Three Logicians: Aristotle, Leibniz, and Sommers and the Syllogistic 9023218159

Citation preview

GEORGE ENGLEBRETSEN

Three Logicians

Aristotle, Leibniz, and Sommers and the Syllogistic

1981 VAN GORCUM, ASSEN, THE NETHERLANDS

© 1981 Van Gorcum & Comp. B.V., P.O.Box 43, 9400 AA Assen, The Netherlands No part of this book may be reproduced in any form, by print, photoprint, microfilm or any other means without written permission from the publishers. The publication of this book was made possible through a grant from the Foundation of Bishop's University, Lennoxville, Que., Canada.

CIP-gegevens Englebretsen, George - Three logicians: Aristotle, Leibniz, and Sommers and the Syllogistic/ George Englebretsen. - Assen: Van Gorcum. - Ill. Reg., bibliogr. ISBN 90-232-1815-9

Printed in The Netherlands by Van Gorcum, Assen

In philosophy. it is syntax, even more than vocabulary. that needs to be corrected Russell, 1959 If we were to devise a logic of ordinary language for direct use on sentences as they come, we would have to complicate our rules of inference in sundry unilluminating ways. Quine, 1960 ...the assumed gap between logical form and natural syntax seems bridgeable. Sommers, 1976

For Nora

VII

Preface

In his Jntroduction to Logical Theory (London, 1952) P.F. Strawson attempted to show that traditional syllogistic logic was more reflective of various features of ordinary language than was modern mathematical logic. P. Geach, the best modern critic of traditional logic, responded to Strawson in HMr. Strawson on Symbolic and Traditional Logic", Mind, 72 ( 1963). His brief remarks there show that Strawson's defense of the old logic is, at best, naive. Geach clearly believes that there just can be no sound defense of traditional logic. He even suggests that those who would persist in their allegiance to the old logic are either irrational or lazy. He says: Many readers will vaguely think Strawson has proved that the traditional system with all its faults is philosophically less misleading than the new-fangled one. Those Colleges of Unreason where the pseudo-Aristotelian logic is presented as the only genuine logic, and those lecturers who would like to teach the philosophy of logic without having to learn any modern logic. may well thus have been supplied with a pretext for supine ignorance.

We believe that syllogistic logic is philosophically defensible. What Geach sees as its faults are either not faults at all or can be remedied. The result of applying such remedies is a new syllogistic - a logic which is broader and stronger than Aristotle's original. It is a logic competative with the "new fangled" logic of today. This new syllogistic was invisaged, but not built, by Leibniz. The hope for such a logic lay dormant during the period when mathematical logic was being born and nurtured through its rapid maturity. But recently that hope has been revitalized, and virtually ful­ filled, in the work of F. Sommers. The best general answer to Geach's overall charge is simply a presentation of this new syllogistic. While the primary motive in presenting this essay is the defense of syllogistic against its modern detractors, we also believe that it is time for a concise introduction to Sommers' logical work. This work is scattered throughout a wide variety of journals and anthologies; and there is now no available account of it. Given the great originality of Sommers' ideas, and the importance of the issues he has chosen to deal with in logic, this void must be filled. Part of this essay is intended as a modest start at that task.

Ylll

I have spent more than a decade learning, doubting, criticizing, defend­ ing, and explaining Sommers' views (both in logic and outside of logic). One result has been a growing conviction that my years as a mathematical logician were years of dogmatic slumber. This essay is the result of many factors and many people have been helpful to me in the production of it. In this regard I should especially mention Professor Sommers himself, W.A. Shearson, J. Fjeld, G. Hunter, and A Reeve. A generous research grant from the Social Sciences and Humanities Research Council of Canada, and a leave from my teaching duties granted by Bishop's Uni­ versity, allowed me to bring this essay to conclusion. George Englebretsen North Hatley, Quebec

.IX

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction Three Logicians

Aristotle Leibniz Sommers

VII I 9 28 42

The Syllogistic

Contemporary Mathematical Logic 67 Syllogistic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Concluding Remarks 109 113 Bibliography 1 16 Index

Introduction

In the history of logic two very general, basic concepts of logic have dominated. The two are mutually exclusive. A war between exponents of each view has gone on for centuries. No blood has been spilled, no ter­ ritories lost; but it has, nonetheless, been a war. Reputations have fallen, traditions have broken, and minds have turned in the name of one side or the other. Though logical matters are subtle and quiet, they cannot be ignored for long by anyone who wishes to be rational. In a sense, logical matters are the most basic of all. A controversy over the very nature of logic is, then, the most pressing kind. It cannot proceed unchecked. Since logical battles have been fought among logicians, we might think of this war as a civil war, an internal rebellion. It has its revolutionaries and its counter­ revolutionaries. But, as in any extremely long war, what looks like a counter-revolution to one generation looks like a revolution to the next. The war started soon after the beginnings of logic itself. There have been as in most wars, both hot and cold periods. We have just come through the hottest period of all. It lasted from the middle of the Nineteenth to the middle of the Twentieth Century. The result is that one side now dominates logic to an extent unknown since that brief period of peace following the birth of logic. The opposing view has lost support over the past century at an ever increasing rate. Today that opposition is almost mute. In this essay we will recount the history of that opposing view through an examination of the three logicians who have most clearly and powerfully developed the logic based upon it, and then examine, explain, and, to some degree, extend that logic. Our hope is that in doing so a sense of continuity, solidarity, and hope will be instilled in our allies, while a challenge and critique is offered to our opponents. It was Aristotle in the Fourth Century B.C. who first gave birth to formal logic. According to him, the validity of an argument is determined by its form alone. The form of an argument is, in turn, determined by the forms of its constituent premises and conclusion. The form of such sentences is fixed by the terms which they involve. Aristotle's logic, then, required an initial investigation into the nature of terms and the ways in which they

2

Three Logicians

could be combined to form sentences. Aristotle's logic was a term logic. Not long after Aristotle had developed his term logic, Stoic logicians began to formulate a quite different notion of logic. While not denying the usefulness of Aristotle's logic, they claimed that there was another sort of logic more basic than his. According to their conception of this more basic logic the validity of an argument is determined by its form, which in turn is determined by the forms of its constituent sentences. But, the form of a sentence is determined not by its term alone. Instead, sentences were conceived as complexes of simpler subsentences, and it was the arrange­ ment of these which constituted the form of the complex sentence. Stoic logic was a sentential logic. It was not long before the battle was joined between the Peripatetics, who claimed term logic as basic logic, and the Stoics, who claimed sen­ tential logic as basic. The Stoic position gained considerable dominance for awhile during the first centuries B.C. and A.D. However, from then until the Nineteenth Century that position was overrun by logicians proclaiming a notion of logic (though not always the logic itself) of Aristotle. By the Fifteenth and Sixteenth Centuries scholastic logicians had developed and codified an Aristotlelian-like logic into a rigid, fixed system. The system dominated the teaching of logic in European universities. It was learned and accepted by philosophers, like Kant, with little question. There was, naturally, an eventual rebellion against such a state of affairs. However, those who sought to overthrow the oppressive dominance of scholastic logic were almost completely unaware of the Stoic alternative. Indeed, what traces of Stoic logic which did remain had, by this time, been incor­ porated in some form into the logic of the schools. The result was that the rebels, such as Locke, simply attacked formal logic (identified now as scholastic logic) in general. And they achieved a notable degree of success. Though they never destroyed formal logic, they did succeed in keeping it in a light of disrepute, and on the defensive, for some time. There were some logicians during this period, however, who feh the urgency of preserving formal logic and also the challenge to strengthen it and make it a living subject once more. Some of these were tempted by the merits of a sentential logic. But one among them, Leibniz, was convinced that only a term logic could do all the tasks required of a formal logic. As a mathematician he saw considerable advantages in adapting mathematical techniques to logical purposes. His work along these lines was brilliant; but, unfortunately, most of it was unpublished for a very long time. Little of Leibniz' logical work was known to the logicians of the Nineteenth Century. By the middle of that century logic was again an

Introduction

3

active battle ground. Mathematicians had become interested in the con­ nection between their subject and logic. Was logic basic to mathematics, or did mathematics underly logic? By the end of the century a large group of mathematicians and logicians, most notable among them being Frege, holding that logic was the foundation of mathematics, had come to dominate the field. Their thesis required a particular concept of logic - the Stoic concept. Logic was basic to mathematics. And sentential logic was the most basic logic of all. From then until now the school of contemporary mathematical logicians has increasingly dictated the course of logical studies and development. Term logicians (including an ever decreasing number of scholastics) have retreated more and more from their positions. The array of mathematical and mathematical-like weapons in the con­ temporary logician's store has been overly intimidating to term logicians. They have almost all admitted of late that a logic such as Aristotle's is, after all, in some important ways inadequate. The speed of conquest, the prestige of so many of the opponents, and the glitter of their arsenal have overwhelmed the term logicians' position. But, of course, the term logicians have hardly put up a fight. They have ignored the potency of their own weapons, failed to explore the weaknesses of their opposition, and have been slow to recognize those who might lead them. Most of what is sound and effective in the term logicians' arsenal was already available and recognized by Aristotle. After him no logician has explored, expanded or exploited the term logicians' position better than Leibniz. Today only one logician has consistently shown the weaknesses in the sentential logicians' defenses and illustrated the corresponding strengths of the logic of terms. He is the American philosopher, F.T. Sommers. We believe that the concept of logic first developed by Aristotle in the Fourth Century B.C., expanded by Leibniz in the Seventeenth Century, and defended by Sommers today, is the proper concept of logic. In the first part of this essay we will review those aspects of the logical work of each of these three philosophers which have contribut­ ed to the strength of term logic. In the second part we will explore, and to a modest degree, extend, that logic. But first it would be helpful to get clear about the nature of formal logic in general. For there is a sense in which term and sentential logicians alike both agree that logic, regardless of how it is conceived, is concerned primarily with form. While they disagree about just what constitutes logical form, they do agree that some notion of form is the heart of logic. Moreover, the search for logical form has a purpose - the discernment of validity and invalidity. When we talk we produce many different kinds of sentences. This is so

4

Three Logicians

because talking is not just doing one thing - it is doing many things. We have different kinds of linguistic tasks to perform. Thus we have different kinds of linguistic tools - sentences - for performing those tasks. We ask questions, make commands, reg uests, prediction, promises, and so forth. But the most obvious and frequent thing we do linguistically is make assertions. We say that something is or is not the case. While commands may be obeyed or disobeyed and promises can be sincere or insincere. assertions are the only sorts of sentences that can be true or false. Thus philosophers sometimes call assertions "truth-claims". Normally (when we are not lying or storytelling, for example) we want our assertions to be true. And, if we cannot be sure our assertions are true, we at least want it to be possible that they are true. But some assertions just cannot be true; and sometimes we make a series of assertions which are sucn that, while some are or can be true, they cannot all be true together. For example, the assertion 'Dogs aren't dogs' cannot be true. Nor can the series of assertions 'When it rains it pours. Today it is not pouring. But today it is raining' all be true together. Such assertions or series of asser­ tions are said to be inconsistent, or contradictory. When we make them we contradict (literally: speak against) ourselves. Since, as we said, we normally want our assertions to at least enjoy the possibility of being true, we normally want to avoid contradiction. Needless to say, the more important we take our assertions to be the more pressing is our demand that contradiction be avoided. As long as our assertions are relatively simple we can easily avoid contradiction. For contradictions which are simple are obvious. However, once our assertions begin to become relatively complex, the contradiction becomes less noticeable. Our assertions become more complex when our discourse turns from the ordinary to the nonordinary - to physics, mathematics, history. theology, metaphysics and so on. Assertions made in such areas are very frequently complex and thus in great danger of harbouring latent con­ tradictions. Here, since contradiction is not obvious, normal intuitions can no longer serve to monitor our assertions. Here we are in need of a more sophisticated, more discerning checking device. It is because we want to avoid contradiction that we also, and equally, need to avoid bad arguments. An argument is a series of assertions premises and a conclusion. In producing an argument we very often are making an implicit claim about the connection between our series of premises and our conclusion. That is: if the premises are accepted as true, then the conclusion must also be accepted as true. We say "must" because if the premises are accepted as true but the conclusion is not, then a

Introduction

5

contradiction arises. So we could rephrase our implicit claim for such arguments as: the series of assertions consisting of the premises and the denial of the conclusion as true is contradictory. For to not accept the conclusion as true is to accept its denial as true. Arguments for which such an implicit claim can be made are deductive arguments. And if the claim does hold, then the argument is valid. The detection of validity amounts, then to the detection of contradiction. We want our arguments to be valid because we want to avoid contradiction. A system of formal logic is the best tool for detecting contradiction and validity. It is often said that logic is the study of reasoning or correct thinking. But it is just not so. Reasoning, indeed, thinking in general, is the domain of the psychologist, not the logician. But there is a connection. Very often when we engage in the process of reasoning the result is that we produce asser­ tions, we talk or write. If assertions are the products of the process of reasoning, then consistency and validity are the products of correct reasoning. The best way to tell whether a person is reasoning correctly is to check his assertions for consistency and his arguments for validity. Thus we could say that logic takes the product rather than the proces of reasoning as its proper domain. The danger of confusing logic with psychology was very real during the last two centuries. Recently it has given way to a different danger - the confusion of logic with linguistics. Logic is a study of sentences, namely assertions. But so is linguistics. Or, at least, one part of linguistics grammar. One of the main tasks of the grammarian is to study the rules which govern the formation of sentences from words. Since there are many languages it follows that there are many different sets of such rules grammars - governing sentences. English grammar differs from French grammar, and differs still more from Chinese grammar. The logician, on the other hand, in devising his logic attempts to discern rules (primarily concerning consistency) which govern any sentence independently of the language to which it belongs. There is no such thing as English logic or French logic or Chinese logic. A contradiction in one language remains a contradiction in any other language into which it is translated. We might say, humbly of course, that while grammar studies languages, logic studies Language.

Three Logicians

9

Aristotle

Our main task in this chapter will be to provide a general picture of Aristotle's logic of terms, the syllogistic, highlighting where necessary those aspects of his theory which are most important with respect to the con­ temporary logician's demotion of such a logic. It should be noted that throughout this essay we are referring by "Aristolelian logic" or "Aris­ tolelian syllogistic", not to the classical logic of the scholastics (however close it does often come to that of Aristotle), but to the logic of Aristotle himself as we have it in his extent major logical works (especially Prior Analytics, On Interpretation, and Topics). There is no doubt that Aristotle is the greatest logician of all time. Among his many accomplishments as a logician there are two in particular, either of which alone would establish his preeminence. He single-handedly discovered formal logic, and he provided a nearly complete development of the logic of terms. He "discovered" formal logic not in the sense that he was the first to become aware of rules governing valid inference, but in the sense that he was the first to see that the description and systematization of such rules was in itself an important and worthwhile enterprise. Aristotle's logical theory, the syllogistic, attempted to account for all inferences whose validity or invalidity depended upon the formal analysis of the constituent sentences into terms. Aristotle was, it must be added, aware that some inferences (the so-called "hypothetical" ones) could be analyzed less deeply into just the subsentences of their constituent sentences. His attitude towards such a sentential analysis was that such hypothetical syllogisms can he translated into the inference forms of his syllogistic, and that such inferences are inferior to those which he had in mind. What is clear is that Aristotle generally believed that valid inferences were either syllogisms or reducible to syllogisms, and the constituent sentences of a syllogism were always categoricals or reducible to categoricals. To understand Aristotle's logic of terms, then, requires an understanding of at least these two notions: that of a categorical and that of a syllogism. The first is provided by a survey of his theory of logical syntax as found mainly in On Interpre­ tation and in various remarks throughout the first book of Prior Analytics.

Three Logicians The second is provided by an examination of his theory of deduction as found in the first twenty-six chapters of the first book of Prior Analytics. In the brief survey of Aristotle's theories of syntax and deduction which follows one very important point should be noted. Throughout Prior Analytics, and Posterior Analytics as well, Aristotle made use of upper case letters as variable symbols (usually for terms sometimes for sentences). This use of, indeed, invention of, logical variables by him allowed him to reveal the logical form of any sentence unencumbered by the material of actual terms. Contrary, then, to what is often taught today, Aristotle is not only the founder of formal logic, but of symbolic logic, too. For Aristotle, sentences are composed of terms. They have meaning and express positive or negative judgments. 1 Propositions, unlike prayers, requests, commandments, promises, etc. are sentences which can be either true or false,2 i.e. assertions. Logic deals with propositions. It studies the inference relations which hold among propositions. All simple, non­ compound, propositions are categorical. That is, they consist of one subject and one predicate. A subject is a noun (a meaningful expression3), while a predicate is a verb. Verbs are nouns which convey a time reference as well. 4 In every such categorical sentence something is said of something. The subject is what something is being said of. The predicate is what is said of the subject. We should perhaps say one or two more things about terms themselves before turning to the ways in which categoricals are formed from subjects and predicates. We note here two of Aristotle's distinctions. Nouns and verbs are either definite or indefinite. An indefinite term is one which is negated. Thus, 'not-man' and 'not-white' are indefinite. 5 Definite terms are unnegated. Aristotle clearly saw that any term could be negated (though he is initially hesitant to give negated terms the full status of terms). Also we must distinguish between ( l ) terms which are never truly predicated of anything else, (2) terms which have nothing else predicated of them, and (3) those terms which are sometimes subjects and sometimes predicates of others. 6 Concerning the second class Aristotle gives no examples. However, See On Interpretation, 16B26-30. I have offered a slightly different account of Aristotle's theory of logical syntax in "On Proposition Form". Notre Dame Journal of Formal Logic, forthcoming. 2 On Interpretation, l 7al-9. 3 On Interpretation, 16a20-22. 4 On Interpretation, 16b6-8, 16b20. 5 On Interpretation, 16a30-34, 16b 12-16. 6 See Prior Analytics, 43a25-43. 1

Aristotle

11

it is clear that he means those terms which are most general, such as 'exists' or 'being'. Aristotle does give examples for the first and third classes. 'Cleon' is a term which is truly predicable of nothing else, and so belongs to the first class. And 'man' is predicable of Callias and predicable by 'animal' so that it belongs to the third. Aristotle holds that the terms of this third class, terms of medium generality, are of primary interest for the syllo­ gistic. 7 How, now, do terms combine to form categorical sentences? Throughout Prior Analytics Aristotle employs notations such as 'AB', 'BC', 'PS', etc. to indicate propositions predicating (in some way or other) the first term of the second term. These might be called proto-propositions. They are not pro­ positions, but simply the two terms required to make up a proposition. The predicate of any proposition is either affirmed or denied of the subject. 8 The first modification, then, of a proto-proposition is by quality - it is an affirmation or a denial. These qualified propositions are then modified in two other ways : by quantity and by modality. Moreover, Aristotle makes it clear9 that these three modes are applied to a proto-proposition in a definite order. A predication is first affirmative or negative (i.e. an affirmation or a denial). Secondly, it is universal, particular, indefinite, or singular. This quantity is determined by the quantity of the subject. Thirdly, it is modified by modality. The affirmation or denial of the predicate is either assertoric, apodeictic, or problematic. We shall say something now about each of these kinds of modification which turn a Prior A nalytics, 43a42-43 . Actually, it is not clear that Aristotle meant to exclude the other two kinds of terms from the syllogistic. W.D. Ross, A ristotle 's Prior and Posterior A nalytics (Oxford, 1949), p.289, claims that only terms of medium generality are suitable for syllogisms in science. J Lukasiewicz, A ristotle 's Syllogistic from the Standpoint of Modern Formal Logic (Oxford, 195 1 ), pp.6- 7, rejects Ross' argument, but goes on to argue that, since in any syllogism there is a term which is once a subject and once a predicate, only terms of medium generality can be included in syllogistic. G . Patzig, A ristotle 's Theory of the Syllogism (Dordrecht, Hol­ land, 1968), pp.5-7, reject both these arguments in favor of a third. Patzig argues that the construction of a syllogism means the search for a middle term; since any such term must have some term universally predicated of it, must itself be universally predicated of some term, and must have a contrary term; only terms of medium generality can be syllogistic terms. None of these arguments are very convincing. A careful reading of this passage shows that Aristotle is not restricting syllogistic exclusively to term� of medium generality. He says, instead, that he is generally speaking (or for the most part) concerned mainly (or chiefly) with such terms. He seems to be going doubly out of his way nor to exclude other kinds of terms from the syllogistic all together.See my "Singular Terms and the Syllogistic", The New Scho/asticism, forthcoming. 8 On Interpretation, l 7a2 l -27. 9 On Interpretation, 2 1 b26-33 . I have discussed Aristotle's notion of propositional formation from proto-propositions in "On Propositional Form". 7

Three Logicians

12

proto-propositions into a full proposition. It should be realized that once each kind of modification which characterizes a given proposition is manifested, along with the two terms of that proposition, then the logical form of that proposition is thereby revealed. Aristotle's theory of logical syntax is his theory of the quality, quantity, and modality of propositions. One of the major and most obvious differences between Aristotelian logic and contemporary logics is that Aristotle countenanced two kinds of negation, while only one (not identifiable with either of Aristotle's) is admitted today. 1 0 As we will see, contemporary logicians usually recognize only the negation of an entire sentence. Negation is a function which operates on an entire sentence - never just a part of a sentence. For Aristotle, a proposition is negated - but only by first modifying a part of it (its predicate). The negation of a proposition is achieved by denying the predicate of the subject. Thus, we might say that for Aristotle propositional negation is merely predicate negation, i.e. denial. The other kind of negation which he recognized is term negation (i.e. the indefinite terms mentioned above). It is crucial, and Aristotle was often at pains to show it, to realize that term negation and predicate negation are never in­ terchangeable or reducible to each other. They are two very different kinds of operation. 1 1 So, a predicate is affirmed or denied of a subject. Furthermore, it is denied by being negated. The sign of affirmation is the copula (e.g. in English, 'is', 'are', 'were', etc.). Thus a proposition like 'A is B' affirms 'B' of 'A'. A predicate is denied (negated) by the negative copula (e.g. "is not', ·are not', 'wasn't'). A proposition like 'A is not B' denies "B' of "A'. A predicate is not denied by negating just the predicate term (i.e. by replacing that term by its corresponding indefinite). For example, 'A is not-B' does not deny "B' of 'A", but rather affirms 'not-B' (the term negation of "B') of •a'. Now if "PS' is a proto-proposition, and '(not-)S' and '(not-)P' indicate that 'S' and 'P' might be negated or unnegated, then we can indicate the first kind of modification, by quality, like this: (not-)S are/ aren't (not-)P Here "are' and 'aren't' simply indicate any positive-negative copula. Note I h � ve discussed ! hese differences extensively in several places. see especially 'Th e Logic of . N egative Theology . Th e New Scho lasticism. 67 ( 1 97 3 ) ; "A Note on Con trariety". Notre Dame Journal of Formal, Logic. 1 5 ( 1 974); "The Square of Opposi tion", No tre Dame Journal of Formal, Logic 1 7 ( 1 976). 1 1 Aristotle's best discussion of this is in the final chapter (xlvi ) of the first book of Prior 10

A n a�ytics.

Aristotle

13

finally that the quality of the predicate determined the quality of the entire proposition. A proposition is an affirmation if its predicate is affirmed, a denial if its predicate is denied. Aristotle uses the characteristics "universal" and "not universal" to describe both terms and entire propositions. 1 2 A term which can be predicated of more than one thing (e.g. 'man', 'white') is universal. A term which can be predicated of just one thing (e.g. 'Callias', 'Felix') is not universal, i.e. singular. 13 Any term, then, is universal or singular in­ dependently of how it might occur in any given proposition. Propositions themselves are either universal or particular or indefinite. These quantities are determined by the signs of quantity attached to the subjects. Words such as 'all', 'every', etc. indicate universality. Words like 'some' indicate particularity. A proposition with no such explicit sign of quantity is in­ definite. A proposition such as 'All S are P' is universal in quantity. A proposition such as 'Some S are P' is particular. 'S are P' is indefinite. Eventually, 14 through his examination of the syllogistic, Aristotle realized that logically all propositions are quantified. He then takes all indefinite propositions to be implicitly particular. As it turns out, of course, such propositions, having no explicit quantifier, are implicity quantified. But sometimes the hidden quantity is universal (' Men are mortal') and sometimes it is particular ('Raindrops are falling on my head'). Aristotle's acceptance of subalternation (a particular can be derived from its corre­ sponding universal) lead him to take all indefinite propositions uniformly as logically particular. We might also note here that, while he does not admit it in any of the extent works, Aristotle must allow that a proposition with a singular subject is also implicitly quantified. For the universality or nonuniversality of a subject term (sans quantifier) is quite independent of the universality or nonuniversality of the proposition in which it occurs. Finally, just as qualifying the predicate results in the overall quality of the proposition as a whole, so too the quantifying of the subject results in the quantity of the proposition as a whole. 1 5 Letting, now, 'all' and 'some' indi cate any universal or particular quantifier respectively, we can illus­ trate the result of quantifying a qualified proposition like this : all/ some (not-)S are/aren't (not-)P See especially On Interpretation, l 7 b2- l 6 13 On Interpretation, l 7a37- l 7b2. N ote that Aristotle has no qualms about singular terms here . A s we have see n such reservations on ly arise later, in Prior A nalytics. 14 Prior A na�ytics, 29a27-29. 15 On Interpretation, 1 7 b l 1 - 1 3 . 12

14

Three Logicians

Each modification of a proposition as a whole has been achieved thusfar by attaching an appropriate sign of modification to either the subj ect term or the predicate term. We could say, then, that a proto-proposition is j ust a subject term and a predicate term. But a genuine proposition is a subj ect (i.e. a quantified subj ect term) and a predicate (i.e. a qualified predicate term). Like quality and quantity, modality modifies a proposition not by add­ ing a third term, but rather by modifying one of the terms already con­ stituting the proposition. Quality applies to the predicate term to form a predicate ; quantity applies to the subj ect term to form a subj ect; and modality applies to the predicate, the qualified predicate term, to form a modalized predicate. Modality modifies the way a predicate applies to a subject. The predicate applies, necessarily applies, or possibly applies to the subj ect. Words like 'necessarily' are signs of apodeictic predication. Words like 'possibly' are signs of problematic predication. And those predications are assertoric which contain no such explicit modifiers. We can now indicate what for Aristotle would be the overall general logical form of any simple categorical proposition. all/some (not-)S are/ aren't (necessarily /possibly) (not-)P

Aristotle's theory of logical syntax as formulated in On Interpretation and Prior Analytics hinges upon such clear contrasts as affirmation/ denial, definite/ indefinite term, universal/ particular. Such contrasts lead naturally to the possibility of a theory of opposition. There is little doubt that Aristotle's thinking about logical opposition played a central role in his subsequent formation of the syllogistic. Before he began to investigate (in Prior Analytics) that special and all-important rela tionship of inference which holds between one pro­ position and one or more others, Aristotle felt the need to look at other. perhaps more primative, relations between pairs of propositions. These were the relations of logical opposition. It was, in fact, his recognition of two kinds of negation which lead him to his two kinds oflogical opposition : contrariety and contradictoriness. Recall that, given Aristotle's theory of logical syntax, both terms and predicates can be negated. And, further­ more, these two kinds of negation are not reducible to each other. Thus, given a proposition like ' S is P' we could negate the predicate term or we could negate the predicate. This would give us 'S is not-P' and ' S is not P', respectively. The first is an affirmation of 'not-P' of 'S'. The second is a denial of 'P' of 'S'. These two propositions are logically unequal. But each has a special relationship with the original proposition, 'S is P'. Note that

Aristotle

15

given any affirmative proposition we can form a corresponding negative by denying (negating) the predicate. Thus, such propositions come, logically at least, in pairs - one affirmative, the other negative (i.e. an affirmation and a denial). Such pairs are, according to Aristotle, logical con­ tradictories. 16 Thus, contradiction between two propositions is charac­ terized syntactically by Aristotle. Propositions which differ only in that the predicate is affirmed in one and denied in the other are logically con­ tradictory. Aristotle also gives a semantic characterization of con­ tradiction. 1 7 Contradictory pairs are such that one must be true and the other false. Contradictory propositions cannot both be true nor both be false. Given 'S is P', then, 'S is not P' is its corresponding contradictory. What of 'S is not-P'? Aristotle has more difficulty here. This is because the relationship in this case is not straightforward, as it is for contradiction. The relation between 'S is P' and 'S is not-P' is contrariety. But unlike the contradictory, the contrary of a proposition can be formed in various ways. Moreover, while every proposition has a contradictory, only affirmations have contraries. The problem Aristotle faces here is best seen by looking at two, seemingly conflicting, things he says about contraries. First he tells us 1 8 that 'Every man is white' and 'No man is white' are contrary pro­ positions. Then, later, 1 9 he tells us that two propositions are contrary whenever they predicate contrary qualities of the same subject. Qualities are contrary whenever they are incompatible, i.e. no subject can have both qualities simultaneously. 2 0 Aristotle says that contraries belong to those things that within the same class differ most. 2 1 He gives several examples: justice/ injustice, black/white2 2 and ill/well. 2 3 He also says that red and yellow are not contrary qualities. 2 4 In Metaphysics he says that the primary contrariety is between a positive and privative. 2 5 We can bring some order to all of this by noting, as Aristotle does, the difference between a thing having a quality naturally or nonnaturally. For 16

17 18 19 20

21 22 23 24 25

See especial ly On In terpretatio n, l 7a3 l -3 5 . Categories, 1 3b l -3 ; On In terpretati on, l 7b26-3 l ; and, of course, M etaphys1c s. On Interpretation, l 7b2ff and l 7b20-23 . On Interpretation, 24b7- I 0. See, for example , Categories, 1 4a9- l 4. O n Interpre tation, 23b23-2 4 and Metaphysics, 1 0 1 8a25 -3 1 . Categories, l 0b l 3- 1 5 . Categories, l 4a l 0- l 2. On In terpretation, l 0b l 8- l 9. Metaphysics, l 05 5a34.

16

Three Logicians

example, neither a blind man nor a stone are sighted (are able to see). Yet sight is not a natural property of stones. Thus, we can deny sigh t of both a blind man and a stone - 'This blind man is not sighted', 'This stone is not sighted'. But a quality is privative to a thing only if it is natural to it. So sighted is privative to a blind man, but not to a stone. I f a quality is privative to a thing we can affirm its privation , i.e. its negative , of that thing. In other words, we can affirm 'not-sighted' o f a blind man, but cannot affirm it of a stone. We could say, now, that two terms which indicate corresponding positive and privative qualities are primarily con­ trary - that a term and its negation (e.g. 'P' and 'not-P') are primary contraries. So 'j ustice' / 'injustice' and 'ill' / 'well' are pairs of primary con­ traries. But what of 'black' / 'white' and 'red' / 'yellow'? Whatever is black is most surely in a state of privation with respect to white . So ' S is black' implies 'S is not-white'. But the converse cannot hold since a thing may be neither black nor white (e.g. red). Yet we can say the same of red , yellow, etc. 'S is red' implies 'S is not-white'. Aristotle takes black and white to be contrary qualities, but not, for example , red and yellow . H e thinks of all colors as arranged on a scale from black to white. Since contraries are those qualities which 'differ most', black and white are most properly contraries. But we can ignore this peculiar view of colors and keep Aristotle's logical insight - that the primary contrariety is between a quality and its privation , or a term and its negation . We could then say that red and yellow are (nonprimary) contrary qualities since each would entail the privation of the other (i.e. they are incompatible). Thus, 'red' and 'yellow' are simple contraries while 'red' and 'not-red' are primary contraries. We can go on to say that propositions which affirm contrary predicates are contrary. Pairs of propositions which affirm primary contraries are themselves primary con traries. We can call the primary con trariety between two propositions logical contrariety since it is the contrariety he has in mind in On Interpre­ tation. We can see now that 'S is P' and 'S is not-P' are logical contraries. But what of 'Every man is white' and 'N o man is white'? It would seem that prima facie the logical contrary of 'Every man is white' would be 'Every man is not-white'. The problem here is really that of interpreting the 'No' of'No man is white'. It is a negation - but what does it negate? We h ave the following choices : it negates the subj ect term, it negates the predicate term, it negates the predicate (i.e. denies it). It is clearly not the first. N or is it the third . For given the third interpretation we get not the contrary of ' Every man is white' but the contradictory. So it must be the second. Propositions of the form 'No S are P' are variations, equivalent to, those of the form 'All

Aristotle

17

S are not-P'. �All S are p' and 'No S are P' are logical contraries. Like contradictoriness, logical contrariety is given a semantic charac­ terization by Aristotle . 26 Logically contrary propositions cannot both be true. If two propositions are contrary, then either one or both are false. They will both be false whenever (i) the quality indicated in the predicate is nonnatural to the subject (e.g. 'This stone is sighted/ not-sighted') or (ii) the subject refers to what does not exist. 2 7 The contrast between contradiction and contrariety, which derived in part from the distinction between term negation and predicate denial, is clearly an important one. More than once in his logical studies Aristotle returned to it. It is the source of some very basic logical relationships. Given any affirmation ( e.g. �All S are P') there are three other corre­ sponding propositions which can be generated from it : the contradictory (by denying the predicate), the logical contrary (by negating the predicate term), and the contradictory of the contrary (by both negating the predicate term and denying the predicate). The relations are all easily seen on a primative square of opposition. 28 all S are P

□

all S are not not-P

all S are not-P (no S are P) all S are not P

The top two propositions are logical contraries. Pairs of propositions at opposite corners are contradictory. Not all logicians today take the extreme negative view of Aristotelian syllogistic which Russell once held. 29 But those who now find something of value in the old system disagree about just what kind of system it is. Some view the syllogistic developed by Aristotle as an axiomatic system, 30 others as a natural deduction system. 3 1 The difference is important, with many 26 27

Especially : Categories. 1 4a l 0- 1 4: On Interpretation, 24b7- 1 0: and Metaphysics, 1 0 1 l b l 5- 1 7 . Categories. 1 3 b l 4-35.

Aristotle offers such squares - not to be con fused with the o;;cholastics' trad itional "squares of oppmition" - in chapter 1 0 of On Interpretation. I have provided a more deta iled discussion of the logical relations exh i bited by such squares in .. The Square of Opposi tion ", Notre Dame Journal of Formal Logic , 1 7 ( 1 976 ). 29 See B. Russe ll. "Aristotle's Logic", in Essays in Logic, ed. R. Jager ( E nglewood Cliffs, N .J . , 1 963). A more reasoned recent attack o n Aristotle i s found in P. G each , " H istory of the Corruptions of Logic", in his L ogic Mailers (Oxford , I 972 ). 3 0 Particularly : J . Lukasiewicz, A ristotle 's Syllogistic from the Standpoint of Modern Formal Logic. and I.M. Bochenski, A ncient Formal Logic ( Amsterdam. 1 968 ). 3 1 See J. Corcora n . "Aristotle's N atural Deduction Syste m", in his A ncien t Logic and its Modern Interp retations ( Dord recht. H olland, 1 974 ). 28

Three Logicians

18

implications. Different views about the nature of arguments, premises, rules, validity and more are involved here. A decision about which of these two views to adopt has to be taken rather early in any attempt to explicate Aristotle's logic. The very notion of what is a syllogism hinges on it. Let us begin by looking at a typical, and important Aristotelian syllogism as found in Prior Analytics. If A is predicated of all B and B is predicated of all C, then necessarily A is predicated of all C. 3 2

This has traditionally been rewritten as: All B are A. All C are B . Therefore, all C are A .

But notice that the syllogism, a s given b y Aristotle, i s just a single, complex, proposition - not a set of two premises and a conclusion. It is a proposition, and as such is either true or false. The traditional syllogism, on the other hand, is an inference (actually a pattern of inference, an inference schema), and as such is not true or false but valid or invalid. If Aristotle's practice of presenting syllogisms as single conditional propositions was meant to in­ dicate his view that syllogisms are propositions, and if, moreover, the ones he accepts as correct are taken as true, and if sentences true by form alone are taken as theses, and if those theses which are self-evident are taken as axioms, then, indeed, Aristotle's system seems to be axiomatic. Are Aristotle's syllogisms propositions or inferences? In the paragraph above we offered an argument or inference. But we used only one pro­ position (admittedly a long and complex one). Now as every logician knows, corresponding to each valid/ invalid inference is a true/ false con­ ditional proposition. We can display the inference (using two or more propositions), or we can, j ust as well, state the corresponding conditional proposition. Aristotle did the latter. Thus, logicians such as Lukasiewicz claim that the syllogistic is an axiomatic system - like such better known axiomatic systems as Euclidian geometry. Now an axiomatic system works generally like this. A small number of self-evident truths concerning cer­ tain obj ects, phenomena , etc. are stated. These are called axioms. Logical rules are applied to these truths in order to derive further truths, called theorems. Logicians who take syllogistic to be axiomatic hold that Aristotle 32

Prior A nalytics, 25b37-38.

Aristotle

19

took the first figure syllogisms a s axioms and all others as theorems derived from them by the rules of conversion and the rules of propositional, or sentential, logic. 33 Our point of view is that Aristotle's syllogistic was a natural, nonaxiomatic, deduction system. We will not argue for this view, 34 but will merely state briefly two reasons which lend weight to it. First of all, the alternative view would require that a propositiona] logic, such as that developed by the Stoics and by contemporary mathematical logicians, is logically prior to, more basic than, a logic of terms. While admittedly Aristotle does seem from time to time aware of certain rules governing a proposition-based logic, he did believe that the logic of analyzed pro­ positions, the logic of terms, is the fundamental logic to be used by the philosopher or scientist. Moreover, as we hope to show in subsequent chapters of this essay, such a term logic can indeed account for all inference - thus eliminating any need for, let alone priority of, a logic of unanalyzed propositions. Secondly, Aristotle developed logic not as a science (though he certainly felt it worthy of independent study), but as a scientific tool (actually, a tool for teaching science). Moreover, he clearly saw each theoretical science (as opposed to practical and productive sciences) as an axiomatic system, whose axioms depend upon the definitions of the objects of the science. Given this view, what was then required was a logic which could be used to deduce theorems from axioms in this or that particular science. 35 His syllogistic was intended as that logic, that system of deduc­ tion. The syllogistic has no axioms in the sense that it states no truths about this or that particular set of objects or phenomena, but rather it displays inferences, patterns of valid deduction, which apply to any subject matter what-so-ever. When Aristotle says that if A is predicated of all B and B is predicated of all C, then necessarily A is predicated of all C, he is merely saying that any inference of a proposition of the form 'All C are A' from Lukasiewicz, A ristotle 's Syllogistic from the Standpoint of Modern Formal Logic, pp.45-46, adds that the laws of identity ('A belongs to all A' and 'A belongs to some A') are additional axioms. 34 But, see Corcoran, "Aristotle's N atural Deduction System" . 35 I n the first book of Posterior A nalytics Aristotle tells us that a science is a sequence of sentences about a common domain, such that each sentence is either an axiom (undemon­ strable) or a theorem (demonstrable). The syllogistic is the theory of such demonstration. Thus logic for Aristotle is not a science but a scientific tool. On the axiomatic nature of science for Aristotle see H . Scholz, "The Ancient Axiomatic Theory", in A rticles on A ristotle, ed. J. Barnes, et al., ( London, 1975). For the argument in favor of the view that syllogistic was meant as a tool for teaching science, rather than for increasing scientific knowledge or making scientific discovery, see J. Barnes, "Aristotle's Theory of Demonstration", Phronesis, 14 ( l 969). 33

Three Logicians

20

propositions of the forms 'All B are A' and 'All C are B' is valid. It is not a statement of a true axiom but a statement of a valid, or correct, rule. In the first chapter of Prior Analytics Aristotle attempts a definition of a syllogism. He says that it is a form of words in which certain assumptions (premises) are made and something else (the conclusion) necessarily follows from them. This seems to be a definition not just of a syllogism but of an inference in general - moreover, a valid inference. According to his practic throughout the Analytics, Aristotle took a syllogism to be an argu­ ment consisting of two premises and a conclusion. The two terms of the conclusion each occur once in different premises and a third term occurs in each premise but not the conclusion. 36 He clearly recognized that the order of the premises is logically irrelevant. 37 The syllogisms are divided by Aristotle, for convenience, into three.figures according, in effect, to the role of the middle term (i.e. the term which does not appear in the con­ clusion). 3 8 In the first figure the middle term is the subject term of the premise containing the predicate term of the conclusion and the predicate term of the premise containing the subject term of the conclusion. In the second figure the middle term is the predicate term of each premise. And in the third figure it is the subject term of each premise. There is a fourth figure, not mentioned by Aristotle, but used by him, in which the middle term is the predicate term of the premise containing the predicate term of the conclusion and the subject term of the premise containing the subject term of the conclusion. 39 Following Aristotle's practice of writing a proto­ proposition in predicate term-subject term order, and letting 'S'. 'P', and 'M' stand for the subject term of the conclusion, the predicate term of the conclusion, and the middle term respectively, we can display the four figures like this. 1st PM MS

2nd MP MS

3rd PM SM

4th MP SM

PS

PS

PS

PS

Aristotle's attempt to characterize these terms for all syllogisms as major. minor. and middle is confused. Lukasiewicz. A ristotle 's Syllogistic from the Standpoint of Modern Formal Logic, pp.28-32, offers an excellent analysis of the problem. 3 7 See Lukasiewicz. A ristotle 's Syllogistic from the Standpoin t of Modern Formal Logic, pp.32-34. 38 _ For example. Prior A na�rtics. 40b30ff and 4 l a 1 3 -20. Each is defined in the following places : First, 25b-32-35 : Second. 26b34-39-39: Third. 28a 1 0- 1 5 . 9 Concern ing the fou rth figure. see. again. Lukasiewicz's excellent a n d important discussion � m A ristotle 's Syllogistic from the Standpoint of Modern Formal Logic. pp.38-42. 36

Aristotle

21

Much of Prior Analytics is devoted to determining just which syllogisms in each of these figures are valid and which are not. Those that are valid are divided by Aristotle into two groups: perfect and imperfect. 40 In effect, the former are those which are obviously valid, while the latter are those which, while valid, are not always obviously so. Imperfect syllogisms require proof, or demonstration, of their validity. He takes the valid syllogisms of the first figure to be perfect. Indeed, just two of the first figure syllogisms are needed to generate the entire syllogistic.4 1 These are, with their traditional Latin mnemonic names: Barbara

Celarent

all M are P all S are M

no M are P all S are M

all S are P

no S are P

Proofs of all other valid syllogisms amount to reduction of them to one of these two. These reductions, or proofs, rely upon the use of the laws governing opposition (e.g. contraries cannot both be true but can both be false, contradictories must be such that one is true and the other false,42 ) the laws of conversion, 43 and the law of subalternation. The laws of con­ version allow, for example, that 'No P are S' entails 'No S are P' and that 'Some S are P' entails 'Some P are S'. The law of subalternation allows that any universal affirmation/ denial entails its corresponding particular affirmation/ denial. Here is a simple example of a reduction of a second figure syllogism (Camestres) to a first figure syllogism (Celarent). 1. 2. 3. 4.

no P are M all S are M no M are P no S are P

Here 1. and 2. are the premises, 3. comes from 1. by conversion, and 2. and 3. constitute the premises of a Celarent syllogism with 4. as its conclusion. Later4 4 Aristotle realized that if syllogisms of any other form are taken as 40 41

Prior A nalytics, 24b23-26. Prior A nalytics, 29a30-33 , 29bl-2.

. . . make use of them in his These are not stated explici tly in Prior A nalytics, but Aristotle does proofs. 43 Prior A nalytics, 25al4-36. 44 Prior A nalytics, xlv. 4

2

Three Logicians

22

perfect, then all remaining forms could equally well be reduced to them. Chapter 8 through 22 of the first book of Prior Analytics are devoted to modal syllogisms - syllogisms containing one or more nonassertoric pro­ positions. There is a great deal of confusion in this account of such syl­ logisms. While Aristotle does correctly reveal numerous valid modal syl­ logisms, he fails to reach a sufficient degree of clarity about the nature of the modal terms themselves (e.g. 'possibly', 'can', 'must', 'may', etc.), and he fails to draw out sufficently general principles governing modality. Nevertheless, he does demonstrate the syntactic character of modality and establishes clearly that a term logic could be constructed in a way which would allow for the systematic study of arguments containing modal pro­ positions. 4 5 There are, of course, arguments which have more than two premises, and more than three terms. So-called compound syllogisms were extensively studies by Aristotle's Peripatetic successors. Aristotle himself mentions them also (calling them prosyllogisms). 46 The reason he gives them little of his attention is because he realizes that such arguments are always reduc­ ible to a series of simple, two-premised syllogisms. What he is searching for is not an account of all kinds of inferences, but rather the underlying principles which could apply to all valid inferences. Thus far we have sketched certain general features of Aristotle's logic of terms. Yet Aristotle was equally aware of the existence of arguments whose validity seemed not to require analysis of their propositions into terms. He called these the hypothetical syllogisms and said that he would analyze them. 4 7 Valid hypothetical arguments are not syllogisms according to Aristotle - even though their conclusions follow necessarily from their premies. 4 8 Such arguments according to him, cannot be reduced to syl­ logisms. 49 Since what Aristotle said about such nonsyllogistic arguments is confused, or even contradictory, it is difficult to know just what his attitude toward them was. But there is little doubt that he took them, and any logic (e.g. that of the Stoics) which would analyze them, to be inferior to his syllogistic. Whether he believed that a logic of unanalyzed propositions is completely independent of a term logic is unclear. It must be kept in mind, however, that Aristotle built the syllogistic as a tool for the theoretical sciences. In doing so he sought the underlying principles governing all 45 46

47 48 49

See S. McCall, A ristotle 's Modal Syllogisms (Amsterdam. 1 963 ). Prior A nalytics, 42bl-26. Prior A na(ytics, 45 b20. Prior A na/ytics, 47a32-35. Prior A na �vtics, 50a I 9-26.

A ristotle

23

syllogistic inference. It would not be surprising to learn that he did not take a broader view. In subsequent chapters we hope to show how a syllo­ gistic-based logic can be extended to cover those logical areas Aristotle usually chose to ignore. It has generally been held since the scholastic period that the entire Aristotelian syllogistic rests upon a single principle: the dictum de omni et nullo. The dictum has been formulated in a variety of generally complex ways over the centuries. What it says in effect, however is simply this : W hatever is affirmed of all of something is likewise affirmed of what that something is affirmed of.

The dictum has been rejected for many reasons. 50 A primary one being that it is nowhere formulated by Aristotle himself. 5 1 Nevertheless, we believe that Aristotle was indeed aware of this principle, that it in fact is the basic principle of syllogistic, and that he refers to it as such (though he does not actually express it in the form of a principle). Acceptance of the dictum is tantamount to acceptance of the view that all syllogisms are reducible to, provable by, the perfect syllogisms of the first figure. We will illustrate this shortly. Aristotle was surely aware of this fact, and actually says that it is so. 52 Here are the first figure valid syl­ logisms. Barbara

Celarent

Darii

Ferio

all M are P all S are M

no M are P all S are M

all M are P some S are M

no M are P some S are M

all S are P

no S are P

some S are P

some S are not P

Our examinations of Aristotle's syntactic theory has shown that all asser­ toric propositions are affirmations or denials of a predicate consisting of a negated or unnegated predicate term of a universally or particularly quantified subject consisting of a negated or unnegated subject term. Propositions of the form 'No A are B' are, as we saw, logically equal to universal affirmations of a negated predicate term, i.e. 'All A are not -B'. In 50 O n e of Lukasiewicz's cri ticisms, A ristotle 's Syllogistic from the Standpoint of Modern Formal Logic, pp.46-57, is that the dictum makes reference to in dividuals, which Aristotle barred from syllogiqic. As we have seen , however, in dividuals are not barred on logical grounds. And, at any rate, the dictum can be form ulated, as we have it above, without reference to individuals. 51 But see Categories, l b9- l 5 . 52 Prior A na(ytics, 29a30-33.

Three Logicians

24

other words the 'no' of such propositions is, logically, the 'not-' of term negation. It is not the 'not' of predicate denial. Thus, the first premises of Celarent and Ferio are universal affirmations of negated predicate term (viz. 'All S are not-P'), and the conclusion of Ferio must be a particular affirmation of a negated predicate term (viz. 'Some S are not-P'). The four syllogisms, then, when regularized according to Aristotle's theory of logical syn tax become: Barbara

Celaren t

all M are P all S are M

all M are not-P all S are M

Darii

Ferio

all S are P

all M are P some S are M some S are P

all S are not-P

all M are not-P some S are M

some S are not-P

Notice, first of all, that all the premises and conclusions here are affirmative. 53 Let us describe, now, each inference. In Barbara what is affirmed of 'all M' (viz. 'P') is likewise affirmed of what 'M' is affirmed of (viz. 'all S'). In Celarent what is affirmed of 'all M' (viz. 'not-P') is likewise affirmed of what 'M' is affirmed of (viz. 'all S'). In Darii what is affirmed of 'all M' (viz. 'P') is likewise affirmed of what 'M' is affirmed of (viz. 'some S'). And in Ferio what is affirmed of 'all M' (viz. 'not-P') is likewise affirmed of what 'M' is affirmed of (viz. 'some S'). In general, then, for each of the four perfect syllogisms of the first figure we can say that what is affirmed of all of something is likewise affirmed of what that something is affirmed of - in other words, the dictum de omni et nu/lo. All syllogisms are reducible to the perfect ones of the first figure. We have seen that there is a characteristic common to all these syllogisms of that figure, i.e. the one expressed in the dictum. T his feature, then, must Given Aristotle's theory of logical syntax, then, every denial of any unnegated/negated predicate of any universally / particularly quantified subject is equivalent to the affirmation of that negated/unnegated predicate of that particularly /universally quantified subject. In other words: all/some S are not P/not-P = some/all S are not-PIP. As we will see later, it was Sommers who first realized the importance of this equivalence.

53

A ristotle

25

actually characterize all valid syllogisms. Such a feature, characterizing all valid syllogisms must be important. When stated in the form we have given it, it becomes a principle underlying the syllogistic. In effect it allows that any invalid syllogism can, after application of the laws of conversion, opposition, and subalternation, be accurately described by the dictum. It seems a simple matter to see that the dictum, as we have it, is the underlying principle of syllogistic. As to why, on the one hand, so many, especially the scholastics, have formulated the dictum in such cumbersome ways, and on the other, others have failed to see it in the syllogistic itself, we will not speculate. Yet we will remark that there is an unwarranted, indeed mis­ leading, tendency among Aristotelian scholars to treat Aristotle's syllogistic and his theory of logical syntax separately and independently of each other. But there is no justifiable reason for doing so. The author of Prior A nalytics also wrote On Interpretation, and an adequate logic of terms requires the discoveries of both. The great beauty and genius of Aristotle's syllogistic nothwithstanding, it is clearly not either an adequate logic of terms nor a complete one. Several important iogical items are either missing or insufficiently developed by Aristotle. We will merely mention, briefly, four such items. Aristotle provided no decision procedure for the syllogistic. Such a pro­ cedure is simply a statement of the necessary and sufficient conditions for inference validity. In chapter 24 of the first book of Prior A nalytics he does tell us that every valid syllogism must have at least one affirmative premise, 54 at least one universal premise, 55 and at least one premise must share a mode of predication with the conclusion. 56 However, these are only necessary conditions for syllogistic validity. Later scholastic logicians sought to rectify this situation by developing the doctrine of distribution, which they saw implicit in Aristotle's logical system. This doctrine is, in a sense, implicit in Aristotle, but was not fully and properly expounded and understood until this century.57 While Aristotle was certainly aware of the need for a logic of relations, he neither developed such a logic nor showed how the syllogistic could handle them. The most he does is devote a brief chapter of Prior A nalytics58 to a

54 55 56 57 58

4 1 b6. 4 l b6- 14, 22-27. 41 b27-3 l . We will examine the doctrine in our discussion of Sommers. Prior A nalytics, xxxvi. Prior A nalytics, Pr;or A nalytics, Prior A nalytics,

26

Three Logicians

few examples of syllogisms containing relations (later called oblique syl­ logisms).59 As we have seen, Aristotle knew the logical syntax of singular pro­ positions. But, though he was able, he was seemingly unwilling to fully incorporate such propositions into the syllogistic. This is mainly due to his view that the syllogistic was primarily a tool for demonstration. And this was so because logic was seen as a tool to be used in theoretical sciences. Demonstrative syllogisms (valid syllogisms with necessarily true premises), as opposed to dialectical and sophistic syllogisms, 60 are the syllogisms proper to theoretical science. Singular terms can never be predicated necessarily. Nevertheless, Aristotle did allow nondemonstrative syllogisms with singular propositions. 6 1 Had he seen that the division of syllogisms into demonstrative and nondemonstrative (based on the distinction be­ tween conclusions which are necessarily true and those which are not necessarily true) deserves no place in a purely formal logic, he would have certainly developed the logic of singular propositions. Finally, and most importantly, Aristotle failed to account for the so­ called hypotheticals either by developing a logic of unanalyzed pro­ positions (like the Stoics') or by incorporating them into an expanded syllogistic. He was, of course, aware of the importance of the logic of hypotheticals and promised to deal with them. But we have no such text. We have only a few brief, confusing remarks to the effect that hypothetical syllogisms do not reduce to the perfect syllogisms. 62 Again, however. we must note that Aristotle has a very limited, specific view about the nature and purpose of the syllogistic. He was attempting to build a tool of logical analysis sufficient for the needs of demonstration, especially in the teach­ ing of the theoretical sciences. He most assuredly did not attempt, ever, to build a universal logic - a logic capable of accounting for every kind of valid inference. There is, in spite of what we have just said above, a temptation to ponder the possibility of either modifying or extending Aristotle's syllogistic in order to make it a universal logic. Yet in all the time from Aristotle to the present only two logicians worthy of mention have had the intellectual capability, and the interest, to attempt such a task. One of them, F. Som­ mers, has in our own day come close to success. The other, whom we will I have discussed Prior Analytics, xxxvi, in .. Aristotle on the Oblique", forthcoming. See Topics, 1 00a27ff. 61 In Prior Analytics, xxxiii, for example. Note, incidentally, that in the first of these examples Aristotle allows for the explicit quantification of the singular term 'Aristomenes'. 62 Prior Ana�ytics, 50a l 6-28. 59

60

Aristotle

27

meet in the next chapter, was the first logician to have a clear concept of what a universal logic should be, saw that Aristotle's logic of terms was its foundation, and provided innumerable insights into a variety of aspect of such a logic. And yet, in the end, Leibniz failed to provide that universal logic itself.

28

Leibniz

G.W. von Leibniz (1646-1716) was interested in logical studies throughout most of his productive life. Indeed, his interest in devising a universal logic, a logic powerful enough to handle any type of inference, was the only one - among very many - which seems to have occupied him constantly. Leibniz produced many logical studies. But most of these are quite brief, and almost all were unpublished until the end of the Nineteenth and beginning of the Twentieth Centuries. His papers, especially after 1680, show him moving closer and closer to his goal. Unfortunately, he was a man of such varied interests and activities that he completed very little, though he began so much. Many of his logical studies are merely notes, fragments, partial essays. He began to build one logical calculus, for example, on l August, 1690. The next day he began all over again. The general assessment of Leibniz (as a logician) today is that he was potentially a great logician, but he was severely limited by one overriding flaw - his unreflective adherence to Aristotelianism. This blind accept­ ance, so it is charged, led him to assume that all sentences must be logically categorical, and to assume the intensional interpretation of categoricals. In this chapter we will suggest that throughout the long period of Leibniz's logical studies he was engaged in the single pursuit of a universal, or general, logic, that in doing so he came to hold certain theories about logic, and that, his great respect for Aristotle notwithstanding, he held these theories and beliefs for sound logical and philosophical reasons. Indeed, throughout his logical writing Leibniz only mentioned or quoted Aristotle a few times. His attitude was clearly that Aristotle's logic was correct but incomplete. He sought to extend syllogistic logic in such a way that it could be used in the analysis of any inference whatsoever. In examining Leibniz's logical works it should always be kept in mind that he was writing during a period of great anti-Aristotelianism (in fact, anti-formalism). The Ramists, in particular had become a powerful intellectual influence in European universities and intellectual circles. It was a time when Aristotle was either completely accepted or completely rejected. The Ramist, following the lead of Peter Ramus, held that virtually everything Aristotle had written

Leibniz

29

was false. Leibniz was one of the handful of intellectuals who wished to both accept Aristotle and at the same time procede along some nonaris­ totelian lines of research. Four kinds of arguments, or inferences, were of concern to Leibniz at various stages in his logical studies : those whose forms depended upon terms alone, those whose forms depended upon sentences, those which contained relational terms, and those which contained singular terms. Leibniz was constantly attempting to devise a logical system, a calculus, which would be able to analyze all such inferences in a uniform fashion. Aristotle had developed an impressive, workable system for dealing with inferences of the first kind. And his theory of logical syntax, at least, could account for the kinds of sentences involved in the fourth kind of inference. So Leibniz naturally started with Aristotle's theory of the syllogistic. His inten tion was to show that sentential functions (viz. those compound sen­ tences investigated by the Stoics), relationals, and sentences containing singular terms were all logically categorical in form. And, furthermore, he intended to show that a logic of categoricals - syllogistic - could be developed which could deal, in a uniform way, with all inferences con­ taining any of those kinds of sentences. In short, Leibniz wished to make term logic a general logic. He failed. But, he came very close to succeeding. He was able to make great progress towards his goal along several important lines. As we will see, he fell short of success in only a few small ways. Leibniz is often cited now as one of the forerunners of symbolic logic. There is more confusion than insight in this. Logic has always been symbolic. Aristotle used symbols in his logic. Those who call Leibniz a symbolic logician, and mean by this that he was in this way different from his predecessors, probably mean that he was a mathematical logician. But, unlike logical systems of the late Nineteenth Century, which were meant to reflect a special dependency relation between logic and mathematics, Leibniz's logic was mathematical only in the sense that he (i) came to adapt certain mathematical symbols to logical ends, and (ii) he sought to build a logical system which would provide a mode of logical reckoning compar­ able in its ease and simplicity, elegance and rigor, to that found in mathematics. Leibniz's concern was logical rather than mathematical, and he was never averse to formulating logical principles which might, prima facie at least, seem contrary to mathematics. We should point out now that one of Leibniz's greatest achievements as a symbolic logician was that he was able to extend Aristotle's symbolization by symbolizing not only the terms of a categorical but the formatives (quantifiers and qualifiers) as well.

30

Three Logicians

Leibniz, like Aristotle (not merely because of Aristotle) believed that natural language was limited in certain ways by logical considerations. Our ordinary discourse is implicitly regulated by logical constraints. Logic logical syntax - specifies certain formal requirements for sentences. If natural language is under logical constraint, then it is reasonable to assume, as Leibniz and Aristotle both did, that natural syntax is, implicitly at least, logical syntax. Leibniz sought not to build a new logic, and with it a new artificial syntax, but to formalize the logical syntax of natural language. The view that natural sentences are logically categorical is an expected result of an inspection of most natural sentences. This, then, was his starting point. All sentences are logically categorical. Contemporary logicians who criticize Leibniz on these grounds have failed to see that he, unlike them, sought not an artificial syntax but the logical syntax under­ lying natural language. Leibniz voiced his view that all sentences are logically categorical many times. For example, in 1 666 : "A proposition is composed of subject and predicate. " 1 And years later, in 1 689 : "Every proposition which is commonly used in speech comes to this, that it is said what term contains what. . . "2 According to Leibniz, a categorical is a sentence in which something is said of something. The two terms of a categorical are in a logical "combination". He argued that affirmative sentences "compound" their subject and predicate and negative sentences "divide" them. This view naturally suggested an arithmetical model: "So just as there are two primary signs of algebra and analytics, + and -, in the same way there are as it were two copulas, 'is' and 'is not'; in the former case the mind compounds, in the latter it divides." 3 Early in his logical studies Leibniz saw that in discerning the logical syntax of natural language sentences it was admissable, even necessary, to ignore those grammatical features which were in no way logically important. A language with only a logical syntax (a "rational grammar") would be a universal language (a "universal polygraphy") with no mere grammatical variations. Thus, for example, "the distinction between sub­ stantive and adjective can be neglected. "4 As "distinction of gender is not relevant to a rational grammar; neither do distinctions of declensions and conjugations have any use in a philosophical grammar. " 5 By ignoring these 1 2 3 4 5

Leibniz: Logical Papers, G.H .R. Parkinson. ed. (Oxford. 1 966) p.3. Parkinson, p.84. Parkinson. p.3. Parkinson. p. 1 2. See also : pp.14. 16, 45, 47. Parkinson, p. 1 3 ; also p.16.

Leibniz

31

purely grammatical features we arrive at the logical form of a sentence. It consists of terms plus formatives: "Words constitute the matter, particles, or formatives, of a sentence are those words or phrases which determine quantity and quality, the copulae and the quantifiers. "Every categorical proposition has a subject, a predicate, a copula, a quality and a quantity. Subject and predicate are called 'terms'."7 In the passage just quoted Leibniz seems slightly confused about wether the copula is or is not the sign of quantity. It is from a paper written in April, 1679. Earlier in that month he had written "Elements of a Calculus", which shows no such confusion. What is most important about this earlier paper is that in it Leibniz makes clear for the first time that what he is seeking is a universal logic. In particular, it is a logic based upon the logic of categoricals, requiring all other kinds of sentences to be reformulated as categoricals. He says there, " . . . the categorical proposition is the basis of the rest, and modal, hypothetical, disjunctive and all other propositions presuppose it."8 In his second paper (mentioned above) of that month Leibniz attempted to devise a decision method for syllogistic based upon a notion that terms could be translated into sets of numhers (so that decision would be merely a matter of arithmetic computation). He eventually abandoned his attempt to reinterpret terms numerically, but his intention was sound. His rules of decision making were uniform and universal. This was partially because he could take all sentences as categorical. He says of his rules that they are "derived from a higher principle, and with certain changes I can adopt them to modal, hypothetical and any other syl­ logisms. " 9 At this stage, however, Leibniz was unable to say clearly just what changes were necessary to adapt his system to all kinds of syllogisms. In 1686 Leibniz wrote "General Inquiries about the Analysis of Con­ cepts and of Truths". In it he struggles to bring together a wide variety of logical concepts and devices into a coherent whole. Though he claimed that he had "made excellent progress" here, he did not succeed in building the complete logic he had invisaged. But he did make important progress in several key areas. Much of the paper is devoted to finding a satisfactory mode of symbolization for categoricals. Leibniz saw that a universal logic would be easier if a uniform means of formulating all sentences could be found. 0

7 8 9

Pa rkuison . Pa rki nson . Parkinson . Parkinson .

p. 1 5 . p.25 . p. 1 7. p.25.

32

Three Logicians If, as I hope, I can conceive all propositions as terms, and hypotheticals as categoricals, and if I can treat all propositions universally, this promises a wonderful ease in my symbolism and analysis of concepts, and will be a discovery of the greatest importance.10

Two things are to be said here. First, Leibniz sees that the major block to his development of a truly universal logic is posed by hypotheticals (by which he means conditionals, disjunctions, etc.). Second, he sees that hypotheticals can be reformulated as categoricals only if their constituent propositions (subsentences) can be reformulated as terms. He attempts to show in this paper how propositions can be treated as terms, and goes on to show how hypotheticals can be treated as categoricals. What Leibniz accomplished here was indeed of the "greatest importance". Early in "General Inquiries" Leibniz formulated some rules, laws, definitions, etc. using 'A' and 'B' as term variables, and said of these symbols that they mean "either a term or a proposition". 1 1 He was certain that propositions could be viewed as terms. But, he felt obliged also to say that terms could be viewed as propositions. Yet, if his logic is to be universal, then all that is required is that both sentences and terms be treated uniformly. And if his logic is to be, as he said, based on the categorical, then all that is required is that sentences be taken as terms. That sentences can be viewed as terms Leibniz had no doubt: "any pro­ position can be conceived as a term." 1 2 Hi�. confidence here not­ withstanding, he was less certain about just how to convert sentences into terms. Sometimes a sentence is correlated with a term naming the fact which makes the sentence true. 1 3 For example, 'Man is an animal' is conceived as the fact of man 's being an animal. Later Leibniz was more specific in offering an alternative: "A proposition itself becomes a term if 'true' or 'false' is added to the term." 1 4 This is not very helpful. He ex­ plained it with examples that fail to show how the term to which 'true' or 'false' is added is a proposition conceived of as a term. While he was justified in believing that sentences could be conceived as terms (which is necessary if categoricals are to be the basis of a universal logic), he was

Parkinson. p.66. Parkinson, p.56. 1 2 Parkinson, p.7 1 ; also see p.86. 1 3 See Parkinson, p. 7 1 . Earlier, p. 57. he seems to use the form: 'that A is B' as a term form of the sentence 'A is B' . 14 Parkinson, p.86. 10

11

Leibniz

33

never able to devise an easy, uniform method for doing so. 1 5 Though Leibniz was unable to provide a satisfactory device for trans­ forming sentences to terms, he did provide a successful account of how hypotheticals, once their constituent sentences are taken as terms, can be viewed as categoricals. He took all categoricals to have the general form: A contains B, where 'A' is the subject and "B' is the predicate. He said that a conditional can be viewed as having this same form since it merely says that the consequent is contained in the antecedent. In other words, a sentence of the form 'A contains B' can be read as a categorical with 'A' and 'B' as its subject term and predicate term respectively, or as a conditional with 'A' the antecedent and 'B' the consequent. Conditionals are categoricals by virtue of the fact that the relationship between an antecedent and a con­ sequent is exactly that between a subject and a predicate, namely, con­ tainment. "U ne proposition categorique est vrai quand le predicat est contenu dans le sujet; un proposition hypothetique est vrai quand le consequent est contenu dans l'antecedent." 1 6 Leibniz used the notion of containment between subject and predicate here according to his usual intensional interpretation of categoricals. On this interpretation, the terms of a categorical are taken as concepts so that to say that A is B is to say that the concept of A contains the concept of B. For example, 'Man is rational' is true since the concept Man (viz. rational animal) contains the concept rational. But Leibniz went further. At the end of "General Inquiries" he suggested that the relation of logical entailment, which holds between some pairs of sentences, can be reduced to this relation of containment as well since" . . . that a proposition follows from a proposition is simply that a consequent is contained in an antecedent, as a term in a term. By this method we reduce inferences to propositions, and propositions to terms." 1 7 It is obvious from this passage that Leibniz strongly believed that a logic of sentences could be reduced completely to a logic of terms. "We have, then, discovered many secrets of great importance· for the analysis of all our thoughts and for the discovery and proof of truths. We have discovered 15

In " Leibniz's Syllogistic- Proposi tional Calculus", Notre Dame Journal of Formal Logic, 1 7 ( 1 976), H . - N . Castaneda offers a n exce llent accou nt of how Leibniz failed here. There is, it might be noted , the possibility that Leih n iz was led to search for a means of incorporating hypotheticals into syllogistic by read ing John Wa llis Jnstitutio Logicae (Oxford 1687). I n an appendix, written some years earlier, Wallis had argued for the inclusion of hypothetical syllogisms in the theory of categorical syllogisms. Wallis was a well-k nown mathe matician whose mathematical work was quite familiar to Leibniz. See W.S. Howell, Eighteenth Century British Logic and Rhetoric. ( Princeton, 1 97 1 ) pp.29-30. 1 6 Q uoted in Cantaneda, p.483. 17 Parkinson, p.87.

34

Three Logicians

how . . . absolute and hypothetical truths have one and the same laws and are contained in the same general theorems, so that all syllogisms become categorical . . . " 1 8 We have seen that while Leibniz was unable to offer an acceptable device for transforming hypotheticals into terms, he was able to give a uniform reading of conditionals and categoricals in terms of the contain­ ment relation. We said above that he extended Aristotle's symbolic logic by finding a means of symbolizing not only the terms of a sentence but the formatives as well. Leibniz investigated a variety of methods for fully symbolizing sentences. None of his methods were completely satisfactory. The one he used most often, and with the most understanding, takes copulae as equal signs (' = ' for 'is', etc. and 'non = ' for 'is not', etc. ). Terms are symbolized by upper case letters. But there are two kinds of terms definite and indefinite. It seems that indefinite terms serve as particular quantifiers. For example, let A be a definite term, say 'man' and let Y be an indefinite term. Then, A Y says: A which is Y. In other words, Y limits A to just some A's (those which are Y). A universal affirmation would be: A = BY. We might read this as 'all A is some B'. Here A is universal since it involves no indefinite term and B is particular since it is limited by Y. Why does Leibniz allow predicates to be quantified. Three reasons are discern­ able here. First, by using the equal sign as the sign of predication he felt forced to formulate the rest of the categorical in such a way that what flanks the equal sign are indeed equivalent. In other words. he took the copula as symetrical. It follows, then, that both sides of the copula must be logically similar; both sides must be quantified terms. Second, in using this mode of symbolization Leibniz had abandoned his usual intensional interpretation of categoricals in favor of an extensional one. On this second interpretation terms are taken not as concepts but as sets of entities. To say that A is B would then be to say that the set of A's is included in the set of B's. It is natural to quantify both terms of a categorical when those terms are viewed as sets of items. Third, Leibniz seems to have confused the distribution of a term with its quantity. Since every term is distributed or undistributed, every term is universally or particularly quantified for him. He adopted this view in a paper written a few years after HGeneral Inquiries". 1 9 He claimed there that predicates of affirmations are particular and the predicates of denials are universa I. 2 0 18 19

20

Park inson, p . 7 8 . "A M athematics of Reason". in Parkinson, pp.9 5- 1 04. Parkinson, p.96.

Leibniz

35

Though Leibniz seems less than completely clear about the quantification of predicates, at least one interesting result does follow from it. In 'A Mathematics of Reason" he pointed out2 1 that by taking the predicates of affirmations as always particular and those of denials as always universal we can ignore the difference between quantity and quality. A sentence becomes a concatenation of two terms, each quantified. What is most important here is that Leibniz has seen that any sentence can be logically construed as a quantified term plus a qualified term - and, more importantly, that quantity and quality can be orthographically similar. A symbol can be used which can indicate indifferently either a quantity or a quality. If we say, for example, that 'O' is a universal quantifier and '6' a particular quantifier, we can formalize the four stan­ dard categoricals as: A E I 0

OS6 P OSOP 6S6 P 6SOP

Though Leibniz has said that quality should be taken as quantity, what is actually the case is that the formatives are indifferently either one. We could read A above as (i) 'all S are P', or (ii) 'all S are some P'; E as (i) 'all S are not P', or (ii) 'all S are all P' ; I as (i) 'some S are P', or (ii) 'some S are some P'; and O as (i) 'some S are not P', or (ii) 'some S are all P'. The real importance of Leibniz's discovery here is that particular quantity and affirmative quality are logically homogeneous, as are universal quantity and negative quality. Had he realized the import of this discovery he could have developed a fully adequate and uniform symbolization for all sen­ tences. But he did not. As we will see, it was almost three centuries before this discovery was made again - and finally understood. Leibniz's goal was a logic which could deal uniformly with all types of sentences and inferences. It would be a syllogistic logic which treated all sentences as categorical. We have examined his attempt to reduce hypotheticals to categoricals. He was also obliged to find a categorical analysis for singular sentences. In particular, the question of how singular subject terms have quantity was yet to be answered. Aristotle had said that denying a term of a singular was the same as affirming its opposite (its logical contrary) of that subject. The seeds of a full theory of singular term quantity were there, but their development required an understanding of 21

In article 24.

36

Three Logicians

the connection between predication and quantification which Leibniz did not fully possess. The scholastics had tended to say that singular subjects were implicitly universal. So did Leibniz in the early stages of his logical studies. But in "General Inquiries" he made a great discovery. He found that a singular term could be taken as either universal or particular indif­ ferently, since the universal quantification of a singular and the particular quantification of a singular are equivalent. He made only a passing remark concerning this there, 22 but a few years later, in "A Paper on 'Some Logical Difficulties'", he was more direct in addressing the question. The first paragraph of that paper is worth quoting in full. Some logical difficulties worth solu tion have occu red to me. How is it that opposition is valid in the case of singular propositions - e.g. 'The Apostle Peter is a soldier' and 'The Apostile Peter is not a soldier' - since elsewhere a universal affirmative and a particular negative are opposed? Should we say that a singular proposition is eq uivalent to a particular and to a u niversal proposition? Yes, we should . So also when it is objected that a sin gular proposition is equivalent to a particular proposition, since the conclusion in the third figure must be particular, and can nevertheless be singular; e.g. ' Every writer is a man, some writer is the Apostle Peter, therefore, the Apostle Peter is a man'. I reply that here also the conclusion is really particular, and it is as if we had drawn the conclusion 'Some Apostle Peter is a man'. For 'some Apostle Peter' and 'every Apostle Peter' coi ncide, since the term is singular. 23

Leibniz's discovery that singulars are indifferently universal or particular, like much of his work, had to await our century for its full appreciation. Much of the criticism directed toward traditional syllogistic has centered on the fact that it cannot adequately account for the logic of relationals. A genuine categorical has exactly two terms - a subject term and a predicate term. A relational, on the other hand, seems to have three or more terms a subject term, a relational predicate term, and one or more object terms. Leibniz was well aware that if a universal logic was possible, then it must be able to analyze relationals categorically. As it turns out, he was unable to provide a satisfactory categorical analysis for relationals. But he did demonstrate a keen understanding of how relationals should fit into the syllogistic. In his early papers on grammar Leibniz had tried to render relationals of the form 'A is R to B' as a conjunction of two categoricals, 'A is an R'er' and 'B is R'ed'. But, of course, the relational term R is still there and still a relational. Leibniz had simply disguised it. He seems to have abandoned the problem after that attempt. But several years later, in 1687, he turned to 22 23

Parkinson, p.65. Park inson, p. 1 1 5 .

Leibniz

37

the problem of g1vmg a syllogistic account of inferences involving relational sentences. The inference he considers is: Painting is an art, therefore he who learns painting learns an art. The proof of the argument depends upon a small number of suppositions. The crucial one is a pri nciple common to all syllogistic inference. Leibniz states it thusly: To be a predicate in a universal affirmative proposition is the same as to be capable of being substituted without loss of truth for the subject in every other affirmative pro­ position where the subject plays the part of predicate. 24

A moment's reflection shows that this is simply a version of the dictum de omni et nullo. Elsewhere Leibniz calls it "the basis of the syllogism"25 and "the foundation of the whole of syllogistic theory". 26 It is used in the proof of 'Painting is an art, therefore he who learns painting learns an art'. 1. Painting is an art. 2. He who learns painting learns a thing which is painting. 3. He who learns painting learns a thing which is an art. 4. 'A thing which is an art' equals 'an art'. 5. He who learns painting learns an art.

Line I is the premise, lines 2 and 4 are suppositions, line 3 follows from I and 2 by the dictum, and 5 follows from 4 by substitution of equals. Two important points to be made here are ( I ) that Leibniz saw that a universal syllogistic ought to be able to deal with relational inferences, and (2) that he saw that the dictum de omni et nullo was the underlying principle of syllogistic. A full understanding of relational inferences requires only one thing more - a method for converting relationals into categoricals. Because Leibniz took all sentences as logically categorical, it was essen­ tial that no sentence be logically complex, such as hypotheticals. This also meant that a negation could never be a complex sentence. The negation of sentence is always another sentence contradictory to it. A sentence is not a (logically) less complex sentence than its negation. As we will see, most contemporary logicians take negation to be an operation on a sentence which results in a new, logically more complex, sentence. With Aristotle, Leibniz held that terms can be negated (each term, say 'A' has a "priv­ ative", 'not-A') and predicates can be denied of subjects. A negative sen­ tence is one in which the predicate is denied. However, as we saw above, in "General Inquiries" Leibniz was looking for a formulation of all sentences in terms of equations. He believed that once all sentences were logically 24

25

Parkinson, p.88. Parkinson, p. 1 05.

Three Logicians

38

reformulated as equations the business o f logical reckoning would simply be a matter of applying the principle of substitutability of equivalences. So, while he was willing to admit negative sentences (denials), he was anxious to show how they could be reduced to (nonnegative) equations. He offers three different ways to do this. O ne method is to make all negation term negation, so that to deny a predicate is to affirm a negated predicate term. Thus, he says that 'is not P' means 'is not-P' . 2 7 On this view, all sentences are affirmations and all logical differences depend upon quantity and term contrariety. A second method is somewhat complex . He suggested in a footnote 28 that we could take 'exists' as the predicate of all sentences. Thus 'some A is not B' would become 'some (A not-B) exists'. He claimed at the same time that 'no A is B' could be transformed as 'A contains not-B'. A uniform treatmen t here, h owever, would take the u niversal negative as affirming nonexistence of all AB's. A third method is to take all denials as affirmations of falsity to their corresponding contradictories. So 'A is not B ' means '(A i s B) i s false'. 2 9 L eibniz w a s obviously willing, i n light o f these last two proposals, to incur the dangers of importing questions of existence into logic and of confusing what we today would call the metalanguage with the object language. H is wish to avoid negation as an operation on an entire sentence was sound. His further attempts to eliminate negative sentences all together was forced on him by his wish to build and equ­ ational logic. We hope to show later that his belief that predicate denial could be replaced uniformly by term negation was correct - though not because it would allow us the equational logic he sough t. We have seen that Leibniz's attempts to eliminate negative sentences in terms of falsity or nonexistence were potentially dangerous. As it happens, however, he was willing to adrl rcss himself, in other contexts. to these problems. Concerning existence, Leibniz saw that different existence assumptions are made on the intensional and extensional interpretations or categoricals. On the intensional interp retation terms are taken as con­ cepts. To say that all men are rational is to say that the concept man includes the concept rational. The concept, rather than the entity. is i n ­ troduced here s o that t h e existence o r nonexistence o f t h e entity. o r entities ( e.g. men) is not raised . From the beginning of his logical studies Leibniz was intent on building a logic free of unnecessary e xistence assumptions. 26

27 28 29

Parkinson, p. 1 1 6. Parkinson, p.84 (article 1 86). Parkinson, p.87. Parkinson, p.58, n. I. and p.84, article 1 85.

Leibniz

39

That is one reason why he usually adopted the intentional approach. "However, I have preferred to consider universal concepts, i.e. ideas, and their combinations, as they do not depend on the existence of in­ dividuals."3 0 In his later paper on "Some Logical Difficulties" Leibniz argued that no sentence has existential import. The accepted view has been, and still is, that subalternation in syllogistic logic forces existential import onto universals since it is unquestionab]y to be found in particular sentences. In other words, since 'some S is P' entails that some S exists, so must 'all S is P' since it entails 'some S is P'. But Leibniz claimed that even 'some S is P' does not entail that some S exists. His suggestion is that terms refer not to actuals but merely to possibles. In other words, the terms of a categorical are not actuals - they do not necessarily refer to existing entities. They are merely possibles - they are just noncontradictory terms.3 1 Thus, for Leibniz, existence is not part of the meaning of either the universal or the particular quantifier. On his view an empty term, one which cannot be used in a sentence to refer, is an impossible term, a contradict.ory term. Modern mathematical logicians use the notion of a universe of discourse for the set of actuals which can be referred to by a subject term in a sentence. This has raised many difficult problems for logicians faced with sentences whose subjects refer to nonexistents. Leibniz, by taking his universe of discourse to be the set of possibles (logically possible entities), rather than actual entities, removed such worries over nonreferring terms from logic all together. One result of such a move is that 'exist' is not a part of the quantifier but is a genuine term which can be affirmed or denied of subjects. In "General Inquiries" Leibniz made several remarks which together constitute a theory about the terms 'true' and 'false'. First of all, he stated clearly that he was committed to the principle of bivalence, "every pro­ position is either true or false."32 He was committed to this by his earlier equation of 'not true' with 'false', 33 since he assumed it would be con­ tradictory to say that a sentence is both true and not true. Every true sentence is either true in itself or derivable from one which is true in itself. Every false proposition is either false in itself or can be derived from one which is. Leibniz went on to argue that contingencies require an infinite

:m Parkinson. p.20. 31 See Parkinson, p. 55, n.1, and p.58, articles 32, 33, 34. 32 Parkinson, p.55. 33 Parkinson, p.54.

Three Logicians

40

deduction from sentences true or false in themselves34 (viz. 'A is A' and 'A is not-A' respectively). 3 5 Finally, in searching for a universal syllogistic Leibniz turned up several rules which govern compound terms. In his studies after 1679 he was intent on discovering the rules necessary for a genuine universal syllogistic. In doing so he shifted the entire concept of syllogistic to a higher plane by allowing compound terms and, what is most important, by formulating rules to govern them. Before this Leibniz's syllogistic studies had almost always been limited to arguments with simple categoricals - sentences whose subjects and predicates were simple monadic terms. An admission of complex terms, however, would be inevitable in a logic which sought to convert entire sentences to terms. Such terms would surely be complex. Actually, Leibniz had started recognizing complex terms in his early grammatical studies. There he had argued that all terms, adjectives, genetives, etc. could be taken logically as nouns or verbs. Thus, a term modified by an adjective, say, becomes a compound of two nouns, or a noun and a verb - a complex term. For example, 'swimming fish' and 'tall man' become 'fish which swims' or 'fish which is swimming', and 'man who is tall' or 'man which is a tall entity'. Leibniz formulated several syllogistic rules which involved compound terms. Some are simple and obvious. For example, in several places36 he stated the law of commutation for compound terms: 'AB' is equivalent to 'BA'. His proof that if painting is an art, then he who learns painting learns an art, shows that he saw that the dictum de omni et nu/lo applies not only to sentences with simple monadic terms but to those with complex terms as well. Thus, a term predicated of a universally quantified term could be substituted for that term wherever it is itself affirmed - even when it is not a predicate but only part of a predicate. Another rule which Leibniz mentioned in several places3 7 states, in effect, that from a categorical can be derived a sentence exactly like it except that both the subject term and the predicate term have been compounded by a common term or two equivalent terms. For example, from �an S are P' we can derive 'all SA are PA', or (if A = B) 'all SA are PB'. Leibniz also noted38 that a term affirmed of another can also be affirmed of a compound including that other term. If A is B, then AX is B. For example, from 'all men are rational' we can infer 'all tall men are rational'. Other 34 35 36 37 38

Parkinson, p.61. Parkinson, pp.55-56. See Parkinson, pp.40, 43, 142. Parkinson, pp. 4 1 , 43 , 1 33 . Parkinson, p.57.

Leibniz

41

rules governing compound terms were not stated explicitly by Leibniz, but were used by him. In "A Specimen of the Universal Calculus" 39 and "Addenda to the Specimen of the U niversal Calcu lus"40 he offered several syllogisms w hich involve compound terms. Leibniz's admission of compound terms into syllogistic is of tremendous importance . A syllogistic, such as Aristotle's limited to simple monadic terms can never be a universal logic. The extension of syllogistic to relationals and hypotheticals alone forces the recognition of complex terms. U nfortunately Leibniz offered no clues for fitting sentences with complex terms into a uniform logical form. What was required was a general theory of logical syn tax which would uniformly formalize all sen ­ tences, even those with complex terms. as categoricals. No logician since Aristotle had contributed so much to the building of a syllogistic term logic. That Leibniz failed now and then to achieve certain goals simply attests to how much he tried to do. Starting, as Aristotle had, with the view tha t a logical calculus s, I : P's, 0 : P's, and validity conditions for any syllogism are stated. These conditions require that any syllogism have exactly three terms, that it contain no denials, and that it is in transitive form. A syllogism is in transitive form when it has the form :

Xy . Yz I Xz

Sommers

45

A syllogism can be put into that form by application of the "rule of inversion". Ps =

Sp ; P's = S'p

and by transposing (a nonaffirmative premise and a nonaffirmative con­ clusion can exchange positions and be thereafter taken as affirmations). Reminsicent of Leibniz's attempts at devising a mathematical analogue for syllogistic reasoning, Sommers suggests a fraction model. By taking a sentence of the form 'Ps' to correspond to the fraction 'sip', and by taking negated terms, e.g. 'not-P', to correspond to algebraic variables raised to the power of minus one ('p- 1 ' ) , and by taking denial to correspond to the raising of an entire fraction to the power of minus one (e.g. 'Some S aren't P' corresponds to '(s/ p)- 1 '), Sommers can treat a syllogism as an algebraic equation. For a valid syllogism the product of the premises must alge­ braically equal the conclusion. To see this consider the Clearent syllogism :

Pm . Ms. ::J .f>s

Its fraction model is m/p- 1

x s/ m

s/p- 1

which is a true algebraic equivalence. So the syllogism is valid. The notion of an algebraic analogue for syllogistic is interesting and, as we have seen, historically respectable. But Sommers' method often turns out to be quite awkward. Fortunately, he is able, eventually, to replace it with a much simpler one. In "Types and Ontology"4 Sommers had defined the universe of dis­ course of any sentence as the intersection of the categories determined by the terms used in that sentence. Thus, the universe of discourse, the domain, for 'Some birds do not fly' is/ bird / n / fly/ . Now some sen­ tences seem to be just about things. For example. 'Everything is material', 'Nothing is im material', 'Something is red'. In 'On a Fregean Dogma' Som mers suggests that in such cases the logical subject is just the universe of discourse itself. So 'Everything is material' says 'Every /material/is material' ; in other words, 'Whatever is spanned by "material" is ma­ terial\ or, 'Whatever is either material or immaterial is material'. Every syllogism will have a domain which is the intersection of all the categories determined by the terms of the syllogisms. The domain of our Clearent above is / S/ n / M / n / P/. By assuming that 'member of the domain' can 4

=

"Types and On tology", p. 362.

Three Logicians

46

be read as 'what there is' or "what exists' (""existents" for Sommers) he concludes that existential statements can be incorporated (contrary to Aristotle's practise) into the syllogistic. Consider the syllogism Humans are mortal Humans exist Therefore, mortals exist Its domain is /human/ n / mortal/ . Sentences of the form 'P's exist' are reformed as "Something is P', which, as we saw above, can be taken as "Something in the domain is P'. If we take, now 'in /human/ n /mortal / ' as "exist', our syllogism becomes Humans are mortal Something which exists is human Therefore, something which exist is mortal Sommer's technique of equating existence with domain membership (something quite familar to contemporary mathematical logicians) clearly shows that the syllogistic need not exclude existential statements. Another advantage which Sommer's theory here seems to have is that by taking singular and nonsingular sentences to be on a logical par, singular terms are given full admittance into the syllogistic. The only difference between singular terms and general terms is one which Aristotle and Sommers both see : for singular sentences there is no difference between denying a predicate and affirming its logical contrary. For Socrates is unhappy if and only if he isn't happy. Suppose we construct a square of opposition for 'S's are P' and its companions. �

s □Ps

P's

P's

When 'S' is singular ( e.g. 'Socrates') the A and I forms are identical, as are the E and O forms. We could reconstruct a square for singular sentences taking note of these equivalences. Ps (P1-s)

Ps (P's)

Leibniz, like Aristotle and Sommers, had noticed the oddity of singular statements. He accounted for it, as we saw, by taking singular subject terms

Sommers

47

to have an ambiguous (thus suppressed) quantity. So, when 'S' is singular, 'All S is P' and 'Some S is P' are equivalent. A (quantified) square, then, for a singular subject term, according to Leibniz, would be All S is P (Some S is P)

All S is not-P (Some S is not-P)

What is common to both Sommers' account of singulars and Leibniz's is that they both recognize the A/ I and E/O equivalences for singular sen­ tences. Later, as we will see, Sommers came to adopt Leibniz's account as more reflective of our ordinary modes of discourse about individuals. Sommers admits that the version of syllogistic which he offers in "On a Fregean Dogma" "is not meant to be a logical instrument of any generality". 5 In particular, it cannot be used to analyze relationals, nor even weakened inferences. But a new syllogistic was not his intention there. What Sommers wanted to do was reinforce the distinction between predicate denial and term negation. A recognition of this permitted him to formulate all categoricals as simple predications (affirmations or denials) of a predicate term or its logical contrary. One result of this is his demon­ stration of the possibility of "dequantifying" the categoricals. Contemporary mathematical logic, following Frege's advice, has ignored the subject-predicate distinction. The result has been that term negation and predicate denial are both eliminated in favor of sentential negation. If the only negation available to the logician is sentential, then, naturally, categoricals can no longer be viewed as subject-predicate sentences. In "On a Fregean Dogma" Sommers did not attempt a full reconstruc­ tion of the traditional syllogistic. But he did lay down some important groundwork for that task, which he would soon carry out. Another important piece of that groundwork was laid down in "Do We Need Identity?" In that brief, but important, paper Sommers sets out to show that statements of identity can be viewed as logically categorical. This results in the virtual elimination of identity as a special binary logical relation between individuals, and shows that the theory of identity (taken in terms of predication) can be included in the syllogistic; and, more generally, it shows how singular sentences can be admitted to the syllo­ gistic. If an identity sentence like 'Tully is Cicero' is to be read as a categorical, then its predicate 'Cicero' must have a sense, since it is being used not to 5

"On a Fregean Dogma", p.78.

Three Logicians

48

refer but to characterize. Modern logicians refuse to recognize the predication of singular terms. So they must take copulae, like 'is', to be systematically ambigous. Thus, while 'Tully is Roman' predicates 'Roman' of Tully ( characterizes him), 'Tully is Cicero' relates Tully and Cicero (it iden tifies them but does not characterize either o f them ). What Sommers wants to do is allow singular terms to occupy all the positions occupied by general terms. S ingular terms, then, can be predicated and, when in subj ect position, can be quantified. Recall that in "On a Fregean Dogma" Sommers had argued that singular sentences, unlike general sentences, collapse together their A and I forms and their E and O forms. Since he was engaged in that paper in an attempt to dequantify all categoricals, he was not led to Leibniz's con­ clusion that what distinguishes singular from general terms is their arbit­ rary quantity when they occur in the subj ect position. In "Do We N eed Identity?" however, Sommers gives an argument in which he derives this conclusion from his previous position. H is argument starts from Aristotle's insigh t that denying a predicate of all/ some of a subject is equivalent to affirming its logical contrary of some/ all of that subj ect. In other words ( 1 ) all /some S aren't P

=

some / all S are not-P

Moreover, whenever 'S' is singular, denying a predicate of it and affirming the logical con trary of that predicate of it are equivalent. Thus, for singular

'S',

(2) S aren't P

=

S are not-P

N ow Sommers suggests that singular terms be allowed to have quantity ( Medieval logicians had permitted this, and, as we saw, Leibniz insisted on it). Consider a singular sentence like 'Socrates is wise'. It is equivalent, by ( 1 ), to 'Some Socrates isn't unwise', which, by (2), is equivalent to "Some Socrates is wise' . What this means, then, is that the (logical) quantity of a singular term in subj ect position is arbitrary, "wild". The quantification of singulars and their admission to predicate position allows us to take all identity as predicational (i.e. identity statements are categoricals). By definition, 'a is identical to b' means "a is b'. Sommers shows that identity, when thought of predicationally, is still reflexive (since 'All a is a' is a tau tology), symmetical (since ' Some a is b' is equivalent by conversion to 'Some b is a'), and transitive (since 'All a is b, all b is c, therefore, all a is c' is a valid syllogism). N otice that the quantity here, since it is wild, can always be assigned as needed. For 'a', ' all a', and 'some a' are all equivalen t. Once iden tity statements are viewed predicationally, as

Sommers

49

predicating a singular term, and singular subjects are viewed as implicitly quantified, there is no reason to bar them from syllogisms. For example, (i) (some) Tully is Roman (all) Tully is Cicero Therefore (some) Cicero is Roman is a third figure valid syllogism (Disamis). (ii) (all) Socrates is mortal (all) Zeus is immortal Therefore, (all) Zeus is not Socrates is a valid second figure syllogism (Camestres). The full incorporation of singular sentences, including identity state­ ments, into the syllogistic is an accomplishment of great importance. The possibility of fulfilling Leibniz's vision of a universal logical calculus based on the categorical form (though not yet shared by Sommers himself) becomes a reality. What remains, however, is to incorporate two important kinds of sentences into syllogistic. One kind is relationals. The other is what Leibniz called ""hypotheticals", what we call today "truth-functions". Recall that in ""General Inquiries" Leibniz had tied the problem of cate­ gorializing hypotheticals to the problem of conceiving all propositions as terms. 6 In ""On Concepts of Truth in N atural Languages" Sommers con­ tinues laying what turns out to be the foundation for a fully generalized, universal syllogistic by showing just how a proposition, or sentence, can be conceived of as a term. Sommers' general concern in ''On Concepts of Truth in Natural Languages" is to defend a version of the correspondence theory of truth. In the process he provides a means for entire sentences to be treated as (logically) terms. In "The Semantic Conception of Truth" 7 A. Tarski had suggested the notion of an entire sentence having, like a term, a denotation. Sentences could be said to designate states of affairs. Indeed, long before this Frege had talked of the denotation, or reference, of a sentence as the circumstances of its truth of falsity. 8 Sommers makes a distinction between what a sentence is about and what it designates, or specifies. Suppose 'S' is a sentence (say 'Nixon is a liar'). Then '[ S]' will be read as 'state of affairs in which S' (e.g. 'sta te of affairs in which Nixoc is a liar'). An expression like Parkinson, p.66. Philosophy and Phenomenological Research , 4 ( I 944 ). 8 See especially "On Sense and Reference", Translations from the Philosophical Writings of Gottlog Frege, ed. P. Geach and M. Black (Oxford. 1 952), pp.62-63.

0

7

50

Three Logicians

'[S]' is a term (a "sentential term"), and the process of forming '[S]' from 'S' is called "nominalization". The nominalization of a sentence always results in a term, a sentential term. The states of affairs in which Nixon is a liar are what is specified by the sentence 'Nixon is a liar'. In general, a sentence 'S' specifies a state of affairs, [S]. Keep in mind that '[S]' is a term which denotes [S], while 'S' is a sentence which specifies [S]. Sommers says that a sentence is about what its terms denote. Thus 'Nixon is a liar' is about Nixon and liars. Moreover, sentential terms, being terms, can be used in other sentences. For example, 'The state of affairs in which Nixon is free is deplorable' contains as one of its terms a sentential term. Let S be the sentence 'Nixon is free' and S. l be the sentence 'The state of affairs in which Nixon is free is deplorable'. S. l says '[S] is deplorable'. One of the things S. l is about is [S]. But what S. l specifies is [S. l ], i.e. [[S] is deplor­ able]. The notion of sentence nominalization, and the resulting distinction between what a sentence is about and what it specifies, proves to be an impressively powerful tool for logical analysis. Sommers shows how it can be used, contra Tarski, to block paradoxes of self-reference in a natural language (e.g. the Liar), while permitting all innocuous self-reference (as in 'This sentence has five words'). But, more importantly for the development of syllogistic, Sommers shows how the nominalization process can be used to convert truth-functions into categoricals. A sentence like 'If Nixon is free then justice has failed' is clearly not a categorical. It is a truth-function of two other sentences (which could themselves be viewed as categoricals). But it can be conceived of as a categorical, given the notion of a sentential term. According to Sommers, 'If Nixon is free then justice has failed' is about two kinds of states of affairs: those in which Nixon is free and those in which justice has failed. What it says about these is that all the states of affairs of the first kind are states of affairs of the second kind. We could rewrite our sentence (using 'S' for 'Nixon is free' and 'P' for 'justice has failed') as: "All [S] are [P]'. Now this sentence is a genuine categorical. It has a subject term, '[S]' and a predicate term, '[P]'. Again sentential terms are terms, not sentences; they denote what are specified by sentences. Since any truth-functions can be transformed into a categorical by the nominalization of its component sentences, Leibniz's quest for a means of categorializing hypotheticals is achieved in a simple and straight-forward manner. Here are some simple examples of such transformations.

Sommers Truth-functional forms If p then q p and q If p then if q then r

51 Categorical forms All [p] are [q] Some [p] are [q] All [p] are [all [q] are [r]]

The nominalizing procedure accounts for how we can talk about states of affairs as well as specify them. Nominalizing, in effect, imbeds one sen­ tence in another. Thus 'p' is embedded in 'All [p] are [q]'. Not all sentences which imbed other sentences are truth-functions. Consider 'John believes that Nixon is innocent'. The object of John's belief is surely not the sentence 'Nixon is innocent', nor even the state of affairs specified by that sentence, i.e. [Nixon is innocent]. In "On Concepts of Truth in Natural Languages" Sommers distinguishes between sentences, statements, states of affairs, and facts. When produced on March 1 0, 1 978, the two sentences 'It rained forty-three days ago' and '11 a plut ii y a quarante-trois jours' likewise both make the same statement and specify the same state of affairs. Suppose it rained on March 10, 1 978. Then it is a fact that it rained on March 1 0, 1 978. The two statements mentioned above are both true. They both express that fact. To summarize : there is a one-many relation between facts and statements, a one-one relation between statements and states of affairs. and a one-many relation between states of affairs and sentences. It is easy now to see how we can extrapolate from Sommers here to show that any nominalized sentence could be taken either as a statement or a state of affairs (again, a statement is not a sentence, it is what is stated by an infinite number of sentences). We could, therefore, say that 'John believes that Nixon is innocent' relates John to the statement made by 'Nixon is innocent'. It says 'John believes Nixon is innocent ' . The one-one relation between states of affairs and statements allows us to take sentential terms as either denoting states of affairs or denoting statements corre­ sponding to (stating) those states of affairs. In "On a Fregean Dogma", "Do We Need Identity?" and "On Concepts of Truth in Natural Languages" Sommers was laying, perhaps uninten­ tionally, the foundation for a universal syllogistic. What is required for such a logic is essentially, that all assertions be formulable as categoricals. Sommers has shown how this is possible for singular sentences, identity statements, and truth-functions. But, of course, the main criticism of traditional syllogistic has always been that it cannot provide a viable logical analysis for relationals and inferences involving them. Sommers answers this criticism in "The Calculus of Terms". Where the first three papers lay the foundations for a new syllogistic, in this paper Sommers sees for the

52

Three Logicians

first time that he is moving toward Leibniz's goal and makes a conscious effort at achieving it. "The ·calculus of Terms" builds an impressive struc­ ture on the foundation already laid down. What makes this structure so impressive is that not only has Sommers built a universal syllogistic ( one which allows for the analysis of all assertions as categoricals), but he has formulated for it a calculus, an arithmetical analogue, which is simple, and which permits the easy determination of valid inference. These are two goals which Leibniz spent a lifetime trying to achieve. Every categorical assertion predicates - affirms or denies - a simple or complex predicate of all or some of a subject. Thus, the logical form of a categorical hinges on its quantity and quality. Quality is always positive or negative, and Sommers recognizes three types of qualitative opposition. A term and its logical contrary (say 'red' and 'nonred') are of opposite term quality. Every term has either a positive or negative term quality. Secondly, an entire predicate has predicate quality. For example, 'is red' and 'isn't red' are predicates having opposite predicate quality. As it turns out, Sommers goes on to claim that term quality and predicate quality really collapse into one another. For 'S' is not-P' says just what 'S isn't P' says. The effect of negating a predicate term and of negating a predicate are always the same - a sentence logically contrary to the sentence sans negation. Sommers calls the kind of opposition, whether term or predicate, which results in logically contrary sentences C- Opposition. The third kind of quality results in a second kind of opposition. Terms may or may not be negated, predicates may or may not be negated, and predicates (negated or un­ negated) may be affirmed or denied of their subjects. Affirmation and denial are predicative qualities, and the opposition of affirmation to denial is called P-opposition. Quantity is also oppositional, for a subject must be either universal or, exclusively, particular. Sommers calls this opposition of quantity Q-opposition. Logically, every categorical is a subject and a predicate. Every subject is a term plus a mark of Q-opposition. Every predicate is a term plus a mark of P-opposition. Moreover, every term is, logically. accompanied by a mark of C-opposition. Sommers has taken care in introducing his three kinds of quality. He shows that none are recognized in today's notion of sentential negation, and even suggests that the distinction between predicate quality and predicative quality was unrecognized by Aristotle and his followers. All opposition, including Q-opposition, is symbolized by a plus or minus. Affirmation is + , denial is -, a term and its logical contrary are given one a + the other a -. And universal quantity is marked - while particular quantity is marked + . Sommers' argument for this

Sommers

53

symbolization of Q-opposition is that it preserves normal logical operations such as conversions. To see this we must first see how a cate­ gorical would be fully symbolized. Let 'S' and 'P' be the terms of a cate­ gorical. N ow ' S' has a mark of C-opposition, as does 'P' (let ' ± ' mean ' + or -'), so ±S... ±P Furthermore, the subject must be marked by Q-opposition (i.e. be quantified), so ± ( ± S) . . . ± p And the predicate must be marked by C-opposition as well. ± ( ± S) ± ( ± P) Finally, the predicate must be affirmed or denied of the subject. The sign of P-opposition is placed in front of the entire sentence. Thus

± ( ± ( ± S) ± ( ± P)) The proper reading of each oppositional sign is determined by its position in the formula. A sentence like 'All S are P' is symbolized as + (-( + S ) + ( + P))

The first plus indicates affirmation, the minus indicates universal quantity, the second and fourth pluses indicate the positive term quality of the two terms, and the third plus indicates that the predicate is positive in predicate quality. Notice now that the choice of minus for universality and plus for parti­ cularity is justified, according to Sommers, by the equivalence + (-( + S ) + ( + P)) = + (- ( - P) + (- S )) which represents the contraposition of an A categorical. Likewise, + ( + ( + S) + ( + P)) = + ( + ( + P) + ( + S)) represents the conversion of an I categorical. Had a plus for universality and a minus for particularity been chosen these equivalences would no longer hold. Sommers claims that a schedule of four general categoricals can be formed using any two kinds of opposition. A PC schedule would take all sentences as affirmations or denials, and all predicates as positive or negative, but would not recognize quantity. A QC schedule would take all

Three Logicians

54

sentences as affirmative or all sentences as negative, and so forth. For example, A: E: I: 0:

All S is P All S is un-P Some S is P Some S is un-P

+ (-( + S) + ( + P)) + (-( + S) + (-P)) + ( + ( + S) + ( + P)) + ( + ( + S) + (-P))

A: E: I: 0:

S aren't un-P S aren't P S are P S are un-P

-( + S + (-P)) -( + S + ( + P)) + ( + S + ( + P)) + ( + S + (-P))

A: E: I: 0:

All S are P Some S aren't P Some S are P All S aren't P

+ (-( + S) + ( + P)) -( + ( + S) + ( + P)) + ( + ( + S) + ( + P)) -(-( + S) + ( + P))

is a QC schedule.

is a PC schedule. (Sommers uses 'is/isn't for C-opposition and 'are/ aren't' for P-opposition.) A PQ schedule would be

Notice that the A formulae in each schedule are arithmetically equal. Likewise for E,I, and 0 formulae. The quantity of any formula will always be determined by the quantity of the subject and the predicative quality of the entire formula. In each schedule above A and E are universal in quantity since the two signs always produce, arithmetically, a minus. I and 0 are all particular. This accords with Sommers' law of immediate inference. If two statements L and M have the same logical quantity. then L entails M if and only if L = M.

So 'L = M' can be read as 'L logically entails M '. All the classical immediate inferences can be seen in the following equations. (As in algebra, we can omit most plus signs and take them as understood : for example 'All S are P' could be formulated as '-S + P'.) A: E: I: 0:

-S + P= -S-(- P) = -(-P + (-S) = -(-P)-( + S) -S-P = -S-( + P) = -P-S = -P-( + S) + S + P = + S-(-P) = + P + S = + P-(-S) + S- P = + S-( + P) = + (- P) + S = + (- P)-(-S)

Here all formulae are affirmations (according to a QC schedule). As Som-

Sommers

55

mers points out, both Aristotelian and modern logicians have a natural inclination to favor affirmation over denial. While a schedule of categoricals which takes all of them to be affirmative is traditionally sound, and easy to understand, a PC schedule is helpful in accounting for singular sentences. Let 'S' in our PC schedule above be a singular term. As we have seen already, 'Socrates is wise' and 'Socrates isn't unwise' are equivalent. Indeed, for singulars the contrast between P­ opposition and C-opposition no longer holds. With 'S' singular, A = I and E = 0. In other words, when 'S' is singular its quantity is wild. A syllogistic alogrithm will countenance both singular and general terms, and nominalized sentences (sentential terms) as well. Since any denial can be converted into an affirmation, we can construct a system which treats all categoricals as positive (affirmative), thus ignoring their predicative quality. A formula representing such a categorical is called elementary. For the purposes of calculation we can safely ignore the brackets on sentential terms. Thus, '-A + B' can be read as either 'All A are B' or 'If A then B'. A syllogism can be written as an equation with the conjunction of premists on the left and the conclusion on the right. A syllogistic inference is valid only if its corresponding equation is true. Barbara, for example, would have the corresponding true equation (-S + M) + (-M + P) = -S + P

Two formulae are called similar if (i) they have the same quantity (a conjunction with one or more particulars is particular, a conjunction of universals is universal), and (ii) they have tht: same extremes. The terms of any two formulae which are not arithmetically eliminable (in the equation above, ' M' is eliminable) are extremes. For a single sentence, its terms are its extremes. For a valid syllogistic inference the two sides of the corre­ sponding equation must be similar. In our Barbara example above 'S' and 'P' are the extremes of both sides of the equation ('M' is a middle term). Both sides of the equation are universal so that the two formulae are simiiar. Any inference whose corresponding equation is true and has similar sides is valid. Weakened syllogisms are admitted by allowing ' + S + S' as a hidden premise. For example, Camestrop:

-P + M-S-M + S + S = + S-P

The crucial test for the efficacy of a syllogistic system 1s its ability to formulate relationals as categoricals, and to account for the validity of inferences involving relationals. Sommers takes relationals to be cate­ goricals with complex predicates. In 'All sophists receive money from some

Three Logicians

56

fools' the subject is 'All sophists' and the predicate is 'receive money from some fools'. Such complex relative predicates consist of a predicate term ('receive . . . from') plus one or more relata, quantified terms ('(some) money', 'some fools'). He would formulate this sentence as

-S + (R + M + F) To show how relationals can fit into Sommers' syllogistic analogue he first draws out what eventually turns out to be the underlying principle of syllogistic. A simple way of stating the principle (Sommers does not state it so simply) is: If to a formula containing ± M we add a formula which has the form + M ± N or ± N + M, we can validly infer a new formula which differs from the first only in that ± N has replaced ± M .

Here ' ± M ' and ' f- M ' have opposite signs of term quality, and ' N ' may be simple or complex. For example, if we add ' + M + N' to '- M + P' (i.e. '-M + P + M + N ') we get, by replacing '-M' by ' + N ', according to the principle, ' + N + P'. So

-M + P + M + N = + N + P which represents a valid inference. Again, if we add '- M + G ' to our relational formula '-S + (R + M + F)', we get '-S + (R + G + F)'. In other words,

-S + (R + M + F) + (-M + G) = -S + (R + G + F) which might represent the valid inference: All sophists receive money from some fools All money is gold Therefore, all sophists receive some gold from some fools

If we add ' + S + S' to '-S + P' we get ' + S + P', i.e. -S + P + S + S = + S + P which might take as an illustration of the syllogistic nature of subalter­ nation - the inference from 'All S is P' to ' Some S is P', which is seen not to be immediate. A careful look at Sommers' syllogistic principle, along with examples, shows that it, in fact, amounts to the traditional dictum de omni et nu/lo. What it says, then, is that Whatever is affirmed of all of something is likewise affirmed of whatever that something is affirmed of.

All syllogistic inferences whose sentences are represented by elementary

Sommers

57

formulae are governed by this principle - including those involving relationals. The familiar inference of 'All who draw circles draw figures' from 'All circles are figures' is not a syllogism, but is an immediate in­ ference. -C + F = -(D + C) + (D + F)

As a practical method for determining validity Sommers' system is hard to beat. It treats all assertions as categoricals (including identity statements, truth functions, and relationals), formulates all categoricals as elementary algebraic formulae, and treats all inference as arithmetical equivalence between formulae. In practice it can easily be used to quickly determine the validity of any inference. The procedure is: ( 1) (2) (3 ) ( 4) (5)

Make explicit all hidden premises, quantifiers, and qualities. Symbolize. Convert all nonelementary formulae to elementary ones. Add the premises arithmetically. Determine whether the sum of the premises equals the conclusion.

Here are a few simple examples which illustrate the system. (a)

Some gods are heroes Some gods aren't mortal Therefore, some heroes are immortal

We symbolize this first as +G+H -( + G + M) + H- M

We next convert the second premise into an elementary formula. +G+H -G-M + H-M

We add the premises to get + H- M, which equals the conclusion. The argument is valid. (b)

All gods love some men All men are fools

Three Logicians

58 Some gods dispise fools So, all gods love some fools -G + (L + M) -M + F + G + (D-F) -G + (L + F)

Here the sum of the premises is + D + L, which does not equal the con­ clusion. The argument is invalid. (c)

If John stays, Mary will leave John will stay So, Mary will leave -[ ± J + S] + [ ± M + L] ±J+S ± M+ L

The argument is valid as long as the wild quantity on J is the same in both premises, the wild quantity on M is the same in the premise and conclusion, and the quantity of the second premise and conclusion are the same. Thus, we could choose the wild quantities to get, for example, -[ + J + S] + [ + M + L ] +J+S

+M+S The sum of the premises equals the conclusion. (d)

Some heroes are faster than some dogs All dogs are faster than some men Therefore, some horses are faster than some men

After adding the suppressed premise that 'faster than' is transitive, we have + H + (F + D) -D + (F + M) -[ - A + (F + B)] + [ - (F + A) + (F + B)] + H + (F + M)

Sommers

59

The argument is valid since the premises add up to the conclusion. (e)

9 = (3.3) 9 = (7 + 2) Therefore, (7 + 2) = (3 .3)

We symbolize this as

-9 + (3 .3) + 9 + (7 + 2)

Remember here that (i) each term is singular, and, so, has wild quantity, (ii) the ' + ' of '(7 + 2)' is the mathematical plus not our logical plus. Since the sum of the premises equals the conclusion the argument is valid. We have merely summarized above the main elements of the new syl­ logistic as presented in "The Calculus of Terms". Yet even a brief sketch should show how far Sommers has come in the Leibnizian quest for a universal syllogistic calculus. In the process Sommers has found additional reasons for establishing an extended and universal syllogistic. In compari­ son with the syntax of modern mathematical logic, the syntax of his syl­ logistic is simple and natural. "The Calculus of Terms" builds the new syllogistic on the foundation laid down in "On a Fregean Dogma", "Do We N eed Identity?", and "On Concepts of Truth in Natural Languages". The last four papers in Som­ mers' logical series attempt to defend, clarify, and refine that system without any extensive additions. In "On a Fregean Dogma" and "The Calculus of Terms" Sommers had argued that in the syllogistic both uni­ versal and particular sentences could be viewed as being neutral as to existence - that if existence is to be attributed to anything it must be done so by an overt predication of an existence predicate. This is contrary to the view held by the modern mathematical logician. On that view existence is part of the sense of the quantifiers. The existential quantifier, ·3 . . . ', is now read by the majority of logicians as 'there exists at least one . . . such that'. In "Existence and Predication" Sommers raises the question of whether 'exists' is a logical or extralogical expression. His general argument is that what distinguishes logical (formative, syncategorematic) expressions from extralogical expressions is the unique oppositional character of formatives. 'Exist' does not exhibit this characteristic. So existence is not a syntactic, or logical, feature, but is a property which can be ascribed only by the use of

+ (7 + 2) + (3.3)

60

Three Logicians

an explicit existence predicate. 9 The most important result of this enter­ prise for us is that Sommers, in showing that existence is not syntactic, is able to demonstrate that the syllogistic as devised by him permits a uniform account of all logical expressions. All logical expressions are oppositional pairs. This contrast markedly with the wide array of heterogeneous logical devices wielded by the contemporary mathematical logician. A less important, but no less interesting feature of this paper is that here, for the first time, Sommers takes note of the precedence of Leibniz. In "The Logical and the Extra-Logical" Sommers repeats his argument that not only are the logical formatives of categoricals to be treated oppositionally, but the truth-functional connectives and identity are to be so treated also. Thus, all formatives have a uniform oppositional character. An important feature of traditional logic, after Aristotle, was its ad­ herence to the doctrine of distribution. According to this well-known doctrine, every term has a "distribution value" (is either distributed or undistributed), and every valid inference must satisfy the rules of dis­ tribution (the middle term of a syllogism must be distributed at least once, a term distribution/undistributed in the conclusion must be distributed/ undistributed in the premises). A term is distributed if and only if it refers to whatever it denotes. In Reference and Generality P. Geach 1 0 denounces this doctrine on the grounds that it rests on the confused distinction be­ tween referring and denoting. In "Distribution Matters" Sommers offers a defense of the doctrine of distribution, and shows that the reference/ denotation distinction is well-made. That distinction is made clear by a prior distinction between terms and logical subjects. While terms are syn­ tactically simple, logical subjects are syntactically complex, consisting of a term plus a quantifier. Logical subjects refer, while terms denote. Thus, 'all men' refers to all men, 'some men' refers to some men, and 'men' denotes all men. In "All men are rational' the term 'men' is distributed since the subject, 'all men' refers to what 'men' denotes (viz. all men). According to the traditional doctrine, a term, 'P', is distributed in a sentence whenever that sentence logically entails a sentence of the form 'All P . . . '. Modern logicians discount the notion of distribution altogether. One � Sommers goes on to argue that, while 'exists' is a predicate. it differs from all other predicates in that it has no logical contrary. One consequence of this is that he is committed to a thesis common to the first order predicate calculus with identity. viz. that everything exists. Sommers' insistence that 'exists' is a special predicate is the direct result of his failure to keep the distinction between 'exists' and · /exists/ ' clearly in mind. One can reject the theory of ·exists' as a unique predicate without in any way impeaching Sommers' general logical theory. I have done this in "Sommers on the Predicate 'Exists' " , Philosophical Studies, 26 ( 1974). 10 ( Ithaca, N . Y. 1962). chapter one.

61

Sommers

reason for this is that 'P' in ' . . . are not P' is traditionally distributed. Now we can infer from ' . . . are not P' (whether our subject is 'all S' or 'some S') that a certain S is not P; i.e. we can instantiate to get 'a is not P'. From this we can derive a sentence of the form 'All P are not a'. But, to do so we must allow two things: (i) that singulars such as 'a' have implicit quantity (in this case universal - '(all) a is not P'), and (ii) that singular terms can occupy predicate position, as in 'All P is not a'. The doctrine of distribution accords with any logic which, like the syllogistic, but unlike modern mathematical logic, ( 1) discerns reference from denotation, (2) admits quantification of singular terms, and (3) allows singular terms to be predicate terms. Sommers goes on in "Distribution Matters" to show that his plus-minus calculus provides a ready way to determine the distribution value of any term. The distribution value of a term is determined by its arithmetical value in any formula. For example, in ( 1 ) -(-S + P)

the value of S is plus and the value of P is minus. In (2) -S + (R + B-C) S is minus, as is C, while R and B are plus. And in (3) -(S + M) + (R + B) S and M are minus in value and R and B are plus. A term with minus value is distribute; a term with plus value is undistributed. Sommers' formulates two conditions, which together are necessary and sufficient for inference validity. ( i ) The conclusion and all premises are universal, or the conclusion and exactly one premise are particular. (ii) Each extreme term has the same value in the premises and the conclusion, and each middle term has opposite values in its two occurances.

A close look at (i) and (ii) reveals that they amount to the condition on a valid inference that the premises arithmetically equal the conclusion. Sommers shows that this amounts to the dictum de omni et nu/lo, which he formulates first as D

What is true of every X is true of what is X.

and then as D * A term that applies to every M may replace any undistributed occurence of M.

62

Three Logicians

No new ground is broken in Sommers' most recent paper in this series. However, in "Logical Syntax in Natural Language" he does give for the first time a general survey of the new syllogistic. In doing so he points out more clearly than ever before that the syntax of natural language is (when sufficiently abstracted, generalized, and simplified) determined by the logical formation rules of a logic of terms. 1 1 Aristotle saw no radical difference between natural and logical syntax. Leibniz sought to give an arithmetical alogrithm which would reflect natural syntax. The syntax of modern mathematical logic, on the other hand, is radically different from the syntax of natural language. Following Frege, it rejects the subject­ predicate form (in favor of functions and arguments), and takes the logic of unanalyzed propositions as prior to, more basic, than the logic of analyzed sentences. Modern logicians have justified their choice of nonnatural syntax by its results. The inference power of the first order predicate calculus, coupled with the identity calculus, is, indeed , impressive. Yet we believe Sommers, following in the tradition of Aristotle and Leibniz, has provided a calculus which is equally powerful for the purposes of in­ ference, and which is at the same time simpler and, most importantly, more natural. Our wonder and respect for Sommers' accomplishments notwithstand­ ing, we believe more work needs to be done before the new syllogistic is completely defensible. In the second part of this essay we will summarize certain parts of the system and suggest ways in which it can be improved or extended. We will, in particular, attempt to say more about the theory of logical syntax required by the new syllogistic. Sommers has suggested in several places that it is this logical syntax which ought to be the object sought for by modern structural linguistics. We will offer an account of logical syntax which more clearly reveals its suitability as the underlying "deep grammar" of natural language. In "The Calculus of Terms" Som­ mers has said that his calculus requires no inference rules. 1 2 We believe this is a source of confusion in the calculus and so will lay down several rules of immediate inference and a single rule (which amounts to the dictum de omni et nu/lo) of syllogistic inference. In that regard. we will reformulate Sommers' version of the dictum in order to remove the notion of truth imported into it by the phrase �true of. We will also attempt to improve I have given a brief appraisal of " Logical Syntax in Natural Language" in my review of Issues in the Philosophy of Language by A. MacKay and D. Merrill in Philosophical Studies ( I re), 25 (1977). 1 2 On page 4 of 'The Calculus of Terms" he says "There are no rules of inference". Yet he goes on, pp.23-25, to offer what amount to rules of immediate inference. 11

Sommers

63

Sommers' account of the syntax of relationals. He, along with most con­ temporary logicians, says that a urelational sentence has two or more subjects". 1 3 However, that contention obliterates the categorical form of relationals. Since a syllogistic must treat all assertions as logically cate­ gorical, and since all categoricals have exactly one subject and exactly one predicate, Sommers' account of the logical form of relationals must be rejected or revised. We will attempt to revise it by enforcing a distinction between logical subjects and logical objects. This will permit a categorical reading of relationals while not ignoring the fact that more than one reference is made in such sentences. There is one final revision of Sommers' program which we will note here. In hi� attempt to axiomatize Aristotle's syllogistic, Lukasiewicz 14 took the Law of Identity as: (a) All A are A (b) Some A are A (a) and (b) are taken as axiomatic in Aristotle's system. Sommers accepts the axiomatic status for (a) ; sentences of that form are tautologies. But he rejects the axiomatic status of (b). He admits that such sentences may be used as suppressed premises (especially for weakened syllogisms), but, while they are true, they are not tautologeous. Sommers argument is this. 1 5 1. 2. 3.

Some A and B is A and B Some A and B is B Some A is B

assumption from I from 2

Since 3 is not a logical truth, what it is derived from 1 , cannot be a logical truth. But I is an instance of (b). So (b) is not a logical truth. However, the argumen t will not work. While it is easy to justify the move from I to 2, there is no justification at all for the move from 2 to 3. For example, we can say that something which is square and round is round, but not that some square is round. There seems to be no sound reason against taking (b) as a logical truth. Since, from both the traditional and Sommersian points of view, ·some' carries no existential import� there could never be a false instance of a sentence having the form "Some A are A'. Consequently, in the next section we will take both (a) and (b) as tautologies. 13 " Logica l Syntax in N atural La nguage", p.23 . 1-l

A ris10tle 's Syllogistic from the Standpoin t of Modern Formal Logic. p . 4 5 .

15 See "Distribution Matte rs". See also my "N otes on the N ew Syl logistic", Logique et A nalyse, forthcoming.

The Syllogistic

67

Contemporary Mathematical Logic

Principia Mathematica was published by Russell and Whitehead from 1 9 1 0 to 1 9 1 3 . It i s the great synthesis of the developments which took place in logic during the last half of the N ineteenth Century. Principia is the "bible" of contemporary mathematical logic. Notice how brief the period of tran­ sition was -- about sixty-five years. After more than two-thousand years of syllogistic logic it was virtually totally replaced by mathematical logic, and it only took sixty-five years! It will be instructive to look, even briefly, at the highlights of this development before offering a sketch of its present day logical theory. The most accepted date for the beginning of the revolution is 1 847 . In that year Boole published his Mathematical Analysis of Logic. 2 Following earlier suggestions by Hamilton and DeMorgan that the predicate (as well as the subject) of a categorical has quantity, Boole conceived of such sentences as equations. By taking this view of categoricals, the extensional interpretation was inevitable. 'All men are bipeds' becomes 'All men are some bipeds', which is read, extensionally, as 'The class of men is part of the class of bipeds'. Letting 'x' be 'the class of men' and 'y' be the class of bipeds' Boole would formulate the sentence by the equation 'x-y = O'. A few years later, in 1 854, Boole published The Laws of Thought. 3 Here he attempted to show how, given his "algebra of logic", a theory of classes (as well as a theory of truth-functions) can be derived using the standard notions of algebra. Boole clearly believed that in deducing conclusions in logic it was proper to use notions and truths which are mathematically sensible, even though they seem to have no logical sense. It is interesting to note that Boole, like Leibniz, had envisioned a common calculus for both truth-functions and categoricals. This vision does not seem to have been shared by those who followed him. (Cambridge-. 1910-1913). G. Boole, The Mathematical A nalysis of Logic, being an essay towards a calculus of deductive reasoning (Cambridge, 1847). 1

2

A n In vestigation of the Laws of Thought, on which are founded the Mathematical Theory of Logic and Probabilities ( London, 1854 ).

3

68

Three Logicians

The development of mathematical logic in the first twenty years after Boole's The Laws of Thought was dominated by attempts, particularly by Jevons, 4 to purify Boole's algebra of logic. It was hoped that an algebra of logic could be built along Boole's lines, but without the great dependence upon strictly mathematical notions which he had permitted. Later Peirce strengthened and extended this logic by the introduction of the notion of class inclusion, a theory of relations, a refined theory of truth-functions, and the notion of quantifiers. 5 Peirce's great achievements are over-shadowed, however, by the work of Frege. He turned the algebra of logic into contemporary mathematical logic. Frege's greatest original contribution to mathematical logic, and the one that in the end distinguishes mathematical from syllogistic logic, was his dismissal of subjects and predicates from logic. As a mathematician he had already contributed to the clarification of the mathematician's notion of a function. In the Begriffsschrift6 and the Grundgesetze 7 he replaced the old subject-predicate distinction with the mathematician's function-argu­ ment distinction. One result has been a radically different theory of logical syntax. While in syllogistic any term may either refer (when, say. it is a subject term) or characterize (for example, when it is a predicate term), in Frege's logic the two roles are assigned to quite different types of logical entities. Logical functions (now predicate variables) alone do the job of characterizing and logical arguments (now individual variables or names) alone do the job of referring. The shift here, from homogeneous terms (able to either refer or characterize) to referring arguments and characterizing functions, is not without its far-reaching philosophical consequences. In our day W.V.O. Quine8 has reinforced the "gulf between meaning and naming" and argued that we are ontologically committed only to those things we allow to be in the range of terms in "purely referential" positions - names, individual variables. 9 4

W . S . Jevons, Pure Logic, or the Logic of Quality apart from Quantity ( London. l 864). and The Principles of Science. a Treatise on Logic and Scientific Met hod ( London. 1 874). 5 He did this in a variety of places. See C.S. Peirce. Collected Works. 8 vols . . ed. C H artshorne.

P. Weiss, and AW. Burks (Cambridge . M ass .. l 93 1 - 1 95 8). G . Frege, Begriffsschrift. eine der arithmetischen nachgebildete Formelsprache des reinen Denkens ( H alle, 1 879). 7 Die Grundgesetze der A rithmetik, begrif fsschr�ftlich abgeleitet. vol i (Jena. 1 893) vol.ii (Jena. 1 903 ). See also Translations from the Philosophical Writings of GoHlob Frege. ed. P. G each and M. Black ( Oxford , 1 952). 8 See "On What There Is", From a Logical Point of View ( N ew York. 1 96 1 ). 9 _ 'W_ e w � ll argue that all terms have both a denotation and a connotation. By failing to d1stmgmsh the denotation of term from its role as a referring term (e.g. when it is a subject 6

Contemporary Mathematical Logic

69

Frege conceived of logic as a new foundation for mathematics (a view inherited by Russell and his followers). Boole had talked of logic as a branch of mathematics, a new branch of algebra. He and his followers thought of it on a par with the other new branches of mathematics developed in the early Nineteenth Century (e.g. vector algebra, matrix algebra, abstract algebra, and Lobachevsky's geometry). Today the question of the proper relation between logic and mathematics is rarely raised. The new logic has been overwhelmingly successful. When one has risen to the top so quickly it is perhaps best not to worry too much about just how one got there. The most obvious difference between logic today and any logic before the last part of the Nineteenth Century is its very obvious symbolic complexity. Traditional logic was, if we ignore the problems of giving a general interpretation of logic and of providing an adequate semantics, fairly simple looking. Aristotle's logic was symbolic, so was the logic of the Stoa, and the scholastics' logic as well. But we tend to refer only to con­ temporary mathematical logic as symbolic logic because it employs so many more kinds of symbols than did the old logics. Contemporary logic is completely symbolized. Every item dealt with is in symbolic form. Why every item? And why so many kinds of symbols? The older logicians began with sentences and arguments expressed in some natural language - Greek, Latin, English, etc. They used symbols to replace the material components of the sentences under examination so that the formal components could be more easily displayed. In other words, symbolization was simply the way to formalize a sentence. Con­ temporary logicians have come to believe that, in a sense, a system of symbolic notation is itself a language. It is not a natural language like Greek or English, but it is a language. It has a vocabulary (which happens to be quite small), and a grammar (theory of syntax). It is special because it is in some way representative of what is common to all languages in general. It is, so to speak, the hidden skeleton within any natural language. It is the underlying logic of language in general. So it is a language, but not term) nominalists, like Quine, chose to enforce the gulf between referring and characterizing so that there would be no danger of general or abstract terms referring. See Quine's Word and Object (Cambridge, M ass. , 1 960), chapter 7. The need to assign the tasks of referring and characterizing of different logical entities harkens back to Russell's notion of a genuine name, in "On Denoting", Mind 14 ( 1 905 ). A genuine name has no descriptive content (cannot charac­ terize, but can-only-refer). In a pu rportedly logically correct language all referring terms would be genuine names (having no descriptive content) and a ll characterizing terms would be 'genuine predicates' (having only descriptive content).

70

Three Logicians

quite like any other language. It is a "logically corrected language" - a "regimented language". It is ordinary civilian language put into uniform , filed in ranks. While the traditional logicians revealed the logical forms of sentences by symbolizing their material components, the contemporary logician does so by translating his sentences from their native language into his formal language. S in ce the new language is completely symbolic, all the elements, material and formal, of a sen tence are symbolized by this process. There are so many more symbols in contemporary logic than there are in traditional logics not only because contemporary logicians require the formal as well as the material components of sentences to be symbolized, but also because they claim to recognize so many more kinds of formal elements. Consequen tly, contemporary logic is really a complex of con­ stituent logics, of calculi. The sentential calculus is a logic of sentences which are not analyzed into their constituent terms, but which may be analyzed into constituen t subsentences. It is claimed as the basic logic, and is a direct descendent from the logic of the Stoics. First order predicate calculus is a logic of sen tences analyzed into their constituent terms. We will see how it has come by the names "first order" and ''predicate" below. The identity calculus, or identity theory, is a logic of sentences which are used to affirm the identity or nonidentity of individuals. The complex of these three calculi is usually referred to as "first order predicate logic. with identity". It is this logic which we are calling "contemporary logic" or "contemporary mathematical logic". Contemporary logicians claim that while the validity of some argu ments depends upon the arrangement of terms which constitute their premises and conclusions, there are others whose validity is determined by a shallower analysis of premises and conclusions. Consider the argument If Socrates was not wise. then Plato was a fool. Socrates was wise. So Plato was not a fool.

Predicate calculus takes the matter of such arguments to be entire sen­ tences. The form is then determined by the ways in which the sentences are combined to form more complex sentences. Thus. the two sentences 'Socrates was not wise' and 'Plato was a fool' are combined by the words 'if and 'then' to form a more complex sentence. A sentence formed from one or more other sentences is called a function of those subsentences, and is formed from them by a sententialfunctor. The words 'if and 'then' together constitute one such functor. A sentence which is not a function of any further subsentence is called atomic. It contains no sentential functor. In our argument above we find the two sentences ' Socrates was a fool' and

Contemporary Mathematical Logic

71

'Socrates was not a fool'. The second is said to be the sentential negation of the first. It is a function of the first one and negation (indicated in this case by the English 'not') is a sentential functor. The sentential calculus usually recognizes a small number of sentential functors. The four usual ones are: (i) negation, which in English is displayed by phrases such as 'not', 'it is not the case that', 'it is false that', etc., (ii) conjunction5 found in English as 'and', 'but', etc., (iii) disjunction, found in 'or' and 'either . . . or', and (iv) conditionalization, exhibited by such English combinations as 'if . . . then', 'only if, 'if etc. A variety of symbols are employed for these functors. The following are reasonably standard. Negation: ,._, conjunction: /'.. or •, disjunction: v, and conditionalization: ::J . Subsentences, taken as the material components of sentential functions, are usually translated in the sentential calculus by lower case letter. Our agument would be symbolized as ,._, s ::J p s

Therefore: ,._, p Notice that this translation reveals the argument-form in terms of sen­ tences alone. No indication is given of the terms of those sentences. The decision procedure for the sentential calculus (the mechanism for determining the validity or invalidity of an inference) is based upon the definitions of the sentential functors. Contemporary mathematical logic takes any sentence to have exactly one of two possible truth-values: true or false. The truth-value of a sentential function is completely determined by the truth-values of its constituent atomic subsentences. The sentential functors are defined operationally (i.e. by the manner in which they determine the truth-values of sentential functions in which they occur from the truth-values of their subsentences). For example, a sentential negation is true when its subsentence is false, and false when its subsentence is true. A conjunction is true only when both of its subsentences are true. A conditional is false only when its first subsentence is true and its second one is false. An assignment of a truth-value to each atomic subsentence of a senten­ tial function is an interpretation of that sentence. An assignment of a truth-value to each subsentence of an argument is an interpretation of that argument. An argument is said to be valid just in case no interpretation of it makes all the premises true and the conclusion false. Our argument is invalid since the two premises are both true while the conclusion is false when both atomic sentences, s and p, are interpreted as having the truth­ value true.

Three Logicians

72

The conclusion of a valid argument can be deduced from its premises by constructing a formal proof, or deduction. A proof is a list of sentential functions consisting of the premises followed by one or more further sentences, the last of which is the conclusion. Each sentence after the premises is justified by one or more of the rules of sentential calculus. These rules are usually taken from a relatively small list of simple valid inference forms. While sentential calculus is a logic of sentences which are not analyzed into terms, the first order predicate calculus is a logic of analyzed sentences. It is obvious that such a logic is required since there are arguments whose validity or invalidity can be determined only after an analysis deeper than that provided by sentential calculus. Here is an example of such an argu­ ment. All philosophers are simple men. All simple men are wise. Therefore, all philosophers are wise.

I f we translate this using only the syntax provided by sentential logic, we will get the following as the logical form of this argument.

p q

Therefore : r

If this were the form of the argument it would be formally invalid (since there is an interpretation which renders the premises all true and the conclusion false). Since the argument is, however, obviously valid, a more sophisticated analysis is in order so that its validity can be revealed in its form. The contemporary logician recognizes a radical logical difference be­ tween two kinds of terms: general and singular. S ingular terms are those used to refer to individual things and persons. General terms do not refer at all. General terms are used to qualify, or characterize, individuals. Singular terms are the logical subj ects of sentences, while general terms are their logical predicates. From the logical point of view being discussed here, every singular term is a logical subject and every logical subj ect is a singular term. Likewise, every general term is a logical predicate and every logical predicate is a general term. The phrases "logical subj ect" and "logical predicate" are used to dis­ tinguish them from the grammatical subj ects and predicates which a sen­ tence might have. For the contemporary logician is convinced that what­ ever grammatical from a sentence might have, its genuine logical form is

Contemporary Mathematical Logic

73

something else - something usually so hidden that only a complete trans­ lation of the sentence into the formal language can reveal it. The "verbal contortions" required to translate a given sentence logically are often quite formid able. Consider the first premise of our argument above ('All philosophers are simple men'). Its grammatical subject is 'philosophers' and its grammatical predicate is 'simple men'. But, as we have seen, for the contemporary mathematical logician, logical subjects, unlike grammatical subjects, must be singular (e.g. pronouns or proper names). Since a sentence such as this one h as neither, it must be translated into one which does. Thus: 'Every thing is such that if it is a philosopher then it is a simple man'. Here, in this translation, there are two atomic subsentences: 'it is a philosopher' and 'it is a simple man'. Each has an appropriate singular term as its subject (viz. 'it') and a general term as its predicate. Moreover, the two are subsentences for the conditional sentence function 'if it is a philosopher then it is a simple man'. Finally, the logical subject of each is governed by an initial phrase 'everything is such that', which is a quantifier. The most important and obvious feature of the contemporary logician's translation of the sentences such as the one above is that all the terms of the original sentence (grammatical subjects and grammatical predicates) are taken as logical predicates (thus the "predicate" of "first order predicate calculus"). The new subsentences are given contrived logical subjects, namely the pronoun 'it'. But, while it was clear what the grammatical subject ('philosophers') of the original sentence was supposed to refer to, what 'it' refers to is not so obvious. The logician accounts for this by claiming that any interpretation of such a sentence must include a specifi­ cation of a domain of discourse. This is any set, often arbitrary, of in­ dividuals. It is what the sentence is about (logically). The pronouns of the translation are said to refer to items in the domain. Whether all or only some of the members of the domain are being referred to by each token of 'it' is determined by the quantifier involved. In our example the quantifier is 'everything is such that' and the 'things' are all members of the domain. The full symbolic translation of our sentence is easily achieved once it has been reparsed as above. Pronouns are translated by individual variables ("individual" because they are logical subjects, which must be singular; and "variable" because their referents - items in the domain - may change from one context to another). Individual variables are usually lower case letters towards the end of the alphabet. The logical predicates are translated by predicate variables, appropriate upper case letters. Since any quantifier must indicate the individual variables over which it governs,

Three Logicians

74

or ranges, quantifier symbols include mention of the appropriate in­ dividual variable type. 'Everything is such that' is translated, usually by '('v . . . )', where the blank is filled - in with an appropriate individual variable. This is a universal quantifier. 'Something is such that' is translated by '(3 . . . )', and is a particular, or existential, quantifier. Consider again our sentence 'All philosophers are simple men'. It is initially reformed as 'Everything is such that if it is a philosopher then it is a simple man'. Using ' x ' as our individual variable, 'P' for the predicate 'philosopher', and 'S' for 'simple man', our two subsentences are now translated as 'P x ' and 'S x ' (' x is P' and ' x is S'). The same individual variable is the subject of each since the same reference is being made in each case. The individual variables are governed by a universal quantifier, and the two atomic subsentences form a conditional sentential function. ('v X ) (P X ::J S X ) Since only the individual variables are quantified, i.e. governed by a quantifier, the calculus is called "first order". A "second order" calculus would allow quantification of predicate variables as well. In the first order predicate calculus each general term is taken as a logical predicate, and each predicate renders an atomic sentence. In our example above, we began with one sentence containing two general terms. Each term was taken as a logical predicate so that we arrive at a new sentence containing two atomic subsentences, each corresponding to a general term. This logic insists upon a one-one correspondence between logical predicates and atomic sentences. However, no such correspondence is admitted for logical subjects and atomic sentences. Any atomic sentence has exactly one logical predicate. but it has one or more logical subjects. Atomic sentences with more than one logical subject are relational sen­ tences and their predicates are relational predicates. As an example look at 'Some men are lovers of all women'. This can be translated as: '(3 X ) (M X •('vy) (Wy ::J Lxy))', which might be read as: 'Something is such that it is a man and everything is such that if it is a woman it loves it.' All the 'its' are properly sorted out by the different individual variables. The important point to notice is that one of the atomic subsentences of a relational must have the relational predicate as its logical predicate, and that atomic sentences with such predicates must have more than one logical subject. Thus, we have not 'L x ', nor 'Ly', but 'Lxy'. A proof of a valid argument using first order predicate calculus makes use of the rules of the sentential calculus plus some new rules governing quantifiers. These are mainly rules for eliminating and for adding

Contemporary Mathematical Logic

75

quantifiers. Generally, in such proofs, the premises are listed, quantifiers removed from them (where permitted by the new rules), the rules of sentential calculus are applied then to deduce a function which, when appropriate quantifiers are added to it, is the conclusion. The rules of sentential logic are applicable here since once the quantifiers of a sentence are removed what remains is either an atomic sentence or sentential func­ tion. Thus, it is clear that the sentential calculus is an essential part of the first order predicate calculus. The first order predicate calculus depends upon, presupposes, the sentential calculus. It is for this reason that con­ temporary logicians claim that the sentential calculus is the basic logic. Consider now the two sentences. ( 1 ) Tully is Roman (2) Tully is Cicero Since 'Tully' is a proper name it is not a general term and, hence, not a logical predicate. It is translated as an individual constant ("constant" rather than "variable" because it always refers to the same individual). The logical predicate of (1) is the general term 'Roman'. So (1) is translated as ( 1 . 1 ) Rt Since 't' is not a variable no quantifier is required to indicate the extent of its reference over the domain of discourse. But, now, what about (2)? Both terms of (2) are singular. There is no apparent general term. Yet there is supposed to be a one-one correspondence between general terms (logical predicates) and atomic sentences. The contemporary logician solves the problem here by proclaiming that the 'is' of ( 1 ) and the 'is' of (2) are different in meaning. When followed by a general term, as in ( 1 ) 'is' is simply a sign that the sentence is not negated. But when followed by a singular term, as in (2), it is itself a special kind of relational predicate identity. The claim is that (2) is an identity sentence while ( 1 ) is a simple predication. Sentence (2) has two subjects, 'Tully' and 'Cicero', and the logical predicate is 'is', or 'is identical with'. Identity is a special relatimn because it poses special problems for interpretation. It is even translated by a special symbol. (2) is translated as (2. 1 )

t=C

Proofs of valid argumen ts involving identity sentences are facilitated by adding one or two simple "rules of identity" to those of the first order predicate calculus. We might think, then, of the identity calculus as a

76

Three Logicians

special appendage to the first order predicate calculus. This brief sketch of contemporary mathematical logic is, of course, far from a complete picture. Many important features of the logic have been blurred over or entirely ignored. Nevertheless, it should serve to illustrate the most basic and general characteristics of that logic. In our examination of the syllogistic logic which follows we will make critical remarks con­ cerning contemporary mathematical logic as presented in our sketch of it here.

77

Syllogistic Logic

The analyzed atomic sentences of first order predicate calculus are cate­ goricals (having one logical predicate and one logical subject), or relationals (having one logical predicate but more than one logical sub­ ject). Logical predicates are always general terms and logical subjects are always singular terms. The difference, according to the contemporary logician, between general terms and singular terms is that singular terms refer while general terms do not. The "Fregean dogma" demands that all reference must be to individuals. The logical subject of any sentence must refer. Individual reference can only be made by singular terms. Therefore, the logical subject of any sentence, whatever its grammatical subject might be, must be a singular term. A logic of terms allows any term, singular or general, to occur as either a logical subject or a logical predicate. Thus, for example, relationals and identities are categoricals. Let us say that a term is any word or phrase (many-worded term), whether singular or plural, active or passive, con­ crete or abstract, which can be used to refer or to characterize. Words and phrases such as 'man', 'dog', 'Socrates', 'happy', 'red', 'ran', 'j ustice', 'the man in the moon', 'under the bed', 'unmarried', and 'happily married', are terms. But 'some', 'no', 'if, 'but', 'are', 'not', 'the', 'of, and 'and' are not. These words ( called functors) are used either to form terms from words 1 complex terms from simpler terms, parts of sentences from terms, or sentences from parts of sentences. At this point it is important to recognize the radical logical difference between terms and functors. It is the dif­ ference found in the contemporary logic between logical variables and logical constants. More helpfully, it is the difference found in medieval logic between categorematic and syncategorematic words. 1 Every term has both a denotation and a connotation. We will take the denotation of a term to consist of all the things to which that term applies (i.e. its extension). A term applies to a thing when it can See the discussion of this distinction i n P. Boeh ner. Medieval Logic ( M a nchester, 1 952), pp. 1 9-26.

1

78

Three Logicians

be used in referring or in characterizing that thing. Thus, 'philosopher' denotes Socrates, Plato, Aristotle, etc., 'red' denotes boiled lobsters, the tie I am now wearing, Mars, etc. ; 'Socrates' denotes Socrates; 'even' denotes 2, 4, 6, the number of ears on my head, etc.; and '2' denotes 2. We take the connotation of a term to consist of the properties which a thing has in virtue of which it is denoted by that term (i.e. its sense or meaning). Thus, 'bachelor' connotes being male, adult, and unmarried; 'red' connotes something like appearing such-and-such a way under con­ ditions thus-and-so; 'Socrates' connotes being a man who lived at a 4. T�s v A v tAA v t 5 . A�R A < . . . 0�> 6. o�qtAT 7. R�qlAt

Note that these rules clearly show how any s is logically analyzable into just qt's, ql's, and t's. Also, by allowing n in rules 3 and 5 to be zero we can take monadic adjectives and predicate as special cases of relationals. Rule 4 permits sentence embedding. These rules determine the underlying logical structure "trees" for any sentence. Nonrelational sentences have the general elementary form (where a line indicates that what is at the bottom of the line is a constituent of what is at the top) :

I discuss this more fully in "Reference and Denotation", Philosophical Studies ( Ire), forthcoming.

3

Three Logicians

86 Relational sentences have the general form: ------ s -------

qt

S ----- ------

T

R

p ----- ------

R

(. · • • · · 0 n )

Indeed, given our seven rules, and their recursive character, any sentence can be generated by the following tree:

!\ /\ t

qi

t

t

qi

A

R')

T

qt

on

qt

/\

T

Finally, note that rule 4 permits adjectives to stand in the position of modified terms. Now when such an adjective is monadic it is just a modified term. Thus 'all men are mortal' is logically indiscernable from 'whatever are men are mortal', where the term 'men' has been replaced, in effect, by the adjective 'are men'. /s�

s

\ qt/ / \ all

men

are

mortal

�11

(whatever)

ql

l

are

ti

men

/ "" qi t p

I

are

mortal

87

Syllo gistic Lo gic

Rep lacing terms by mond aic adjectives, then, is logica lly excessive. How­ ever, the replacement of terms by relations is logically impor tant. It allows us to make reference to things having some relation. For example, we can say that whatever is bigger than the sun is bigger than the moon.

/

0

p------

/\ / \ t

qI

are (is)

are

all/ some (the)

bigger than

sun

qt

t

moons all/ some (the)

bigger than

Let us look at a few more examples to illustrate our theory.

1.

s

/s� qt

all

t

men

are

mortal

Three Logicians

88 s

2.

/ � t ql p

s

� �

qt

t

Socrates

(all/ some)

wise

lS

�s�

3.

qt

t

q( (all/ some)

/, \0 RI '

/ p ----------

-------- s ---....._____

I

Plato

R\

t

(did)

I I\

qt

I

I teach

o f)

(all/ some)

t

qt

t

(some)

Aristotle metaphysics 4.

s

s

qt

(all/some)

� � t ql p

/ � t

Tully

IS

Cicero

Syllogistic Logic 5.

s

s

� �

qt

t

all

I

6.

� � t qI p

T\

phi losophers q l

� all

89

A

/ \t I

(who) are

dead

�

s

q t � "'

respected

are

s

/ "'t " A I / � men qi

I

qI

T

/ ""'

(who) (do)

t

I

read

I are

fools

/ """"

qt

o Rl

I (some)

/ � T

A

I R( \ R' l boo / � "'- t

ql

I

(which) are

I

"" ls foo qt I written \ by (some) t

Three Logicians

90 7.

s

qt/ � t

I

I

(all/some) N arcissue

8.

qi

love

(did)

Narcissus (all/some)

s

� �p

�s"

all

1 phy s icians

R//

qi / � t

I

(do)

I

�0 I � R

s

qt

(all/ some

know

/ \

/\

;\

qt

t

ql

all

men

are

t

I I I l

mortal

Notice here that the embedded sentence ('all men are mortal') is what is known by all physicians. It is the logical object. To be more accurate, it is what that sentence can be used to assert (viz. that all men are mortal),

Syllogistic Logic

91

which, since i t is a singular (a single state of affairs) has arbitrary quantity, is the logical object. We could safely read the arbitrary quantity of such embedded sentences as 'that'.

9.

s

are

s (who) (do)

think (all/ some) (that)

s

/\

l l \

all money , / 9

I

1s

1

I

evil

unhappy

Three Logicians

92 1 0.

all (i f)

s

[P] p

are (then)

Here we use Sommers' notation for exhibiting nominalization.

We saw earlier that the modern logician's insistence on a wide variety of logical forms is reflected in his requirements of a large number of devices for exhibiting such forms: predicate variables, proper names, individual variables (pronouns), identity, quantifiers, sentential negation, and one or more binary sentential connectives. In contrast the new syllogistic makes use of a relatively small numb�r of devices : terms, quantifiers, qualifiers, and term-forming functors such as term negation and sentence nominalization.

It would be expected that any simplification of syntax would have as its most important pay-off a simplification in proof theory. In fact, the development of the new syllogictic by Sommers over the past few years substantiates this. However, the logic of complex terms, which as Leibniz saw, must be essential to a general logic of terms such as the new syllogistic, has yet to be adequately explored. What is now required is an examination of how complex terms are formed by negation, conjunction, and the like. For a proof theory must account for inferences such as 'All men are mortal. Therefore, all wise men are mortal', which depends solely upon the internal structure of the complex term 'wise men' (in this case, a term plus an adj ective).

Syllogistic Logic

93

In the account of our proof theory which follows we will make use of the system of symbolization which Sommers has developed. This system admirably reflects the simple, uniform syntactic interpretation for all sen­ tences. All compound terms will be transposed into conjunctive terms. Thus 'tall or short' will be read as 'non (nontall and nonshort)'. For what is tall or short fails to be neither tall nor short. However, not all many-worded terms are compoun d terms. Often what is grammatically a compound term is logically simple. For example, 'fake money' does not denote money which is fake. So while 'money and rare' is a compound term, and 'green money' is a modified term (i.e. 'money which is green'), 'money', 'green', 'fake', 'rare ', and 'fake money' are simple terms. Let us look b riefly at some examples of our symbolization. We will symbolize the ten sentences which we analyzed syntatically above. Note, that, with Sommers, we can often drop some of our plus and minus signs with impunity. l . All men are mortal. M + R 2 . Socrates is wise. ± s + w 3 . Plato taught Aristotle metaphysics. ± P + T ± A + M 4. Tully is Cicero. ± T + C 5 . All philosophers who are dead are respected. (P + D) + R 6. All men who read books written by fools are fools. (M + R + B + W + F) + F 7. Narcissus loved Narcissus. + N + (L ± N) 8. All physicians know that all men are mortal. P + (K ± [- M + R]) 9. Some men who think that all money is evil are unhappy. + (M + T ± [- N + E]) + -H 1 0. If p then q. [p] + [q]

We add to these : 1 1 . Socrates is unperplexed. + S + (-P)

94

Three Logicians

12. Socrates is not perplexed. p ± s 13. Jones is tall and thin. ± J + (T + H) 14. Whatever is bigger than the sun is bigger than the moon. ( + B ± S) + (B ± M) 15. All geometrical figures are two demensional or three. ( F + G) + (-(-T + H) 16. Zeno is happy if puzzled. ± Z + (-(P + -H) 1 7. If Zeno is puzzled then he is happy. -[ ± Zi + P ] + [i + H] Here we use Sommers device of indexing terms for later pronominal replacement. In looking at such formalization it is important to note that only the position of a plus or minus will indicate whether it is a quantifier, qualifier, or term functor. Nevertheless, we will, following Sommers again, take all pluses as logically univocal and all minuses as logically univocal. Recall that Leibniz, too, had entertained the notion of letting quantity and quality be orthographically similar. Once a sentence is formulated, then, we can ignore any differences of interpretation indicated by position. Doing so allows us to algebraically compute an "ultimate- value" of either plus or minus for each term of a sentence. For example, the ultimate value of 'P' in 1 7 above is minus and that of 'H' is plus. The point to see here, of course, is that negative terms, denied nonnegative terms, and universailized nonne­ gative terms all are logically co-valued (as are nonnegative terms, affirmed nonnegative terms and particularly quantified nonnegative terms). Any sentence will uniquely determine an ultimate value for each of its terms. The fact that we can compute such a value for any term is guaranteed by our used of pluses and minuses for heterogeneous logical roles, and (as we will soon see) allows us to construct a relatively simple system of proofs and decisions for inferences. The logician's aim is to build a tool for detecting (and thus avoiding) contradiction. In particular, this tool is intended to detect the con­ tradictoriness which involves inference - i.e. it discerns validity from invalidity. A test for determining the validity or invalidity of any inference, then, is an integral part of any system of logic. Once it is decided that an inference is valid, it is often necessary to show how it is valid - i.e. it must

Syllogistic Logic

95

be proved. We will outline our theory of proof, or deduction, here and then makes some remarks about how validity decisions are to be made. Like the proof procedure in Aristotle's syllogistic, ours will depend completely upon our acceptance of a very small number of elementary valid arguments, all of which are simple and prima facie valid. No decision procedure is first required to determine their validity. As we saw at the outset of this study, logic deals with the product of reasoning (sentences) rather than the process (which is an object of psychological studies). Nevertheless, the logician is interested in giving proofs of valid inferences which adequately approximate and illustrate the reasoning processes which take place when we draw the conclusions of such inferences from their premises. If a logic does this well, then the length and complexity of a given proof should correspond to a reasonably high degree, to the length and complexity of the actual reasoning process which would occur when we, prelogically, draw the conclusion from the premises. Naturally, our only access to that process is through an inspection (intro­ spection or public survey) of actual attempts to draw the conclusions of valid inferences. If, normally, persons find little time and effort required to see the validity of a particular inference, then our logic ought to be able to provide a brief and simple proof for it. Some valid inferences are so simple and obvious that no proof of their validity is required. A cursory examination of them is sufficient to reveal their validity. Other valid inferences, which are either longer or more complex, or both, need their validity demonstrated. Such inferences are in need of proof. Following Aristotle, we will say that an inference which is clearly valid and not in need of a proof is a perfect inference. One in need of proof is imperfect. Keep in mind that an imperfect inference is not invalid. It is not flawed in any way, but is simply long enough or complex enough to require a proof of its validity. A proof, or deduction, is a series of sentences (usually symbolized so that their forms are obvious) such that the first members of the set are the inference's premises, the last is the conclusion, and all members after the premises are justified by a rule of the deduction system . All such rules are perfect inferences (or, to be more exact, general patterns for perfect inferences). To illustrate what a proof would be like consider an imperfect inference with premises P1 and P2 and conclusion C. Since our inference is imperfect no rule says that C follows directly from P 1 and P2 . But we might have a rule which says that sentence P3 follows from P 1 and P2 ( call it rule R 1 ) . We might then have another rule (R2 ) which says that C follows directly from P 3 . Our proof would then look like this.

Three Logicians

96

premise premise from P 1 and P2 by R 1 from P3 by R2

As we said, every rule is a perfect inference pattern. Rules (indeed, all valid inferences) will be written in the following standard manner, where C is the conclusion and P l through Pn are the premises. P l , . . . Pn . · . C

This can be read in a variety of ways: 'C logically follows from P l through Pn' ; 'if P l through Pn are true then C is true' ; ' P l through Pn, therefore C' ; 'the inference from P 1 through Pn to C is valid'. Naturally, if an inference is perfect it must not only be simple, but it must be brief as well. A perfect inference has a very small number of premises. Our deduction system limits perfect inferences to those with two or less premises. We have, then, just three kinds of perfect inferences: those with no premises, those with one premise, and those with two premises. An inference which is obviously valid and which has no premise is simply a sentence which is prima facie true. It is one which could not be denied (without contradiction). In other words, if a sentence is prima facie contradictory, then its denial will be a sentence which must be true (a tautology). A zero-premise perfect inference is simply a tautology. There are two unquestionably contradictory sentence forms: (a) All S aren't S (b) Some S aren't S (No S are S)

Their denials are :

(a') All S are S (b') Some S are S

Now these are tautologies, and in each case the subject may or may not be modified, so that (c) and (d) are likewise tautologies. (c) All S which are M are S (d) Some S which are M are S

These four tautologies are patterns for zero-premise perfect inferences.

Syllogistic Logic

97

Fully formulated, they constitute the first rule of our deduction system. 4 Rule of Identity (abbreviated: Id) . · . ± (S ± M) + S Here M may be empty, i.e. not occur at all. The rule says, in effect, that any affirmation whose subject and predicate terms are identical is a perfect zero-premise inference. Such sentences can be incorporated at any time in a proof and are often the hidden premises in enthymemes. No other tautology will be rule of our system. Before continuing we should point out an assumption common to all logical system, including ours. It is that any two symbols which are graphically indistinguishable (to a reasonable degree) are identical. Thus, in 'p => (q v,._, p)' or ' + S + S' the two 'p's' are identical, as are the two 'S's'. In other words, no symbol is equivocal throughout a given sentence, nor throughout a given inference or proof. Before formulating single-premise rules let us examine once more our basic pattern for sentence forms. all/someS/nonS are/aren't P/nonP This is only a basic pattern since it does not take into account, does not reveal, the possible logical adjectives or objects of a sentence, nor the possible compoundness of its terms. However, since in selecting rules for our system we are interested only in inferences which are brief and simple, we can restrict our search for rules to inferences involving only sentences which are fully formulated according to our basic sentence pattern. Latter we can develop rules involving more complex sentence forms. Now our basic pattern yields sixteen sentence forms. (a) all S are P (b) all S are non P (c) all S aren't P (d) all S aren't nonP (e) some S are P some S are nonP ( f) (g) some S aren't P

Leib n iz took sentences of these form to be true in themselves. See. for example, Parkinson, pp. 33, 41 and 42. See also I . M . Bochenski, "On the Categorica l Syllogism", Logico -Philosophical Studies, ed. A Menne ( Dord recht, Holland, 1 962).

4

Three Logicians

98 (h) (i) (j ) (k )

(l) (m) (n) (o) (p)

some S aren ·1 nonP all nonS are P all nonS are nonP all nonS aren't P all nonS aren't nonP some nonS are P some nonS are nonP some nonS aren't P some nonS aren't nonP

Notice that the list of assertion forms can be easily divided in half : (a) through (h) and (i) through (p). As we will see, important logical relations hold among the members of each half. Since the only difference between the two halves of our list is that the subject terms in the first are unnegated and in the second half negated, we can ignore the negation or nonnegation of the subject term (for now) and thus simply take (a) through (h) as our standard assertion forms (allowing 'S' to be negated or unnegated). In examining this list of standard forms we see some interesting features. Two assertions are contradictory whenever one affirms of a subject just what the other denies of that same subject. Given this, we can see that the following pairs are contradictory: Contradictories (a)/(c) (b)/ (d) (e)/(g) ( f) / (h)

It is important to remember again here the difference between denying a predicate of a subject and affirming the negation of that predicate term of the same subject (the predicate denial/term negation distinction). While (c) denies P of all S, (b) affirms nonP of all S. Many important logical relations between sentences will depend upon such differences among predicates. A perfect inference with one premise is called an immediate inference. We introduce now several rules of immediate inference. In each case we omit the plus or minus on simple terms whenever it makes no logical difference. Thus our rule of identity could expand to . · . ± ( ± S ± M) + ( ± S)

Syllogistic Logic

99

So, unless a term is explicitly marked, we take it to be either negated or unnegated. Now, since any assertion can clearly be validly drawn from itself we have Rule of Self-inference (SI)

± S±P . · . ± S±P A second rule of immediate inference can be formulated by considering the obvious fact that the following pairs are prima facie equivalent. all humans are mortal I all immortals are nonhuman all gods are immortal I all mortals are nongods all nongods are mortal / all immortals are gods all nonhumans are immortal / all mortals are human Each is a universal affirmation. They show that given any such assertion an equivalent sentence can be constructed by replacing the subject term with the negation of the predicate term and the predicate with the negation of the subject term. Thus we have Rule of Contraposition (Contrap) - ( ± S)

+ ( ± P) . · . - ( + P) + ( + S )

The mle of contraposition holds for universal affirmations. A comparable rule governs particular affirmations. Thus the following paus are equivalent. some men are immortal I some immortals are men some men are animals / some animals are men For any particular affirmation there is an equivalent particular affirmation which results from simply exchanging, (without further alteration) the subject and predicate terms. Rule of Converswn (Conv)

+ ( ± S) + ( ± P) . · . + ( ± P) + ( ± S) (or, more simply : + (S) + (P) . · . + (P) + (S))

The rules of contraposition and conversion govern only affirmations. Rules governing denials could be incorporated in one of two ways: (i) formulate corresponding rules for denials, or (ii) formulate a rule for converting deniais into affirmations so that separate rules for denial are unnecessary. We choose the second alternative. As we have seen, Sommers, claiming Aristotle's lead, claims that any denial is equivalent to an affirmation. To

Three Logicians

1 00

see that this is so let us return to our list of standard forms. We note (a) and (b) are logical contraries; (a) affirms P of just what (b) affirms nonP. Their respective contradictories are (c) and (d). For (c) denies what (a) affirms and (d) denies what (b) affirms. The following square illustrates these logical relations. all S are nonP

(a) all S are P

(b)

(d) all

(c) all S aren't P

S

aren't nonP

Let us call this the universal square since all the assertions displayed on it are universal. Compare it to the traditional square : (a) all S are P

(e) some S are P

D

(b)

all S are nonP (i.e. no S are P)

(t) some S are nonP

The traditional square is an affirmation square. It takes all of its assertions in the affirmative form. Since an assertion can have only one logical contrary and only one contradictory, we must conclude that, at least, (c) and (t) are equivalent and that (d) and (e) are equivalent. But, of course. the contradictories of (e) and (t), then, must be equivalent to (b) and (a) respectively (i.e. (g) must be equivalent to (b) and (h) must be equivalent to (a)). To summarize (a), ( h) .-----------. (b), (g) (e), (d)

I

(f ), (c)

So, any denial can safely be transformed into an affirmation. Our rule for this is Rule of Equivalence (Eq) ± S- ( ± P) . · .

+ S + ( + P)

Two additional types of immediate inference involve just a single term. Since any term is the negation of its own negation, the negation of any term is equivalent to that term itself. Let ' . . . T . . . ' be a sentence containing 'T'. Then ' . . . nonnonT . . .' is an equivalent sentence. Thus we have

Syllogistic Logic

101

R ule of Double Negation (DN)

. . . T . . . . · . . . . (-(-T)) . . . . . . (-(-T)) . . . . · . . . T . . .

While term negation is a functor which forms terms from terms, denial is a functor which forms predicates and adjectives from terms. So there is no corresponding "rule of double denial". Another rule derives from the obvious fact that compound terms are equivalent to one another independently of the order of their common constituent terms. Remember that all compound terms are now taken in their conjunctive form. For example 'tall and thin' is equivalent to 'thin and tall'. We can formulate the following rule to represent this. R ule of Commutation (Com)

. . . (P + Q) . . . . · . . . . (Q + P) . . .

Finally, consider a universal affirmation. It is clear that a similar affirmation with the subject and predicate terms both modified in the same way will follow. For example, since all men are mortal, all rational men are rational mortals. Moreover, any modifying term, as we have seen, could be a relation. For example, since whatever is a circle is a figure, whatever draws a circle draws a figure. Letting 'Q' be any adjective term (monadic or relational), we can formulate these facts as our final rule of immediate inference. Rule of Composition (Comp)

-S + P . · . - (S + Q) + (P + Q)

We turn now to an examination of two-premised inferences. Such in­ ferences are called mediate inferences. We saw earlier that any assertion is logically equivalent to an affirmation. Indeed, this is the import of our Rule of Equivalence, Using only affirmations, then, we can list the following valid mediate inferences as representing all such inferences. 1. 2. 3. 4. 5. 6.

all M are P, all S are M; therefore all S are P all M are nonP, all S are M ; therefore all S are nonP all M are P, some S are M ; therefore some S are P all M are nonP, some S are M; therefore some S are nonP all P are nonM, all S are M ; therefore all S are nonP all P are M, all S are nonM; therefore all S are nonP

Three Logicians

102 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

all P are nonM, some S are M; therefore some S are nonP all P are M, some S are nonM; therefore some S are nonP all M are P, all M are S; therefore some S are P some M are P, all M are S; therefore some S are P all M are P, some M are S; therefore some S are P all M are nonP, all M are S; therefore some S are nonP some M are nonP, all M are S; therefore some S are nonP all M are nonP, some M are S; therefore some S are nonP all P are M, all M are S; therefore some S are P all P are M, all M are nonS; therefore all S are nonP some P are M, all M are S; therefore some S are P all P are nonM, all M are S; therefore some S are nonP all P are nonM, some M are S; therefore some S are nonP

While all of these are valid, they are not all perfect. Only the first four are prima facie valid. It takes more (sometimes much more) than a casual glance to see that the others are valid. A careful examination of the first four inferences reveals the old dictum de omni et nu/lo : Whatever is affirmed of all of something is likewise affirmed of what that something is affirmed of.

When formulated as a rule, it becomes our rule of mediate inference.

Rule of Mediate Inference (Ml)

-M + P, ± S + M . · . ± S + P The inferences I through 19 above are the traditional standard valid (unweakened) syllogisms. Simple proof of 5 through 19, as well as for the weakened syllogisms, can be given using the rule of mediate inference, based on I through 4, the rules of immediate inference, and the rule of identity. Here are some examples of such proofs. Cesare: (5)

1. -( + P) + (-M 2. -S + M 3. -( + M) + (-P) 4. -S + (-P)

premise premise from I by Contrap from 2 and 3 by MI

Darapti: (9)

1. -M + P 2. -M + S 3. + M + M

premise premise Id

Syllogistic Logic

Fresion : ( 1 9)

Barbari :

103 +M+P + P+ M + P+ S +S+P

from I and 3 by MI from 4 by Conv from 2 and 5 by MI from 6 by Conv

1 . -M + P 2. -S + M 3 . -S + P 4. + s + s 5. + S + P

premise premise from I and 2 by MI Id from 3 and 4 by MI

4. 5. 6. 7.

1 . -( + P) + (-M) 2. + M + S 3 . -( + M) + (-P) 4. + S + M 5 . + S + (-P)

premise premise from I by Contrap from 2 by Conv from 3 and 4 by MI

The following imperfcct arguments will help illustrate more fully our system of deduction.

(i) All gods are immortal; therefore no gods are happy mortals. premise 1 . -G + (-M) from I by Con trap 2. -( + M) + (-G) Id 3 . -(M + H) + (M) from 2 and 3 by MI 4. -(M + H) + (-G) from 4 by Contrap. 5. -( + G) + (-(M + H))

(ii) All circles and figures ; therefore whatever draws a circle draws a figure. 1 . -C + F premise from I by Comp 2. -(C + D) + (F + D) from 2 by Com 3 . -(D + C) + (D + F) (iii) All men are rational ; therefore some men are rational. I . -M + R premise Id 2. + M + M 3. + M + R from 1 and 2 by MI

(iv) Tully is Cicero; therefore Cicero is Tully. 1. +T+C premise 2. + C + T from 1 by Conv

104

Three Logicians

Notice here that we take advantage of the fact that a singular term is quantified indifferently. Since the quantity of the two subjects here is wild, we take them both particularly. (v) Tully is Cicero, Cicero is Roman; 1. ± T + C 2. -C + R 3. ± T+ R

therefore Tully is Roman. premise premise from I and 2 by MI

Here we take the second premise universally and the first premise and conclusion are indifferently either both universal or both particular. (vi) John is mortal; Therefore some man is mortal. (There is the obvious hidden premise here: John is a man) premise 1. -J + M hidden premise 2. + J + N from 2 by Conv 3. + N + J 4. + N + M from I and 3 by MI (vii) If p then q, if q then r; therefore, if p then r. 1. -[p] + [q] premise premise 2. -[q] + [r] from I and 2 by MI 3 . -[p] + [r] (viii) If p then q, not q; therefore not p. 1. -[p] + [ q] premise 2. -[q] + (-E) premise 3 . - [p ] + (- E) from I and 2 by MI Here we read 2 and 3 as : 'No state of affairs in which q holds exist' and 'No state of affairs in which p holds exists' (or: 'There are no states of affairs in which q holds' and 'There are no states of affairs in which p holds'). Because of inferences like (vi) contemporary logicians have charged that traditional term logic must accept the "existential import" of universal sentences. The charge is, in effect, that universal sentences could only be true if something satisfying the subject term exists. So 'All men are rational' cannot be true unless some men exist. The claim is that such a view is forced on the term logician by his acceptance of the validity of inferences like (vi). Most contemporary logicians believe this only because they are committed to an interpretation of the particular quantifier which can be

Syllogistic Logic

1 05

rendered as: 'there exists some thing which is such that'. Since a particular sentence does follow logically from its corresponding universal (this is the relation of subaltemation in traditional logic), both quantifiers are charged with importing covertly into the sentence the existence of the quantified terms. Yet, there is no good reason for reading even the particular quantifier as having existential import. Like those syllogistic logicians before us, we can take both quantifiers as void of an existential sense whatsoever. Recall that Leibniz had simply taken all referents to be just logically possible things, existent or nonexistent. This means that if exist­ ence is to be affirmed or denied it must be done explicitly - not hidden in a quantifier but openly in a predicate term of existence. Now, from the point of view of the first order predicate calculus with identity, every quantified sentence is actually about everything or some­ thing ( e.g. 'All men are mortal' is read as 'Everything is such that if it is a man then it is mortal' and 'Some fish fly' is taken as 'Some thing is such that it is a fish and it flies' - with 'some' rtad as 'there exists at least one'). But what are these things? The contemporary logician, in order to interpret a quantified sentence must first specify a domain of discourse, a set of things to be referred to in the sentence. To be in the domain is to exist. No existence predicate is necessary for such a logic. Whatever can be referred to (i.e. is the value of a variable) exists. Our logic has no need for specifying domains of discourse in order to interpret quantified sentences. We are not committed to the view that everything exists. Some things exist and some do not. Thus, while 'some' is a quantifier, 'exists' is a predicate. Easy proofs of this are possible. To see them, however, we need first to show how a logic such as ours can analyze sentences which seem to have as their subjects everything or something. We have said that the syllogists' logic of terms has no need for a domain of discourse (taken as a set of things) in order to interpret quantified sentences. But, it does recognize a domain of discourse for each sentence. Following Sommers, we can say that the domain of discourse of a sentence is the intersection of all the categories determined by each term of the sentence (each term used in the sentence). Every term determines a cate­ gory which consists of whatever can be truly characterized by that term or its logical negation. Thus the category determined by 'red' (written '/red/') consists of whatever is red or nonred (i.e. apples, Mars, firetrucks, lemons, chalk, etc., but not the number two or my headache, since these are neither red nor nonred). So the domain of '-S + P' is the intersection of I S i and / P / . Sentences whose grammatical subjects are just 'everything' or 'some­ thing' have as their logical subjects 'every member of the domain' and

1 06

Three Logicians

'some member of the domain'. This could be said for the contemporary logician's usual interpretation of such quantifiers also. The important difference, however, is that for us the specification of the domain does not need to be established first, is never just an arbitrary set, and is uniquely determined by the (terms of the) sentence. Sentences of the grammatical forms: ( I ) Everything is P (2) Something is P

have the logical forms:

( I . I) All / PI are P (2. 1 ) Some I P/ are P

The word 'thing' and its synonyms, then, is never a genuine term. We might call it a pseudo-term. It is always a place-holder for either a genuine term which is understood (because of the context, etc.) or the term which specifies the domain of discourse for the sentence in which it is used. We can show now that, contrary to the belief held by most logicians today, 5 something exists and something is nonexistent. Let us define ' / E / ' (for 'I exists/ ') a s follows: / E / = df -(-E + E)

In other words, what is /E/ is by definition what is either E or nonE.

Proof that something exists: . · . + / E / + E I . -(-E + E) + (-E) Id 2. -E + (-(-E + E)) from I by Contrap 3. --E + / E / from 2 b y df of /E/ Id 4. + E + E 5. + E + /E/ from 3 and 4 by M I 6. + /E/ + E from 5 by Conv

There are, of course, some important exceptions.See especially: N. Rescher, ·'Definitions of 'Existence', •· Philosophical Studies, 7 ( l 957) ; N. Rescher. Topics in Philosophical Logic ( Dordrecht, Holland, 1968), chapter 9; R. Routley, "Some Things do not Exist", Notre Dame Journal of Formal Logic, 7 ( 1 966). The recent literature concerning the definition of 'exists' is extensive, but see especially H.S. Leonard, "The Logic of Existence", P hilosophical Studies, 7 ( 1956) which is the starting point for many of these studies. Attempts to give a formal definition of 'exists' have recently been assessed by H. Sarlet, "La Formalisation de ' Existe"', Logique et A nalyse, 74-76 ( 1976). I have offered a critique of Rescher's definitions in " Rescher on 'E'," Notre Dame Journal of Formal Logic, 16 ( 1975). 5

Syllogistic Logic

1 07

Proof that something is nonexistent: . · . + / El + (-E) Id 1 . -(E + (-E)) + E 2. -(-E) + (-(E + (-E))) from I by Contrap from 2 by Com 3. -(-E) + (-(- E + E)) 4. -(-E) + / E/ from 3 by df of / E/ 5. + (-E) + (-E) Id 6. + (-E) + ! El from 4 and 5 by MI from 6 by Conv 7. + / E/ + (- E) A system of deduction or proof is usually accompanied by a decision procedure. A decision procedure is a technique for determining, prior to proof, the validity or invalidity of any inference. For example, a decision procedure for the sentential calculus is provided by the truth-tables. On the other hand, it is well known that no comparable procedure exists for all of the first order predicate calculus. This is due partly to the fact that an interpretation of a formula in that calculus depends upon the specification of a domain. While such domains are restricted to nonempty, finite sets of individuals, there are, of course, an infinite number of such domains. And validity for the predicate calculus is defined in terms of every such domain. The syllogistic logic outlined here is a single, uniform calculus. There is a single decision procedure governing all inferences. It is a brief series of criteria, constituting the necessary and sufficient conditions for syllogistic validity (the validity of any inference of one logical categorical from one or more logical categoricals). They are: (a) (b )

(l) (2 )

A universal conclusion is validly drawn only from universal premises. A particular conclusion i s validly drawn only from premises, exactly one of which is particular. Any term occuring i n a valid inference has a t total ultimate value (the sum of its ultimate values) in the premise equal to its ultimate value in the conclusion.

A few brief examples will illustrate our procedure. (i) - M -(-S),-(-P) + S . · . -M + P This inference is invalid since it satisfies (a) but not (b). To see this we compute the ultimate values of all terms. Thus: -M + S, + P + S . · . -M + P We can see that the total ultimate value of S in the premises does not equal its ultimate value in the conclusion, where it has the ultimate value of zero (i.e. does not occur). ( ii) -S + P, - P + Q, -Q + (-R) . · . -R + (- S)

l 08 This valid inference satisfies both (a) and (b).

(iii) + S + (-M), + M + P . · . + S + P This is invalid since it satisfies (b) but not (a). (iv) -A + B, + B + (-(C + (-D)), -A + (-D) . · . -C + B This invalid inference satisfies neither (a) nor (b). (v) - (A + B) + C, - C + (-B) . · . -A + (-B) This is invalid since it fails to satisfy (b).

(vi) -p + q, + s-p . · . + s -q This inference also fails to satisfy (b).

Three Logicians

1 09

Concludin g Remarks

M athematical logic is the logic of the schools today. It replaced the old logic there for a variety of reasons. Not the least of these was its over­ whelming inference power. Nonetheless, as we ha":e tried to illustrate in this brief essay, there are solid grounds for expecting syllogistic to exhibit at least that same degree of inference power. These expectations do not arise from an irrational reaction against the modern logic, but were held strongly by Leibniz long before Frege's revolution. We h ave attempted to reveal how the new syllogistic provides a more uniform and simple calculus than the amalgam of calculi usually found in mathematical logic. But, most importantly, this simplicity is a result of its syntactical insights. Aristotle's logic grew out of his considerations of the natural constraints operating on ordinary Greek. The syntax of syllogistic is intended to reflect the syntax of natural language. The syntax of mathematical logic, on the other hand, is artificial, intended to contrast with the (logically) misleading and inadequate syntax of natural language. There is a reason for this important difference between the two logics. Syllogistic logic was developed by a philosopher, for philosophical reasons. Aristotle took logic to be a tool for demonstrating the correct inference of a conclusion from premises ( especially in the teaching of science). Leibniz attempted to extend and strengthen that logic. As well, he sought a mathematical analogue which would reduce inference to calculation. And Sommers has carried Leibniz's program close to its completion. Mathematical logic, however, finds its origin in the search by Nineteenth Century mathematicians (e.g. Boole, Frege, Peano) for a logical foundation for mathematics. The developments in that century of new branches of algebra, and of noneuclidean geometry, cast new light on the conception of matliematical systems as systems of deduction. Thus it was natural that some mathematici ans search for the logic underlying mathematics. Mathematical logic is well-suited for its original task (though even this is debated in our century). But, unlike syllogistic, it has never been tied to an interest in accounting for, and preserving, the syntax of natural language.

1 10

Three Logicians

That is why, when mathematical logic is applied _to sentences and in­ ferences in natural language, the first result is a radical translation from the ordinary into the extra-ordinary. The syntax of subjects and predicate of natural language is replaced by the syntax of functions and arguments of mathematics. Why abandon natural syntax for mathematical syntax? For one thing, in the absence of a rigorous calculus for syllogistic, mathematics simply looked better. It was rigorous and, internally at least, clear. Moreover, rapid progress had been made in mathematics, and the mathematical sciences (viz. physics and astronomy) during the Nineteenth Century and early Twentieth Century. Philosophical defenders of the new logic (primarily Russell) saw this progress in marked contrast to the lack of progress in (nonmathematical) philosophy. The wedding of logic to mathematics would bring the riches of scientific progress to philosophy. 1 But was the marriage worth the dowry? One obvious result (especially among English speaking philosophers) has been a division of philosophers into two quite separate camps: logicians and nonlogicians. Those philos­ ophers who have felt doubts about the value of mathematical logic for philosophy (e.g. the latter Wittgenstein, Ryle, Strawson) as well as those who have feared the "mathematics" of "mathematical logic", have seen no alternative but to abandon formal logic altogether. N. Rescher has written2 that in a few years mathematical logic will be a branch of study completely separate from philosophy. Already the majority of those who have an interest in and understanding of the kinds of studies found, say, in the Journal of Symbolic Logic are mathematicians - few are philosophers. Until the advent of the new logic few philosophers imagined that logic was anything other than either a part of philosophy itself or a fundamental tool of philosophy. Logic for Aristotle, Aquinas, Descartes, Leibniz. Kant, Hegel, was philosophical logic. In particular the syntax of sentences and inferences in philosophy is the syntax of natural language. If logic is to account for inferences in philosophy then its syntax must be natural. A logic which borrows its syntax from the mathematics of functions is not natural. It may work well when applied to the task of explicating mathematical systems, building artificial systems, etc., but it is only by torturous contortions that it is made to fit the usual kinds of inferences of interest to the philosopher. Syllogistic logic is a natural logic in the sense that its syntax is natural 1 P . H . Nidditch, The Developm ent of Mathematical Logic ( London, 1962), says, "Only when _ _ �og 1c �as _mam_ed to mathemati cs did it bec ome fertile" . (p.9). Topics m Pht!osophical Logic ( Dordre cht, Hollan d, 1968), p.4.

Concluding Remarks

III

syn tax. In this sense, then, syllogistic logic is philosophical logic. It is important here to distinguish mathematical logic from a logic with a mathematical analogue. The former makes logic serve mathematics, the latter makes mathematics serve logic. Contemporary logicians often take Leibniz to have been a distant pioneer in the development of mathematical logic only because they confuse these two. Sommers, as we have said, has carried out Leibniz's program for ex­ tending syllogistic to a universal logic and for providing it with a simple arithmetical analogue for logical reckoning. He has made great progress in this. We have merely added here certain emendations and extensions in a modest way. Yet there is still more to be done. Much of the interest among philosophers with respect to modern logic is centered around what might be called the "nonstandard" areas of mathematical logic. Modal logics, existence logics, epistemic logics, significance logics, and so on, are all results of attempts to stretch the standard mathematical logic beyond its normal bounds, or, indeed, to radically alter some part of the standard logic. 3 Often the development of such a nonstandard logic is motivated by a dissatisfaction with some part of the standard logic. We believe logics of existence and free logic are so motivated. Presumably, a replacement of the current standard logic by a different standard logic (the new syllogistic) could dispel such motives. The demands made on logic by those attempt­ ing to build nonstandard logics are often well-founded. The most obvious and important of these is the call for an adequate logic of modal terms. It is appropriate that the first modal logician was Aristotle. What is clearly called for, then, is first, a study of Aristotle's own modal sylJogistic by someone familiar with and sympathetic to syllogistic in general. 4 Secondly, the extension of the new syllogistic of Sommers to include modals is essential. Whoever chooses to engage in such a task would do well to start with Aristotle 's ancient insight that modality is a mode of predication. Kant held that in his day logic was a virtually complete science. He was mistaken. In our own day there are those who think that logic is a virtually complete science. A better knowledge of the history of logic would have convinced Kant otherwise. The same still holds today. Logic has had a long development. In the course of that development there have been numerous 3 See the survey by S. H aack . Deviant Logic (Cambridge, 1 974 }. 4 Too often studies of Aristotle's logic have been offered either by scholars ignorant of logic or by those with an obvious pred ilection for mathematical logic. Examples of the latter are S. McCall, A ristotle ·s Modal Syllogisms (Amsterdam, 1 963 }. ;ind J . M . Bochenski. A ncien t Formal Logic (Amsterdam, 1 968). I have defended Aristotle against a more recent attempt in my "Aristotle on the Subj ect of Predication". Notre Dame Journal of Formal Logic, 19 ( 1 978).

1 12

Three Logicians

insights and valuable suggestions, as well as real accomplishments. We who seek to continue towards the goal of a complete logic would do well, as Lukasiewicz taught us, not to ignore logics' own history. 5

5

See Bochenski's excellent brief appraisal of the current state of history studies in Ancient

Formal Logic, pp. 4-8 .

1 13

Bibliography

Aristotle, The Basic Works ofAristotle, ed. R. McKean (New York, 194 1 ). Aristotle, Prior and Posterior Analytics, ed. W.D. Ross (Oxford, 1957) Aristotle, Categories and De lnterpretatione, ed.J.L. Ackrill (Oxford, 1963) Aristotle, The Categories, On Interpretation, and Prior Analytics, ed. H.Cook and H . Tredennick ( London, 1962). Barnes, J. "Aristotle's Theory of Demonstration", Phronesis, 14 ( 1969). Bochenski, I. M . Ancient Formal Logic (Amsterdam, 1968). Bochenski, I. M . , "On the Categorical Syllogism". Logico-Philosophica/ Studies, ed. A. Mennes (Dordrecht, Holland, 1962) Boehner, P., Medieval Logic ( Manchester, 1952) Boole, G., The Mathematical Analysis of Logic (Cambridge, 1 847). Boole, G., An Investigation of the Laws of Thought ( London, 1 854). Castaneda, H . - N . , "Leibniz's Syllogistico-Propositional Calculus", Notre Dame Journal of Formal Logic, 1 7 ( 1976). Corcoran, J . , "Aristotle' s N atural Deduction System", Ancient Logic and its Modern Inter­ pretations, ed., J. Corcoran (Dordrecht, Holland, 1974). Englebretsen, G., "Elgood on Sommers' Rules of Sense", Philosophical Quarterly, 2 1 ( 197 1 ). Englebretsen, G., "The Incompatibility of God's Existence and Omnipotence," Sophia, 1 0 ( 197 1 ) . Englebretsen, G., "Sommers' Theory and the Paradox of Confirmation", Philosophy of Science, 38 ( 197 1 ). Englebretsen, G., "On the Nature of Sommers' Rule", Mind, 80 ( 197 1 ). Englebretsen, G., "Vacuousity", Mind, 8 1 ( 1972). Englebretsen, G., "On van Straaten's Modification of Sommers' Rule", Philosophical Studies, 23 ( 1972). Englebretsen, G . , "True Sentences and True Propositions", Mind, 8 1 ( 1972). Englebretsen, G., "Persons and Predicates", Philosophical Studies, 23 ( 1972). Englebretsen, G., "The Logic of Negative Theology", The New Scholasticism, 67 ( 1973 ). Englebretsen, G ., "A Note on Contrariety", Notre Dame Journal of Formal Logic, 1 5 ( 1974). Englebretsen, G., "Sommers on the Predicate 'Exists'," Philosophical Studies, 26 ( 1974). Englebretsen, G., "Rescher on 'E"', Notre Dame Journal of Formal Logic, 16 ( 1975). Englebretsen, G., Speaking of Persons ( H alifax, 1975). Englebretsen, G., "The S4uare of Opposition", Notre Dame Journal of Formal Logic, 1 7 ( 1976). Englebretsen, G., "Sommers' Tree Theory and Possible Things", Philosophical Studies ( Ire), 24 ( 1976 ). Englebretsen, G., "Review: Issues in the Philosophy of Language ed. A. Mackay and D.Mer­ rill," Philosophical Studies ( Ire) 25, ( 1977). Englebretsen, G., "Aristotle on the Subject of Predication". Notre Dame Journal of Formal Logic, 19 ( 1978 ).

1 14

Three Logicians

Englebretsen, G., "Notes on the New Syllogistic", Logique et Analyse, forthcoming. Englebretsen, G., "On Propositional Form", Notre Dame Journal of Formal Logic, forthcoming. Englebretsen, G., "Singular Terms and the Syllogistic", The New Scholasticism, forthcoming. Englebretsen, G., "Reference and Denotation", Philosophical Studies ( Ire), forthcoming. Englebretsen, G., "Aristotle on the Oblique," Philosophical Studies ( Ire), forthcoming. Frege, G., Begriffsschrift (Halle, 1879). Frege, G . , Die Grundgesetze der Arithmetik, vol i (Jena, 1893 )., vol.ii (Jena, 1903). Frege, G., "On Sense and Reference", Translations from the Philosophical Writings of Gottlob Frege, ed. P. Geach and M. Black (Oxford, 1952). Geach, P., R eference and Generality (Ithaca, N .Y. 1962). Geach, P., "Mr.Strawson on Symbolic and Traditional Logic", Mind, 72 ( 1963). Geach, P., "History of the Corruptions of Logic", Logic Matters (Oxford, 1972). Haack, S., Deviant Logic (Cambridge, 1974). Howell, W.S., Eighteenth Century British Logic and Rhetoric ( Princeton, 197 1). Jevons, W.S. Pure Logic ( London, 1 864). Jevons, W.S. The Principles of Science ( London, 1874). Leibniz, G.W. Leibniz: Logical Papers, ed. G.H.R. Parkinson (Oxford, 1966). Leonard, H.S. "The Logic of Existence", Philosophical Studies, 7 ( 1956). Lukasiewicz, J., Aristotle 's Syllogistic from the Standpoint of Modern Formal Logic (Oxford, 195 1). McCall, S., Aristotle 's Modal Syllogisms (Amsterdam, 1963). Nidditch, P.H. The Development of Mathematical Logic ( London, 1962). Patzig, G., Aristotle 's Theory of the Syllogism ( Dordrecht, Holland, 1968). Peirce, C.S. Collected Works, ed. C.Hartshorne, P. Weiss, and A.W. Burks (Cambridge, Mass. 193 1- 1958). Quine, W.V.O., "On What There Is", From a Logical Point of View (New York, 196 1). Quine, W.V.O., Word and Object (Cambridge, Mass., 1960). Russell, B., "On Denoting", Mind, 14 ( 1905). Russell, 8., and Whitehead, A.N.Principia Mathematica (Cambridge, 19 10- 1913). Russell, B., "Aristotle's Logic", Essays in Logic, ed. R.Jager ( Englewood Cliffs, N .J. 1963). Rescher, N., "Definitions of 'Existence'," Philosophical Studies 7 ( 1957). Rescher, N ., Topics in Philosophical Logic ( Dordrecht, Holland, 1968). Routley, R., "Some Things Do Not Exist", Notre Dame Journal of Formal Logic. 7 ( 1966). Sarlet, H., " La Formalisation de 'Existe'," Logique et Ana(rse, 74-76 ( 1976). Sayward, C.and Voss, S., "Absurdity and Spanning", Philosophia. 2 ( 1972). Scholz, H., "The Ancient Axiomatic Theory", Articies on Aristotle. ed., J. Barnes et.al ( London. 1975). Sommers, F., "The Ordinary Language Tree", Mind, 68 ( 1959). Sommers, F., "Types and Ontology", Philosophical Review. 72 ( 1963). Sommers, F., " Predicability", Philosophy in America, ed. M. Black (Ithaca. N .Y . 1965). Sommers, F., "What We Can Say About God", Judaism, 15 ( 1966). Sommers, F., "On a Fregean Dogma". Problems in the Philosophy of Mathematics, ed. I. La­ katos (Amsterdam, 1967). Sommers, F., "Do We Need Identity?" Journal of Philosophy, 66 ( 1969). Sommers, F., "On Concepts of Truth in Natural Languages", The Review of Metaphysics, 23 ( 1969). Sommers, F., "The Calculus of Terms", Mind. 79 ( 1970). Sommers, F., "Structural Ontology", Philosophia, 1 ( 197 1 ).

Bibliography

1 15

Som mers, F., " Existence and Predication", Logic and Ontology, ed. M.K. Munitz (New York, 1 973). Sommers, F., "The Logical and the Extra-Logical", Boston Studies in the Philosophy of Sciences, 1 4 ( 1 973). Som mers, F., "Distribution M atters", Mind, 84 ( 1 975). Sommers, F., " Logical Syntax in N atural Language", Issues in the Philosophy of Language, ed. A. MacKay and D, Merrill (Oberlin, 1 976). Strawson, P.F., Introduction to Logical Theory ( London, 1 952). Strawson, P.F. Subject and Predicate in Logic and Grammar ( London, 1 974). Tarski, A., "The Semantic Conception of Truth", Philosophy and Phenomenological Research, 4 ( 1 944).

1 16

Index

algebraic logic 67 Aquinas 110 argument 4, 29 Aristotle vii, I , 2, 3, 9ff, 28, 29, 30, 35, 43, 44, 46, 48, 52,55, 60, 62,63,69, 78, 80, 83, 84, 95, 99, 109, 110, I l l atomic sentence 70, 74, 77, 83 axiom system I 8ff, 63 Barnes, J. 19n "basic logic" 75 Bochenski, l. M. 17n, 97n, I I I n, 112n Boehner, P. 77n Boole, G. 67ff, 109 Castaneda, H.-N. 33n categoricals 9ff, 28, 29, 42ff, 55, 77, 83ff, 107 category 42, 45 category mistake 42ff characterization 48, 68, 77ff, 84 connotation 77ff, 84 contradictory 4, 5, l4ff, 22, 28ff, 94 , 98, 100 contraposition 53 contrariety 14, l5ff, 28ff, 44, 46, 48, 52 , 100 conversion 25, 53 copula 12, 31, 34 Corcoran, J. 17n, 19n decision procedure 25, 31, 57, 71, 94, 95. 107 deduction 5, 9, 19, 40. 67, 94ff deep grammar 62 DeMorgan, A. 67 denotation 49, 60ff, 77ff, 84 Descartes, R. I I O dictum de omni e t nu/lo 23ff, 37, 40. 56ff. 61, 62, 102 distribution 25, 34. 60ff Englebretsen, G. I I n, 12n. 17n, 26n. 42n. 43n . 60 n, 62n . 63n, 85n. 106n, I I I n enthymeme 97

equational logic 38 existence 38ff, 46, 59ff. l 04ff existential import 38ff, 63, l 04ff existential (particular) quantifier 39, 59, 74, 104 fact 32, 51 Frege, G. 3, 49, 62. 68ff, 109 "Fregean dogma" 44, 77 function/ argument 68 functor (formative) 29, 59. 70. 77, 83ff. 92 Geach, P. vii, 17n. 60 grammar 5, 30ff, 36. 40. 62 Haack, S. 11 l n Hamilton, W . 67 Hegel, W. 110 Howell, W. S. 33n hypotheticals 26. 3 I ff. 49 hypothetical syllogism 9. 22 identity 47ff. 51, 57. 75. 77. 82ff. 92 law of 63 immediate inference 54. 98. 99 inference 14. 23, 28. 47. 62. 95ff. 109 weakened 47. 55. 63. 102 interpretation 34. 38ff. 71 inversion 45 Jager, R. 17n Jevons, W. S. 68 Kant. I. 2. I I O. 111 Leibniz. G. vii. 2. 3. 27. 28ff. 43. 44. 46. 47. 48. 49. 52. 59, 60, 62. 67. 79. 80. 83, 92. 97n . 105. 109. I IO. 111 Leonard. H. S. 106n Liar Paradox 50 linguistics 5 Locke. J. 2 logical adjective 80. 97. I O I logical obj ect 63. 80ff. 9 I , 97 Lukasiewicz. J. I I n, 17n. 18, 19n. 20n, 23n. 63. 112

Index mathematical logic 3, 29, 39, 67ff, 82ff, 109, 111 mathematics 3, 43, 68, 109 M cCall, S. 22n, 111n meaning 68 mediate inference 10 l ff modality 11, 14, 22, 111 naming 68, 92 negation 12, 23 predicate (denial) 12, 14ff, 24, 28ff, 46, 47, 52, 78, 84ff, 98ff sentential (propositional) 12, 15ff, 30, 38, 47, 52, 71, 92 Nidditch, P. H. 1 1 0 norninalization 50ff, 83ff, 92 ontology 42 opposition 25, 52 C-opposition 52ff P-opposition 52ff Q-opposition 52ff square of 17, 46, 100 theory of 14 Parkinson, G. H. R. 30nff, 49n, 97n Patzig, G. I I n Peano, G . I 09 Peirce, C. S. 68 predicate 10, 12ff, 23, 30ff, 42, 45, 47, 63, 68, 70, 72, 73, 77, 84ff, I 05 privation 16 pronoun 73, 92, 94 prosyllogism 22 proto-proposition 11, 12, 20 pseudo-term 106 psychology 5, 95 quality 11, 13ff, 23, 29ff, 52, 56, 78ff, 84 quantified predicate 34, 52 quantity 11, 13, 23, 29ff, 47, 52, 56, 73ff, 78ff, 84, 92 singular 36, 48. 55, 59ff, 75ff, 84 Quine, W. V. 68, 69n Ramus, P. 28 "rational grammar" 30 reference 48, 49, 60ff, 63, 68, 74, 77ff, 84 relationals 25, 26, 29ff, 36ff, 43, 47, 49, 55ff, 63ff, 68, 74, 77, 80ff, 86ff, 101 Resch er, N . 106n, I 10 rewrite rules 85ff Ross, W. D. I I n Routley, R. 106n

1 17 rule of inference/ deduction 62, 74ff, 95ff Commutation l O l ff Composition 10 I ff Contraposition 99ff Conversion 99ff Double Negation l O l ff Equivalence I OOff Identity 97ff M ediate Inference l 02ff Self-Inference 99fT Russell, B. 17, 67, 69n, 80, 110 Ryle, G. 110 Sarlet, H. 106n Sayward, C. 42n scholastic logic 2, 9, 25, 69 Scholz, H. I 9n semantics 69 sense-relations 42 similarity 55 Sommers, F. vii, viii, 3, 26, 41, 42ff, 83, 92ff, 99, 105, 111 spanning 42, 45 specification 49 state of affairs 49ff, 83ff, 91, 104 Stoics 2, 3, 26, 69, 70 Strawson, P. F. vii, 78, 101 structure trees, logical 85ff subaltemation 21, 25, 56 subject 10, 12, l4ff, 23, 30ff, 42, 45, 47, 60, 63, 68, 72, 73, 77, 84ff, I 05 syllogism 17fT compound 22 demonstrative 26 perfect/imperfect 21, 23, 24, 26 sophistic 26 syllogistic vii, 9, 11, 28, 42ff, 68, 77ff, 105, 101 "new syllogi�tic" vii, 42ff, 51, 62, 92 syntax (form) logical 9, 10, 14, 23ff, 29ff, 44ff, 59, 62, 68, 72, 82ff, 85ff, 92, 109ff mathematical 59, 62, l l Off natural 30, 59, 62, 69, 82, 109ff Tarski, A. 49, 50 term 9, 10, 11, 42, 45, 60, 70, 72, 77ff, 83, 101 compound 40, 41, 93 definite/indefinite 10, 34 general 46, 48, 55, 72ff, 77, 84

1 18 modified 80, 85, 86, I O I propositional (sentential) 32, 33, 49ff, 84ff singular 1 3n, 26, 29, 35, 46ff, 55, 59ff, 72ff, 77ff, 84, 1 04 term logic 2, 3, 9, I O, 80 "tree theory", Sommers' 42ff truth-claims 4 truth/falsity 4, 18, 32, 39ff, 49, 55, 7 1 , 1 04 truth-functions (sentential-functions) 29, 43, 49ff, 57, 67, 68, 70ff, 74

Three Logicians universal logic (syllogistic) 26ff, 43, 49, 5 1 , 52, 59, 92 universe (domain) of discourse 39, 45, 46, 73, 1 05ff validity 4, 5, 9, 2 1 , 25, 44, 45, 56ff, 6 1 , 70ff, 94ff, 1 07 Voss, S 42n Wallis, J. 33n Whitehead, A. N. 67 Wittgenstein, L. I I O