Since the product of two different constituents is zero, every other term drops out, and we have axy=0, and so, since a is not zero, xy=0, This result does not depend at all on what the other coefficients are. Giving x and y their logical meaning, it reads: "No individuals exist that belong both to the class represented by x and to the class represented by y."

But if the coefficient a is zero, the term axy does not appear in the development (1), and we cannot deduce xy=0 from it.

In the same way, if the coefficient b is not zero, we have x(1−y)=0, which reads: "There are no individuals that belong to the class x and do not belong to the class y."

Either of these readings can also be put in a different form, as we will show later.

Taken together, the separate readings we get from all the terms of the expansion whose coefficients are not zero make up the complete interpretation of the equation V=0. The argument does not depend on how many logical symbols the function V contains. So the aim of the proposition is always reached by the following Rule:

Rule.—Develop the function V, and set equal to 0 every constituent whose coefficient is not zero. The readings of these results, taken together, make up the interpretation of the given equation.

6. As an example, take the definition of "clean beasts" in the Jewish law (Leviticus 11): "Clean beasts are those which both divide the hoof and chew the cud." Let

x = clean beasts;
y = beasts dividing the hoof;
z = beasts chewing the cud;

Then the proposition is represented by the equation x=yz that is, the clean beasts are exactly the beasts that both divide the hoof and chew the cud. We bring it to the form x−yz=0, and look for the kind of interpretation this method leads to. Developing the left-hand side fully, we get 0xyz+xy(1−z)+x(1−y)z+x(1−y)(1−z)−(1−x)yz+0(1−x)y(1−z)+0(1−x)(1−y)z+0(1−x)(1−y)(1−z). The terms whose coefficients are not zero give xy(1−z)=0,xz(1−y)=0,x(1−y)(1−z)=0,(1−x)yz=0. These equations deny that certain classes of things exist. They deny:

First, beasts that are clean and divide the hoof but do not chew the cud.

Second, beasts that are clean and chew the cud but do not divide the hoof.

Third, beasts that are clean and neither divide the hoof nor chew the cud.

Fourth, beasts that divide the hoof and chew the cud but are not clean.

All of these denials are really contained in the original proposition. And the other way round: grant these denials, and the original proposition follows necessarily. They are, in fact, the separate pieces of that proposition. In this way every primary proposition can be broken down into a series of denials that certain defined classes of things exist, and can be rebuilt from that set of denials. Someone may ask how an assertion can be made out of a series of denials. Where does the positive part come from? My answer is that the mind assumes the existence of a universe. It does not assume it à priori, as a fact independent of experience. It assumes it either à posteriori, as something learned from experience, or hypothetically, as the ground that makes assertive reasoning possible at all. Take the proposition "There are no men who are not fallible." It is a denial: it denies that "infallible men" exist. From it we may infer either hypothetically, "All men (if men exist) are fallible," or absolutely, since experience has assured us that the human race exists, "All men are fallible."

The form in which this Proposition's method presents its conclusions may be called the form of "Single or Conjoint Denial."

Form II.

7. The previous form came from developing and interpreting an equation whose right-hand side is 0. The present form, which complements it, comes from developing and interpreting an equation whose right-hand side is 1. The analysis of the last Proposition points straight to it.

In the example just discussed, we deduced from the equation x−yz=0 a joint denial that the classes represented by the constituents xy(1−z),xz(1−y),x(1−y)(1−z),(1−x)yz, exist, since their coefficients were not 0. It follows that the remaining constituents stand for classes that together make up the universe. So we have xyz+(1−x)y(1−z)+(1−x)(1−y)z+(1−x)(1−y)(1−z)=1. This amounts to saying that everything that exists belongs to one or other of the following classes:

First, clean beasts that both divide the hoof and chew the cud.

Second, unclean beasts that divide the hoof but do not chew the cud.

Third, unclean beasts that chew the cud but do not divide the hoof.

Fourth, things that are not clean beasts, do not chew the cud, and do not divide the hoof.

This form of conclusion may be called the form of "Single or Disjunctive Affirmation": single when only one constituent appears in the final equation, disjunctive when, as here, more than one does.

Any equation V=0 in which V obeys the law of duality can also be made to yield this form of interpretation. We rewrite it as 1−V=1 and develop the left-hand side. But this case is really included in the next general form. Both of the forms so far are of little importance next to the one that follows.

Form III.

8. In the two cases so far, the function to be developed was set equal to 0 and to 1 respectively. Now I will suppose it set equal to any logical symbol w. Our task is then to interpret the equation V=w, where V is a function of the logical symbols x, y, z, and so on. First, though, I think it necessary to show how the equation V=w, or, as it will usually appear, w=V, comes about.

Let us go back to the definition of "clean beasts" used in the previous examples: "Clean beasts are those which both divide the hoof and chew the cud." Suppose we want to find how "beasts chewing the cud" are related to "clean beasts" and "beasts dividing the hoof." The equation expressing the proposition is x=yz, and we will have what we want if we can find z as an interpretable function of x and y.

Treating x, y, z as symbols of quantity subject to their special law, we can solve the equation above to get z=xy.

But this equation is not yet in a form we can read. If we can bring it to such a form, it will give us the relation we want.

Developing the right-hand side of the equation, we get z=xy+10x(1−y)+0(1−x)y+00(1−x)(1−y), and we will show later (Prop. 3) that it can be read as follows:

"Beasts which chew the cud consist of all clean beasts (which also divide the hoof), together with an indefinite remainder (some, none, or all) of unclean beasts which do not divide the hoof."

9. This is one case of a problem of the widest generality in logic, which can be stated like this: "Given any logical equation connecting the symbols x, y, z, w, find an interpretable expression for the relation of the class represented by w to the classes represented by the other symbols x, y, z, and so on."

In every case the solution consists in finding, from the given equation, an expression for the symbol w in terms of the other symbols, and then making that expression readable by development. Now the given equation is always of the first degree in each of its symbols (no symbol appears squared, since the square of each equals the symbol itself). So the expression for w can always be found. In fact, whatever the form of the given equation with respect to w, if we develop it we get an equation of the form Ew+E′(1−w)=0,(1) where E and E′ are functions of the remaining symbols. In words: E is what multiplies the class w, and E′ what multiplies everything outside it. From this we have E′=(E′−E)w. Therefore w=E′E′−E(2) and once the right-hand side is expanded by the rule of development, all that is left is to interpret the result in logic by the next proposition.

If the fraction E′E′−E has factors common to its top and bottom, we are not allowed to cancel them unless they are mere numbers. The reason is that the symbols x, y, and so on, taken as quantities, may take such values of 0 and 1 as make the common factors equal to 0, and then the algebraic rule of cancelling fails. This is the case we had in mind when we remarked on the failure of the algebraic axiom of division (II. 14). The course this book follows is strictly in line with its general principles: we express the solution in the form (2), attempt no unauthorized cancelling, and interpret the result by the theorem of development.

If we want the relation of the class expressed by 1−w to the other classes x, y, and so on, we deduce from (1), just as above, 1−w=EE−E′, and the method of the following Proposition applies to interpreting this as well.

Proposition III.

10. To find the interpretation of any logical equation of the form w=V, where w is a class symbol and V is a function of other class symbols, of any form whatever.

Expand the right-hand side of the equation fully. Each coefficient in the result will fall into one of four kinds. We now take them in turn, with what each means.

First, let the coefficient be 1. This is the symbol of the universe, and the product of two class symbols stands for the individuals found in both classes. So any constituent with 1 as its coefficient must be read without limit: the whole of the class it stands for is included.

Second, let the coefficient be 0. In logic, just as in arithmetic, this is the symbol of Nothing. No part of the class represented by its constituent is to be taken.

Third, let the coefficient have the form 00. In arithmetic the symbol 00 stands for an indefinite number, unless some special circumstance fixes it. So analogy suggests that in the system of this book the same symbol should stand for an indefinite class. The following example will make clear that this is its true meaning.

Take the proposition "Men not mortal do not exist." Let us put it into symbols, and then, obeying the laws those symbols have been shown to follow, look for a reverse definition: a definition of "mortal beings" in terms of "men."

If we let y stand for "men" and x for "mortal beings," the proposition "Men who are not mortals do not exist" is expressed by the equation y(1−x)=0, from which we are to find the value of x. The equation above gives y−yx=0, or yx=y. that is, the men who are mortal are all the men there are. If this were an ordinary algebraic equation, our next step would be to divide both sides by y. But we remarked in Chap. II. that the operation of division cannot be performed with the symbols we are now using. What we can do instead is express the division, and then develop the result by the method of the last chapter. So we have, first, x=yy, and, expanding the right-hand side as directed, x=y+00(1−y). This says that mortals (x) consist of all men (y), together with whatever remainder of beings who are not men (1−y) the coefficient 00 indicates. Now let us ask what remainder of "not men" the premise implies. The remainder might include all the beings who are not men; it might include some of them and not others; or it might include none. Any one of these suppositions fits our premise perfectly. In other words, the premise in effect asserts that all men are mortal, and its truth is untouched whether all, some, or none of the beings who are not men are mortal. So the expression 00 here means that all, some, or none of the class it is attached to are to be taken.

This account of the meaning of the symbol 00 rests on one particular case. But the principle behind the argument is general, and the symbol can turn up in no circumstances where the same analysis would not apply. We may properly call 00 an indefinite class symbol, and, where convenient, replace it by a single symbol v, subject to the fundamental law v(1−v)=0.

Fourth, a coefficient of a constituent in an expansion may belong to none of the kinds above. To find its true meaning in that case, we must first prove the following theorem.

11. Theorem.—If a function V, meant to represent some class or collection of objects w, is expanded, and if the numerical coefficient a of any constituent in its development does not satisfy the law a(1−a)=0, then the constituent in question must be set equal to 0. In words: if a coefficient is anything other than 0 or 1, its constituent must be empty.

To prove the theorem in general, write the given expansion in the form w=a1t1+a2t2+a3t3+&c.,(1) where t1, t2, t3, and so on, are the constituents and a1, a2, a3, and so on, the coefficients. Suppose also that a1 and a2 do not satisfy the law a1(1−a1)=0,a2(1−a2)=0; but that all the other coefficients do obey it, so that we have a32=a3, &c. Now multiply each side of equation (1) by itself. The result is w=a12t1+a22t2+ &c.(2) This is clear because it must be the development of the equation w=V2, (and the square of w is just w). But it can also be proved by actually squaring (1) and noting that t12=t1,t22=t2,t1t2=0, &c. by the properties of constituents. Now subtracting (2) from (1), we have (a1−a12)t1+(a2−a22)t2=0. or, which is the same thing, a1(1−a1)t1+a2(1−a2)t2=0. Multiply this last equation by t1. Then, since t1t2=0, we have a1(1−a1)t1=0, whence t1=0. In the same way, multiplying the same equation by t2, we have a2(1−a2)t2=0, whence t2=0. (In each case the number in front is not zero, so the constituent itself must be.)

In this way it can be shown in general that any constituent whose coefficient does not obey the same fundamental law as the symbols themselves must be set equal to 0 on its own. Such coefficients usually appear in the form 10. This is the algebraic symbol of infinity. And the closer a number comes to infinity (if we may put it that way), the further it is from satisfying the fundamental law just mentioned.

The symbol 00, whose meaning we discussed earlier, does not necessarily break the law we are now considering, since it can take the numerical values 0 and 1 equally well. But its actual meaning, as an indefinite class symbol, cannot in my view be deduced from its arithmetical properties, except on the ground of analogy. It has to be established by trial, as we did above.

Editor's note: Boole's division, and the symbols 0/0 and 1/0, are formal devices, not arithmetic. A coefficient 0/0 marks an indefinite portion of a class (the v of Chapter IV), and a coefficient 1/0 marks a constituent that must be empty. His contemporaries found this the hardest part of the book, and Jevons, Venn and Peirce rebuilt the algebra without any division at all; modern Boolean algebra has none. His conclusions are correct; only his route to them was abandoned.

12. We can now gather the results we have reached into the following summary.

First, the symbol 1, as the coefficient of a term in a development, means that the whole of the class that constituent stands for is to be taken.

Second, the coefficient 0 means that none of the class is to be taken.

Third, the symbol 00 means that a completely indefinite portion of the class is to be taken: that is, some, none, or all of its members.

Fourth, any other symbol as a coefficient means that the constituent it stands in front of must be set equal to 0.

It follows that if the solution of a problem, found by development, has the form w=A+0B+00C+10D, that solution can be split into the two following equations: w=A+vC,(3)D=0,(4) where v is an indefinite class symbol. Equation (3) says that the class w is all of A together with some, none, or all of C; equation (4) says that the class D has no members. The reading of (3) shows what elements make up, or may make up, the class w whose definition we want. The reading of (4) shows what relations hold among the elements of the original problem, quite independently of w.

These are the rules of interpretation. I may add that they apply universally, and that using them never runs into an exception or a failure.

13. Corollary.—If V is an independently interpretable logical function, it will satisfy the symbolic law V(1−V)=0.

By an independently interpretable logical function I mean one that can be interpreted without assuming any relation among the things its symbols stand for. So x(1−y) is independently interpretable, but x−y is not. The second function assumes, as a condition of its having any meaning, that the class represented by y is wholly contained in the class represented by x. The first makes no such demand.

Now suppose V is independently interpretable, and let w stand for the collection of individuals it contains. Then the equation w=V holds true without forcing any of the constituents in the development of V to vanish, since a vanishing constituent would mean some relation among the classes of things the symbols in V denote. So the development of V will have the form a1t1+a2t2+&c. with the coefficients a1, a2, and so on, all satisfying the condition a1(1−a1)=0,a2(1−a2)=0,&c. that is, every coefficient is 0 or 1.

So, by the reasoning of Prop. 4, Chap. v., the function V will obey the law V(1−V)=0. that is, nothing is both in the class V and outside it.

This result is obvious à priori, since V is supposed to stand for a class or collection of things. But we now see that it also follows from the properties of the constituents that make it up. The condition V(1−V)=0 may be called "the condition of interpretability of logical functions."

14. The general form of solution, or of logical conclusion, set out in the last Proposition may be called a "Relation between terms." As before, I use the word "terms" for the parts of a proposition, simple or complex, that are joined by the copula "is" or "are." The classes of things represented by the single symbols may be called the elements of the proposition.

15. Ex. 1.—Going back to the definition of "clean beasts" (VI.6), we want a description of "unclean beasts."

As before, with x standing for "clean beasts," y for "beasts dividing the hoof," and z for "beasts chewing the cud," we have x=yz;(5) so 1−x=1−yz; and developing the right-hand side, 1−x=y(1−z)+z(1−y)+(1−y)(1−z); which reads as the following proposition: Unclean beasts are all those which divide the hoof without chewing the cud, all those which chew the cud without dividing the hoof, and all those which neither divide the hoof nor chew the cud.

Ex. 2.—With the same definition, we want a description of beasts that do not divide the hoof.

From the equation x=yz we have y=xz; therefore, 1−y=z−xz; and developing the right-hand side, 1−y=0xz+−10x(1−z)+(1−x)z+00(1−x)(1−z). Here, by the Rule, the term whose coefficient is −10 must be set equal to 0 on its own, and so we have 1−y=(1−x)z+v(1−x)(1−z),x(1−z)=0; The first of these equations reads as the proposition: Beasts which do not divide the hoof consist of all unclean beasts which chew the cud, and an indefinite remainder (some, none, or all) of unclean beasts which do not chew the cud.

The second equation reads as the proposition: There are no clean beasts which do not chew the cud. This is one of the independent relations mentioned above. We were looking for the direct relation of "beasts not dividing the hoof" to "clean beasts" and "beasts which chew the cud." But it happens that the premise also sets up a separate relation between clean beasts and beasts that chew the cud, quite apart from any relation to beasts not dividing the hoof. The process necessarily gives us this relation too.

Ex. 3.—Take the following definition: "Responsible beings are all rational beings who are either free to act, or have voluntarily sacrificed their freedom," and apply the same analysis to it.

Letxstand forresponsible beings.
y"rational beings.
z"those who are free to act,
w"those who have voluntarily sacrificed their freedom of action.

In expressing this definition, I will assume that its two alternatives, "rational beings free to act" and "rational beings whose freedom of action has been voluntarily sacrificed," exclude each other, so that no one is found in both. This lets us translate the proposition literally into the language of symbols, as follows: x=yz+yw.(6) that is, the responsible beings are the rational beings who are free, together with the rational beings who have given up their freedom.

First let us find from this the relation of "rational beings" to responsible beings, beings free to act, and beings who have voluntarily given up their freedom of action. Perhaps it is better to put our aim this way: we want to express the relation among the elements of the premise in a form that lets us decide how far rationality can be inferred from responsibility, freedom of action, a voluntary sacrifice of freedom, and their opposites.

From (6) we have y=x(z+w), and developing the right-hand side, but leaving out the terms whose coefficients are 0, y=12xzw+xz(1−w)+x(1−z)w+10x(1−z)(1−w)+00(1−x)(1−z)(1−w), so, setting equal to 0 the terms whose coefficients are 12 and 10, we have y=xz(1−w)+xw(1−z)+v(1−x)(1−z)(1−w);(7) xzw=0;(8) x(1−z)(1−w)=0;(9) and these read as follows.

Direct Conclusion.—Rational beings are all responsible beings who are either free to act, not having voluntarily sacrificed their freedom, or not free to act, having voluntarily sacrificed their freedom, together with an indefinite remainder (some, none, or all) of beings not responsible, not free, and not having voluntarily sacrificed their freedom.

First Independent Relation.—No responsible beings are at the same time free to act, and in the condition of having voluntarily sacrificed their freedom.

Second.—No responsible beings are not free to act, and at the same time in the condition of not having sacrificed their freedom.

The independent relations just found can, however, be put in another and handier form. Thus (8) gives xw=0z=0z+00(1−z), on development; or, xw=v(1−z);(10) and in the same way (9) gives x(1−w)=01−z=00z+0(1−z); or, x(1−w)=vz;(11) and (10) and (11), interpreted, give the following propositions:

First, Responsible beings who have voluntarily sacrificed their freedom are not free.

Second, Responsible beings who have not voluntarily sacrificed their freedom are free.

These, however, are only different forms of the relations found before.

16. In looking at these results, the reader must remember that the only business of a method of inference or analysis is to find the relations required by the connexion of the terms in the original proposition, that is, by the way the proposition joins them. So in judging how completely the method does its job, we have nothing at all to do with other relations that the meaning of the words may suggest to us, as distinct from their stated connection. For example, it seems obvious that "They who have voluntarily sacrificed their freedom are not free." That relation is implied in the very meaning of the words. So it might seem that the first of the two independent relations the method gave us is, on the one hand, needlessly narrow, and on the other, unnecessary. But if we look only at how the terms are connected in the original premise, we will see that the relation is open to neither charge. The solution, given by the direct conclusion and the independent relations together, is perfectly complete, and nothing in it is superfluous.

If we want to take into account the unstated relation just mentioned, "They who have voluntarily sacrificed their freedom are not free," we can do so by making it a separate proposition. Its proper expression would be w=v(1−z). that is, those who have sacrificed their freedom are some portion of those who are not free.

We would then have to use this equation together with the one expressing the original premise. How such an examination is to be carried out will appear when we come to the theory of systems of propositions in a later chapter. The only difference this makes to the result is that the first of the independent relations found above is replaced.

17. Ex. 4.—Taking the same definition as in Example 2, we want a description of irrational persons.

We have 1−y=1−xz+w=z+w−xz+w =12xzw+0xz(1−w)+0x(1−z)w−10x(1−z)(1−w)+(1−x)zw+(1−x)z(1−w)+(1−x)(1−z)w+00(1−x)(1−z)(1−w)=(1−x)zw+(1−x)z(1−w)+(1−x)(1−z)w+v(1−x)(1−z)(1−w)=(1−x)z+(1−x)(1−z)w+v(1−x)(1−z)(1−w), together with xzw=0,x(1−z)(1−w)=0.

The independent relations here are the same ones we reached before, as they clearly ought to be. Whatever relations hold regardless of whether a given class of things y exists must also hold regardless of whether the opposite class 1−y exists.

The direct solution given by the first equation is: Irrational persons consist of all irresponsible beings who are either free to act, or have voluntarily sacrificed their liberty and are not free to act; together with an indefinite remainder of irresponsible beings who have not sacrificed their liberty and are not free to act.

18. The propositions analyzed in this chapter have all been of the kind called definitions. I have discussed none whose second term, the predicate, is particular (that is, speaks only of "some"). The general type of such propositions is Y=vX, where Y and X are functions of the logical symbols x, y, z, and so on, and v is an indefinite class symbol. Their analysis is made much easier, though the step is not essential, by eliminating the symbol v, and that process depends on the method of the next chapter. I also put off another important problem needed to complete the theory of single propositions, because its analysis really depends on the method of reducing systems of propositions, which will be developed later in this work.

Chapter 7: On Elimination

1. In the examples of the last chapter, every element of the original premise came back in the conclusion, only in a different order and with a different connection. But in ordinary reasoning, and especially when there is more than one premise, it is more usual for some of the elements to be kept out of the conclusion. Such elements are commonly called "middle terms." We may think of them as brought into the original propositions only to help set up a connection among the other elements, and those other elements alone are meant to appear in the conclusion.

2. Some mistaken ideas are current about these intermediate elements, or middle terms. It is a general opinion, though the examples of the last chapter contradict it, that inference consists above all in eliminating such terms (combining the premises so that a term drops out). On this view, the basic pattern of the process is the elimination of one middle term from two premises, giving a single conclusion in which that term no longer appears. So it is commonly held that the syllogism is the basis, or at least the common type, of all inference, and that any inference, however complex its form and structure, can be broken down into a series of syllogisms. Whether this view is right will be considered in a later chapter. For now I want to point to a difference, important but so far unnoticed, between the system of logic expressed in symbols and the system of ordinary algebra, in the matter of elimination. In algebra we can eliminate one symbol from two equations, two symbols from three equations, and in general n−1 symbols from n equations. So there is a fixed connection between the number of independent equations we are given and the number of number symbols we can eliminate from them. In logic it is otherwise. There is no fixed connection between the number of equations representing propositions or premises and the number of class symbols that can be eliminated. From a single equation any number of such symbols may be eliminated. On the other hand, from any number of equations, only a single class symbol may be eliminated. We may say that in this unusual system the problem of elimination can be solved in every case alike. This follows from that remarkable law of duality which the symbols of logic obey. To the equations supplied by the given premises, another equation, or system of equations, is added. It is drawn from the fundamental laws of thought themselves, and it supplies what is needed to solve the problem. Of the many consequences of the law of duality, this is perhaps the one that most deserves attention.

3. In algebra, eliminating symbols from a system of equations often leads to a mere identity of the form 0=0, which means that no independent relations connect the symbols that remain. The same thing can happen in logic, and it means the same thing there. This does not weaken the general principle just stated. The method we are about to take up aims to eliminate any number of symbols from any number of logical equations, and to show in the result whatever relations actually remain. It may be that none remain. In that case the method proves its soundness by leading us to a proposition that says nothing, a mere identity.

4. The notation in the Propositions that follow is like that of the last chapter. By f(x) is meant any expression containing the logical symbol x, with or without other logical symbols. By f(1) is meant what f(x) becomes when x is changed into 1 throughout it; by f(0), what the same function becomes when x is changed into 0.

Proposition I.

5. If f(x)=0 is any logical equation containing the class symbol x, with or without other class symbols, then the equation f(1)f(0)=0 will be true, whatever x is taken to mean; and it will be the complete result of eliminating x from the equation we started with. In words: the function with x put equal to 1, multiplied by the function with x put equal to 0, gives nothing.

In other words, we eliminate x from any given equation, f(x)=0, by changing x into 1 in that equation, then changing x into 0, and multiplying the two resulting equations together.

In the same way, the complete result of eliminating any class symbols, x, y, etc.,from any equation of the form V=0, is obtained as follows. Fully expand the left-hand side of the equation in constituents of those symbols; multiply together all the coefficients of those constituents (the numbers standing in front of the compartments); and set the product equal to 0.

Developing the left-hand side of the equation f(x)=0, we have (V. 10) f(1)x+f(0)(1−x)=0;or, [f(1)−f(0)]x+f(0)=0.(1)∴x=f(0)f(0)−f(1);and 1−x=−f(1)f(0)−f(1). Now put these expressions for x and 1−x into the fundamental equation x(1−x)=0, which says that nothing is at once an x and not an x. The result is −f(0)f(1)[f(0)−f(1)]2=0;or, f(1)f(0)=0,(2) which is the form we wanted.

6. This process shows that the elimination really takes place between the given equation f(x)=0 and the equation x(1−x)=0, which is true always and everywhere, because it expresses the fundamental law of logical symbols, qua logical (that is, insofar as they are logical). So we never need more than one premise, or equation, to make it possible to eliminate a term. The necessary law of thought in effect supplies the other premise or equation. The proof of this conclusion can be set out in other forms, but the same element, supplied by the mind itself, will still be present in effect. For instance, we could proceed as follows.

Multiply (1) by x, and we have f(1)x=0,(3) and let us try to eliminate x from this equation and (1) by the methods of ordinary algebra.

Now if we have two algebraic equations of the form ax+b=0, a′x+b′=0; it is well known that the result of eliminating x is ab′−a′b=0(4) But comparing that pair of equations with (1) and (3) respectively, we find a=f(1)−f(0),b=f(0); a′=f(1)b′=0; and these, put into (4), give f(1)f(0)=0, as before. In this form of the proof, the fundamental equation x(1−x)=0 shows up in the step that takes us from (1) to (3).

7. I shall add one more form of the proof. It is half logical in character, and it may show this important theorem in a clearer light.

We have, as before, f(1)x+f(0)(1−x)=0. Multiply this equation first by x, and then by 1−x, and we get f(1)x=0f(0)(1−x)=0. From these, by solving and developing, we have f(1)=0x=00(1−x), on development,f(0)=01−x=00x. Read directly, these equations say two things.

First, whatever individuals are included in the class represented by f(1) are not x's.

Second, whatever individuals are included in the class represented by f(0) are x's.

So, by ordinary logic, there are no individuals that are at once in the class f(1) and in the class f(0); that is, there are no individuals in the class f(1)f(0). Therefore f(1)f(0)=0.(5) Or it would be enough to multiply the two developed equations together, and the result would follow at once.

8. The theorem (5) gives us the following Rule:

To Eliminate any Symbol from a Proposed Equation.

Rule.—First bring all the terms of the equation over to the left-hand side, if necessary. Then give the symbol the value 1, and then the value 0, and multiply the two resulting equations together.

This proves the first part of the Proposition.

9. Next, consider the general equation f(x,y)=0; whose left-hand side stands for any function of x, y, and other symbols.

By what we have shown, the result of eliminating y from this equation will be f(x,1)f(x,0)=0; for that is the form we reach by changing y into 1 in the given equation, then y into 0, and multiplying the results together.

Now take this result, change x into 1, then x into 0, and multiply the results together. We get f(1,1)f(1,0)f(0,1)f(0,0)=0;(6) as the final result of elimination. But the four factors on the left-hand side of this equation are exactly the four coefficients of the complete expansion of f(x,y), the left-hand side of the original equation. So the second part of the Proposition is plain.

Examples.

10. Ex. 1.—We are given the Proposition "All men are mortal," and its symbolic expression in the equation y=vx, where y stands for "men" and x for "mortals." The task is to eliminate the indefinite class symbol v and to interpret the result.

Bringing the terms over to the left-hand side, we have y−vx=0. When v=1 this becomes y−x=0; and when v=0 it becomes y=0; and these two equations, multiplied together, give y−yx=0, or y(1−x)=0, since y2=y.

This last equation is the result of elimination we wanted. It says that there are no men who are not mortals, and its interpretation is: Men who are not mortal do not exist. That is an obvious conclusion.

Now suppose we use this last equation to look for a description of beings who are not mortal. We have x=yy, ∴1−x=0y. So, by expansion, 1−x=00(1−y): the not-mortals are some, none, or all of the things that are not men. Interpreted, this gives: They who are not mortal are not men. This is an example of what ordinary logic calls conversion by contraposition, or negative conversion (turning "All men are mortal" into "Whatever is not mortal is not a man").

Footnote: Whately’s Logic, Book II. chap. II. sec. 4.

Ex. 2.—Take the Proposition "No men are perfect," represented by the equation y=v(1−x), where y stands for "men" and x for "perfect beings." The task is to eliminate v, and from the result to find a description both of perfect beings and of imperfect beings. We have y−v(1−x)=0. So, by the rule of elimination, {y−(1−x)}×y=0, or y−y(1−x)=0, or yx=0; that is, the class of things both men and perfect is empty. This reads as the Proposition: Perfect men do not exist. From that equation we have x=0y=00(1−y)by development; in words, the perfect beings are some, none, or all of the things that are not men. So, by interpretation: No perfect beings are men. Similarly, 1−x=1−0y=yy=y+00(1−y), that is, the imperfect beings are all the men, plus some, none, or all of what is not a man. Interpreted, this gives: Imperfect beings are all men with an indefinite remainder of beings, which are not men.

11. In handling propositions, it will generally be most convenient to eliminate the indefinite class symbol v first, wherever it appears in the equations. This only changes their form; it does not change what they mean. Let us apply this step to one of the examples of Chap. IV. For the Proposition "No men are placed in exalted stations and free from envious regards," we found the expression y=v(1−xz), and for the equivalent Proposition "Men in exalted stations are not free from envious regards," the expression yx=v(1−z); and we observed that these equations, with v an indefinite class symbol, were themselves equivalent. To prove it, we need only eliminate the symbol v from each. The first equation is y−v(1−xz)=0, so, putting first v=1 and then v=0, and multiplying the results, we have (y−1+xz)y=0,or yxz=0. Now the second of the given equations becomes, when its terms are moved across, yx−v(1−z)=0;whence (yx−1+z)yx=0,or yxz=0, the same as before. The reader will easily interpret the result: there is nothing that is at once a man, in an exalted station, and free from envious regards.

12. Ex. 3.—As material for the general method of this chapter, we will take up again Mr. Senior's definition of wealth, namely: "Wealth consists of things transferable, limited in supply, and either productive of pleasure or preventive of pain." As remarked earlier, we shall take this definition to include all things that have both of the qualities named in its last part. On that assumption, our equation is w=st{pr+p(1−r)+r(1−p)},or w=st{p+r(1−p)}, where

wstands forwealth.
s"things limited in supply.
t"things transferable.
p"things productive of pleasure.
r"things preventive of pain.

From this equation we can eliminate any symbols we do not wish to take into account. We can then express the result, by solving and developing, with whatever arrangement of subject and predicate we choose.

Let us first ask what the expression for w, wealth, would be if we eliminated the element r, which refers to the prevention of pain. Bringing the terms of the equation over to the left-hand side, we get w−st(p+r−rp)=0. Putting r=1, the left-hand side becomes w−st, and putting r=0 it becomes w−stp. So, by the Rule, we have (w−st)(w−stp)=0(7) or w−wstp−wst+stp=0;(8) and so w=stpst+stp−1; and developing the right-hand side of this equation gives w=stp+00st(1−p).(9) This gives us the conclusion: Wealth consists of all things limited in supply, transferable, and productive of pleasure, and an indefinite remainder of things limited in supply, transferable, and not productive of pleasure. That is obvious enough.

Notice that we do not have to carry out the multiplication shown in (7), and reduce that equation to the form (8), in order to find the expression for w in terms of the other symbols. Development can always take the place of multiplication. For instance, if we develop (7) with respect to w, we find (1−st)(1−stp)w+stp(1−w)=0, and so w=stpstp−(1−st)(1−stp); and this equation, developed, gives the same result as before: w=stp+00st(1−p).

13. Suppose next that we want a description of things limited in supply, in terms of how they stand to wealth, to transferability, and to the tendency to produce pleasure, leaving out all reference to the prevention of pain.

From equation (8), which is the result of eliminating r from the original equation, we have w−s(wt+wtp−tp)=0; and so s=wwt+wtp−tp=wtp+wt(1−p)+10w(1−t)p+10w(1−t)(1−p)+0(1−w)tp+00(1−w)t(1−p)+00(1−w)(1−t)p+00(1−w)(1−t)(1−p). We will first give the direct interpretation of this solution, term by term. Then we shall make some general remarks it suggests. Finally, we shall show how the conclusion can be stated a little more briefly.

First, then, the direct interpretation is this. Things limited in supply consist of all wealth that is transferable and productive of pleasure; all wealth that is transferable and not productive of pleasure; and an indefinite amount of what is not wealth, but is either transferable and not productive of pleasure, or intransferable and productive of pleasure, or neither transferable nor productive of pleasure.

To this, the terms whose coefficient is 10 let us add the following independent relations, namely:

First, Wealth that is intransferable, and productive of pleasure, does not exist.

Second, Wealth that is intransferable, and not productive of pleasure, does not exist.

14. I expect the following remarks to be made about this solution.

First, it may be said that in the expression we found for "things limited in supply," the term "All wealth transferable," and so on, is partly redundant. For all wealth is necessarily transferable: the original proposition implies it, and the independent relations state it outright.

My answer is this. In ordinary speech, once another part of our reasoning had shown that there is no wealth that is not transferable, we would not think it necessary to add the word "transferable" to "wealth." But part of what makes this method perfect is that it always defines completely the objects named by each term of the conclusion. It does so by stating how they stand to every quality, every mark of distinction, that we have chosen to use. This is needed to keep the different parts of the solution truly distinct and independent, and it actually prevents redundancy. Suppose the pair of terms we have been looking at had not contained the word "transferable," and had together made up just "All wealth." We could then logically split the single term "All wealth" into two terms, "All wealth transferable" and "All wealth intransferable." But the "independent relations" show that the second of these vanishes. So it is no part of the description we want, and is therefore redundant. The remaining term matches the conclusion we actually obtained.

Solutions that cannot, by any logical division, be made to yield superfluous or redundant terms may be called pure solutions. All the solutions given by the method of development and elimination explained above are of this kind. It is worth noticing that if we used the ordinary algebraic method of elimination, in the cases where that method can be used in this system, we could not count on the solutions being pure. Its lack of generality would not be its only defect.

15. Second, it will be remarked that the conclusion contains two terms whose combined meaning would be more conveniently expressed by a single term. Instead of "All wealth productive of pleasure, and transferable" and "All wealth not productive of pleasure, and transferable," we could simply say "All wealth transferable." This remark is quite fair. But notice that whenever such simplifications are possible, the form of the equation we have to interpret points them out at once; and if that equation is brought to its simplest form, the interpretation it leads to will be in its simplest form too. Thus in the original solution, the terms wtp and wt(1−p), which have unity (that is, 1) as their coefficient, add up to wt; the terms w(1−t)p and w(1−t)(1−p), which have 10 as their coefficient, add up to w(1−t); and the terms (1−w)(1−t)p and (1−w)(1−t)(1−p), which have 00 as their coefficient, add up to (1−w)(1−t). So the complete solution is s=wt+00(1−w)(1−t)+00(1−w)t(1−p), with the independent relation w(1−t)=0, or w=00t. The interpretation would now read like this.

First, things limited in supply consist of all wealth that is transferable, together with an indefinite remainder of what is not wealth and not transferable, and of transferable articles that are not wealth and are not productive of pleasure.

Second, All wealth is transferable.

This is the simplest form in which the general conclusion, with the condition that goes with it, can be stated.

16. When we need to eliminate two or more symbols from an equation, we can either use (6) Prop. I., or eliminate them one after another; the order does not matter. From the equation w=st(p+r−pr), we have already eliminated r, and found the result w−wst−wstp+stp=0. Suppose we had been asked to eliminate both r and t. Taking the step above as the first stage of the process, it remains to eliminate t from this last equation. When t=1, the left-hand side of that equation becomes w−ws−wsp+sp, and when t=0, the same side becomes w. So we have w(w−ws−wsp+sp)=0, or w−ws=0, as the result of elimination we wanted: there is no wealth that is not limited in supply.

If we use this last result to find w, we have w=01−s=00s, so "All wealth is limited in supply." Since p does not appear in the equation, this clearly holds no matter how the elements of the conclusion stand to the quality "productive of pleasure."

Going back to the original equation, suppose we are asked to eliminate s and t. We have w=st(p+r−pr). Instead of eliminating s and t one at a time by the Rule, however, it is enough to treat st as a single symbol, since it satisfies the fundamental law of the symbols, by the equation st(1−st)=0. So we write the given equation in the form w−st(p+r−pr)=0; and, putting st equal first to 1 and then to 0 and taking the product of the results, we have (w−p−r+pr)w=0,or w−wp−wr+wpr=0, as the result we were looking for. In words: there is no wealth that is neither productive of pleasure nor preventive of pain.

As a particular illustration, suppose we are asked to find an expression for "things productive of pleasure" (p) in terms of "wealth" (w) and "things preventive of pain" (r).

Solving the equation, we have p=w(1−r)w(1−r) =00wr+w(1−r)+00(1−w)r+00(1−w)(1−r) =w(1−r)+00wr+00(1−w). This gives the following conclusion: Things productive of pleasure are all wealth not preventive of pain, an indefinite amount of wealth that is preventive of pain, and an indefinite amount of what is not wealth.

From the same equation we get 1−p=1−w(1−r)w(1−r)=0w(1−r), and this, developed, gives 1−p=00wr+00(1−w)⋅r+00(1−w)⋅(1−r)=00wr+00(1−w). So: Things not productive of pleasure are either wealth, preventive of pain, or what is not wealth.

Any similar case could be worked out just as easily.

17. In the last example we eliminated the compound symbol st from the given equation by treating it as a single symbol. The same method works for any combination of symbols that satisfies the fundamental law obeyed by each single symbol. For instance, the expression p+r−pr, multiplied by itself, gives itself back. So if we stand for p+r−pr by a single symbol, say y, the fundamental law will be obeyed, since the equation y=y2, or y(1−y)=0, will be satisfied. The rule of elimination for symbols rests on the assumption that each single symbol obeys that law. So whenever a function or combination of such symbols obeys the law too, it can be eliminated from an equation in a single operation.

The forms of interpretation used in this chapter and the last show, perhaps better than any others, what the symbols 1 and 00 directly mean. But other ways of putting things, closer to everyday speech, are just as true and just as proper. For example, equation (9) can be read like this: Wealth is either limited in supply, transferable, and productive of pleasure, or limited in supply, transferable, and not productive of pleasure. And, the other way round: Whatever is limited in supply, transferable, and productive of pleasure, is wealth. Reverse readings like this one are always available when the final development brings in terms whose coefficient is unity.

18. Note.—The fundamental equation f(1)f(0)=0, which gives the result of eliminating the symbol x from any equation f(x)=0, admits of a remarkable interpretation.

Remember that the equation f(x)=0 stands for some proposition that speaks of the individuals in the class x, say "men," perhaps together with other individuals. Our aim is to find out whether the proposition implies any relation among those other individuals that does not depend on the class men. Now the equation f(1)=0 says what the original proposition would become if men made up the whole universe, and the equation f(0)=0 says what it would become if men ceased to exist. So the equation f(1)f(0)=0 says what, given the original proposition, would be true on either assumption alike: true whether "men" were "all things" or "nothing." The theorem therefore says that whatever is true both when a given class of objects takes in the whole universe and when it vanishes from existence is independent of that class altogether, and vice versâ. Here we have another example of formal results, drawn directly from the mathematical laws of thought, being interpreted as general axioms of philosophy.

Chapter 8: On the Reduction of Systems of Propositions

1. In the preceding chapters we worked out, well enough for the most essential purposes, the theory of single primary propositions; or, to put it more exactly, of primary propositions expressed by a single equation. On that theory we built a method that does the job. We showed how to eliminate any element from the given system of equations, and how to deduce the relation that connects the elements left over, in whatever form is asked for: a denial, an affirmation, or the more usual relation of subject and predicate. What remains is to turn to systems of propositions, and to carry out the same series of investigations for them. We want to know three things. Can we eliminate, ad libitum (as many as we please), any number of the symbols from the equations that express a system of propositions? Can we then deduce, by interpreting the result, every relation that holds among the symbols that remain? And can we find, in particular, the expression of any single element, or of any meaningful combination of elements, in terms of the others, so as to state the conclusion in whatever acceptable form is wanted? The answer to all three will come from showing that any system of equations, or any of the equations in a system, can be reduced to a single equation that is equivalent to it. To that single equation the methods of the previous chapters can be applied at once. We will see, too, that this reduction brings with it an important extension of the theory of single propositions, one we had to do without in the earlier discussion. There is nothing unusual about this. Many special branches of science cannot be fully surveyed from the inside. They have to be studied from an outside point of view as well, and seen in connection with other, related subjects, before their full proportions can be understood.

This chapter will set out two distinct ways of reducing a system of equations to a single equivalent equation. The first uses arbitrary constant multipliers (numbers with no fixed value, multiplied into the equations). It is simple enough in theory, but it makes the later steps of elimination and development, when they come, rather tedious. It was, however, the method of reduction I discovered first, and partly for that reason, partly for its simplicity, I have thought it right to keep it. The second method needs no arbitrary constants, and is better than the first in nearly every respect. So it is the one I will generally use in the rest of this book.

2. We turn to the first method.

Proposition I.

Any system of logical equations can be reduced to a single equivalent equation. Multiply each equation after the first by a different arbitrary constant, and add all the results together, the first equation included.

By Prop. 2, Chap, VI., the interpretation of any single equation f(x,y..)=0 is found by setting equal to 0 those constituents in the development of the left-hand side whose coefficients are not zero. So if we are given two equations, f(x,y..)=0, and F(x,y..)=0, their combined meaning is contained in the set of results we get by setting equal to 0 every constituent that shows up in this way in both, or in either, of the given equations, each developed by the Rule of Chap. VI. For instance, suppose we have the two equations xy−2x=0,(1) x−y=0;(2) and see what each yields.

The development of the first gives −xy−2x(1−y)=0; whence, xy=0,x(1−y)=0.(3) That is, nothing is both x and y, and nothing is x without being y.

The development of the second equation gives x(1−y)−y(1−x)=0; whence, x(1−y)=0,y(1−x)=0.(4) The constituents whose coefficients are not zero, in one development or the other, are xy, x(1−y), and (1−x)y. Together these give the system xy=0,x(1−y)=0,(1−x)y=0;(5) which is equivalent to the two systems the developments give separately, since the equation x(1−y)=0 simply appears in both. Keeping to the case of a pair of equations, then, what we have to find is a single equation which, when developed, yields the same constituents with nonzero coefficients as the given equations do. Now suppose the given equations are V1=0,V2=0, V1 and V2 being functions of the logical symbols x,y,z, &c. Then the single equation V1+cV2=0,(6) c being an arbitrary constant, will do what we want. For let At stand for any term in the full development V1, where t is a constituent and A its numerical coefficient, and let Bt stand for the matching term in the full development of V2. Then the matching term in the development of (6) will be (A+cB)t. The coefficient of t is zero if A and B are both zero, and not otherwise. To see this, suppose A and B are not both zero, and at the same time suppose A+cB=0,(7) then only the following cases can arise.

First, A is zero and B is not. Then the equation above becomes cB=0, and this requires c=0. But that contradicts the assumption that c is an arbitrary constant.

Second, B is zero and A is not. Then (7) reduces to A=0, which breaks the very assumption we started from.

Third, neither A nor B is zero. Then equation (7) gives c=−AB which is a definite value, and so again conflicts with the assumption that c is arbitrary.

So the coefficient A+cB is zero when A and B are both zero, and not otherwise. It follows that exactly the same constituents appear with nonzero coefficients in the development of (6) as in the equations V1=0,V2=0, taken singly or together. And so the equation V1+cV2=0, is equivalent to the system V1=0,V2=0.

By the same kind of reasoning, the general system of equations V1=0,V2=0,V3=0, &c.; can be replaced by the single equation V1+cV2+c′V3+ &c.=0, c,c′, &c., being arbitrary constants. The equation formed this way can be handled in every respect like the ordinary logical equations of the previous chapters. The arbitrary constants c1,c2, &c., are not logical symbols. They do not obey the law c1(1−c1)=0,c2(1−c2)=0. that is, they are not restricted to the values 0 and 1. But bringing them in is justified by the general principle stated in (II. 15) and (V. 6), and used in nearly all our investigations since: equations in the symbols of logic may be treated in every respect as if those symbols were symbols of quantity, subject to the special law x(1−x)=0, until, at the final stage of the solution, they take a form that can be read in the system of thought that logic deals with.

3. The following example will show the method at work.

Ex. 1.—Suppose that a study of the properties of some particular class of substances has led to the following general conclusions.

First, wherever the properties A and B occur together, either the property C or the property D is present as well, but not both together.

Second, wherever the properties B and C occur together, the properties A and D are either both present with them or both absent.

Third, wherever the properties A and B are both absent, the properties C and D are both absent too; and the other way round, wherever C and D are both absent, A and B are both absent too.

From these, suppose we are asked what can be concluded in any particular case from the presence of the property A, about whether the properties B and C are present or absent, leaving the property D out of account. Let us set up the symbols as follows.

Representthe property A by x;
"the property B by y;
"the property C by z;
"the property D by w.

Then the symbolic expression of the premises will be xy=v{w(1−z)+z(1−w)};yz=v{xw+(1−x)(1−w)};(1−x)(1−y)=(1−z)(1−w). In words: the things that are x and y are some of those that are w but not z, or z but not w; the things that are y and z are some of those that are both x and w, or neither; and the things that are neither x nor y are exactly the things that are neither z nor w.

From the first two of these equations, eliminating the indefinite class symbol v from each separately, we have xy(1−w(1−z)−z(1−w))=0; yz(1−xw−(1−x)(1−w))=0. Now we notice that by development 1−w(1−z)−z(1−w)=wz+(1−w)(1−z), and 1−xw−(1−x)(1−w)=x(1−w)+w(1−x), and in these expressions we write, for simplicity, a bar over a letter for "one minus" that letter, replacing 1−x by x‾,1−y by y‾,&c., so that a barred letter means "not" that class. We then get from the last three equations xy(wz+w‾z‾)=0;(1)yz(xw‾+x‾w)=0;(2)x‾y‾=w‾z‾;(3) and from this system we have to eliminate w.

Multiplying the second of these equations by c, and the third by c′, and adding the results to the first, we have xy(wz+w‾z‾)+cyz(xw‾+x‾w)+c′(x‾y‾−w‾z‾)=0. When w is set equal to 1, and so w‾ to 0, the left-hand side of this equation becomes xyz+cx‾yz+c′x‾y‾. And when in the same expression w is set to 0 and w‾ = 1, it becomes xyz‾+cxyz+c′x‾y‾−c′z‾. So the result of eliminating w can be written as the product of these two, set equal to zero: (xyz+cx‾yz+c′x‾y‾)(xyz‾+cxyz+c′x‾y‾−c′z‾)=0;(4) and from this equation we have to find x.

If we went on as in earlier examples, we would multiply out the factors on the left-hand side of this equation. But it is worth showing that this is not needed at all. Let us develop the left-hand side of (4) with respect to x, the symbol whose expression we want. We find yz(yz‾+cyz−c′z‾)x+(cyz+c′y‾)(c′y‾−c′z‾)(1−x)=0; or, cyzx+(cyz+c′y‾)(c′y‾−c′z‾)(1−x)=0; from which we find x=(cyz+c′y‾)(c′y‾−c′z‾)(cyz+c′y‾)(c′y‾−c′z‾)−cyz; and, developing the right-hand side with respect to y and z, x=0yz+00yz‾+c′2c′2y‾z+00y‾z‾; or, x=(1−y)z+00y(1−z)+00(1−y)(1−z); or, x=(1−y)z+00(1−z); Here the coefficient that is a number divided by itself counts as 1, so the things that are z but not y are all among the x's, while of the things that are not z, an indefinite portion are x's. The interpretation is: Wherever the property A is present, either C is present and B absent, or C is absent. And the other way round: Wherever the property C is present and the property B absent, the property A is present.

These results can be reached much more easily by the method I will explain next. Still, it is reassuring to have different routes to the same conclusion, so that each can check the other.

4. We turn to the second method.

Proposition II.

Suppose some equations, V1=0, V2=0, &c., are such that when their left-hand sides are developed, every constituent has a positive coefficient. Then those equations can be combined into a single equivalent equation simply by adding them.

For, as before, let At stand for any term in the development of the function V1, Bt the matching term in the development of V2 and so on. Then the matching term in the development of the equation V1+V2+&c.=0,(1) formed by adding the given equations together, will be (A+B+&c.)t. But by assumption none of the coefficients A, B, &c. is negative. So the combined coefficient A+B, &c. in the new equation can be zero only when the separate coefficients A, B, &c. are all zero together: positive numbers cannot cancel one another out. So the same constituents will appear in the development of equation (1) as in the equations V1=0, V2=0, &c. of the original system taken together. And therefore the interpretation of equation (1) is equivalent to the combined interpretations of the separate equations it was built from.

Proposition III.

5. Let V1=0,V2=0,&c. be any system of equations whose terms have all been moved to the same side. Then the combined interpretation of the system is contained in the single equation V12+V22+&c.=0, formed by adding together the squares of the given equations.

For let any equation of the system, say V1=0, produce on development an equation a1t1+a2t2+&c.=0 in which t1,t2,&c. are constituents, and a1,a2,&c. their coefficients. Then the equation V12=0 will produce on development the equation a12t1+a22t2+&c.=0, with each coefficient squared. This can be proved either from the law of development, or by squaring the function a1t1+a2t2,&c. under the conditions t12=t1, t22=t2,, t1t2=0 laid down in Prop. 3, Chap. v.; that is, each constituent times itself is itself, and any two different constituents multiplied together give zero. So the constituents that appear in the expansion of the equation V12=0, are the same as those that appear in the expansion of the equation V1=0, and now they have positive coefficients, since a square is never negative. The same holds for the equations V2=0,&c. So, by the last Proposition, the equation V12+V22+&c.=0 will be equivalent in interpretation to the system of equations V1=0, V2=0, &c.

Corollary.—Any equation V=0, whose left-hand side already satisfies the condition V2=V, or V(1−V)=0, does not need squaring, since squaring would leave it unchanged. Such equations, in fact, develop at once into a series of constituents with coefficients equal to 1, Chap. v. Prop. 4.

Proposition IV.

6. Suppose the equations of a system have been brought, by the squaring process above or by any other process, to a form in which every constituent in their development has a positive coefficient. Then any equations derived from them by elimination will have the same character, and may be combined with the other equations by addition.

Suppose we have to eliminate a symbol x from an equation V=0, in which none of the constituents in the full development of the left-hand side has a negative coefficient. That expansion can be written in the form V1x+V0(1−x)=0 V1 and V0 each being of the form a1t1+a2t2...+antn, in which t1t2...tn are constituents of the other symbols, and a1a2...an are in each case positive numbers or zero. The result of elimination is V1V0=0; that is, the part multiplying x times the part multiplying one minus x, set equal to zero. And since none of the coefficients in V1 and V0 is negative, there can be no negative coefficients in the product V1V0. So the equation V1V0=0 can be added to any other equation whose constituents all have positive coefficients, and the resulting equation will carry the full meaning of the equations it came from.

Proposition V.

7. To draw from the previous Propositions a practical rule, or method, for reducing systems of equations that express propositions in logic.

The investigations so far have established two points.

First, any equations of the form V=0, where V satisfies the fundamental law of duality V(1−V)=0, can be combined by simple addition.

Second, any other equations of the form V=0 can be brought, by squaring, to a form where the same principle of combining by mere addition applies.

All that remains, then, is to settle which equations, among those that actually express propositions, belong to the first class and which to the second.

Now the general types of propositions were set out at the end of Chap. IV. The division of propositions they represent is as follows.

First, propositions whose subject is universal and whose predicate is particular.

The symbolic type (IV. 15) is X=vY, that is, the X's are some of the Y's, with X and Y satisfying the law of duality. Eliminating v, we have X(1−Y)=0,(1) that is, nothing is an X that is not a Y; and this, we will find, also satisfies the same law. No further reduction by squaring is needed.

Second, propositions both of whose terms are universal, of the symbolic type X=Y, with X and Y each satisfying the law of duality. Writing the equation in the form X−Y=0, and squaring, we have X−2XY+Y=0,or, X(1−Y)+Y(1−X)=0.(2) that is, there is nothing that is X but not Y, and nothing that is Y but not X. The left-hand side of this equation satisfies the law of duality, as its very form makes plain.

We can reach the same equation another way. The equation X=Y is equivalent to the two equations X=vY, Y=vX, (for to say that the X’s are identical with the Y’s is to say both that All X’s are Y’s, and that All Y’s are X’s). Now these equations give, on eliminating v, X(1−Y)=0, Y(1−X)=0, and these, added together, produce (2).

Third, propositions both of whose terms are particular. Such propositions have the form vX=vY, but here v is not entirely arbitrary, and so must not be eliminated. For v stands for some, and though its meaning can include all, it does not include none. So we must move the right-hand side over to the left and square the resulting equation, according to the rule. The result will plainly be vX(1−Y)+vY(1−X)=0. It may be convenient to sum up these conclusions in a Rule, to guide us from now on.

8. Rule.— Express the equations so that the terms X and Y in the following typical forms obey the law of duality. Then change the equations X=vY into X(1−Y)=0, X=Y into X(1−Y)+Y(1−X)=0. vX=vY into vX(1−Y)+vY(1−X)=0.

An equation given in the form X=0 needs no transformation. An equation in the form X=1 can be replaced by 1−X=0, as the second of the transformations above shows.

Once the equations of the system have been reduced in this way, any of them, together with any equations derived from them by elimination, can be combined by addition.

9. Note.—We saw in Chapter IV. that if we translate the terms of a proposition into symbols word for word, without attending to its real meaning, we may produce equations whose terms X and Y do not obey the law of duality. The equation w=st(p+r), given in (3) Prop. 3 of that chapter, is of this kind. Such equations, as we saw, do have a meaning. If, out of curiosity or for any other reason, someone decides to use them, it is best to reduce them by the Rule (VI. 5).

10. Ex. 2.—Let us take the following propositions of elementary geometry.

First, similar figures consist of all those whose corresponding angles are equal and whose corresponding sides are proportional.

Second, triangles whose corresponding angles are equal have their corresponding sides proportional, and vice versâ (the other way round).

To represent these premises, let

s=similar.
t=triangles.
q=having corresponding angles equal.
r=having corresponding sides proportional.

Then the premises are expressed by the following equations: s=qr,(1)tq=tr.(2) that is, the similar figures are exactly those with equal angles and proportional sides, and the triangles with equal angles are exactly the triangles with proportional sides. Reducing by the Rule (or, what comes to the same thing, moving the terms of each equation to the left-hand side, squaring each equation, and then adding), we have s+qr−2qrs+tq+tr−2tqr=0.(3) Suppose we are asked for a description of dissimilar figures in terms of the elements named by the words triangles, having corresponding angles equal, and having corresponding sides proportional.

From (3) we have s=tq+qr+rt−2tqr2qr−1,∴1−s=qr−tq−rt+2tqr−12qr−1.(4) And developing the right-hand side in full, we find 1−s=0tqr+2tq(1−r)+2tr(1−q)+t(1−q)(1−r)+0(1−t)qr+(1−t)q(1−r)+(1−t)r(1−q)+(1−t)(1−q)(1−r).(5)

In this development two of the terms have the coefficient 2. By the Rule these must be set equal to 0. Then, dropping the terms whose coefficients are 0, we have 1−s=t(1−q)(1−r)+(1−t)q(1−r)+(1−t)r(1−q)+(1−t)(1−q)(1−r);(6)tq(1−r)=0;(7)tr(1−q)=0;(8) and the direct interpretation of these is as follows.

First, dissimilar figures consist of all triangles that do not have both their corresponding angles equal and their sides proportional; and of all figures that are not triangles and that have either their angles equal and sides not proportional, or their corresponding sides proportional and angles not equal, or neither their corresponding angles equal nor their corresponding sides proportional.

Second, there are no triangles whose corresponding angles are equal and whose sides are not proportional.

Third, there are no triangles whose corresponding sides are proportional and whose angles are not equal.

11. These are the immediate interpretations of the final equation. We see, just as the general theory says, that when we work out a description of one particular class of objects (here, dissimilar figures) in terms of certain other elements of the original premises, we also get the independent relations that hold among those elements because of the same premises. It is easy to show that this is not wasted information, even for the immediate question we asked. For one thing, the independent relations can always be used, if we wish, to shorten the expression of the relation we were directly looking for. Thus if we write (7) in the form 0=tq(1−r), and add it to (6), we get, since t(1−q)(1−r)+tq(1−r)=t(1−r),1−s=t(1−r)+(1−t)q(1−r)+(1−t)r(1−q)+(1−t)(1−q)(1−r), which, interpreted, gives as the first part of the description of dissimilar figures "Triangles whose corresponding sides are not proportional," in place of the fuller description we had at first. Whether such a reduction is worth making is always a matter of convenience.

12. One reduction is always worth making (VII. 15): gathering the terms of the description we want, such as the right-hand side of (5) or (6), into as few groups as possible. Thus the third and fourth terms on the right-hand side of (6) add up to the single term (1−t)(1−q). If we combine this reduction with the last one, we have 1−s=t(1−r)+(1−t)q(1−r)+(1−t)(1−q), and the interpretation of this is:

Dissimilar figures consist of all triangles whose corresponding sides are not proportional, and of all figures that are not triangles and that have either their corresponding angles unequal, or their corresponding angles equal but their sides not proportional.

So the fullness of the general solution is not something extra we could do without. It gives us all the information we are looking for, and it also gives us the means of stating that information in whatever way suits us best.

13. One more observation remains to be made, illustrating a principle already stated. Two of the terms in the full development of 1−s in (5) have 2 as their coefficient, instead of 10. It will be shown later that this means the two premises were not independent of each other. To check this, let us go back to the premises in their reduced forms, namely s(1−qr)+qr(1−s)=0,tq(1−r)+tr(1−q)=0. Now if the left-hand sides of these equations have any constituents in common, those will show up when we multiply the equations together. Doing so, we get stq(1−r)+str(1−q)=0. From this follow stq(1−r)=0, str(1−q)=0, and these are equations that can be deduced from either of the original premises. Their interpretations are as follows.

Similar triangles that have their corresponding angles equal have their corresponding sides proportional.

Similar triangles that have their corresponding sides proportional have their corresponding angles equal.

And these conclusions can be deduced from either premise on its own. In this respect, by the definitions laid down, the premises are not independent.

14. To finish, let us go back to the problem we used to illustrate the first method of this chapter, and try to find, by the present method, what can be concluded from the presence of the property C about the properties A and B.

On eliminating the symbols v we found the following equations: xy(wz+w‾z‾)=0,(1)yz(xw‾+x‾w)=0,(2)x‾y‾=w‾z‾.(3) From these we have to eliminate w and find z. Now (1) and (2) already satisfy the condition V(1−V)=0. The third equation, when we move its terms to the left-hand side and square, gives x‾y‾(1−w‾z‾)+w‾z‾(1−x‾y‾)=0.(4) Adding (1) (2) and (4) together, we have xy(wz+w‾z‾)+yz(xw‾+x‾w)+x‾y‾(1−w‾z‾)+w‾z‾(1−x‾y‾)=0. Eliminating w, we get (xyz+yzx‾+x‾y‾){xyz‾+yzx+x‾y‾z+z‾(1−x‾y‾)}=0. Now when we multiply the terms of the second factor by those of the first in turn, remembering that xx‾=0, yy‾=0, zz‾=0, (nothing is both a class and its opposite), nearly all of them vanish, and we are left with only xyz+x‾y‾z=0;(5) that is, nothing is z together with both x and y, or z with neither. From this, z=0xy+x‾y‾ =0xy+00xy‾+00x‾y+0x‾y‾ =00xy‾+00x‾y, which gives the interpretation: wherever the property C is found, either the property A or the property B will be found with it, but not both of them together.

From equation (5) we can easily deduce the result we reached earlier by the method of arbitrary constant multipliers, as well as any other requested form of the relation between x, y, and z. For example: If the property B is absent, either A and C will be present together, or C will be absent. And conversely, If A and C are present together, B will be absent. The converse part of this conclusion rests on the presence of a term xz with unity for its coefficient in the developed value of y‾.

Chapter 9: On Certain Methods of Abbreviation

1. The three fundamental methods set out and illustrated in the previous chapters, development, elimination, and reduction, are enough for every practical purpose of logic. Still, there are cases where they can be simplified a great deal, and this is especially true of elimination. Those cases are the subject of this chapter. I will first prove some propositions that contain the principles of these shortcuts, and afterwards apply them to particular examples.

Let us call any terms that satisfy the fundamental law V(1−V)=0 class terms. Each such term, taken alone, is a constituent. But when several occur together, they need not all involve the same symbols, as the terms of a development must. So ax+bxy+cyz can be described as an expression made of three class terms, x, xy, and yz, multiplied by the coefficients a, b, c respectively. The principle used in the next two Propositions, which in some cases shortens elimination a great deal, is the rejection of superfluous class terms. A term counts as superfluous if it adds nothing to the constituents of the final result.

Proposition I.

2. Take any equation V=0 in which V is a series of class terms with positive coefficients. From it we may throw out any term that contains another term as a factor, and we may change every positive coefficient to unity (that is, 1).

Here is why. What a series of positive terms means depends only on the number and kind of the constituents in its final expansion, that is, its expansion with respect to all the symbols it contains. It does not depend at all on the actual values of the coefficients (VI. 5). Now let x be any term of the series, and xy any other term that has x as a factor. The expansion of x with respect to the symbols x and y is xy+x(1−y), and the expansion of the sum of the terms x and xy is 2xy+x(1−y).

By what has just been said, these two expressions are equivalent when they stand on the left-hand side of an equation whose right-hand side is 0 and whose left-hand coefficients are all positive. Either way, the final expansion contains just the two constituents xy and x(1−y), and from them come simply the equations xy=0, x(1−y)=0. That is, nothing is both x and y, and nothing is x without being y. So the pair of terms x+xy can be replaced by the single term x.

The same reasoning covers every case the Proposition describes. For instance, if the term x is repeated, the sum 2x can be replaced by x, because in these circumstances the equation x=0 is bound to appear in the final reduction anyway.

Proposition II.

3. Suppose that in the course of an elimination we have to multiply together two factors, each made up only of positive terms that obey the fundamental law of logical symbols. Then we may throw out of both factors any term they have in common, or throw out of either factor any term that is divisible by a term in the other factor. The condition is that every term thrown out must be added to the product of the factors that remain.

In stating this Proposition, the word "divisible" is a term of convenience, used in the algebraic sense. In that sense xy and x(1−y) are both said to be divisible by x.

To make the meaning of the Proposition clearer, suppose the factors to be multiplied together are x+y+z and x+yw+t. The Proposition says that we may throw the term x out of both factors, and the term yw out of the second factor, as long as these terms are carried over to the final product. The factors that remain are then y+z and t. If to their product yt+zt we add the terms x and yw, we get x+yw+yt+zt, as an expression equivalent to the product of the given factors x+y+z and x+yw+t. It is equivalent, that is, for the purposes of elimination.

First take the case where the two factors share a term x. Write the factors as x+P and x+Q, where P in the first factor and Q in the second stand for the sum of the positive terms besides x.

Now, (x+P)(x+Q)=x+xP+xQ+PQ.(1) But elimination consists in multiplying certain factors together and setting the result equal to 0. So there are two possibilities. Either the right-hand side of the equation above is itself to be set equal to 0, or it is a factor of some larger expression that is to be set equal to 0.

Take the first possibility. By the last Proposition, we may throw out the terms xP and xQ, since they are positive terms that have another term, x, as a factor. What is left is x+PQ, which is exactly what we get by throwing x out of both factors and adding it to the product of the factors that remain.

Take the second possibility. The only way the right-hand side of (1) can affect the final result of the elimination is through the number and kind of its constituents. Throwing out the terms xP and xQ changes neither. The reason is that the development of x already includes every possible constituent that has x as a factor.

Finally, take the case where one factor contains a term, such as xy, that is divisible by a term, x, in the other factor.

Let the factors be x+P and xy+Q. Now (x+P)(xy+Q)=xy+xQ+xyP+PQ. By the reasoning of the last Proposition, the term xyP may be thrown out, because it contains another positive term, xy, as a factor. So we have xy+xQ+PQ=xy+(x+P)Q. But this is just what we get by throwing the term xy out of the second factor and carrying it over to the final product. So the Proposition is plain.

Proposition III.

4. Let t be any symbol that is kept in the final result when other symbols are eliminated from some system of equations. Then the result of that elimination can be written in the form Et+E′(1−t)=0, in which E is found by putting t=1 in the given system and eliminating the same other symbols, and E′ is found by putting t=0 in the given system and eliminating the same other symbols.

For let ϕ(t)=0 stand for the final result of the elimination. Expanding this equation, we have ϕ(1)t+ϕ(0)(1−t)=0. Now whatever process takes us from the given system of equations to the function ϕ(t), the same process would take us to ϕ(1) if t were changed into 1 in those equations. And the same process would take us to ϕ(0) if t were changed into 0 in them. So the truth of the Proposition is plain.

5. A remark about the three propositions just proved. They are not needed at all for the strict development or application of the general theory. But they serve important practical ends. By Prop. 1 we can simplify the results of addition. By Prop. 2 we can simplify the results of multiplication. And by Prop. 3 we can break any long, tedious elimination into two separate processes, each of which will usually be much less complicated. We will use this method very often when the final aim is to find the value of t in terms of the other symbols that remain after the elimination.

6. Ex. 1.—In the Nicomachean Ethics, Book II. Cap. 3, Aristotle first settles that actions are virtuous not because of some character they have in themselves, but because they imply a certain state of mind in the person who does them. That person must act knowingly, by deliberate choice, for the action's own sake, and on fixed principles of conduct. In the next two chapters he goes on to ask whether virtue belongs to the genus of Passions, or of Faculties, or of Habits, along with some related points. He bases his inquiry on the following premises. From them he also draws the general doctrine and definition of moral virtue that the rest of the treatise lays out.

Premises.

1. Virtue is either a passion (πάθος), or a faculty (δύναμις), or a habit (ἕξις).

2. Passions are not things according to which we are praised or blamed, or in which we exercise deliberate preference.

3. Faculties are not things according to which we are praised or blamed, and which are accompanied by deliberate preference.

4. Virtue is something according to which we are praised or blamed, and which is accompanied by deliberate preference.

5. Whatever art or science makes its work to be in a good state avoids extremes, and keeps the mean in view relative to human nature (τὸ μέσον … πρὸς ἡμᾶς)

6. Virtue is more exact and excellent than any art or science.

This last is an argument à fortiori (from the stronger case). If science and true art avoid falling short and going too far alike, then virtue keeps to the straight line of moderation all the more. If they make their work to be in a good state, we have all the more reason to say that virtue makes her own particular work to be "in a good state." Let the final premise be read in that way. Let us also leave out every mention of praise and blame. In the premises they are mentioned only alongside deliberate preference, and deliberate preference is an element we mean to keep. We may then take as our symbols:

v = virtue.
p = passions.
f = faculties.
h = habits.
d = things accompanied by deliberate preference.
g = things causing their work to be in a good state.
m = things keeping the mean in view relative to human nature.

Using q as the symbol of an indefinite class, we can express our premises by the following equations: v=q{p(1−f)(1−h)+f(1−p)(1−h)+h(1−p)(1−f)}.p=q(1−d).f=q(1−d).v=qd.g=qm.v=qg. Eliminating the symbol q from each of these separately, we get v{1−p(1−f)(1−h)−f(1−p)(1−h)−h(1−p)(1−f)}=0.(1)pd=0.(2)fd=0.(3)v(1−d)=0.(4)g(1−m)=0.(5)v(1−g)=0.(6) We will first eliminate the symbol d from (2), (3), and (4), and then find v in terms of p, f, and h. Adding (2), (3), and (4) gives (p+f)d+v(1−d)=0. Eliminating d from this in the ordinary way, we find (p+f)v=0.(7) That is, nothing is both virtue and either a passion or a faculty. Adding this to (1), and solving for v, we find v=0p+f+1−p(1−f)(1−h)−f(1−p)(1−h)−h(1−f)(1−p). So by development, v=00h(1−f)(1−p). This equation reads: Virtue is a habit, and not a faculty or a passion.

Next we will eliminate f, p, and g from the original system of equations, and then find v in terms of h, d, and m. This time we will eliminate p and f together. Adding (1), (2), and (3), we get v{1−p(1−f)(1−h)−f(1−p)(1−h)−h(1−p)(1−f)}+pd+fd=0.

Developing this with respect to p and f, we have (v+2d)pf+(vh+d)p(1−f)+(vh+d)(1−p)f+v(1−h)(1−p)(1−f)=0. So the result of the elimination will be (v+2d)(vh+d)(vh+d)v(1−h)=0. Now v+2d=v+d+d, and by Prop. I. this can be reduced to v+d. Its product with the second factor is (v+d)(vh+d), which by Prop. II. reduces to d+v(vh) or vh+d.

In the same way, multiplying this result by the third factor gives simply vh+d again. Finally, multiplying it by the fourth factor, v(1−h), gives the final equation vd(1−h)=0(8) that is, there is no virtue accompanied by deliberate preference that is not a habit. It remains to eliminate g from (5) and (6). The result is v(1−m)=0(9) that is, all virtue keeps the mean in view.

Finally, adding the equations (4), (8), and (9) gives v(1−d)+vd(1−h)+v(1−m)=0 from which we have v=01−d+d(1−h)+1−m. And the development of this result gives v=00hdm, of which the interpretation is: Virtue is a habit accompanied by deliberate preference, and keeping in view the mean relative to human nature.

Strictly speaking, this is not a definition of virtue but a description of it. It is, however, all that can correctly be inferred from the premises. Aristotle ties to it the need for prudence, to find the safe middle line of action. And there is no doubt that the ancient theories of virtue were, in general, more intellectual in character than the theories that have mostly prevailed in modern times (the theory of utility excepted). Virtue was seen as the right state and habit of the whole mind, rather than as the sole rule of conscience or of the moral faculty. To some extent those theories were surely right. Unconditional obedience to what conscience commands is an essential part of virtuous conduct. But another part is that those commands should agree with the unchanging principles of right (αἰώνια δίκαια), which are grounded in the constitution of things, or rather are themselves its foundation. And such agreement, at least in any high degree, cannot go along with ignorance and dullness of mind. To come back to Aristotle's own theory: most people, I think, will find it too negative. Merely avoiding extremes does not give enough room for the nobler energies of our nature. Aristotle seems to have been dimly aware of this weakness in his system when, at the opening of his seventh book, he spoke of a "heroic virtue" that rises above the measure of human nature.

Footnote: τὴν ὑπὲρ ἡμᾶς ἀρετὴν ἡρωϊκήν τινα καὶ θείαν—Nic. Eth. Book vii.

7. I have already remarked (VIII. 1) that the theory of single equations or propositions includes questions that cannot be fully answered except in connection with the theory of systems of equations. We see this when we are asked to take a single equation and find, in terms of the remaining elements, the relation not of some single elementary class but of some compound class, one whose expression involves more than one element. The particular example that follows, and the general problem after it, are of this kind.

Ex. 2.—Let us go back to the symbolic expression of the definition of wealth used in Chap. VII., namely, w=st{p+r(1−p)}, where, as before,

w = wealth,
s = things limited in supply,
t = things transferable,
p = things productive of pleasure,
r = things preventive of pain;

and suppose we are asked to find from it how things that are transferable and productive of pleasure are related to the other elements of the definition, namely wealth, things limited in supply, and things preventive of pain.

The expression for things transferable and productive of pleasure is tp. Let us stand for this by a new symbol, y. We then have the equations w=st{p+r(1−p)},y=tp, and if we eliminate t and p from them, we can find y as a function of w, s, and r. That result, interpreted, will give the relation we want.

Bringing all the terms of these equations over to the left-hand side, we have w−stp−str(1−p)=0.y−tp=0.(3) Adding the squares of these equations together gives w+stp+str(1−p)−2wstp−2wstr(1−p)+y+tp−2ytp=0.(4) To eliminate t and p, we develop the left-hand side with respect to them, and get (w+s−2ws+1−y)tp+(w+sr−2wsr+y)t(1−p)+(w+y)(1−t)p+(w+y)(1−t)(1−p);(5) and the result of eliminating t and p is found by setting equal to 0 the product of the four coefficients of tp,t(1−p),(1−t)p, and (1−t)(1−p).

Or, by Prop. 3, the result of eliminating t and p from the equation above will have the form Ey+E′(1−y), where E is what we get by changing y into 1 in the given equation and then eliminating t and p, and E′ is what we get by changing y into 0 in the same equation and then eliminating t and p. In each case, the way to eliminate t and p is to multiply together the coefficients of the four constituents tp, t(1−p), and so on.

If we put y=1, the coefficients become as follows.

The first is w(1−s)+s(1−w).

The second is 1+w(1−sr)+s(1−w)r, which is equivalent to 1 by Prop. I.

The third and fourth are 1+w, which is equivalent to 1 by Prop. I.

So the value of E is w(1−s)+s(1−w). Again, putting y=0 in (5), we have for the coefficients:

The first is 1+w(1−s)+s(1−w), which is equivalent to 1.

The second is w(1−sr)+sr(1−w).

The third and fourth are w.

Multiplying these coefficients together gives E′=w(1−sr).

So the equation from which y is to be found is {w(1−s)+s(1−w)}y+w(1−sr)(1−y)=0,∴y=w(1−sr)w(1−sr)−w(1−s)−s(1−w); and expanding the right-hand side, y=00wsr+ws(1−r)+10w(1−s)r+10w(1−s)(1−r)+0(1−w)sr+0(1−w)s(1−r)+00(1−w)(1−s)r+00(1−w)(1−s)(1−r); and reducing this, we get y=ws(1−r)+00wsr+00(1−w)(1−s),(6)with w(1−s)=0.(7)

Read in words, this says two things.

First, things transferable and productive of pleasure consist of all wealth (limited in supply and) not preventive of pain, an indefinite amount of wealth (limited in supply and) preventive of pain, and an indefinite amount of what is not wealth and not limited in supply.

Second, all wealth is limited in supply.

In the solution above I have put in parentheses the part of the full description that is already implied by the separate relation (7) that comes with it.

8. The following problem is more general. It will give us an easy practical rule for problems like the last one.

General Problem.

Given any equation connecting the symbols x, y..w, z..

Required: to find the logical expression of any class described in any way by the symbols x, y.. in terms of the remaining symbols, w, z, and so on.

Let us keep to the case of just two symbols, x and y, on one side, and two symbols, w and z, on the other. That case is enough to establish the general Rule.

Let V=0 be the given equation, and let ϕ(x,y) stand for the class whose expression we want to find.

Put t=ϕ(x,y). Then x and y are to be eliminated from these two equations.

Now the equation V=0 can be expanded in the form Axy+Bx(1−y)+C(1−x)y+D(1−x)(1−y)=0,(1) where A, B, C, and D are functions of the symbols w and z.

Again, since ϕ(x,y) stands for a class or collection of things, it must consist of a constituent, or a series of constituents, whose coefficients are 1.

So if the full development of ϕ(x,y) is written in the form axy+bx(1−y)+c(1−x)y+d(1−x)(1−y), each of the coefficients a, b, c, d must be 1 or 0.

Now by transposing and squaring, we reduce the equation t=ϕ(x,y) to the form t{1−ϕ(x,y)}+ϕ(x,y)(1−t)=0; and expanding with respect to x and y, we get {t(1−a)+a(1−t)}xy+{t(1−b)+b(1−t)}x(1−y)+{t(1−c)+c(1−t)}(1−x)y+{t(1−d)+d(1−t)}(1−x)(1−y)=0; and adding this to (1), we have {A+t(1−a)+a(1−t)}xy+{B+t(1−b)+b(1−t)}x(1−y)+&c.=0.

Let the result of eliminating x and y have the form Et+E′(1−t)=0, Then, by what has been said, E will be the reduced product of what the coefficients of the expansion above become when t=1 , and E′ the product of the same factors, reduced in the same way, when t=0.

So E will be the reduced product (A+1−a)(B+1−b)(C+1−c)(D+1−d). Look at any one factor of this expression, say A+1−a. When a=1 it becomes A, and when a=0 it becomes 1+A, which reduces to 1 by Prop. I. From this we may conclude that E is the product of the coefficients of those constituents in the development of V whose coefficients in the development of ϕ(x,y) are 1.

Moreover, E′ will be the reduced product (A+a)(B+b)(C+c)(D+d). Look at any one of these factors, say A+a. It becomes A when a=0, and reduces to 1 when a=1 ; and the same goes for the others. So E will be the product of the coefficients of those constituents in the development of y, whose coefficients in the development ϕ(x,y) are 0. Putting these two cases together, we can lay down the following Rule.

9. To find, from a logical equation, how any class expressed by a given combination of the symbols x, y, and so on, is related to the classes represented by the other symbols in that equation.

Rule.—Expand the given equation with respect to the symbols x, y. Then form the equation Et+E′(1−t)=0, in which E is the product of the coefficients of all those constituents in the development whose coefficients in the expression of the given class are 1, and E′ is the product of the coefficients of those constituents in the development whose coefficients in the expression of the given class are 0. The value of t found from this equation, by solving it and interpreting the result, will be the expression we want.

Note.—In proving this Rule we assumed that V consists only of positive terms. But it is easy to show that this condition is unnecessary, that the Rule is general, and that the given equation really needs no preparation at all.

10. Ex. 3.—Take the same definition of wealth as in Example 2. We want an expression for things transferable, but not productive of pleasure, t(1−p), in terms of the other elements, represented by w, s, and r.

The equation w−stp−str(1−p)=0, gives, when squared, w+stp+str(1−p)−2wstp−2wstr(1−p)=0; and developing the left-hand side with respect to t and p, (w+s−2ws)tp+(w+sr−2wsr)t(1−p)+w(1−t)p+w(1−t)(1−p)=0.

It is best to write the coefficients as in the following equation: {w(1−s)+s(1−w)}tp+{w(1−sr)+sr(1−w)}t(1−p)+w(1−t)p+w(1−t)(1−p)=0

Let the function to be found, t(1−p), be represented by z. Then the full development of z with respect to t and p is z=0tp+t(1−p)+0(1−t)p+0(1−t)(1−p). So, by the last problem, we have Ez+E′(1−z)=0;where E=w(1−sr)+sr(1−w);E′={w(1−s)+s(1−w)}×w×w=w(1−s);∴{w(1−sr)+sr(1−w)}z+w(1−s)(1−z)=0. So, z=w(1−s)2wsr−ws−sr=00wsr+0ws(1−r)+10w(1−s)r+10w(1−s)(1−r),+0(1−w)sr+00(1−w)s(1−r)+00(1−w)(1−s)r+00(1−w)(1−s)(1−r).Or, z=00wsr+00(1−w)s(1−r)+00(1−w)(1−s),with w(1−s)=0.

So things transferable and not productive of pleasure are either wealth (limited in supply and preventive of pain); or things that are not wealth, but are limited in supply and not preventive of pain; or things that are not wealth and are unlimited in supply.

The following results, found in the same way, are easy to check:

Things limited in supply and productive of pleasure which are not wealth are intransferable (cannot be transferred).

Wealth that is not productive of pleasure is transferable, limited in supply, and preventive of pain.

Things limited in supply which are either wealth or productive of pleasure, but not both, are either transferable and preventive of pain, or intransferable.

11. A great many curious examples could be taken from natural history. But I do not think such applications would have any value of their own. For instance, they would throw no light on the true principles of classification in zoology. To discover those, we need some foundation of positive knowledge: some acquaintance with the structure of living things, and with how their parts are fitted to their purposes. That kind of knowledge can only come from the outside means of observation and analysis. Still, take any collection of propositions in natural history, and a great many logical problems present themselves, whatever system of classification is used. In making up such examples, it is perhaps better to leave out, as superfluous, the property of a class or species that its name already suggests. An example is the ring-structure of the annelida, the class of animals that includes the earthworm and the leech.

Ex. 4.—1. The annelida are soft-bodied, and either naked or enclosed in a tube.

2. The annelida consist of all invertebrate animals having red blood in a double system of circulating vessels.

Assumea = annelida;
s = soft-bodied animals;
n = naked;
t = enclosed in a tube;
i = invertebrate;
r = having red blood, &c.

Then the propositions are expressed by the equations a=vs{n(1−t)+t(1−n)};(1)a=ir;(2) and to these we may add the condition they imply, that nothing is both naked and enclosed in a tube, nt=0.(3)

Eliminating v, and reducing the system to a single equation, we have a[1−sn(1−t)−st(1−n)]+a(1−ir)+ir(1−a)+nt=0.(4)

Suppose we want to find how soft-bodied animals enclosed in tubes stand, by virtue of the premises, with respect to the following elements: having red blood, having an outer covering, and having a backbone.

We must first eliminate a. The result is ir{1−sn(1−t)−st(1−n)}+nt=0.

Then (IX. 9), developing with respect to s and t, and reducing the first coefficient by Prop. 1, we have nst+ir(1−n)s(1−t)+(ir+n)(1−s)t+ir(1−s)(1−t)=0.(5)

So if we put st=w, we find nw+ir(1−n)×(ir+n)×ir(1−w)=0; or, nw+ir(1−n)(1−w)=0;∴w=ir(1−n)ir(1−n)−n=0irn+ir(1−n)+0i(1−r)n+00i(1−r)(1−n)+0(1−i)rn+00(1−i)r(1−n)+0(1−i)(1−r)n+00(1−i)(1−r)(1−n);or, w=ir(1−n)+00i(1−r)(1−n)+00(1−i)(1−n).

So soft-bodied animals enclosed in tubes consist of all invertebrate animals that have red blood and are not naked, together with an indefinite remainder made up of invertebrate animals that do not have red blood and are not naked, and of vertebrate animals that are not naked.

In exactly the same way, the following reduced equations have been drawn from the development (5). Their interpretation is left to the reader. s(1−t)=irn+00i(1−n)+00(1−i)(1−s)t=00(1−i)r(1−n)+00(1−r)(1−n)(1−s)(1−t)=00i(1−r)+00(1−i).

In none of these examples have I tried to show off the power of the method in any special way. That power, I believe, can only be fully displayed in connection with the mathematical theory of probability (in the chapters on probability, not included in this edition). But I would make a suggestion to anyone who wants to form a fair opinion on this point. Work through the following problem by the rules of ordinary logic before looking at its solution. And remember that whatever complexity it has could be multiplied without limit. The only effect would be to make its solution by the method of this book more laborious, but no less certain.

Ex. 5. Suppose that observing a certain class of natural productions has led to the following general results.

First, in whichever of these productions the properties A and C are missing, the property E is found, together with one of the properties B and D, but not with both.

Second, wherever the properties A and D are found while E is missing, the properties B and C will either both be found or both be missing.

Third, wherever the property A is found together with either B or E, or both, there either the property C or the property D will be found, but not both. And conversely, wherever the property C or D is found on its own, there the property A will be found together with either B or E, or both.

We are asked to find two things. First, what can be concluded in any particular case from the known presence of the property A, with respect to the properties B, C, and D; and also whether any relations hold among the properties B, C, and D on their own. Second, what can be concluded in the same way about the property B and the properties A, C, and D.

Notice that in each of the three data, what we are told about the properties A, B, C, and D is tangled up with another element, E, about which we want to say nothing in our conclusion. So we will need to eliminate the symbol for the property E from the system of equations that expresses the given propositions.

Let us represent the property A by x, B by y, C by z, D by w, and E by v. The data are then x‾z‾=qv(yw‾+wy‾);(1)v‾xw=q(yz+y‾z‾);(2)xy+xvy‾=wz‾+zw‾;(3) where x‾ stands for 1−x, and so on, and q is a symbol of an indefinite class. We eliminate q from the first and second equations separately, and add the results to the third equation, reduced by (5), Chap. VIII. We get x‾z‾(1−vyw‾−vwy‾)+v‾xw(yz‾+zy‾)+(xy+xvy‾)(wz+w‾z‾)+(wz‾+zw‾)(1−xy−xvy‾)=0.(4)

From this equation v must be eliminated, and then the value of x found from the result. For this purpose it will be convenient to use the method of Prop. 3 of the present chapter.

So let the result of the elimination be represented by the equation Ex+E′(1−x)=0.

To find E, put x=1 in the left-hand side of (4). We find v‾w(yz‾+zy‾)+(y+vy‾)(wz+w‾z‾)+(wz‾+zw‾)v‾y‾.

Eliminating v, we have (wz+w‾z‾){w(yz‾+zy‾)+y(wz+w‾z‾)+y‾(wz‾+zw‾)}; and when this is actually multiplied out, subject to the conditions ww‾=0, zz‾=0, and so on, it gives E=wz+yw‾z‾

Next, to find E′, put x=0 in (4). We have z(1−vyw‾−vy‾w)+wz‾+zw‾. Eliminating v from this, and reducing the result by Propositions 1 and 2, we find E′=wz‾+zw‾+y‾w‾z‾;

and so, finally, we have (wz+yw‾z‾)x+(wz‾+zw‾+y‾w‾z‾)x‾=0;(5) from which x=wz‾+zw‾+y‾w‾z‾wz‾+zw‾+y‾w‾z‾−wz−yw‾z‾ and so, by development, x=0yzw+yzw‾+yz‾w+0yz‾w‾+0y‾zw+y‾zw‾+y‾z‾w+y‾z‾w‾; or, collecting the terms that stand in the same vertical column, x=zw‾+z‾w+y‾z‾w‾;(6) which reads as follows.

In whatever substances the property A is found, there will also be found either the property C or the property D, but not both; or else the properties B, C, and D will all be missing. And conversely, wherever the property C or the property D is found on its own, or the properties B, C, and D are all missing together, there the property A will be found.

It also turns out that there is no separate relation among the properties B, C, and D on their own.

Second, we are to find y. Developing (5) with respect to that symbol, we get (xwz+xw‾z‾+x‾wz‾+x‾zw‾)y+(xwz+x‾wz‾+x‾zw‾+x‾z‾w‾)y‾=0; and proceeding as before, y=x‾w‾z‾+00(x‾wz+xwz‾+xzw‾),(7) together with xzw=0;(8)x‾z‾w=0;(9)x‾zw‾=0;(10)

Solving (10), we reduce it to the form x‾z=00w; and from it we have the separate relation: If the property A is absent and C present, D is present. Again, adding (8) and (9) and solving gives xz+x‾z‾=00w‾.

So, for the general solution and the remaining separate relation, we have the following.

First, if the property B is present in one of the productions, then either the properties A, C, and D are all absent, or exactly one of them is absent. And conversely, if they are all absent it may be concluded that the property B is present (7).

Second, if A and C are both present or both absent, D will be absent, quite independently of whether B is present or absent (8) and (9).

I have not attempted to verify these conclusions.

Chapter 10: Of the Conditions of a Perfect Method

1. We have now discussed primary propositions at length, and we are about to take up secondary propositions. This point of transition between the two great divisions of the science of logic is a good place to pause. While we look back over some of the steps we have taken, we can ask what it is that makes a method perfect in a subject like the one we have been studying. I do not mean only the perfection that lies in power. I mean also the perfection that rests on our sense of what is fitting and beautiful. A careful analysis of the question would probably lead us to something like this conclusion. A perfect method should not only be efficient, doing well the work it was designed to do. It should also show a certain unity and harmony in all its parts and processes. This ideal would be most fully reached if the very forms of the method pointed back to the fundamental principles they are built on, and, if possible, to the one fundamental principle beneath them all. When we apply these thoughts to the science of reasoning, it may be wise to look beyond the analytical processes alone. We should ask what is best not only in the manner or form of deduction, but also in the system of data, or premises, from which the deduction is made.

2. As far as power alone goes, there is no doubt that the first of the methods developed in Chapter VIII. is, within its proper range, a perfect one. Bringing in arbitrary constants frees us from depending on the forms of the premises, and from any conditions among the equations that represent them. But it seems to bring in a foreign element. It is a more laborious form of solution than the second method of reduction proved in the same chapter, and a less elegant one too. There are, however, conditions under which that second method takes on a more perfect form than it usually has. Suppose we made the one fundamental condition, expressed by the equation x(1−x)=0, the universal pattern for every form, so that every expression we work with obeyed the law of duality just as a single class symbol does. That would give a unity of character to both the processes and the results which could not be had in any other way. If brevity or convenience were the only things that made a method valuable, adopting such a principle would bring no advantage. To force this character onto every step of a solution would sometimes take no small labour of preliminary reduction. Still, it is interesting to know that it can be done. It is even of some importance to know the conditions under which a solution of this kind would turn up on its own. Some of these points are considered in this chapter.

Proposition I.

3. To reduce any equation among logical symbols to the form V=0, in which V satisfies the law of duality, V(1−V)=0. It was shown in Chap. V. Prop. 4, that this condition holds whenever V is the sum of a series of constituents. It is also clear from Prop. 2, Chap. VI. that certain equations are all equivalent. These are the equations which, when we move everything to one side to get the form V=0 and then develop the left-hand side, produce the same series of constituents with coefficients that are not zero. The particular numbers those coefficients happen to be make no difference.

So the aim of this Proposition can always be reached in three steps. Bring all the terms of an equation over to the left-hand side. Expand that side fully. Then, in the result, change every coefficient that is not zero into unity (that is, 1), leaving alone those that are 1 already.

But developing functions that contain many symbols leads to expressions inconveniently long. It is worth showing, then, how this source of complexity can be avoided in the only cases that actually come up in practice.

The great primary forms of equations were discussed in Chapter VIII. They are— X=vY,X=Y,vX=vY. We have seen that whenever the conditions X(1−X)=0,Y(1−Y)=0, hold, that is, whenever X and Y each obey the law of duality, the first two of these equations lead us to the forms X(1−Y)=0,(1)X(1−Y)+Y(1−X)=0;(2) the first saying that there is nothing which is X and not Y, the second that there is nothing which is X and not Y, and nothing which is Y and not X. Under the same circumstances it can be shown that the last of them gives v(X(1−Y)+Y(1−X))=0;(3) and all of these results clearly satisfy, on their left-hand sides, the condition V(1−V)=0.

These are the forms and conditions in which the equations of a properly expressed logical system actually present themselves. So it is always possible, by the method above, to bring them under the required law. But although each separate equation may satisfy the law this way, their equivalent sum (VIII. 4) may not. It remains to show how the condition can be imposed on the sum too.

Let us write the equation formed by adding together the separate reduced equations of the system as v+v′+v″,&c.=0,(4) an equation which by itself is equivalent to the whole system it came from. We suppose v,v′,v″, &c. to be class terms (IX. 1) satisfying the conditions v(1−v)=0,v′(1−v′)=0,&c. Now the full interpretation of (4) would be found by developing its left-hand side with respect to all the elementary symbols x, y, &c. that it contains, and setting equal to 0 every constituent whose coefficient is not zero. In other words, we would set equal to 0 every constituent found in any one of v, v′, v″, &c. Those constituents fall into groups. First come those found in v. Second, those not found in v but found in v′. Third, those found in neither v nor v′ but found in v″. And so on. So they will be exactly the constituents found in the expression v+(1−v)v′+(1−v)(1−v′)v″+&c.,(5) an expression in which no constituent appears twice, and which plainly satisfies the law V(1−V)=0.

For example, suppose we had the expression (1−t)+v+(1−z)+tzw, in which the terms 1−t, 1−z are bracketed to show that each is to be taken as a single class term. Following (5), we would reduce it to an expression satisfying the condition V(1−V)=0 in three moves. Multiply every term after the first by t. Then multiply every term after the second by 1−v. Last, multiply the term that remains after the third by z. The result is 1−t+tv+t(1−v)(1−z)+t(1−v)zw.(6)

4. So all logical equations can be reduced to the form V=0, with V satisfying the law of duality. But it would clearly be a higher degree of perfection if equations always came in this form without any preparation at all. Better still if they not only showed this form when first stated, but kept it intact after the additions we must make to reduce a system of equations to a single equivalent one. That they do not do this on their own is not really a defect of the method. It comes from the fact that our premises are not always complete, accurate, and independent. They are not complete when they involve material relations (as distinguished from formal ones) that are left unstated. They are not accurate when they imply relations that were not intended. Let us set those two points aside, since they concern us less here, and consider in what sense premises may fail to be independent.

5. A system of propositions may be called independent when no conclusion that can be drawn from one portion of the system can also be drawn from any other portion of it. Suppose the equations representing those propositions are all reduced to the form V=0, then this condition means that no constituent which can be made to appear in the development of one function V of the system can be made to appear in the development of any other function V′ of the same system. When the condition fails, the equations of the system are not independent. This can happen in various ways. Let all the equations satisfy the law of duality on their left-hand sides. Then if a positive term x appears in the expansion of one equation, and a term xy in the expansion of another, the equations are not independent. The term x can be developed further into xy+x(1−y), and so the equation xy=0 is contained in both equations of the system, that is, both of them say that there is nothing which is both x and y. Again, let a term xy appear in one equation and a term xz in another. Both can be developed so as to give the shared constituent xyz. Other cases are easy to imagine in which premises that look quite independent at first sight are not really so. Whenever equations of the form V=0 are not truly independent in this way, then even though each one may satisfy the law of duality, V(1−V)=0, the equivalent equation we get by adding them together will not satisfy that condition, unless enough reductions have been carried out by the method of this chapter. When, on the other hand, the equations of a system both satisfy this law and are independent of one another, their sum will satisfy the same law. I have dwelt on these points longer than would otherwise be necessary. I think it important to form for ourselves, and to keep before us in all our investigations, the pattern of an ideal perfection, the goal and the guide of future efforts. In inquiries of this kind, the chief aim in improving the method should be to make the transformation of equations easier, as far as brevity allows, so that the fundamental condition spoken of above becomes universal.

In connexion with this subject the following Propositions deserve attention.

Proposition II.

If the left-hand side of any equation V=0 satisfies the condition V(1−V)=0, and if the expression of any symbol t of that equation is found as a developed function of the other symbols, then the coefficients of the expansion can take only the forms 1, 0, 00, 10.

For if we expand the equation with respect to t, we get Et+E′(1−t),(1) where E and E′ are what V becomes when t is changed in it first to 1 and then to 0. So E and E′ will themselves satisfy the conditions E(1−E)=0,E′(1−E′)=0.(2) Now (1) gives t=E′E′−E, and the right-hand side of this is to be expanded as a function of the remaining symbols. Clearly the only numerical values that E and E′ can take in calculating the coefficients are 1 and 0. So only the following cases can arise:

First, E′=1, E=1, then E′E′−E=10.

Second, E′=1, E=0, then E′E′−E=1.

Third, E′=0, E=1, then E′E′−E=0.

Fourth, E′=0, E=0, then E′E′−E=00.

This makes the truth of the Proposition plain.

6. It is worth noting that the forms 1, 0, and 00 appear in the solution of equations whether or not the condition V(1−V)=0 holds. The coefficient 10 is different. The terms that carry this coefficient when the condition is satisfied may take any value other than the three values 1, 0, and 00 when it is not. It is allowed, and it would make for uniformity, to change any coefficient of a development that does not come in one of the four forms named in this Proposition into 10. We treat this as the proper symbol to show that the term it is attached to must be set equal to 0. I shall often do this.

Proposition III.

7. The result of eliminating any symbols x, y, &c. from an equation V=0, whose left-hand side identically satisfies the law of duality, V(1−V)=0, can be obtained by developing the given equation with respect to the other symbols, and setting equal to 0 the sum of those constituents whose coefficients in the expansion are equal to unity.

Suppose the given equation V=0 involves only three symbols, x, y, and t, and that x and y are to be eliminated. Let the development of the equation with respect to t be At+B(1−t)=0,(1) with A and B free of the symbol t.

By Chap. IX. Prop. 3, the result of eliminating x and y from the given equation will have the form Et+E′(1−t)=0,(2) in which E is the result of eliminating the symbols x and y from the equation A=0, and E′ the result of eliminating them from the equation B=0.

Now A and B must satisfy the condition A(1−A)=0,B(1−B)=0 So A (we look at this coefficient alone for now) will be either 0, or 1, or a single constituent, or the sum of some of the constituents that involve the symbols x and y. If A=0, clearly E=0. If A is a single constituent, or the sum of only some of the constituents involving x and y, then E will be 0 as well. The reason is that the full development of A with respect to x and y will contain terms with zero coefficients, and E is the product of all the coefficients. So when A=1, E is equal to A, but in every other case E is equal to 0. In the same way, when B=1, E′ is equal to B, but in every other case E′ is zero. So the expression (2) will consist of that part of (1), if there is any, in which the coefficients A, B are unity. And this reasoning is general. Suppose, for instance, that V involved the symbols x, y, z, t, and that we had to eliminate x and y. Then if the development of V with respect to z and t were zt+xz(1−t)+y(1−z)t+(1−z)(1−t), the result we want would be zt+(1−z)(1−t)=0, this being the part of the development whose coefficients are unity.

Hence, if from any system of equations we derive a single equivalent equation V=0, with V satisfying the condition V(1−V)=0, the ordinary processes of elimination can be dropped entirely, and the single process of development can take their place.

8. There may be no practical advantage in the method pointed out here. But it has a theoretical unity and completeness that make it worth attention, and so I shall give a later chapter (XIV.) to illustrating it. The progress of applied mathematics has offered other striking examples of whole systems of problems or equations being brought under the rule of some central law that runs through all of them.

9. From what has gone before, we can see that there is one class of propositions for which none of the special devices of preparation above are needed. It is the class marked by the following conditions:

First, That the propositions are of the ordinary kind, the kind implied by the copula is or are, with the predicates particular.

Secondly, That the terms of the proposition are meaningful without assuming any unstated relation among the elements that go into expressing those terms.

Thirdly, That the propositions are independent.

We may, if such speculation is not altogether idle, allow ourselves a guess. These may be the conditions that language would obey, as an instrument of expression and of thought, in the hands of unerring beings who said simply what they meant, without holding anything back on the one hand and without repeating themselves on the other. Considered both in relation to the idea of a perfect language and in relation to the processes of an exact method, these conditions deserve the student's attention equally.

Chapter 11: Of Secondary Propositions, and of the Principles of Their Symbolical Expression

1. We established in Chap. IV. that every logical proposition belongs to one of two great classes, Primary Propositions and Secondary Propositions. The preceding chapters have dealt with the first class. We now turn to Secondary Propositions, that is, propositions about other propositions, considered as true or false. The inquiry we are beginning will follow the same general order and course as the one we have already carried out. The two inquiries differ in the subjects of thought they deal with. They do not differ in the formal and scientific laws they reveal, or in the methods and processes built on those laws. We might, to some degree, have expected this. It fits everything we know of the uniformity of Nature, and everything we believe of the unchanging constancy of the Author of Nature, to suppose that the mind should have a harmony and uniformity no less real than the one the physical sciences reveal to us. This is a mind, after all, that has been given high powers, not only for dealing with the world around it, but for knowing itself and reflecting on the laws of its own make-up. Expectations like this must never become the main rule of our inquiries. Nor should they ever draw us away from the patient research by which we find out how things actually are within the particular field we are investigating. But once the grounds of a resemblance have been properly and independently established, it is not at odds even with purely scientific aims to reflect on that resemblance, to trace how far it reaches, and to receive whatever hints it seems to give us of truths not yet discovered. This does not set aside the need for a final appeal to fact. Nor does it stretch analogy beyond its proper job, which is to suggest relations that independent inquiry must then either confirm or reject.

2. Secondary Propositions are those which concern or relate to Propositions considered as true or false. We express the relations of things by primary propositions. But we can also make propositions themselves the subject of thought, and express our judgments about them. The expression of any such judgment is a secondary proposition. There is no proposition at all about which enough knowledge would not let us make one of two assertions: either that the proposition is true, or that it is false. Each of these assertions is a secondary proposition. "It is true that the sun shines" and "It is not true that the planets shine by their own light" are examples. In the first, the proposition "The sun shines" is asserted to be true. In the second, the proposition "The planets shine by their own light" is asserted to be false. Secondary propositions also include every judgment that expresses a relation or dependence among propositions. Conditional propositions belong here, such as "If the sun shine the day will be fair." So do most disjunctive propositions, such as "Either the sun will shine, or the enterprise will be postponed." The first example expresses how the truth of the proposition "The day will be fair" depends on the truth of the proposition "The sun will shine." The second expresses a relation between the two propositions "The sun will shine" and "The enterprise will be postponed": the truth of one shuts out the truth of the other. To the same class of secondary propositions we must also assign every proposition that asserts that several propositions are true together or false together, such as "It is not true both that 'the sun will shine' and that 'the journey will be postponed.'" These different elements can even be mixed in a single secondary proposition. It may contain both the disjunctive element, expressed by either, or, and the conditional element, expressed by if; and the propositions it connects may themselves be compound. Take this: If "the sun shine," and "leisure permit," then either "the enterprise shall be commenced," or "some preliminary step shall be taken." Here several propositions are joined together, not at random or without meaning, but so as to express a definite connection among them, a connection that concerns whether each is true or false. So by our definition this combination is a Secondary Proposition.

The theory of Secondary Propositions deserves careful study, both for its many applications and for the close and harmonious analogy, already mentioned, that it bears to the theory of Primary Propositions. I want to say a little more about each of these points.

3. First, the reasonings of everyday life take the form of secondary propositions at least as often as the form of primary ones. The moralist and the metaphysician, too, perhaps talk less often about things and their qualities than about principles and hypotheses, about truths and how truths are connected and related to one another. Our experience is narrow, and the conclusions it suggests on the great unsolved questions of morals and society show their human origin in more ways than one. Universal principles certainly exist. But the partial formulas that make up our knowledge of how those principles apply are subject to conditions, exceptions, and failures. So in the fields of inquiry that ought, by their subject matter, to be the most interesting of all, much of what we actually know is hypothetical. We shall see later that writers on speculative philosophy have tended strongly to adopt these same forms of thought. A general method for handling hypothetical propositions, and the other kinds of secondary propositions, will therefore open a more interesting field of applications than any we have met so far.

4. Second, the theory of Secondary Propositions is interesting for the close and remarkable analogy it bears to the theory of Primary Propositions. We shall see that the formal laws governing the operations of the mind are expressed in exactly the same way in both cases. So the mathematical processes built on those laws are the same as well, and the methods worked out in the earlier part of this book will still serve in the new applications we are about to take up. But while the laws and processes of the method stay the same, the rule of interpretation must be adapted to new conditions. In place of classes of things we shall have propositions. In place of the relations of classes and individuals we shall consider the connections of propositions, or of events. Still, however much the two systems differ in what they mean and how they are read, a harmonious relation will be seen to run through both. This analogy helps us overcome every difficulty that remains. It is also interesting to study for its own sake, and it is conclusive proof of the unity of character that marks the make-up of the human faculties.

Proposition I.

5. To investigate the nature of the connexion of Secondary Propositions with the idea of Time.

To begin this inquiry, we must state clearly what the analogy is that links Secondary Propositions to Primary ones.

Primary Propositions express relations among things. They treat those things as parts of a universe, and the matter of our discourse is confined within that universe's limits, whether or not those limits are the same as the limits of the actual universe. The relations expressed are essentially substantive, that is, relations among things themselves: some, all, or none of the members of one class are also members of another class. Every subject that primary propositions refer to, and every relation among those subjects that they express, is of this kind.

With secondary propositions we find ourselves dealing with a different kind both of subject and of relation. The subjects we deal with are themselves propositions. So we may ask: can we regard these subjects, too, as things, and assign them, by analogy with the earlier case, to a universe of their own? Again, the relations among these propositions are relations of being true or false at the same time, not of being the same thing. When we express the connection between two different propositions, we do not say that one is the other. Instead, depending on the meaning we want, we use forms of speech like these: "Either the proposition X is true, or the proposition Y is true"; "If the proposition X is true, the proposition Y is true"; "The propositions X and Y are jointly true"; and so on.

In considering relations like these, we do not need to explore the whole range of their possible meaning. That could draw us into metaphysical questions about causation, which lie beyond the proper limits of science. It is enough to find some meaning they certainly have, and which is adequate for logical deduction. Take as an example the conditional proposition "If the proposition X is true, the proposition Y is true." One meaning it certainly has is that the time in which the proposition X is true is time in which the proposition Y is true. This is only a relation of existing together, and it may or may not exhaust the meaning of the proposition. But it really is part of what the proposition states, and it is enough for every purpose of logical inference.

Everyday language supports this view, that secondary propositions are essentially connected with the notion of time. We limit a primary proposition with the word "some," but a secondary proposition with the word "sometimes." To say "Sometimes injustice triumphs" amounts to saying that there are times when the proposition "Injustice now triumphs" is true. There are, it is true, propositions whose truth is not limited to particular periods or occasions in this way. These are true through all time, and have been given the name "eternal truths." Every reader of Plato and Aristotle will know the distinction. Aristotle in particular uses it to contrast the abstract truths of science, such as the propositions of geometry, which are always true, with the chance relations of things as they appear, which are sometimes true and sometimes false. But the forms of language in which both kinds of proposition are expressed show that both depend on the idea of time: in the one case limited to some finite stretch, in the other stretched out to eternity.

6. Someone may object that in ordinary reasoning we are often quite unaware of this notion of time, even though it is built into the very language we use. The remark is fair, but it only shows that we usually reason with the help of words and the forms of a well-built language, without attending to the deeper grounds on which those forms were established. The course of this investigation will illustrate the same principle. I shall use the notion of time to work out the laws for expressing secondary propositions, and the laws by which the symbols that express them combine. But once those laws and forms are settled, the notion of time (essential, as I believe it is, for that purpose) can in practice be dropped. We can then pass from the forms of common language to the closely matching forms of the symbolic instrument of thought developed here, use its processes, and interpret its results, without being aware of the idea of time at all.

Proposition II.

7. To establish a system of notation for the expression of Secondary Propositions, and to show that the symbols which it involves are subject to the same laws of combination as the corresponding symbols employed in the expression of Primary Propositions.

Let the capital letters X, Y, Z, stand for the elementary propositions about whose truth or falsehood we want to assert something, or among which we want to express some relation in the form of a secondary proposition. And let the matching small letters x, y, z, stand for operations of the mind, in the following sense. Let x represent an act of the mind by which we fix our attention on that portion of time during which the proposition X is true; and let this be what we mean when we say that x denotes the time during which the proposition X is true. Let us also use the connecting signs +, -, =, and so on, in the following sense. Let x+y stand for the total of the portions of time during which the propositions X and Y are respectively true, those times being entirely separate from each other. In the same way, let x−y stand for the time that is left when we take away, from the portion of time during which X is true, the portion during which Y is true, that portion being (by supposition) included in the first. And let x=y mean that the time during which the proposition X is true is the same as the time during which the proposition Y is true. We shall call x the representative symbol of the proposition X, and so on for the others.

From these definitions it follows that we shall always have x+y=y+x, since each side will stand for the same total of time.

Next, let xy represent performing one after the other the two operations represented by y and x. That is, it stands for the whole mental operation made up of these steps. First, the mind selects the portion of time during which the proposition Y is true. Second, out of that portion of time, it selects whatever part of it is time during which the proposition X is true. The result of these two steps is to fix the mind's attention on the whole of the portion of time during which the propositions X and Y are both true.

From this definition it follows that we shall always have xy=yx.(1) For suppose we first mentally select the portion of time during which the proposition Y is true, and then, out of that, the part during which X is true. Or suppose we first select the portion of time during which the proposition X is true, and then, out of that, the part during which the proposition Y is true. Either way we reach the same final result: the portion of time during which the propositions X and Y are both true.

By carrying this reasoning further we can show that the laws by which the symbols x, y, z, and so on, combine, under the interpretation given them here, are expressed in exactly the same way as the laws by which the same symbols combine under the interpretation given them in the first part of this book. The reason for this identity is plain. In both cases we are studying the operations of the same faculty, or the same combination of faculties. The essential character of those operations does not change, whether we suppose them to be at work on the universe of things, which contains all existence, or on the whole of time, in which all events take place and to some part of which, at least, every assertion, truth, and proposition refers.

So, besides the laws already stated, we shall have, by (4), Chap. II., the law whose expression is x(y+z)=xy+xz;(2) that is, the time when X is true and either Y or Z is true is the time when X and Y are true together with the time when X and Z are true. And above all we shall have the fundamental law of duality, (2) Chap. II., whose expression is x2=x,or,x(1−x)=0;(3) that is, there is no time when X is both true and not true. This law separates the system of thought in logic from the system of thought in the science of quantity, and it gives the processes of logic a completeness and generality they could not otherwise have.

8. Again, this law (3), like the other laws, is satisfied by the symbols 0 and 1. So we are led, as before, to ask whether those symbols can be given a meaning in the present system of thought. The same reasoning we followed before shows that they can, and justifies the following two statements.

First, in the expression of secondary propositions, 0 represents nothing with respect to time; that is, no time at all.

Second, in the same system, 1 represents the universe, or the whole of time, that the discourse is taken to be about in any way.

In primary propositions the universe of discourse is sometimes limited to a small part of the actual universe of things, and sometimes is the whole of that universe. In the same way, in secondary propositions the universe of discourse may be limited to a single day, or to the passing moment, or it may take in the whole of time. It may, in the most literal sense, be "eternal." Indeed, unless the nature of the discourse states or implies some limit, the proper meaning of the symbol 1 in secondary propositions is "eternity," just as its proper meaning in the primary system is the whole universe of what actually exists.

9. Instead of using the symbols x, y, z, to represent the truth of propositions, we could just as properly use them to represent the occurrence of events. In fact, the occurrence of an event both implies and is implied by the truth of a proposition, namely the proposition which asserts that the event occurs. Either meaning of the symbol x carries the other with it. It will be very convenient to be able to use our symbols in whichever of these two equivalent meanings the circumstances of a problem make most suitable, and I shall take that liberty whenever it is needed. In problems of pure logic I shall treat the symbols x, y, and so on, as representing elementary propositions, among which the premises express some relation. In the mathematical theory of probability, which, as I said before (I. 12), rests on a foundation of logic, and which this work goes on to treat in a later portion (in the chapters on probability, not included in this edition), I shall use the same symbols for the simple events whose implied or required frequency of occurrence is among the things that theory reckons with.

Editor's note: Boole reads the symbol of a secondary proposition as the portion of time during which that proposition is true. Modern logic drops the idea of time altogether. Instead it gives each proposition one of just two truth values, true or false, written 1 or 0, which is exactly where Boole's own algebra of 0 and 1 lands. This chapter is the ancestor of what is now called propositional logic.

Proposition III.

10. To deduce general Rules for the expression of Secondary Propositions.

In the questions that arise from this Proposition, full proof is less necessary, because they match exactly the questions we have already settled for primary propositions. We shall first consider how to express terms, and then how to express the propositions that connect them.

Since 1 stands for the whole of time, and x for the portion of it during which the proposition X is true, 1−x will stand for the portion of time during which the proposition X is false.

Again, since xy stands for the portion of time during which the propositions X and Y are both true, combining this with the last remark leads us to the following interpretations.

The expression x(1−y) will represent the time during which the proposition X is true and the proposition Y false. The expression (1−x)(1−y) will represent the time during which the propositions X and Y are both false at once.

The expression x(1−y)+y(1−x) will express the time during which either X is true or Y true, but not both; for that time is the sum of the times when each is true on its own, without the other. The expression xy+(1−x)(1−y) will express the time during which X and Y are either both true or both false.

If another symbol z comes in, the same principles still apply. So xyz stands for the time when the propositions X, Y, and Z are all true at once; (1−x)(1−y)(1−z) for the time when they are all false at once; and the sum of these two expressions would stand for the time when they are either all true together or all false together.

The general principles of interpretation at work in these examples need no further illustration or fuller statement.

11. We can now set out and study the laws for expressing propositions, case by case. First, though, I want to draw attention to one principle of fundamental importance. The principles of expression laid down so far are perfectly general. They let us limit our assertions that propositions are true or false to any particular parts of the whole of time that makes up the universe of our discourse, whether that whole is an unlimited eternity, or a period with a fixed beginning and end, or the passing moment. Yet in the actual practice of human reasoning such limits are not usually used. When we assert that a proposition is true, we generally mean that it is true throughout the whole of the time our discourse refers to. And when several assertions that propositions are simply true or false are made together as the premises of a logical proof, they all refer to the same universe of time, not to particular, limited parts of it. In the realm of necessary truth that the exact sciences study, every assertion of a truth may be the assertion of an "eternal truth." In reasoning about passing events (some state of affairs in society, for instance), each assertion may be tied directly to the present time, "Now." But in both cases, unless something is stated to the contrary, each separate proposition refers to the same stretch of time. The cases we then have to consider are the following.

First, to express the proposition, "The proposition X is true."

Here we must express that, within the limits of time to which the matter of our discourse is confined, the proposition X is true. The time during which the proposition X is true is denoted by x, and the stretch of time our discourse refers to is represented by 1. So we have x=1(4) as the expression required: the time when X is true is the whole of time.

Second, to express the proposition, "The proposition X is false."

Here we must express that, within the limits of time our discourse refers to, the proposition X is false; in other words, that within those limits there is no portion of time during which it is true. The portion of time during which it is true is x. So the equation required will be x=0.(5) We could also get this result by setting the expression for the time during which the proposition X is false, namely 1−x, equal to the whole of time, 1. This gives 1−x=1,whence x=0.

Third, to express the disjunctive proposition, "Either the proposition X is true or the proposition Y is true," where it is implied that the two propositions exclude each other, that is, that only one of them is true.

The time during which either the proposition X is true or the proposition Y is true, but not both, is represented by the expression x(1−y)+y(1−x). So we have x(1−y)+y(1−x)=1,(6) as the equation required.

Suppose instead that in this proposition the words either, or, are not meant to be strictly exclusive, so that the propositions X and Y might both be true at once. Then we must add the term xy to the left-hand side of the equation above. We shall then have xy+x(1−y)+(1−x)y=1,or x+(1−x)y=1.(7) that is, at every time either X is true, or, if it is not, Y is.

Fourth, to express the conditional proposition, "If the proposition Y is true, the proposition X is true."

Since whenever the proposition Y is true the proposition X is true, it is necessary and sufficient here to express that the time during which the proposition Y is true is time during which the proposition X is true. In other words, it is some indefinite portion of the whole time during which the proposition X is true. Now the time during which the proposition Y is true is y, and the whole time during which the proposition X is true is x. Let v be a symbol of time indefinite. Then vx will represent an indefinite portion of the whole time x. So we shall have y=vx as the expression of the given proposition: the time when Y is true is some of the time when X is true.

12. When v is taken in this way as a symbol of time indefinite, vx may be understood to represent the whole of the time x, or an indefinite part of it, or no part of it at all; for any of these meanings can be produced by fixing the arbitrary symbol v in a particular way. If v is fixed to represent a time that includes the whole time x, then vx will represent the whole time x. If v is fixed to represent a time some part of which lies inside the time x, but which does not fill up that time, then vx will represent a part of the time x. Lastly, if v is fixed to represent a time no part of which is shared with any part of the time x, then vx will take the value 0, and will amount to "no time," or "never."

Now notice that the proposition "If Y is true, X is true" does not assert that either of the propositions X and Y is true. It fits equally well with several suppositions. It fits the supposition that the truth of the proposition Y is a condition the truth of the proposition X cannot do without; in that case we shall have v=1. It fits the supposition that Y expresses a condition which, when it is met, assures us that X is true, but that X can be true without that condition being met; in that case v stands for a time some part of which lies within the whole time x. And lastly it fits the supposition that the proposition Y is never true at all; in that case v represents some time no part of which is shared with any part of the time x. All these cases are covered by the general supposition that v is a symbol of time indefinite.

Fifth, to express a proposition that is both conditional and disjunctive.

The general form of a conditional proposition is "If Y is true, X is true," and by the last section its expression is y=vx. Following the usage we set up for primary propositions, we may properly call Y and X the terms of the conditional proposition they belong to. We may also adopt the language of ordinary logic, which calls the term Y, the one the word if is attached to, the "antecedent" of the proposition, and the term X the "consequent."

Now suppose that the terms, instead of being simple propositions as in the case above, are each, or either of them, disjunctive propositions, joining different terms with the words either, or. Here are some examples, in which X, Y, Z, and so on, stand for simple propositions.

First, if either X is true or Y is true, then Z is true.

Second, if X is true, then either Y is true or Z true.

Third, if either X is true or Y is true, then either Z and W are both true, or they are both false.

Clearly, in these cases the relation of the antecedent to the consequent is not affected by the fact that one or both of those terms is disjunctive. So all we need to do is find, by the principles already established, the proper expressions for the antecedent and the consequent, attach the indefinite symbol v to the consequent, and set the two equal. For the propositions above, this gives the following equations, in order: 1st. x(1−y)+(1−x)y=vz.2nd. x=v{y(1−z)+z(1−y)}.3rd. x(1−y)+y(1−x)=v{zw+(1−z)(1−w)} The rule shown here applies in general.

One can imagine cases in which the disjunctive and conditional elements enter a compound proposition in some other way than this. But I do not know of any that the natural needs of human reasoning ever actually present to us, so I shall not discuss them. Leaving them out will cause no serious difficulty. The principles underlying the applications above are perfectly general, and a little thought will adapt them to any case one can imagine.

13. The laws of expression just stated already contain the laws of interpretation. The equation x=1 must be understood to say that the proposition X is true; the equation x=0, that the proposition X is false. The equation xy=1 will say that the propositions X and Y are both true together; and the equation xy=0 that they are not both true together.

In the same way the equations x(1−y)+y(1−x)=1,x(1−y)+y(1−x)=0, will assert, respectively, the truth and the falsehood of the disjunctive proposition "Either X is true or Y is true." The equations y=vxy=v(1−x) will express, respectively, the propositions "If the proposition Y is true, the proposition X is true" and "If the proposition Y is true, the proposition X is false."

In the chapters that follow we shall often meet a case in which some terms on one side of an equation have the indefinite symbol v attached and others do not. The following example will illustrate it. Suppose we have y=xz+vx(1−z). This says that the time during which the proposition Y is true is made up of all the time during which X and Z are true together, plus an indefinite portion of the time during which X is true and Z false. From this we can see, first, that if Y is true, then either X and Z are true together, or X is true and Z false; and second, that if X and Z are true together, Y is true. The second of these may be called the reverse interpretation. It takes the antecedent from the right-hand side of the equation, and the consequent from the left-hand side. What makes this reverse reading possible is that the right-hand side has a term whose coefficient is unity (that is, 1). The general principle involved may be stated as follows.

14. Principle.—Any constituent term or terms in a particular member of an equation which have for their coefficient unity, may be taken as the antecedent of a proposition, of which all the terms in the other member form the consequent.

In plainer words: any compartment on one side of an equation that has 1 in front of it can serve as the "if" part of a proposition whose "then" part is everything on the other side. So the equation y=xz+vx(1−z)+(1−x)(1−z) would have the following interpretations.

Direct Interpretation.—If the proposition Y is true, then either X and Z are true, or X is true and Z false, or X and Z are both false.

Reverse Interpretation.—If either X and Z are true, or X and Z are false, Y is true.

These partial interpretations, taken together, express the whole meaning of the given equation.

15. Let me repeat a remark made earlier. The idea of time seems to be an essential element in the theory of how secondary propositions are interpreted. Yet in practice it can be set aside as soon as the laws of expression and of interpretation have been firmly established. Indeed, the forms those laws produce seem to match the forms of a perfect language. Imagine any known language freed from its idioms and stripped of everything superfluous. Then imagine expressing some proposition in that language in the simplest and most literal way, the way most faithful to the principles of pure and universal thought on which all languages are founded, which all of them display, but from which all of them have strayed more or less. Passing from such a language to the notation of algebra would mean nothing more than swapping one set of signs for another, with no essential change of form or character. For the elements among which a relation is expressed, whether things or propositions, we would write letters; for the disjunctive conjunction "or" we would write +; for the connecting copula, the sign of relation, we would write =. I need not follow this analogy further. How real and complete it is will become clearer as we study the forms of expression that turn up in later applications of this theory, and compare them more directly with that imperfect yet noble instrument of thought, the English language.

16. Before leaving the subject of this chapter, I want to add a few remarks on the general analogy between the theory of Primary Propositions and that of Secondary Propositions.

We could no doubt have founded the theory of Primary Propositions on the simple notion of space, in the same way that the theory of secondary propositions has been founded on the notion of time. If we had, the analogy we are considering might have agreed a little more closely with the view of those who regard space and time as merely "forms of the human understanding," conditions of knowledge that the very make-up of the mind imposes on everything it takes in. But this view cannot be proved. And it ties us to accepting "place," τὸ ποῦ (the Greek "where"), as an essential category of existence. Whether it is one or not lies, I suspect, beyond the reach of our faculties. But it can be shown, and I believe it has been shown, that the formal processes of reasoning in primary propositions do not require, as an essential condition, that the things we reason about appear in space. Those processes would apply just as strictly to forms of existence, if there are any, that lie beyond the realm of things extended in space. Something perhaps similar happens in many known examples in geometry and dynamics: we can set out the formal analysis of problems that rest on some intellectual conception of space different from the one our senses present, or the one our imagination can picture. So I believe that the idea of space is not essential to developing a theory of primary propositions. But I am inclined to think, though I wish to speak cautiously on a question so very difficult, that the idea of time is essential to establishing a theory of secondary propositions. There seem to be grounds for thinking that space might have appeared to the human mind differently from the way it does, without any change in the faculties involved in reasoning. There do not seem to be the same grounds, at least, for supposing that time could have appeared to us otherwise than it does. But let me set these speculations aside, since they may not be entirely free from presumption. Let it simply be affirmed that the real reason the symbol 1 represents, in primary propositions, the universe of things and not the space they occupy is this: the sign of identity =, joining the two sides of the equations, implies that the things the two sides represent are identical, not merely that they are found in the same portion of space. Let it be affirmed, in the same way, that the reason the symbol 1 in secondary propositions represents, not the universe of events, but the eternity in whose successive moments and periods they unfold, is this: the same sign of identity, joining the logical sides of the equations, implies, not that the events the two sides represent are identical, but that the times at which they occur are the same. These reasons seem to me to settle the immediate question of interpretation. In an earlier treatise on this subject (Mathematical Analysis of Logic, p. 49), I followed Wallis's theory of the Reduction of Hypothetical Propositions, and was led to interpret the symbol 1 in secondary propositions as the universe of "cases" or "conjunctures of circumstances." But that view requires us to define what a "case," or a "conjuncture of circumstances," means; and whatever that term contains beyond the notion of time is certainly foreign to the aims of formal logic, and restricts its processes.

Footnote: Our perception presents space to us as having three dimensions: length, breadth, and depth. But in a large class of problems, those about the properties of curved surfaces, the rotation of solid bodies about axes, the vibrations of elastic media, and so on, this limit to three looks arbitrary in the mathematical working. If one paid attention only to the processes of solution, one could find no reason why space should not exist in four dimensions, or in any greater number. The intellectual procedure in the imaginary world this suggests can be grasped by the clear light of analogy. Aristotle sets out the existence of space in three dimensions, and the views on it held by the religious and philosophical mind of antiquity, as follows:— Μέγεθος δὲ τὸ μὲν ἐφ’ ἓν, γραμμή τὸ δ’ ἐπὶ δύο ἐπίπεδον, τὸ δ’ ἐπὶ τρία σῶμα· καὶ παρὰ ταῦτα οὐκ ἔστιν ἄλλο μέγεθος, διὰ τὸ τρία πάντα εἶναι καὶ τὸ τρὶς πάντῃ. Καθάπερ γάρ φασι καὶ οἱ Πυθαγόρειοι, τὸ πᾶν καὶ τὰ πάντα τοῖς τρισὶν ὥρισται. Τελευτὴ γὰρ καὶ μέσον καὶ ἀρχὴ τὸν ἀριθμὸν ἔχει τὸν τοῦ παντός· ταῦτα δὲ τὸν τῆς τριάδος. Διὸ παρὰ τῆς φύσεως εἰληφότες ὥσπερ νόμους ἐκείνης, καὶ πρὸς τὰς ἁγιστείας χρώμεθα τῶν θεῶν τῷ ἀριθμῷ τούτῳ.—De Caelo, 1. (In English: "Of magnitude, what extends one way is a line, two ways a plane, three ways a body; and besides these there is no other magnitude, because three is all, and 'three ways' is 'every way.' For, as the Pythagoreans also say, the whole and all things are bounded by threes: end, middle, and beginning hold the number of the whole, and that number is the triad. So, having taken it from nature as one of her laws, we use this number in the worship of the gods as well.")

Chapter 12: Methods in Secondary Propositions

1. We have already seen (XI. 7) that the literal symbols of logic combine by the same laws whether they express primary propositions or secondary ones. The only difference between the two cases is a difference of interpretation. We have also established (V. 6) a general principle: whenever two different systems of thought and interpretation are tied to the same system of formal laws (that is, of laws governing how the symbols are combined and used), then everything that happens in between is the same in both. By "in between" I mean the processes that lie between writing down the first conditions of a problem and interpreting its symbolic solution. Primary and secondary propositions share exactly this kind of common formal law. So the processes we established and illustrated for the first kind of proposition will apply to the second kind without any change at all.

2. In particular, the two fundamental processes of elimination and development obey the same laws for secondary propositions as for primary ones. Again, we saw (Chap. VI. Prop. 2) how to interpret any equation among primary propositions that contains no fractions. We develop it into a series of constituents, and set equal to 0 every constituent whose coefficient is not zero. The same method works for the equations of secondary propositions. And the interpreted result it finally leads to is, as before (VI. 6), a system of denials that hold together. There is one difference. In the primary case those denials deny that certain classes of things exist. In the secondary case they deny the truth of certain combinations of the elementary propositions that make up the terms of the premises. And just as in primary propositions the system of denials could be converted into various other forms of proposition (VI. 7), &c., we shall find that the same conversion is possible here. Again the only difference lies not in the form of the equations but in the nature of their interpretation.

3. Further, in primary propositions we can find the expression of any one element of a system of equations in terms of the remaining elements (VI. 10), or in terms of any chosen few of them, and interpret that expression as a logical inference. The same thing can be done by the same means for secondary propositions, the difference of interpretation alone excepted. The steps are no different from the corresponding steps for primary propositions: eliminating the elements we want to banish from the final solution, reducing the system to a single equation, solving it algebraically, and developing the solution into a form that can be interpreted.

To clear away any possible difficulty, however, it may help to gather into a general Rule the different cases that arise in working with secondary propositions.

Rule.—Express the given propositions in symbols (XI. 11).

From each equation in which it appears, eliminate separately the indefinite symbol v (VII. 5).

Eliminate the remaining symbols that you want to banish from the final solution. Before each elimination, always reduce to a single equation all the equations in which the symbol or symbols to be eliminated appear (VIII. 7). Collect the resulting equations into a single equation V=0.

Then proceed according to the particular form in which you want to express the final relation, as follows.

First, if you want it in the form of a denial, or a system of denials, develop the function V, and set equal to 0 all those constituents whose coefficients are not zero.

Second, if you want it in the form of a disjunctive proposition, set equal to 1 the sum of those constituents whose coefficients are zero.

Third, if you want it in the form of a conditional proposition whose antecedent is a simple element, such as x or 1−x, find the algebraic expression of that element, and develop that expression.

Fourth, if you want it in the form of a conditional proposition whose antecedent is a compound expression, such as xy, xy+(1−x)(1−y), &c., set that expression equal to a new symbol t. Then find t as a developed function of the symbols that are to appear in the consequent, either by the ordinary methods or by the special method of (IX. 9).

Fifth, interpret the results by (XI. 13, 14).

Suppose we only want to find out whether one particular elementary proposition x is true or false. Then we must eliminate every symbol except x. If the result is the equation x=1, the proposition is true; if x=0, it is false; if 0=0, the premises are not enough to decide whether it is true or false.

4. Ex. 1.—Cicero's treatise On Fate (De Fato), which survives only in fragments, contains a curious discussion of the following prediction: "Si quis (Fabius) natus est oriente Canicula, is in mari non morietur." That is, "If anyone (Fabius) was born at the rising of the dogstar, he will not die at sea." I shall apply the method of this chapter to it. Let y represent the proposition, "Fabius was born at the rising of the dogstar," and x the proposition, "Fabius will die in the sea." When I say that x represents the proposition "Fabius will die in the sea," I mean only that x is a symbol assigned (XI. 7) to that proposition in such a way that the equation x=1 asserts its truth and the equation x=0 denies it. Reading the symbols as times, as in the last chapter, x stands for the time during which "Fabius will die in the sea" is true, and y for the time during which "Fabius was born at the rising of the dogstar" is true. The equation we have to discuss will be y=v(1−x).(1) that is, the time for which the first proposition is true is some part of the time for which the second is false: if Fabius was born at the dogstar's rising, he will not die at sea. First, let us reduce the given proposition to a denial, or a system of denials (XII. 3). Moving the right-hand side across, we have y−v(1−x)=0. Eliminating v, y{y−(1−x)}=0,or, y−y(1−x)=0,or, yx=0.(2) This result means: "It is not true that Fabius was born at the rising of the dogstar and will die in the sea." Cicero calls this form of proposition "Conjunctio ex repugnantibus" (a conjunction of things that cannot go together, denied). He remarks that Chrysippus hoped by this form to escape a difficulty he believed he saw in contingent statements about the future, that is, statements about things that may or may not happen: "Hoc loco Chrysippus aestuans falli sperat Chaldaeos caeterosque divinos, neque eos usuros esse conjunctionibus ut ita sua percepta pronuntient: Si quis natus est oriente Canicula is in mari non morietur; sed potius ita dicant: Non et natus est quis oriente Caniculâ, et in mari morietur. O licentiam jocularem! ... Multa genera sunt enuntiandi, nec ullum distortius quam hoc quo Chrysippus sperat Chaldaeos contentos Stoicorum causa fore."—Cic. De Fato, 7, 8. In English: here Chrysippus, sweating over the problem, hopes that the Chaldaean astrologers and the other diviners can be talked out of announcing their findings as "if" statements, such as "If anyone was born at the rising of the dogstar, he will not die at sea," and will say instead, "It is not the case both that someone was born at the rising of the dogstar and that he will die at sea." What playful license! ... There are many ways of stating a thing, and none more twisted than this one, which Chrysippus hopes the Chaldaeans will accept for the Stoics' sake.

5. Next, let us reduce the given proposition to a disjunctive form. The constituents that do not appear on the left-hand side of (2) are x(1−y),(1−x)y,(1−x)(1−y). So we have y(1−x)+x(1−y)+(1−x)(1−y)=1.(3) This means: either Fabius was born at the rising of the dogstar and will not perish in the sea; or he was not born at the rising of the dogstar and will perish in the sea; or he was not born at the rising of the dogstar and will not perish in the sea.

In cases like this one, however, where some constituents differ from each other in only a single factor, it is most convenient, as we have seen (VII. 15), to gather those constituents into a single term. If we combine the first and third terms of (3) in this way, we have (1−y)x+1−x=1; and if we combine the second and third in the same way, we have y(1−x)+1−y=1. These two forms of the equation give, respectively, the interpretations:

Either Fabius was not born under the dogstar and will die in the sea, or he will not die in the sea.

Either Fabius was born under the dogstar and will not die in the sea, or he was not born under the dogstar.

Clearly these interpretations say exactly the same as the earlier one.

Now let us find, in the form of a conditional proposition, what follows from the supposition that "Fabius will perish in the sea."

In equation (2), which is what remains of the original equation once v has been eliminated, we must try to find x as a function of y.

We have x=0y=0y+00(1−y) on expansion, or x=00(1−y); that is, the time in which Fabius dies at sea lies somewhere within the time in which he was not born at the dogstar's rising. Interpreted: If Fabius dies in the sea, he was not born at the rising of the dogstar.

These examples go some way to illustrate the connection we have established in the previous chapters between primary and secondary propositions. That connection has two marks: the processes are identical, and the interpretations are analogous.

6. Ex. 2.—The second book of Plato's Republic contains a remarkable argument meant to prove that the Divine Nature cannot change. It is a very fine example of two things at once. It shows the careful induction from familiar instances by which Plato arrives at general principles, and the clear, connected logic by which he then deduces from them the particular conclusions he wants to establish. The argument is contained in the following dialogue:

"Must not that which departs from its proper form be changed either by itself or by another thing? Necessarily so. Are not things which are in the best state least changed and disturbed, as the body by meats and drinks, and labours, and every species of plant by heats and winds, and such like affections? Is not the healthiest and strongest the least changed? Assuredly. And does not any trouble from without least disturb and change that soul which is strongest and wisest? And as to all made vessels, and furnitures, and garments, according to the same principle, are not those which are well wrought, and in a good condition, least changed by time and other accidents? Even so. And whatever is in a right state, either by nature or by art, or by both these, admits of the smallest change from any other thing. So it seems. But God and things divine are in every sense in the best state. Assuredly. In this way, then, God should least of all bear many forms? Least, indeed, of all. Again, should He transform and change Himself? Manifestly He must do so, if He is changed at all. Changes He then Himself to that which is more good and fair, or to that which is worse and baser? Necessarily to the worse, if he be changed. For never shall we say that God is indigent of beauty or of virtue. You speak most rightly, said I, and the matter being so, seems it to you, O Adimantus, that God or man willingly makes himself in any sense worse? Impossible, said he. Impossible, then, it is, said I, that a god should wish to change himself; but ever being fairest and best, each of them ever remains absolutely in the same form."

The premises of this argument are the following.

First, if the Deity suffers change, He is changed either by Himself or by another.

Second, if He is in the best state, He is not changed by another.

Third, the Deity is in the best state.

Fourth, if the Deity is changed by Himself, He is changed to a worse state.

Fifth, if He acts willingly, He is not changed to a worse state.

Sixth, the Deity acts willingly.

Let us express the elements of these premises as follows.

Let x represent the proposition, "The Deity suffers change."

Let y represent the proposition, "He is changed by Himself."

Let z represent the proposition, "He is changed by another."

Let s represent the proposition, "He is in the best state."

Let t represent the proposition, "He is changed to a worse state."

Let w represent the proposition, "He acts willingly."

Written in symbols, and with the indefinite class symbols v eliminated, the premises then give the following equations:

xyz+x(1−y)(1−z)=0,(1)sz=0,(2)s=1,(3)y(1−t)=0,(4)wt=0,(5)w=1.(6)

Read aloud, they say: (1) it is never true that He suffers change and is changed both by Himself and by another, nor that He suffers change and is changed by neither; (2) He is never both in the best state and changed by another; (3) He is in the best state; (4) He is never changed by Himself without being changed to a worse state; (5) He never both acts willingly and is changed to a worse state; (6) He acts willingly.

Keeping x, I shall eliminate in turn z, s, y, t, and w (the order in which those symbols occur in the system above), and interpret each result as it comes.

Eliminating z from (1) and (2), we get xs(1−y)=0.(7) Eliminating s from (3) and (7), x(1−y)=0.(8) Eliminating y from (4) and (8), x(1−t)=0.(9) Eliminating t from (5) and (9), xw=0.(10) Eliminating w from (6) and (10), x=0.(11)

From equation (8) onward, these give the following results.

From (8) we have x=00y, so If the Deity suffers change, He is changed by Himself.

From (9), x=00t: If the Deity suffers change, He is changed to a worse state.

From (10), x=00(1−w): If the Deity suffers change, He does not act willingly.

From (11): The Deity does not suffer change. This is Plato's result.

I have remarked before that the order of elimination makes no difference. Let us test that fact here by eliminating the same symbols in the reverse order, beginning with w. The resulting equations are

t=0,y=0,x(1−z)=0,z=0,x=0;

and they give the following interpretations, one for each equation:

God is not changed to a worse state. He is not changed by Himself. If He suffers change, He is changed by another. He is not changed by another. He is not changed.

So we reach the same conclusion by a different route.

As a display of the power of the method, these examples are of little value. But they illustrate its nature and character just as well as more complicated cases would.

7. One last instance of the analogy between the system of primary propositions and that of secondary propositions deserves mention. In the secondary system, too, the fundamental equation, x(1−x)=0, can be interpreted. It says that there is no time during which the proposition is both true and false; that is, it expresses the axiom, A proposition cannot be true and false at the same time. Compare this with the corresponding interpretation for classes of things (III. 15). If we solve the equation in the form x=01−x=00x, and develop it, it gives, in the two systems, the axioms "A thing is what it is" and "If a proposition is true, it is true." Both are forms of what has been called "the principle of identity." People have held the most opposite opinions about the nature and value of these axioms. Some have treated them as the very core and essence of philosophy. Locke gave them a chapter headed "On Trifling Propositions." Both views, I think, mix truth with error. If such propositions are put in place of experience, or used as material for the empty, wordy quarrels of the schools, they are worse than trifling. But seen as closely bound up with the very laws and conditions of thought, they rise to an importance that is at least speculative.

Footnote: Essay on the Human Understanding, Book IV. Chap. viii.

Chapter 13: Clarke and Spinoza

1. In this chapter and the next I mean to follow a definite plan. I shall examine the actual premises involved in proving some of the general propositions of the two treatises named above, whether those premises are stated outright or only implied. By the actual premises I mean every proposition that the argument assumes without proving and uses as part of the foundation on which its final conclusion is built. Once I have found these premises, I shall express them in the language of symbols. Then, by the methods developed in the earlier chapters of this work, I shall deduce from them the most important inferences they contain, beyond the particular inferences the authors themselves drew. In some cases I shall change the premises: I shall leave out some fact or principle they contain, or add or substitute some new proposition, and then work out how the final conclusions are affected by the change. In all this it is not my task to ask, except in passing, how far the metaphysical principles laid down in these famous works deserve our trust. My task is only to find out what conclusions may rightly be drawn from given premises. And in doing that, I want to show the complete freedom we have in choosing both the elements of the final propositions and the order in which they appear: that is, in deciding which elementary propositions to declare true or false, and which to declare true or false under given restrictions, or in given combinations.

2. The main practical difficulty of this inquiry will not lie in applying the method once the premises are fixed. It will lie in finding out what the premises are. Even in what are regarded as the most rigorous examples of reasoning on metaphysical questions, we shall sometimes find that different trains of thought are blended together. We shall find that some particular but essential parts of a proof are given in parentheses, or off to the side of the main argument; that the meaning of a premise is to some degree ambiguous; and, not rarely, that arguments judged by the strict laws of formal reasoning are incorrect or inconclusive. In such cases it may be very hard to determine the true premises of a proof and set them out clearly. But this is a difficulty that everyone must overcome who wants to know whether a particular conclusion is proved or not, whatever form they intend to give the reasoning that follows. So it is not a difficulty peculiar to the method of this work, though it shows itself more plainly with this method than with any other. The connection is in fact so close that, using the method of this treatise, we cannot even guess whether a conclusion is valid until we have clearly grasped and exactly stated every premise it rests on. In the more usual way of proceeding, on the other hand, nothing is more common than to examine a few steps of a train of argument and from them form a vague general impression of the whole, without any such preliminary and thorough analysis of the premises it involves.

This need to pin down exactly the real premises of a proof should not be thought of as a drawback, especially because, once it is done, every source of doubt or ambiguity is gone. With the method of this treatise, the order in which the premises are arranged, the way they are connected, and every similar circumstance may be treated as a matter of indifference. The process of inference is then carried out with a precision that might almost be called mechanical.

3. The "Demonstration of the Being and Attributes of God" consists of a series of propositions, or theorems. Each is proved from premises that can mostly be sorted into two distinct kinds. The first kind is facts of observation, such as the existence of a material world, the phenomenon of motion, &c. The second kind is hypothetical principles, whose authority and universal reach are supposed to be recognized à priori. It is of course on the truth of the second kind that the validity of the proof really depends, assuming the reasoning is correct. But whatever may be thought of its claims on that score, there is no question that, as an intellectual achievement, the work stands very high. Its trains of argument are not generally very clearly arranged, but they are almost always examples of correct logic, and they show a subtlety of understanding and a force of reasoning that have seldom been equalled and perhaps never surpassed. In them we see the culmination of the intellectual efforts that were stirred up in the field of metaphysical inquiry at a time when the rule of hypothetical principles was less questioned than it is now, and when the rigorous proofs of the newly risen school of mathematical physics seemed to offer a model for such efforts to follow. For this reason, not to mention the dignity of their subject, they seem to me to deserve serious attention, and I do not think it a vain or pointless task to spend a careful analysis on some of them.

4. The Ethics of Benedict Spinoza is a treatise whose object is to prove that God and the universe are identical, and to build on that doctrine a system of morals and of philosophy. Its main argument is extremely hard to analyse. The difficulty is not that its separate propositions are complicated. It is that it uses vague definitions, and axioms so unclear that it is hard to decide whether we ought to accept them or reject them. The reasoning of Dr. Samuel Clarke is partly verbal; that of Spinoza is verbal to a much greater degree. Perhaps that is why, to some minds, Spinoza's reasoning has seemed to have a formal force to which in reality it has no fair claim. These points, however, will be considered in their proper place.

Clarke's Demonstration.

Proposition I.

5. "Something has existed from eternity."

The proof is as follows:—

"For since something now is, 'tis manifest that something always was. Otherwise the things that now are must have risen out of nothing, absolutely and without cause. Which is a plain contradiction in terms. For to say a thing is produced, and yet that there is no cause at all of that production, is to say that something is effected when it is effected by nothing, that is, at the same time when it is not effected at all. Whatever exists has a cause of its existence, either in the necessity of its own nature, and thus it must have been of itself eternal: or in the will of some other being, and then that other being must, at least in the order of nature and causality, have existed before it."

Let us now analyse this proof. Its first sentence breaks down into the following propositions:

1st. Something is.

2nd. If something is, either something always was, or the things that now are must have risen out of nothing.

The next part of the proof argues that the second of these alternatives, namely "The things that now are have risen out of nothing," is impossible. Formally, it breaks down as follows:

3rd. If the things that now are have risen out of nothing, something has been effected, and at the same time that something has been effected by nothing.

4th. If that something has been effected by nothing, it has not been effected at all.

This second part of the argument seems to be simply an assumption of the point to be proved, or an attempt to make that point clearer by stating it in different words.

The third and last part of the proof gives a separate proof of one of two things: either of the original proposition to be proved, namely "Something always was," or of the point argued in the second part, namely that the hypothesis "the things that now are have risen out of nothing" cannot stand. It breaks down as follows:—

5th. If something is, either it exists by the necessity of its own nature, or it exists by the will of another being.

6th. If it exists by the necessity of its own nature, something always was.

7th. If it exists by the will of another being, then the proposition, that the things which exist have arisen out of nothing, is false.

This last proposition is not stated in this form in Dr. Clarke's text. But his stated conclusion, that another Being existed first, is plainly meant to amount to a denial of the proposition that the things which now are have risen out of nothing.

So the proof consists of two distinct trains of argument. One is made up of what I have called the first and second parts of the proof; the other is made up of the first and third parts. Let us consider the second of these trains.

The premises are:—

1st. Something is.

2nd. If something is, either something always was, or the things that now are have risen out of nothing.

3rd. If something is, either it exists in the necessity of its own nature, or it exists by the will of another being.

4th. If it exists in the necessity of its own nature, something always was.

5th. If it exists by the will of another being, then the hypothesis, that the things which now are have risen out of nothing, is false.

We must now express these propositions in symbols.

Letx =Something is.
y =Something always was.
z =The things which now are have risen from nothing.
p =It exists in the necessity of its own nature (that is, the something spoken of above).
q =It exists by the will of another Being.

It should be understood that writing Let x= "Something is" means no more than that x is the representative symbol of that proposition (XI. 7). The equations x=1 and x=0 then declare, respectively, that it is true and that it is false.

The equations of the premises are:—

1st.x = 1;
2nd.x=v{y(1−z)+z(1−y)};
3rd.x=v{p(1−q)+q(1−p)};
4th.p=vy;
5th.q=v(1−z);

and when we eliminate the various indefinite symbols v, we get 1−x=0;(1)x[yz+(1−y)(1−z)]=0;(2)x[pq+(1−p)(1−q)]=0;(3)p(1−y)=0;(4)qz=0.(5)

6. First, I shall examine whether any conclusions can be deduced from these equations about the truth or falsehood of the single propositions represented by the symbols y, z, p, q, namely of the propositions "Something always was;" "The things which now are have risen from nothing;" "The something which is exists by the necessity of its own nature;" "The something which is exists by the will of another being."

To do this, we must eliminate in turn every symbol except y, then every symbol except z, &c. The equation that results each time will show whether any such separate relation exists.

To eliminate x from (1), (2), and (3), we need only put the value of x given by (1) into (2) and (3). The results are yz+(1−y)(1−z)=0.(6)pq+(1−p)(1−q)=0.(7) That is, of the two propositions in each pair, exactly one is true.

To eliminate p, we add (4) and (7), which gives p(1−y)+pq+(1−p)(1−q)=0;(8) and from this we find (1−y)(1−q)=0.(9) that is, it is not the case both that nothing always was and that the something does not exist by the will of another Being.

To eliminate q from (5) and (9), we have qz+(1−y)(1−q)=0; and from this we find z(1−y)=0.(10) Only the two equations (6) and (10) now remain. Adding them gives yz+1−y=0. Eliminating z from this equation, we have 1−y=0,or, y=1.(11) Eliminating y from the same equation, we have z=0.(12) The interpretation of (11) is:

Something always was.

The interpretation of (12) is:

The things which are have not risen from nothing.

Next let us go back to the system (6), (7), together with the two equations (4) and (5), and find the two equations that involve p and q respectively.

To eliminate y, we have from (4) and (6) p(1−y)+yz+(1−y)(1−z)=(0); and from this (p+1−z)z=0,or, pz=0.(13) that is, it is not the case both that the something exists in the necessity of its own nature and that the things which now are have risen from nothing.

To eliminate z from (5) and (13), we have qz+pz=0; and from this we get 0=0. So there remains only equation (7). Eliminating q from it, we get 0=0 as the final equation in p.

So no conclusion can be drawn from the premises asserting simply that the proposition "The something which is exists in the necessity of its own nature" is true, or that it is false. And since eliminating p gives the same result, 0=0, as the final equation in q, it follows also that no conclusion can be drawn from the premises about the simple truth or falsehood of the proposition "The something which is exists by the will of another Being."

As for relations connecting more than one of the propositions represented by the elementary symbols, we need only consider the one expressed by equation (7), which connects p and q. The propositions represented by the other symbols are absolutely true or false on their own, independent of any connection of this kind. Put into the form p(1−q)+q(1−p)=1, is, the interpretation of (7) is:

The something which is, either exists in the necessity of its own nature, or by the will of another being.

I have set out the details of this analysis at a length that is perhaps needlessly full and wordy. I did so because in the examples that follow I mean rather to indicate the steps by which the results are reached than to risk repeating myself tiresomely. The conclusions the method has produced here are easily checked by ordinary reasoning.

The reader will have no difficulty in applying the method to the other train of premises in Dr. Clarke's first Proposition, and deducing from them the first two of the conclusions this analysis has reached.

Proposition II.

7. Some one unchangeable and independent Being has existed from eternity.

The premises from which this proposition is proved are the following:

1st. Something has always existed.

2nd. If something has always existed, either there has existed some one unchangeable and independent being, or the whole of existing things has been comprehended in a succession of changeable and dependent beings.

3rd. If the universe has consisted of a succession of changeable and dependent beings, either that series has had a cause from without, or it has had a cause from within.

4th. It has not had a cause from without (because it includes, by hypothesis, all things that exist).

5th. It has not had a cause from within (because no part is necessary, and if no part is necessary, the whole cannot be necessary).

Leaving out, purely for brevity, the supporting proofs given in parentheses in the fourth and fifth premises, we may represent the premises as follows: Let x= Something has always existed.y=There has existed some one unchangeable and independent being.z=There has existed a succession of changeable and dependent beings.p=That series has had a cause from without.q=That series has had a cause from within. Then we have the following system of equations, namely: 1st. x=1;2nd. x=v{y(1−z)+z(1−y)};3rd. z=v{p(1−q)+(1−p)q};4th. p=0;5th. q=0: which, when we eliminate each indefinite symbol v separately, gives 1−x=0;(1)x{yz+(1−y)(1−z)}=0;(2)z{pq+(1−p)(1−q)}=0;(3)p=0;(4)q=0.(5) Eliminating x, p, q, and y from this system leads to the equation z=0. And eliminating x, p, q, and z leads in the same way to the equation y=1. The interpretations of these equations are, respectively:

1st. The whole of existing things has not been comprehended in a succession of changeable and dependent beings.

2nd. There has existed some one unchangeable and independent being.

The second of these is the proposition Dr. Clarke proves. This analysis has settled every one of the propositions represented by the literal symbols x, y, z, p, q as absolutely true or absolutely false. So there is no need to look for any further relations connecting them.

Clarke gives another proof of Prop. II., which for brevity I pass over. It is worth noticing that the "impossibility of infinite succession," which Clarke proves as part of his argument, has commonly been assumed as a fundamental principle of metaphysics and carried over to questions other than causation. Aristotle uses it to establish that there must be first principles of demonstration, and that there must be an end (the good) in human actions, &c. There is perhaps no principle he appeals to more often in his writings. The schoolmen (the philosophers of the medieval universities) used it in the same way to prove that genera and species cannot be ranked one under another without end, and so that universals must exist. It seems that our inability to form a definite and complete conception of an infinite series, that is, to grasp it as a whole, has been confused with a logical inconsistency, or a contradiction in the idea itself.

Footnote: Metaphysics, III. 4; Anal. Post. I, 19, et seq.

Footnote: Nic. Ethics, Book I. Cap. II.

8. The analysis of the next argument depends on the theory of Primary Propositions.

Proposition III.

That unchangeable and independent Being must be self-existent.

The premises are:—

1. Every being must either have come into existence out of nothing, or it must have been produced by some external cause, or it must be self-existent.

2. No being has come into existence out of nothing.

3. The unchangeable and independent Being has not been produced by an external cause.

To express these in symbols, let us take x=Beings which have arisen out of nothing.y=Beings which have been produced by an external cause.z=Beings which are self-existent.w=The unchangeable and independent Being. Then we have x(1−y)(1−z)+y(1−x)(1−z)+z(1−x)(1−y)=1,(1)x=0,(2)w=v(1−y),(3) and eliminating v from the last of these, wy=0.(4) that is, the unchangeable and independent Being is not among the beings produced by an external cause.

Whenever, as here, a symbol's value is given as 0 or 1, the best way to eliminate it is by simple substitution. So eliminating x gives y(1−z)+z(1−y)=1;(5)or, yz+(1−y)(1−z)=0.(6) Now adding (4) and (6), and eliminating y, we get w(1−z)=0,∴w=vz; and the interpretation of this is,—The unchangeable and independent being is necessarily self-existing.

The interpretation of (5), in the form it has as it stands, is,—Every being has either been produced by an external cause, or it is self-existent.

9. In Dr. Samuel Clarke's remarks on this proposition there is a remarkable argument meant to prove that the material world is not the self-existent being just spoken of. The passage I mean is the following:

"If matter be supposed to exist necessarily, then in that necessary existence there is either included the power of gravitation, or not. If not, then in a world merely material, and in which no intelligent being presides, there never could have been any motion; because motion, as has been already shown, and is now granted in the question, is not necessary of itself. But if the power of gravitation be included in the pretended necessary existence of matter: then, it following necessarily that there must be a vacuum (as the incomparable Sir Isaac Newton has abundantly demonstrated that there must, if gravitation be an universal quality or affection of matter), it follows likewise, that matter is not a necessary being. For if a vacuum actually be, then it is plainly more than possible for matter not to be."—(pp. 25, 26).

A careful examination shows that the actual premises involved in this proof are the following:

1st. If matter is a necessary being, either the property of gravitation is necessarily present, or it is necessarily absent.

2nd. If gravitation is necessarily absent, and the world is not subject to any presiding Intelligence, motion does not exist.

3rd. If the property of gravitation is necessarily present, the existence of a vacuum is necessary.

4th. If the existence of a vacuum is necessary, matter is not a necessary being.

5th. If matter is a necessary being, the world is not subject to a presiding Intelligence.

6th. Motion exists.

Of these premises, the first four are stated in the proof. The fifth is implied in the way its first and second sentences are connected. The sixth states a fact that the author does not seem to have thought it necessary to state, but which is clearly part of the ground of his reasoning. Let us represent the elementary propositions as follows: Let x=Matter is a necessary being.y=Gravitation is necessarily present.t=Gravitation is necessarily absent.z=The world is merely material, and not subject to any presiding Intelligence.w=Motion exists.v=A vacuum is necessary. Then the system of premises is represented by the following equations, in which q is used as the symbol of an indefinite time: x=q{y(1−t)+(1−y)t}.tz=q(1−w).y=qv.v=q(1−x).x=qz.w=1. If we eliminate the symbols q from these, we get the following system, namely: x{yt+(1−y)(1−t)}=0.(1)tzw=0.(2)y(1−v)=0.(3)vx=0.(4)x(1−z)=0.(5)1−w=0.(6) Now if we eliminate w, v, z, y, and t from these equations, we obtain the equation x=0, which expresses the proposition, Matter is not a necessary being. This is Dr. Clarke's conclusion. If we try to eliminate any other set of five symbols (except the set v, z, y, t, and x, which would give w=1), we get a result of the form 0=0. So there are no other conclusions stating that any of the elementary propositions marked by single symbols is absolutely true or absolutely false.

Of conclusions expressed by equations in two symbols, there is only the following one, namely:— If the world is merely material, and not subject to a presiding Intelligence, gravitation is not necessarily absent. This conclusion is expressed by the equation tz=0, whence z=q(1−t). Suppose that in this analysis we drop the last premise, the one stating the fact that motion exists, and leave untouched the hypothetical principles contained in the other premises. Then some remarkable conclusions follow. I shall turn to these in the next chapter.

10. Of the rest of Dr. Clarke's argument I shall give only the substance and the connections, briefly. I shall dwell only on certain parts of it that are more complex than the others and better illustrate the method of this work.

In Prop. IV. it is shown that the substance or essence of the self-existent being is incomprehensible. The drift of the reasoning is that we do not know the essential nature of anything else, and so, much less, the essence of the self-existent being.

In Prop. V. it is argued that "though the substance or essence of the self-existent being is itself absolutely incomprehensible to us, yet many of the essential attributes of his nature are strictly demonstrable, as well as his existence."

In Prop. VI. it is argued that "the self-existent being must of necessity be infinite and omnipresent;" and it is maintained that his infinity must be "an infinity of fulness as well as of immensity." The ground the proof rests on is that an absolute necessity of existence must be independent of time, place, and circumstance, free from limitation, and so exclude all imperfection. And from this it is inferred that the self-existent being must be "a most simple, unchangeable, incorruptible being, without parts, figure, motion, or any other such properties as we find in matter."

The premises actually used may be set out as follows:

1. If a finite being is self-existent, it is a contradiction to suppose it not to exist.

2. A finite being may, without contradiction, be absent from one place.

3. That which may without contradiction be absent from one place may without contradiction be absent from all places.

4. That which may without contradiction be absent from all places may without contradiction be supposed not to exist.

Let us take x=Finite beings.y=Things self-existent.z=Things which it is a contradiction to suppose not to exist.w=Things which may be absent without contradiction from one place.t=Things which without contradiction may be absent from every place. Expressing the premises and eliminating the indefinite symbols, we have xy(1−z)=0.(1)x(1−w)=0.(2)w(1−t)=0.(3)tz=0.(4) Eliminating t, w, and z in turn, we get xy=0, ∴y=00(1−x); and the interpretation of this is,—Whatever is self-existent is infinite.

In Prop. VII. it is argued that the self-existent being must of necessity be One. The proof runs in this order: the self-existent being is "necessarily existent"; "necessity absolute in itself is simple and uniform, and without any possible difference or variety"; all "variety or difference of existence" implies dependence; and therefore "whatever exists necessarily is the one simple essence of the self-existent being."

The conclusion is also made to follow from these premises:—

1. If there are two or more necessary and independent beings, either of them may be supposed to exist alone.

2. If either may be supposed to exist alone, it is not a contradiction to suppose the other not to exist.

3. If it is not a contradiction to suppose this, there are not two necessary and independent beings.

Let us represent the elementary propositions as follows:— x=there exist two necessary independent beings.y=either may be supposed to exist alone.z=it is not a contradiction to suppose the other not to exist. Proceeding as before, we then have x(1−y)=0.(1)y(1−z)=0.(2)zx=0.(3) Eliminating y and z, we have x=0. So, There do not exist two necessary and independent beings.

11. It is well known that objections to the premises on which these last two propositions rest were raised by Bishop Butler, in some celebrated letters. When the "Demonstration" was published, Butler was a student at a nonconformist academy (a college for Protestants outside the Church of England). His letters, with Dr. Clarke's replies, are usually printed at the back of editions of the work. The real question at issue is whether the principle holds that "whatsoever is absolutely necessary at all is absolutely necessary in every part of space, and in every point of duration." Clarke's reasoning assumes this principle, and he states it outright in his reply to Butler's first letter. In his second letter Butler says: "I do not conceive that the idea of ubiquity is contained in the idea of self-existence, or directly follows from it, any otherwise than as whatever exists must exist somewhere." That is to say, necessary existence implies existence in some part of space, but not in every part. Dr. Clarke never seems to have managed to answer this objection effectively. The whole correspondence is extremely curious and interesting. Butler's objections are exactly the ones that would occur to a sharp mind convinced that the successful search for truth depends mainly on sifting first principles, not on any mechanical skill in reasoning. And Dr. Clarke's replies, though they cannot be accepted as satisfactory, show to a remarkable degree the particular intellectual power that is evident in the work from which the discussion arose.

12. In Prop. VIII. it is argued that the self-existent and original cause of all things must be an Intelligent Being.

The main argument offered for this proposition runs as follows. The cause is more excellent than the effect. So the self-existent being, as the cause and origin of all things, must contain in itself the perfections of all things; and intelligence is one of the perfections shown in a part of the creation. It is further argued that this perfection is not a modification of shape, divisibility, or any of the known properties of matter, for these are not perfections but limitations. To this is added the à posteriori argument, the argument from the evidence of design in the structure of the universe.

There is also attached, however, a separate argument for the existence of an intelligent self-existent being, based on the observed existence of motion in the universe. I shall set out this proof briefly and apply the method of this treatise to it.

Leaving out unimportant explanations, the argument is as follows:—

"’Tis evident there is some such a thing as motion in the world; which either began at some time or other, or was eternal. If it began in time, then the question is granted that the first cause is an intelligent being.... On the contrary, if motion was eternal, either it was eternally caused by some eternal intelligent being, or it must of itself be necessary and self-existent, or else, without any necessity in its own nature, and without any external necessary cause, it must have existed from eternity by an endless successive communication. If motion was eternally caused by some eternal intelligent being, this also is granting the question as to the present dispute. If it was of itself necessary and self-existent, then it follows that it must be a contradiction in terms to suppose any matter to be at rest. And yet, at the same time, because the determination of this self-existent motion must be every way at once, the effect of it would be nothing else but a perpetual rest.... But if it be said that motion, without any necessity in its own nature, and without any external necessary cause, has existed from eternity merely by an endless successive communication, as Spinoza inconsistently enough seems to assert, this I have before shown (in the proof of the second general proposition of this discourse) to be a plain contradiction. It remains, therefore, that motion must of necessity be originally caused by something that is intelligent."

The premises of this argument may be arranged as follows:

1. If motion began in time, the first cause is an intelligent being.

2. If motion has existed from eternity, either it has been eternally caused by some eternal intelligent being, or it is self-existent, or it must have existed by endless successive communication.

3. If motion has been eternally caused by an eternal intelligent being, the first cause is an intelligent being.

4. If it is self-existent, matter is at rest and not at rest.

5. That motion has existed by endless successive communication, and that at the same time it is not self-existent, and has not been eternally caused by some eternal intelligent being, is false.

To express these propositions, let us take— x=Motion began in time (and therefore)1−x=Motion has existed from eternity.y=The first cause is an intelligent being.p=Motion has been eternally caused by some eternal intelligent being.q=Motion is self-existent.r=Motion has existed by endless successive communication.s=Matter is at rest. The equations of the premises are then— x=vy.1−x=v{p(1−q)(1−r)+q(1−p)(1−r)+r(1−p)(1−q)}.p=vy.q=vs(1−s)=0.r(1−q)(1−p)=0. The fourth equation gives q=0, since nothing can be both at rest and not at rest. Putting this value for q into the other equations, we get the system x=vy,p=vy,1−x=v{p(1−r)+r(1−p)},r(1−p)=0, and eliminating the indefinite symbols v from this, we have the final reduced system, x(1−y)=0,(1)(1−x){pr+(1−p)(1−r)}=0,(2)p(1−y)=0.(3)r(1−p)=0.(4) We shall first look for the value of y, the symbol that appears in Dr. Clarke's conclusion. First, eliminating x from (1) and (2), we have (1−y){pr+(1−p)(1−r)}=0.(5) Next, to eliminate r from (4) and (5), we have r(1−p)+(1−y){pr+(1−p)(1−r)}=0,∴{1−p+(1−y)p}×(1−y)(1−p)=0; and so (1−y)(1−p)=0.(6) Last, eliminating p from (3) and (6), we have 1−y=0,∴y=1, which expresses the conclusion we were looking for, The first cause is an intelligent being.

Let us now examine what other conclusions can be deduced from the premises.

If we put the value just found for y into equations (1), (2), (3), (4), they reduce to the following pair of equations, namely, (1−x){pr+(1−p)(1−r)}=0,r(1−p)=0.(7) Eliminating x from these equations, we have r(1−p)=0, whence r=vp, which expresses the conclusion, If motion has existed by endless successive communication, it has been eternally caused by an eternal intelligent being.

Again, eliminating r from the same pair, we have (1−x)(1−p)=0,or, 1−x=vp, which expresses the conclusion, If motion has existed from eternity, it has been eternally caused by some eternal intelligent being.

Last, eliminating p from the same original pair, we get (1−x)r=0, which, solved in the form 1−x=v(1−r), gives the conclusion, If motion has existed from eternity, it has not existed by an endless successive communication.

Solved in the form r=vx, the same equation leads to the equivalent conclusion, If motion exists by an endless successive communication, it began in time.

13. The reader will now notice that the first and the last of these four conclusions do not agree. From the hypothesis that motion exists by an endless successive communication, two consequences were drawn. First, that it has been eternally caused by an eternal intelligent being. Second, that it began in time. These are plainly at odds. And yet both are rigorous deductions from the original premises. The conflict between them is not logical. It is what is technically called a material conflict, one that lies in the meaning of the words rather than in the form of the argument. Still, the conflict could have been stated formally in the premises. We could have added a formal proposition saying that "whatever is externally caused by an eternal intelligent being, does not begin in time." Had we done so, no such conflict as now appears in our conclusions could have arisen. Formal logic can take account only of relations that are formally expressed (VI. 16). So in particular cases it may become necessary to state, in a formal way, some connection among the premises that is already contained in the very meaning of the language used, even though nobody has said it out loud.

To show what this means, let us add to the equations (2) and (4) the equation px=0, which expresses the condition just mentioned. We then have (1−x){pr+(1−p)(1−r)}+r(1−p)+px=0.(8) Eliminating p from this, we find simply r=0, which expresses the proposition, Motion does not exist by an endless successive communication. If we now put the value of r back into (8), we have (1−x)(1−p)+px=0, or, 1−x=p; and this gives the interpretation, If motion has existed from eternity, it has been eternally caused by an eternal intelligent being, together with the converse of that proposition.

In Prop. IX. Clarke argues that "the self-existent and original cause of all things is not a necessary agent, but a being endued with liberty and choice." The proof rests mainly on two grounds: that this cause possesses intelligence, and that there exist final causes (purposes in nature), which imply design and choice. To the objection that the supreme cause acts by necessity to produce what is best, Clarke replies that this is a necessity of fitness and wisdom, not a necessity of nature.

14. In Prop. X. he argues that "the self-existent being, the supreme cause of all things, must of necessity have infinite power." The ground of the demonstration is that, since "all the powers of all things are derived from him, nothing can make any difficulty or resistance to the execution of his will." He sets limits to this infinite power. It does not extend to the "making of a thing which implies a contradiction," or to doing that "which would imply imperfection (whether natural or moral) in the being to whom such power is ascribed." But it does extend to the creation of matter, and to the creation of an immaterial, thinking substance, gifted with a power of beginning motion and with a liberty of will or choice. On the strength of this doctrine of liberty, Clarke holds that we can give a satisfactory answer to "that ancient and great question, πόθεν τὸ κακὸν, what is the cause and original of evil?" I shall set out his argument on this point briefly. It runs as follows:

"All that we call evil is either an evil of imperfection, as the want of certain faculties or excellencies which other creatures have; or natural evil, as pain, death, and the like; or moral evil, as all kinds of vice. The first of these is not properly an evil; for every power, faculty, or perfection, which any creature enjoys, being the free gift of God,…it is plain the want of any certain faculty or perfection in any kind of creatures, which never belonged to their natures is no more an evil to them, than their never having been created or brought into being at all could properly have been called an evil. The second kind of evil, which we call natural evil, is either a necessary consequence of the former, as death to a creature on whose nature immortality was never conferred; and then it is no more properly an evil than the former. Or else it is counterpoised on the whole with as great or greater good, as the afflictions and sufferings of good men, and then also it is not properly an evil; or else, lastly, it is a punishment, and then it is a necessary consequence of the third and last kind of evil, viz., moral evil. And this arises wholly from the abuse of liberty which God gave to His creatures for other purposes, and which it was reasonable and fit to give them for the perfection and order of the whole creation. Only they, contrary to God's intention and command, have abused what was necessary to the perfection of the whole, to the corruption and depravation of themselves. And thus all sorts of evils have entered into the world without any diminution to the infinite goodness of the Creator and Governor thereof."—p. 112.

The main premises of this argument can be stated as follows.

First, all reputed evil (what is called evil) is either evil of imperfection, or natural evil, or moral evil.

Second, evil of imperfection is not absolute evil.

Third, natural evil is either a consequence of evil of imperfection, or it is compensated with greater good, or it is a consequence of moral evil.

Fourth, whatever is either a consequence of evil of imperfection, or is compensated with greater good, is not absolute evil.

Fifth, all absolute evils are included among reputed evils.

To express these premises, let us take w= reputed evil.x= evil of imperfection.y= natural evil.z= moral evil.p= consequence of evil of imperfection.q= compensated with greater good.r= consequence of moral evil.t= absolute evil. We treat the premises as primary propositions whose predicates are all particular, so that each subject is only some part of its predicate class. And we take the words either, or as absolutely disjunctive, meaning that each alternative excludes the others. We then have the following equations: w=v{x(1−y)(1−z)+y(1−x)(1−z)+z(1−x)(1−y)}x=v(1−t).y=v{p(1−q)(1−r)+q(1−p)(1−r)+r(1−p)(1−q)}p(1−q)+q(1−p)=v(1−t).t=vw. If we eliminate the symbol v from each of these separately, we have w{1−x(1−y)(1−z)−y(1−x)(1−z)−z(1−x)(1−y)}=0,(1)xt=0,(2)y{1−p(1−q)(1−r)−q(1−p)(1−r)−r(1−p)(1−q)}=0,(3){p(1−q)+q(1−p)}t=0,(4)t(1−w)=0.(5)

Suppose we want to find, first, what conclusion the premises allow us to draw about absolute evils, in their dependence on moral evils and on the consequences of moral evils.

For this we must find t in terms of z and r.

So the symbols w, x, y, p, q must be eliminated. The work is easy, because any set of these equations can be reduced to a single equation simply by adding them.

Eliminating w from (1) and (5), we have t{1−x(1−y)(1−z)−y(1−x)(1−z)−z(1−x)(1−y)}=0.(6) Eliminating p from (3) and (4) gives yqr+yqt+yt(1−r)(1−q)=0.(7) Eliminating q from this gives yt(1−r)=0.(8) that is, nothing is at once natural evil, absolute evil, and not a consequence of moral evil. Eliminating x between (2) and (6) gives t{yz+(1−y)(1−z)}=0.(9) Eliminating y from (8) and (9) gives t(1−z)(1−r)=0. This is the only relation that holds among the elements t, z, and r. It says that there is no absolute evil which is neither moral evil nor a consequence of moral evil. From it we get t=0(1−z)(1−r)=00zr+00z(1−r)+00(1−z)r+0(1−z)(1−r)=00z+00(1−z)r; and its interpretation is, Absolute evil is either moral evil, or it is, if not moral evil, a consequence of moral evil.

Any of the results reached along the way to this solution will also yield interpretations. From (8), for example, we might deduce t=0y(1−r)=00yr+00(1−y)r+00(1−y)(1−r)=00yr+00(1−y); which means, Absolute evils are either natural evils, which are the consequences of moral evils, or they are not natural evils at all.

Many other conclusions can be drawn from the given equations, in answer to any questions we care to put. I shall give a few examples of these, without showing the intermediate steps of the solution.

Quest. 1.—Can any relation be deduced from the premises connecting the following elements: absolute evils, consequences of evils of imperfection, and evils compensated with greater good?

Ans.—No relation exists. If we eliminate all the symbols but t, p, q, the result is 0=0, an identity that says nothing.

Quest. 2.—Is any relation implied between absolute evils, evils of imperfection, and consequences of evils of imperfection?

Ans.—The final relation between x, t, and p is xt+pt=0; so that t=0p+x=00(1−p)(1−x). Therefore, Absolute evils are neither evils of imperfection, nor consequences of evils of imperfection.

Quest. 3.—What is the relation of natural evils to evils of imperfection and to evils compensated with greater good?

We find pqy=0,∴y=0pq=00p(1−q)+00(1−p). Therefore, Natural evils are either consequences of evils of imperfection which are not compensated with greater good, or they are not consequences of evils of imperfection at all.

Quest. 4.—How do those natural evils which are not moral evils stand in relation to absolute evils and to the consequences of moral evils?

If y(1−z)=s, we find, after elimination, ts(1−r)=0;∴s=0t(1−r)=00tr+00(1−t). Therefore, Natural evils, which are not moral evils, are either absolute evils, which are the consequences of moral evils, or they are not absolute evils at all.

The following conclusions were found in the same way. In each one, the subject shows what things a description was asked for, and the predicate shows which elements the description was meant to involve:

Absolute evils, which are not consequences of moral evils, are moral and not natural evils.

Absolute evils which are not moral evils are natural evils, which are the consequences of moral evils.

Natural evils which are not consequences of moral evils are not absolute evils.

Last, let us look for a description of the evils which are not absolute, in terms of natural and moral evils.

The final equation we obtain is 1−t=yz+00y(1−z)+00(1−y)z+(1−y)(1−z). Read directly, this equation states a necessary truth. Read in reverse, it gives something remarkable: Evils which are both natural and moral, and evils which are neither natural nor moral, are not absolute evils.

This conclusion may not express a truth. But it is certainly contained in the given premises, as formally stated.

15. Let us now take from the same argument a somewhat fuller set of premises. And this time let us suppose that the words either, or are not absolutely disjunctive. So by the phrase "either evil of imperfection, or natural evil, or moral evil," we will mean whatever has one or more of these qualities.

Let the premises be these:

1. All evil (w) is either evil of imperfection (x), or natural evil (y), or moral evil (z).

2. Evil of imperfection (x) is not absolute evil (t).

3. Natural evil (y) is either a consequence of evil of imperfection (p), or it is compensated with greater good (q), or it is a consequence of moral evil (r).

4. Whatever is a consequence of evil of imperfection (p) is not absolute evil (t).

5. Whatever is compensated with greater good (q) is not absolute evil (t).

6. Moral evil (z) is a consequence of the abuse of liberty (u).

7. That which is a consequence of moral evil (r) is a consequence of the abuse of liberty (u).

8. Absolute evils are included in reputed evils.

We express these premises in the usual way and eliminate the indefinite symbols v. This gives the following equations: w(1−x)(1−y)(1−z)=0,(1)xt=0,(2)y(1−p)(1−q)(1−r)=0,(3)pt=0,(4)qt=0,(5)z(1−u)=0,(6)r(1−u)=0,(7)t(1−w)=0.(8) Each of these equations satisfies the condition V(1−V)=0.

The following results are easily deduced:

Natural evil is either absolute evil, which is a consequence of moral evil, or it is not absolute evil at all.

All evils are either absolute evils, which are consequences of the abuse of liberty, or they are not absolute evils.

Natural evils are either evils of imperfection, which are not absolute evils, or they are not evils of imperfection at all.

Absolute evils are either natural evils, which are consequences of the abuse of liberty, or they are not natural evils, and at the same time not evils of imperfection.

Consequences of the abuse of liberty include all natural evils which are absolute evils and are not evils of imperfection, together with an indefinite remainder of natural evils which are not absolute, and of evils which are not natural.

16. These examples are enough to illustrate the method. The reader can easily supply others if they are wanted. We now go on to examine the most essential parts of Spinoza's demonstration. His definitions and axioms, from the opening of the first book of the Ethics, are these.

Definitions.

1. By a cause of itself (causa sui), I understand that of which the essence involves existence, or that of which the nature cannot be conceived except as existing.

2. That thing is said to be finite or bounded in its own kind (in suo genere finita) which may be bounded by another thing of the same kind; e. g. Body is said to be finite, because we can always conceive of another body greater than a given one. So thought is bounded by other thought. But body is not bounded by thought, nor thought by body.

3. By substance, I understand that which is in itself (in se), and is conceived by itself (per se concipitur), i.e., that whose conception does not require to be formed from the conception of another thing.

4. By attribute, I understand that which the intellect perceives in substance, as constituting its very essence.

5. By mode, I understand the affections of substance, or that which is in another thing, by which thing also it is conceived.

6. By God, I understand the Being absolutely infinite, that is the substance consisting of infinite attributes, each of which expresses an eternal and infinite essence.

Explanation.—I say absolutely infinite, not infinite in its own kind. For to whatever is only infinite in its own kind we may deny the possession of (some) infinite attributes. But when a thing is absolutely infinite, whatsoever expresses essence and involves no negation belongs to its essence.

7. That thing is termed free, which exists by the sole necessity of its own nature, and is determined to action by itself alone; necessary, or rather constrained, which is determined by another thing to existence and action, in a certain and determinate manner.

8. By eternity, I understand existence itself, in so far as it is conceived necessarily to follow from the sole definition of the eternal thing.

Explanation.—For such existence, as an eternal truth, is conceived as the essence of the thing, and therefore cannot be explained by mere duration or time, though the latter should be conceived as without beginning and without end.

Axioms.

1. All things which exist are either in themselves in se or in another thing.

2. That which cannot be conceived by another thing ought to be conceived by itself.

3. From a given determinate cause the effect necessarily follows, and, contrariwise, if no determinate cause be granted, it is impossible that an effect should follow.

4. The knowledge of the effect depends upon, and involves, the knowledge of the cause.

5. Things which have nothing in common cannot be understood by means of each other; or the conception of the one does not involve the conception of the other.

6. A true idea ought to agree with its own object. (Idea vera debet cum suo ideato convenire.)

7. Whatever can be conceived as non-existing does not involve existence in its essence.

Other definitions are implied, and other axioms are quietly assumed, in some of the demonstrations. Take Prop. I., "Substance is prior in nature to its affections." Its proof consists of nothing but a reference to Defs. 3 and 5. Yet it seems to assume a further axiom: "That by which a thing is conceived is prior in nature to the thing conceived." Again, the demonstration of Prop. V. assumes that the converse of this axiom is true. Many other examples of the same kind occur. So it is impossible, by the processes of logic alone, to deduce all the conclusions of the first book of the Ethics from the axioms and definitions set at its head, which are the ones given above. In the brief analysis that follows, I shall try to set out in their proper order what seem to me to be the real premises, whether stated or implied, and to show how they lead to the conclusions Spinoza reached.

17. My view, then, is that in the course of his demonstration Spinoza divides the universe of possible existence in several parallel ways, as follows.

First, into things which are in themselves, x, and things which are in some other thing, x′. Since these two classes together make up the universe, we have x+x′=1; (Ax. I.)or, x=1−x′.

Second, into things which are conceived by themselves, y, and things which are conceived through some other thing,y′; so that y=1−y′. (Ax. II)

Third, into substance, z, and modes, z′; so that z=1−z′. (Def. III. V.)

Fourth, into things free, f, and things necessary,f′; so that f=1−f′. (Def. VII.)

Fifth, into things which are causes and self-existent, e, and things caused by some other thing, e′; so that e=1−e′. (Def. I. Ax. VII.) And his reasoning proceeds on the principle, sometimes stated and sometimes assumed, that these divisions are not only parallel but equivalent: that they cut the universe along the same line. Thus in Def. III., substance is made equivalent to that which is conceived by itself; so that z=y. Again, Ax. IV., as Spinoza actually applies it, makes a cause the same thing as that by which a thing is conceived; so that y=e. Again, in Def. VII., things free are identified with things self-existent; so that f=e. Last, in Def. V mode is made identical with that which is in another thing; so z′=x′, and therefore z=x. All these results can be gathered into one series of equations: x=y=z=f=e=1−x′=1−y′=1−f′=1−z′=1−e′. Any two members of this series joined by the sign of equality express a conclusion that legitimately follows from his system, whether or not Spinoza drew it himself. Thus the equation z=1−e′, that is, substance is the same as what is not caused by another thing, expresses the sixth proposition of his system: One substance cannot be produced by another. In the same way the equation z=e, expresses his seventh proposition, "It pertains to the nature of substance to exist." There is no need to follow this train of deduction further. Spinoza uses it chiefly to deduce, according to his views, the properties of the Divine Nature, having first tried to prove that the only substance is God. In the steps of this process there seem to me to be some fallacies, which depend chiefly on the ambiguous use of words. I must draw attention to them here.

18. In Prop. V. Spinoza tries to show that "There cannot exist two or more substances of the same nature or attribute." The proof is in effect as follows. If there are more substances than one, they are told apart either by their attributes or by their modes. If by attributes, then there is only one substance with any given attribute. If by modes, then once we set these aside as not essential, no real ground of distinction is left. So there exists only one substance of the same attribute. But the assumptions made here do not agree with those found in other parts of the treatise. By Def. IV., the intellect grasps substance by means of attribute. By Def. VI., a substance may have many attributes. So one substance could conceivably be told apart from another by differing in some of its attributes, while the others stay the same.

In Prop. VIII. Spinoza tries to show that all substance is necessarily infinite. The proof goes like this. There exists only one substance of one attribute, Prop. V.; and it belongs to its nature to exist, Prop. VII. So it will be of its nature to exist either as finite or as infinite. But not as finite. For by Def. II. it would then have to be bounded by another substance of the same nature, which would also have to exist necessarily, Prop. VII. Then there would be two substances of the same attribute, which is absurd, Prop. V. Therefore substance is infinite.

In this demonstration the word "finite" is confused with the expression "finite in its own kind," Def. II. The proof thus assumes that nothing can be finite unless it is bounded by another thing of the same kind. That is not the ordinary meaning of the word. Spinoza's use of "finite" tends to make space the only form of substance, and all existing things mere affections (states) of space. This, I think, is really one of the ultimate foundations of his system.

The first scholium (explanatory remark) attached to this Proposition is remarkable. I give it in the original words: "Quum finitum esse revera sit ex parte negatio, et infinitum absoluta affirmatio existentiae alicujus naturae, sequitur ergo ex sola Prop. VII. omnem substantiam debere esse infinitam." That is: since being finite is in truth partly a negation, and being infinite the absolute affirmation of the existence of some nature, it follows from Prop. VII. alone that every substance must be infinite. Now this is really an assertion of the principle that Clarke affirmed and Butler disputed (XIII. 11): that necessary existence implies existence in every part of space. This principle will probably be found at the base of every attempt to prove, à priori, the existence of an Infinite Being.

From the general properties of substance stated above, together with the definition of God as the substance consisting of infinite attributes, Spinoza's particular doctrines about the Divine Nature follow necessarily. Substance is self-existent, free, and causal in its very nature; it is the thing in which other things are, and by which they are conceived. So the same properties are asserted of the Deity. He is self-existent, Prop. XI.; indivisible, Prop. XIII.; the only substance, Prop. XIV.; the Being in which all things are, and by which all things are conceived, Prop. XV.; free, Prop. XVII.; the immanent cause of all things (the cause that acts from within them), Prop. XVIII. The proof that God is the only substance is drawn from Def. VI., which is read as declaring that "God is the Being absolutely infinite, of whom no attribute which expresses the essence of substance can be denied." Every conceivable attribute is thus assigned to Him by definition. And Prop. V. has settled that there cannot exist two substances of the same attribute. It follows that God is the only substance.

The "Ethics" of Spinoza, like much of his other writing, is set out in geometrical form, as a chain of definitions, axioms, and propositions. Even so, it does not give the symbolic method of this book much useful practice. Of course any train of reasoning can be handled by that method once its ultimate premises have been correctly identified. But here such treatment hardly differs from the processes of the original demonstrations, except in using letters for words. Reasoning that consists so largely of a play on terms defined as equivalent is not often met with. I have given a few pages here to setting it out more because of the interest of its subject than because of the merits of its demonstrations, however highly some people value them.

19. I do not think anyone can come away from reading the arguments of Clarke and Spinoza without a deep conviction that every attempt is futile to establish, entirely à priori, the existence of an Infinite Being, His attributes, and His relation to the universe. The basic principle of all such speculations is that whatever we can clearly conceive must exist. Even if that principle is granted to be true, it fails to do what it is meant to do. For how shall the finite comprehend the infinite? Yet before any solid ground can be laid for knowing, à priori, things infinite and eternal, we must grant that such a conception is possible, and in some sense more than simply removing the limits from the existence we experience through the senses. Spinoza affirms the reality of such knowledge plainly and explicitly: "Mens humana adaequatum habet cognitionem aeternae et infinitae essentiae Dei" (Prop. XLVII., Part 2nd), that is, the human mind has an adequate knowledge of the eternal and infinite essence of God. Compare this with Prop. XXXIV., Part 2nd: "Omnis idea quae in nobis est absoluta sive adaequata et perfecta, vera est;" (every idea which in us is absolute, or adequate and perfect, is true); and with Axiom VI., Part 1st, "Idea vera debet cum suo ideato convenire" (a true idea must agree with its object). What is more, he makes this kind of knowledge the essential part of all other knowledge: "De natura rationis est res sub quadam aeternitatis specie percipere" (Prop. XLIV., Cor. II., Part 2nd), that is, it is of the nature of reason to perceive things under a certain form of eternity. Suppose it were said that the human mind tends to rise in contemplation from the particular toward the universal, from the finite toward the infinite, from the passing toward the eternal; and that this tendency suggests to us, with high probability, that more exists than the senses perceive or the understanding grasps. That statement might be accepted as true, at least for a great many minds. But there is a class of speculations whose character must be explained partly by other causes. One is impatience with knowledge that is only probable or limited, which is so often all we can really reach. Another is a desire for absolute certainty, where we have been granted only hints enough to mark out the path of duty before us, but not enough to satisfy the demands of the speculative intellect. A third, perhaps, is dissatisfaction with the present state of things. When these motives gain too much weight, the more sober method of analogy and probable induction falls into neglect. Yet that method is, beyond all question, the one best suited to our present condition. To infer an intelligent cause from the abundant evidence of design all around us; to rise to the idea of a moral Governor of the world from studying the make-up and the moral provisions of our own nature: these are only the feeble steps of an understanding limited in its faculties and in its materials of knowledge. But they are worth more than the ambitious attempt to reach a certainty that cannot be had on the ground of natural religion. And as these were the most ancient foundations, so they are still the most solid ones, Revelation being set apart, for the belief that the course of this world is not abandoned to chance and inexorable fate.

Chapter 14: Example of Analysis

1. Let us go back to the remarkable system of premises used in the previous chapter to prove that "Matter is not a necessary being." This time we will drop the sixth premise, namely "Motion exists," and look at some of the consequences that follow from the premises that remain. In effect, we accept Dr. Clarke's hypothetical principles as true, but suppose ourselves ignorant of the fact that motion exists. There may be cases where picking out part of the premises of an argument in this way leads to interesting consequences. That is not, however, my reason for taking up this example again. The premises we will actually use are these—

1. If matter is a necessary being, either the property of gravitation is necessarily present, or it is necessarily absent.

2. If gravitation is necessarily absent, and the world is not subject to any presiding intelligence, motion does not exist.

3. If gravitation is necessarily present, a vacuum is necessary.

4. If a vacuum is necessary, matter is not a necessary being.

5. If matter is a necessary being, the world is not subject to a presiding intelligence.

As before, we represent the elementary propositions by letters, as follows: x=Matter is a necessary being.y=Gravitation is necessarily present.w=Motion exists.t=Gravitation is necessarily absent.z=The world is merely material, and not subject to a presiding intelligence.v=A vacuum is necessary.

When we express the premises in symbols and eliminate the indefinite class symbols (q), we get the following system of equations: xyt+xy‾t‾=0,tzw=0,yv‾=0,vx=0,xz‾=0; Here, to save space, y‾ stands for 1−y, t‾ for 1−t, and so on; so also 1−t‾=t, 1−y‾=y, and so on. Read aloud, the five equations say: matter is never a necessary being while gravitation is both necessarily present and necessarily absent, or while it is neither; gravitation is never necessarily absent in a merely material world where motion exists; gravitation is never necessarily present without a vacuum being necessary; a vacuum is never necessary while matter is a necessary being; and matter is never a necessary being without the world being merely material.

The left-hand sides of these equations contain only positive terms, so we can combine them into a single equation by adding them together (VIII. Prop. 2), namely: xyt+xy‾t‾+yv‾+vx+xz‾+tzw=0, What remains is to reduce the left-hand side so that it satisfies the condition V(1−V)=0, the law of duality, which holds when the left-hand side is a sum of constituents with no constituent counted twice.

To do this, we first develop the left-hand side with respect to the symbols x and y. The result is— (t+v‾+v+z‾+tzw)xy+(t‾+v+z‾+tzw)xy‾+(v‾+tzw)x‾y+tzwx‾y‾=0. Our goal will be reached if we reduce the four coefficients of this development to equivalent forms that themselves satisfy the required condition.

Take the first coefficient. Since v+v‾=1, it is 1+t+z‾+tzw, and this reduces to unity (IX. Prop. 1), because a sum of class terms that contains the whole universe is the whole universe.

The second coefficient is t‾+v+z‾+tzw; and its reduced form (X. 3) is t‾+tv+tv‾z‾+tv‾zw. Here each term after the first has been cut down to the part not already covered by the terms before it.

The third coefficient, v‾+tzw, reduces by the same method to v‾+tzwv. The last coefficient, tzw, needs no reduction. So the development becomes xy+(t‾+tv+tv‾z‾+tv‾zw)xy‾+(v‾+tzwv)x‾y+tzwx‾y‾=0;(1) and this is the reduced form we were looking for.

2. Now by the principle stated in Prop. III., Chap. X., we can find the whole relation connecting any particular set of the symbols in this equation. We develop the equation with respect to the symbols in question, and keep in the result only those constituents whose coefficients are unity. So if we choose the symbols x and y, we are led at once to the equation xy=0, that is, matter is never a necessary being while gravitation is necessarily present. From this we have y=00(1−x), that is, the cases in which gravitation is necessarily present are some of the cases in which matter is not a necessary being. In words: If gravitation is necessarily present, matter is not a necessary being.

Next let us look for the relation between x and w. Developing (1) with respect to those symbols, we get (y+t‾y‾+tvy‾+tv‾z‾y‾+tv‾zy‾)xw+(y+t‾y‾+tvy‾+tv‾z‾y‾)xw‾+(v‾y+tzvy+tzy‾)x‾w+v‾yx‾w‾=0. The coefficient of xw, and that one alone, reduces to unity. For tv‾z‾y‾+tv‾zy‾=tv‾y‾, and tvy‾+tv‾y‾=ty‾, and t‾y‾+ty‾=y‾, and lastly, y+y‾=1. Each step joins two terms that differ only in one letter being present in one and absent in the other, and that letter drops out. This is always how such reductions go. So we get xw=0,∴w=00(1−x), that is, the cases in which motion exists are some of the cases in which matter is not a necessary being. In words: If motion exists, matter is not a necessary being.

If we develop (1) in the same way with respect to x and z, we get the equation xz‾=0,∴x=00z, which says: If matter is a necessary being, the world is merely material, and without a presiding intelligence.

This, admittedly, is just the fifth premise over again. But it shows that there is no other relation connecting the two elements it involves.

If we look for the whole relation connecting the elements x, w, and y, we find, by developing (1) with respect to those symbols and proceeding as before, xy+xwy‾=0. That is, matter is never a necessary being while gravitation is necessarily present; nor is it ever a necessary being while motion exists and gravitation is not necessarily present. Suppose we are asked to find from this what follows from the hypothesis "Motion does not exist," as it bears on two questions: whether matter is necessary, and whether gravitation is necessarily present. We find w=−xyxy‾,∴1−w=xxy‾=10xy+xy‾+00x‾;or, 1−w=xy‾+00x‾, with xy=0. The direct interpretation of the first equation is this: If motion does not exist, then either matter is a necessary being and gravitation is not necessarily present, or matter is not a necessary being.

The reverse interpretation is: If matter is a necessary being, and gravitation not necessary, motion does not exist.

In exactly the same way, if we looked for the full relation between x, z, and w, we would find xzw+xz‾=0. That is, matter is never a necessary being in a merely material world where motion exists, nor in a world that is not merely material. From this we can deduce z=xw‾+00x‾, with xw=0.

So: If the world is merely material, and not subject to any presiding intelligence, then either matter is a necessary being and motion does not exist, or matter is not a necessary being.

And, reversely, If matter is a necessary being, and there is no such thing as motion, the world is merely material.

3. We could of course extend the same method to find the consequences of any complex hypothesis u. Take, for instance, "The world is merely material, and without any presiding intelligence (z), but motion exists" (w), and ask what it implies about any other open questions raised in the original premises, such as "Matter is a necessary being" (x), or "Gravitation is a necessary quality of matter" (y). To do this we would join to the general equation (1) a new equation, u=wz, which says that u stands for the cases where the world is merely material and motion exists. We would then reduce the system thus formed to a single equation, V=0, in which V satisfies the condition V(1−V)=0. Then we would proceed as above to find the relation between u, x, and y, and finally u as a developed function of x and y. But it is much better to use the methods of Chapters VIII. and IX. Here I will simply point out a few results, with the main steps of how they are deduced, and leave checking them to the reader, if the reader wishes.

In the problem just mentioned, we find the relation connecting x, y, w, and z to be xw+xw‾y+xw‾y‾z‾=0. That is, matter is never a necessary being while motion exists; nor while motion does not exist and gravitation is necessarily present; nor while motion does not exist, gravitation is not necessarily present, and the world is not merely material. If we now write u=wz, and then eliminate the symbols w and z by the general problem of Chap. IX., we find xu+xyu‾=0, and from this u=10xy+0xy‾+00x‾; and so wz=00x‾ with xy=0. That is: If the world is merely material, and without a presiding intelligence, and at the same time motion exists, matter is not a necessary being.

Now we have already shown that if motion exists, matter is not a necessary being. So this conclusion actually tells us less than we had already found (by inference) to be true. All the same, it is the proper and complete answer to the question that was asked, which was simply to find the consequences of a certain complex hypothesis.

4. So even from the limited set of premises before us, it would be easy to deduce a great variety of further inferences, taking as the given condition any combination of the elementary propositions we like. If the condition is inconsistent with the premises, the form of the solution will show it: the method will assign the value 0 to the combination of symbols that expresses the proposed condition. If, on the other hand, the condition puts no restriction at all on the propositions whose relation we are looking for, so that every combination of those propositions is equally possible, the form of the solution will show that too. Here are examples of each case.

Suppose we look, in the usual way, for the consequences of the condition that matter is a necessary being and at the same time motion exists, as they bear on the propositions The world is merely material, and without a presiding intelligence, and Gravitation is necessarily present. We get the equation xw=0, which shows that the proposed condition is inconsistent with the premises, and so can never be met.

Now suppose we look for the consequences of the condition that Matter is not a necessary being and at the same time Motion does exist, with respect to the same elements as above, namely the absence of a presiding intelligence and the necessity of gravitation. We get the following result, (1−x)w=00yz+00y(1−z)+00(1−y)z+00(1−y)(1−z), in which every one of the four constituents carries the indefinite coefficient, so that each of them may or may not hold. Read literally, it says this:

If matter is not a necessary being, and motion exists, then either the world is merely material and without a presiding intelligence, and gravitation is necessary; or one of these two holds without the other; or both fail to be true. So of the four possible combinations, one of which must be true and only one of which can be true, we are told that any one may be true. Such a result is a truism, a merely necessary truth. Still, it is the only answer that can be given to the question asked.

I do not think I need to defend these applications against the charge of laborious trifling. To build confidence in general principles and methods, it may be necessary to go fairly fully into details that are useless in themselves. When that end has been reached in the subject of this inquiry, let everything that helped to reach it, but later proved unnecessary, be forgotten.

Chapter 15: Of the Aristotelian Logic

1. The logical system of Aristotle, altered in its details but unchanged in its essentials, holds such an important place in university education that this book ought to give some account of it, and briefly discuss the main problems it raises. I hope I come to the task in no narrow or harshly critical spirit. My aim is not to set up a direct comparison between the time-honoured system of the schools and the system of this book. It is, putting truth above every other consideration, to show what the ancient doctrine really is, and to clear away one or two common misunderstandings about how far it reaches and whether it is sufficient.

What is essential in the spirit and method of Aristotle's logic, and of every system related to it, is this: it tries to classify the allowable forms of inference, and it refers those forms, together or one by one, to some general principle that is taken as an axiom. One such principle is the "dictum of Aristotle:" Whatsoever is affirmed or denied of the genus may in the same sense be affirmed or denied of any species included under that genus. About such general principles I think we can say this. Either they state directly, but in abstract form, the very argument they are supposed to explain, and in stating it declare it valid; or they use technical terms which, once defined, bring us back to the same place, namely the abstract statement of the forms of inference that are supposed to be allowed. So the idea of classification runs all through these systems. They also divide logic into two great branches. One deals with categorical propositions, the other with hypothetical or conditional ones. This distinction is nearly the same as the one this book draws between primary and secondary propositions. In all the ordinary textbooks of logic, the theory of categorical propositions is treated far more fully and carefully than the theory of hypothetical ones. Part of that treatment is taken up with old scholastic distinctions, and part with the rules of deductive inference. Only the second part needs our attention here.

2. Categorical propositions are sorted under the following four heads:

Type
1st.Universal affirmativePropositions:All Y’s are X’s.
2nd.Universal negative"No Y’s are X’s.
3rd.Particular affirmative"Some Y’s are X’s.
4th.Particular negative"Some Y’s are not X’s.

Four more forms have recently been added to these, making eight in all (see the next article). The eight can, however, be reduced to six, and the relations among them have been discussed very fully and ably by Professor De Morgan in his Formal Logic. Sir W. Hamilton has published a somewhat different scheme, and has made it the basis of a method of syllogistic inference that authorities on logic speak of with very high respect.

Footnote: Thomson’s Outlines of the Laws of Thought, p. 177.

The textbooks describe the processes of formal logic, as applied to this system of propositions, as being of two kinds, "Conversion" and "Syllogism." Conversion means stating a proposition of one of these kinds in an equivalent form, but with its terms in reverse order. Syllogism means taking two such propositions that share a common term, whether as subject or as predicate, and deducing from them a third proposition that the two contain between them, called the "conclusion." Most writers on logic hold that these two processes, or, according to some, the single process of syllogism, provide the universal patterns of all reasoning. On their view, in any chain of proof the mind's business is to conform itself, knowingly or not, to the particular models of these processes that logicians have classified in their books.

3. What I mean to do is this. First, I will show how syllogism and conversion can be carried out in the most general way on the principles of this book. Then, viewing them as part of a system of logic whose foundations, I believe, have been laid in the ultimate laws of thought, I will try to settle their true place and their essential character.

In the language of symbols, the eight basic types of proposition are written as follows:

1.All Y’s are X’s,y=vx.
2.No Y’s are X’s,y=v(1−x).
3.Some Y’s are X’s,vy=vx.
4.Some Y’s are not-X’s,vy=v(1−x).
5.All not-Y’s are X’s,1−y=vx.(1)
6.No not-Y’s are X’s,1−y=v(1−x).
7.Some not-Y’s are X’s,v(1−y)=vx.
8.Some not-Y’s are not-X’s,v(1−y)=v(1−x).

When we refer to these forms, it will be convenient to use the old words for logical quantity, "universal" and "particular," and for quality, "affirmative" and "negative," in a sense I will explain in a moment. But we will apply them to the terms of propositions, not to the propositions themselves. So we will call the term "All Y’s" universal-affirmative; the term "Y’s," or "Some Y’s," particular-affirmative; the term "All not-Y’s" universal-negative; and the term "Some not-Y’s" particular-negative. "No Y’s" is not really a term of a proposition at all. For the true meaning of "No Y’s are X’s" is "All Y’s are not-X’s." The subject of that proposition is therefore universal-affirmative, and its predicate particular-negative. That there is a real difference between the ideas "men" and "not men" is obvious. That difference is all I have in mind when I call terms affirmative and negative in this way; I do not insist that the words are etymologically the right ones for terms. The words positive and privative would have fitted better, but "positive" is already used in a fixed sense elsewhere in this book.

4. The laws of conversion follow at once from these symbolic forms. Take the equation y=vx, which stands for the proposition "All Y’s are X’s." Eliminating v, we get y(1−x)=0, that is, nothing is a y that is not an x. Solving this for 1−x gives 1−x=00(1−y); and this reads All not-X’s are not-Y’s.

This is an example of what is called "negative conversion." In the same way, the equation y=v(1−x), which stands for "No Y’s are X’s," gives x=00(1−y), and this reads "No X’s are Y’s," since every x falls among the things that are not y. This is an example of what is called simple conversion, though in reality it is the same kind of conversion as the one in the previous example. Every example of conversion that logicians have noticed is of one of two kinds. Either it is of the kind just shown, or it simply swaps the terms of a proposition around without changing their quality, as when we change vy=vx, representing, Some Y’s are X’s, into vx=vy, representing, Some X’s are Y’s; Or else it combines these processes with a further step of limitation, as when from the equation y=vx, representing, All Y’s are X’s, we multiply both sides by v and deduce vy=vx, representing, Some Y’s are X’s, and from this vx=vy, representing, Some X’s are Y’s. In this example the limitation (narrowing "all" to "some") comes first, and the swapping of terms second.

Editor's note: Here Boole passes from "All Y's are X's" to "Some Y's are X's", the step traditional logic calls conversion per accidens, or conversion by limitation. Modern logic does not allow it. It counts "All X are Y" as true even when there are no X's at all, so the statement cannot imply that any X exist. For the same reason some of Aristotle's moods (Darapti, Felapton, Bramantip, Fesapo, and the weakened moods) need an extra premise saying that the class in question is not empty. Boole's v quietly supplies it: to read v times y as "some Y's" is already to assume there are Y's to take some of.

These cases show that conversion is one particular use of a much more general process in logic, a process this book has illustrated many times. Its aim is to find any element of any proposition, however complex, as a logical function of the remaining elements. We need not confine ourselves to the subject and the predicate, treated as simple terms. We can take any element, or any combination of elements, appearing in either of them; make that element or combination the "subject" of a new proposition; and work out what its predicate must be from the data we are given. Even the simple forms of proposition listed above leave room for using such a method beyond what the accepted laws of conversion seem to recognise. So the equation y=vx, representing, All Y’s are X’s, gives us, besides the proposition we deduced from it before, the following three:

1st.y(1−x)=0.There are no Y’s that are not-X’s.
2nd.1−y=00x+(1−x).Things that are not-Y’s include all things that are not-X’s, and an indefinite remainder of things that are X’s.
3rd.x=y+00(1−y).Things that are X’s include all things that are Y’s, and an indefinite remainder of things that are not-Y’s.

These conclusions, it is true, only put the given proposition into other, equivalent forms. But that, and no more, is exactly what the accepted method of "negative conversion" does.

Furthermore, these processes of conversion are not elementary. They are combinations of simpler processes, which depend more directly on the ultimate laws and axioms that govern the use of the symbolic instrument of reasoning. The same remark applies to syllogism, which we now go on to consider.

5. The nature of the syllogism is best seen in a particular case. Suppose we have the propositions All X’s are Y’s,All Y’s are Z’s.

From these we can deduce the conclusion All X’s are Z’s. This is a syllogistic inference. The terms X and Z are called the extremes, and Y is called the middle term. We can now define what a syllogism does in general. We are given two propositions of the kinds listed in (1), sharing one middle or common term Y. In one proposition it is connected with an extreme X, in the other with an extreme Z. We are asked for the relation connecting the extremes X and Z. The term Y may appear in its affirmative form, as All Y’s or Some Y’s, or in its negative form, as All not-Y’s or Some not-Y’s, in either proposition, whatever form it takes in the other.

Nothing is easier than to resolve a particular syllogism by the method of this book. Resolving it is in fact just one application of the process for reducing systems of propositions. Taking the example above, we have x=vy,y=v′z; so, substituting, x=vv′z, which reads All X’s are Z’s. Or, keeping strictly to the method set out in (VIII.7), we deduce x(1−y)=0,y(1−z)=0. Adding these equations and eliminating y, we get x(1−z)=0; so x=00z, or All X’s are Z’s. Any other case can be handled in the same way.

6. But let us leave particular examples aside and examine the general forms to which every syllogism can be reduced.

Proposition I.

To deduce the general rules of Syllogism.

By the general rules of syllogism I mean here the rules that apply to premises in which the subjects and predicates may have every variety of quantity and quality, except that no proposition may join two universal terms. The forms of proposition allowed are therefore those set out in the table at (1).

Let X and Y be the things in the first premise, and Z and Y those in the second. Two cases, fundamentally different in character, then arise. The terms involving Y will be either of like or of unlike quality. They are of like quality when both speak of "Y’s," or both of "Not-Y’s"; of unlike quality when one speaks of "Y’s" and the other of "Not-Y’s." Any pair of premises of the first sort can be written as the equations vx=v′y,(1)wz=w′y;(2) For we can use the symbol y to stand either for "All Y’s" or for "All not-Y’s," since what the symbol means is purely a matter of convention. If we let y stand for "All not-Y’s," then 1−y will stand for "All Y’s," and nothing else changes. We have the same freedom with the symbols x and z. So by choosing the meanings of x, y, and z properly, equations (1) and (2) can be made to represent every combination of premises that depends on the quality of their terms. Again, by giving suitable meanings to the symbols v, v′, w, w′ in those equations, every variety of quantity can be represented as well. For instance, if we take v=1, and let v′ stand for an indefinite class, equation (1) will represent a universal proposition in the ordinary sense of the word, that is, a proposition with a universal subject and a particular predicate. In fact we may give the subject and the predicate in either premise whatever quantities we like (using the word in its scholastic sense), except that, as we assumed, they must not both be universal. The system (1), (2) therefore represents, in complete generality, every possible combination of premises with like middle terms.

7. So that our analysis is as general as the equations it works on, let us eliminate y from (1) and (2) by the method of this book, and look for the expressions for x, 1−x, and vx in terms of z and the symbols v, v′, w, w′. These cover every possible form of the subject of the conclusion. The form v(1−x) is left out, because the premises give us the reading vx= Some X’s, and that does not let us read v(1−x) as Some not-X’s. When the symbol v is used to mean "some," it applies only to the term it is attached to in the premises.

The results of the analysis are these: x=[vv′ww′+00{vv′(1−w)(1−w′)+ww′(1−v)(1−v′)+(1−v)(1−w)}]z+00{vv′(1−w′)+1−v}(1−z),(I.) 1−x=[v(1−v′){ww′+(1−w)(1−w′)}+v(1−w)w′+00{vv′(1−w)(1−w′)+ww′(1−v)(1−v′)+(1−v)(1−w)}]z+[v(1−w)w′+00{vv′(1−w′)+1−v}](1−z),(II.) vx={vv′ww′+00vv′(1−w)(1−w′)}z+00(1−w′)(1−z).(III.)

Each of these expressions has two terms on its right-hand side: one with z as a factor, the other with 1−z. But syllogistic inference, as a matter of form, does not allow opposite classes in its conclusion, such as Z’s and not-Z’s together.

So, to find the rules of that kind of inference, we must find out under what conditions the right-hand side of any of our equations shrinks to a single term.

The simplest is (III.), and it shrinks to a single term if w′=1. The equation then becomes vx=vv′wz,(3) Its left-hand side is the same as the extreme in the first premise. Its right-hand side has the same quantity and quality as the extreme in the second premise. Here is why. Since w′=1, the right-hand side of (2), which contains the middle term y, is universal. So, by our assumption, the left-hand side of (2) is particular; and so the right-hand side of (3), which has the same symbol w in its coefficient, is particular too. From this we get the following law.

Condition of Inference.—One middle term, at least, universal.

Rule of Inference.—Equate the extremes.

Examining equations (I.) and (II.) further shows that this is the only condition of syllogistic inference when the middle terms are of like quality. The right-hand side of (I.) shrinks to a single term if w′=1 and v=1; the right-hand side of (II.) shrinks to a single term if w′=1, v=1, w=1. In each of these cases w′=1 is required, and that is the one sufficient condition already found.

Consider next the case in which the middle terms are of unlike quality. The premises can then be written in the forms vx=v′y,(4)wz=w′(1−y);(5) and if, as before, we eliminate y and find the expressions for x, 1−x, and vx, we get x=[vv′(1−w)w′+00{ww′(1−v)+(1−v)(1−v′)(1−w)+v′(1−w)(1−w′)}]z+[vv′w′+00{(1−v)(1−v′)+v′(1−w′)}](1−z).(IV.) 1−x=[ww′v+v(1−v′)(1−w)+00{ww′(1−v)+(1−v)(1−v′)(1−w)+v′(1−w)(1−w′)}]z+[v(1−v′)+00{v′(1−w′)+(1−v)(1−v′)}](1−z).(V.) vx={vv′(1−w)w′+00vv′(1−w)(1−w′)}z+{vv′w′+00vv′(1−w′)}(1−z).(VI.)

Now the right-hand side of (VI.) shrinks to a single term in z if w=1, giving vx={vv′w′+00vv′(1−w′)}(1−z); and the right-hand side of this is opposite, in both quantity and quality, to the corresponding extreme wz in the second premise. For since w=1, wz is universal. But the factor vv′ shows that the term it is attached to is particular, since by our assumption v and v′ are not both equal to 1. From this we get the following law of inference in the case of unlike middle terms:

First Condition of Inference.—At least one universal extreme.

Rule of Inference.—Change the quantity and quality of that extreme, and equate the result to the other extreme.

Again, the right-hand side of (V.) shrinks to a single term if v′=1, w′=1; it then gives 1−x={vw+00(1−v)w}z. Now since v′=1, w′=1, the middle terms of both premises are universal, and so the extremes vx, wz are particular. But in the conclusion, the extreme involving x is opposite in both quantity and quality to the extreme vx in the first premise, while the extreme involving z agrees in both quantity and quality with the corresponding extreme wz in the second premise. From this we get the following general law:

Second Condition of Inference.—Two universal middle terms.

Rule of Inference.—Change the quantity and quality of either extreme, and equate the result to the other extreme unchanged.

When the middle terms are unlike, there are no other conditions or rules of syllogistic inference than these. Equation (IV.), for instance, can be brought to the form of a syllogistic conclusion when w=1 and v=1, but that does not give a new condition of inference, since, by what we have already seen, the single condition v=1, or w=1, would be enough.

8. The following examples will be enough to illustrate the general rules of syllogism just given: 1. All Y’s are X’s. All Z’s are Y’s. This belongs to the first case, like middle terms. All Y’s is the universal middle term. Equating the extremes gives the conclusion

All Z’s are X’s;

with the universal term, All Z’s, as the subject, and the particular term, (some) X’s, as the predicate. 2. All X’s are Y’s. No Z’s are Y’s. The proper way to express these premises is All X’s are Y’s.All Z’s are not-Y’s. They belong to the second case, unlike middle terms, and they satisfy the first condition of inference. The middle term Y’s in the first premise is particular-affirmative; the middle term not-Y’s in the second premise is particular-negative. If we take All Z’s as the universal extreme and change its quantity and quality as the rule says, we get the term Some not-Z’s. Equating this with the other extreme, All X’s, gives

All X’s are not-Z’s, that is, No X’s are Z’s.

If we start from the other universal extreme and do the same, we get the equivalent result

No Z’s are X’s.

3. All Y’s are X’s. All not-Y’s are Z’s. Here too the middle terms are unlike in quality. So the premises belong to the second case, and since both middle terms are universal, the second condition of inference is met. If, following the rule, we change the quantity and quality of the first extreme, (some) X’s, we get All not-X’s, and equating this with the other extreme gives

All not-X’s are Z’s.

Going the other way round would give the equivalent result

All not-Z’s are X’s.

The scholastic system of logic would not accept the conclusions of these last two examples as valid, because in effect it requires the subject of a proposition to be affirmative. They are, however, perfectly legitimate in themselves, and the rules that produce them are without doubt the most general rules of syllogistic inference. The investigation that led to them will probably look needlessly complicated. Certainly they could have been reached more easily, and without any symbolic instrument at all. But my aim was to carry out the investigation in the most general way, by an analysis that left nothing out. With that aim, whether the method is short or long does not matter. Indeed the analysis is not really an analysis of the syllogism, but of a much more general combination of propositions, since we are free to give the symbols v, v′, w, w′ any class-meanings we like. To illustrate this, I will apply the solution (I.) to the following imaginary case.

Suppose a number of pieces of cloth, striped in different colours, were brought for inspection, and the following two observations were made about them:

1st. Every piece striped with white and green was also striped with black and yellow, and vice versâ (and the other way round).

2nd. Every piece striped with red and orange was also striped with blue and yellow, and vice versâ. Suppose we are then asked how the pieces marked with green stand with respect to the colours white, black, red, orange, and blue.

Here, if we take v= white, x= green, v′= black, y= yellow, w= red, z= orange, w′= blue, our premises are written vx=v′y,wz=w′y, which matches the system (1) (2). Equation (I.) then leads to the following conclusion:

Pieces striped with green are of one of these kinds. Either they are striped with orange, white, black, red, and blue all together, and every piece with that combination is among those striped with green. Or they are striped with orange, white, and black, but not with red or blue. Or they are striped with orange, red, and blue, but not with white or black. Or they are striped with orange, but not with white or red. Or they are striped with white and black, but not with blue or orange. Or they are striped with neither white nor orange.

Given what kind of conclusion this is, neither the symbolic expression (I.) that conveys it nor the analysis that produced that expression can fairly be called needlessly complicated.

9. The way the doctrine of the syllogism has been presented in this chapter leads to an important observation. We have seen that in each of its two great divisions the whole discussion, at least as far as finding rules and methods goes, comes down to analysing a pair of equations: the system (1), (2) when the premises have like middle terms, and the system (4), (5) when the middle terms are unlike. Moreover, that analysis was actually carried out by a method founded on certain general laws. Those laws were drawn directly from the structure of language, Chap. II., confirmed by the study of the operations of the human mind, Chap. III., and proved to apply to the analysis of every system of equations whatever by which propositions, or combinations of propositions, can be represented, Chap. VIII. So here we have the means of settling definitely whether the syllogism really is the fundamental type of reasoning, that is, whether the study of its laws covers the whole of deductive logic. For if it does, some sign of that must show up in the systems of equations we have been analysing. It is impossible to believe that the syllogism could be the one essential process of reasoning and yet, when the process is laid out, show nothing of that high rank. But no such sign appears. Nothing suggests that the treatment of every system of equations expressing propositions is contained in the treatment of the particular system examined in this chapter. And yet writers on logic have been all but unanimous in claiming for syllogistic inference not just supremacy in deductive reasoning, but complete sufficiency. Archbishop Whately, whose language is always clear and definite, and who deserves special attention on the subject of logic, is very explicit on this point. "For Logic," he says, "which is, as it were, the Grammar of Reasoning, does not bring forward the regular Syllogism as a distinct mode of argumentation, designed to be substituted for any other mode; but as the form to which all correct reasoning may be ultimately reduced." And Mr. Mill, in a chapter of his System of Logic entitled "Of Ratiocination or Syllogism," lists the ordinary forms of syllogism and then remarks, "All valid ratiocination, all reasoning by which from general propositions previously admitted, other propositions, equally or less general, are inferred, may be exhibited in some of the above forms." And again: "We are therefore at liberty, in conformity with the general opinion of logicians, to consider the two elementary forms of the first figure as the universal types of all correct ratiocination." In line with these views, it has been argued that the science of logic is exempt from the imperfection and the progress to which every other science is subject. Its birth from the labour of one mighty mind of long ago has even been compared, in a rather daring metaphor, to the mythological birth of Pallas (the goddess Athena, who sprang fully armed from the head of Zeus).

Footnote: Elements of Logic, p. 13, ninth edition.

Footnote: Introduction to Kant’s “Logik.”

Since the syllogism is a kind of elimination, the question before us clearly splits into two. First, can all elimination be reduced to syllogism? Second, can deductive reasoning properly be regarded as consisting of elimination and nothing else? After careful examination, I believe the true answer to the first question is this. In theory it is always possible to break down and recombine propositions so that the elimination can then be done by the rules of the syllogism. But in many cases that reduction would be forced and unnatural, and it would involve operations that are not syllogistic. To the second question I answer that reasoning cannot be confined to elimination, except by arbitrarily narrowing what the word means. No definition will do that makes reasoning anything less than the whole collection of methods founded on the laws of thought as they apply to propositions. Among those methods elimination, important as it is, holds only one place.

Much of what I can only regard as error about the nature of the syllogism and the scope of its work seems to come from a habit of mind. It is the habit of treating as primary every truth in logic that is simple and intuitively certain, without asking how it stands to the other truths of the science, or to the general methods of the art, of reasoning. Aristotle's dictum de omni et nullo is a self-evident principle. But it is not one of those ultimate laws of the reasoning faculty to which all its other laws, however plain and self-evident, can be traced back, and from which they can be deduced in strict scientific order. In every science the fundamental truths are usually the simplest to grasp. Yet that simplicity is not the test by which we judge their claim to be fundamental. The test must be found in the nature and size of the structure they can support. On this view, Leibnitz seems to me to have judged rightly when he gave the "principle of contradiction" a fundamental place in logic. For we have seen the consequences of the law of thought of which that principle is the axiom-like statement (III. 15). But enough has been said about the nature of deductive inference and its elements. The subject of induction will perhaps get some attention in another part of this work.

Footnote: Nouveaux Essais sur l’entendement humain. Liv. IV. cap. 2. Theodicée Pt. I. sec. 44.

10. I remarked earlier in this chapter that the ordinary treatment of hypothetical propositions is much more defective than that of categorical ones. What is usually called the hypothetical syllogism seems, in fact, to be no syllogism at all. Take the argument

If A is B, C is D,
But A is B,
Therefore C is D,

and put it in the form

If the proposition X is true, Y is true,
But X is true,
Therefore Y is true;

where X means the proposition A is B, and Y the proposition C is D. We then see that the premises contain only two terms or elements, while a syllogism essentially involves three. The following would be a genuine hypothetical syllogism:

If X is true, Y is true;
If Y is true, Z is true;
∴If X is true, Z is true.

After the discussion of secondary propositions earlier in this book, it is clear that the forms of hypothetical syllogism must match those of categorical syllogism exactly, in every respect. Particular propositions, such as "Sometimes if X is true, Y is true," can be brought in, and the conditions and rules of inference worked out in this chapter for categorical syllogisms can be read, without any loss, to cover the corresponding cases among hypotheticals.

11. What final conclusions, then, are we led to about the nature and reach of the scholastic logic? I think these. It is not a science, but a collection of scientific truths, too incomplete to form a system by themselves, and not fundamental enough to serve as the foundation on which a perfect system can rest. It does not follow, however, that because the logic of the schools has been credited with qualities it has no right to, it does not deserve our regard. A system that has been bound up with the very growth of language, and has left its stamp on the greatest questions and the most famous proofs of philosophy, cannot be wholly unworthy of attention. Memory, too, and habit, it must be admitted, have a great deal to do with how the intellect works; and some of the rules of the ancient logic have become almost woven into the very fabric of thought in educated minds. But are the memory-aid forms in which the particular rules of conversion and syllogism have been set out of any real use? Might not the very skill they are supposed to give be gained, with more benefit to the powers of the mind, by the unaided efforts of a mind left to its own resources? These are questions that might still be worth examining. As for the particular results worked out in this chapter, it should be noted that they are meant only to help the inquiry into the nature of the ordinary or scholastic logic, and into its relation to a more perfect theory of deductive reasoning.

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