Something To Reckon With: The Logic of Terms (Philosophica)

. GEORGE ENGLEBRETSEN

SOMETHING TO RECKON WITH

THE LOGIC OF TERMS FOREWORD BY FRED SOMMERS

····----···--·-····,_-]

UNIVERSITY

OF OTTAWA PRESS

SOMETHING TO RECKON WITH THE LOGIC OF TERMS At the turn of the centwy, mathematical logic replaced the more traditional term logic, which had dominated the philosophical realm for hundreds of years. Today, however, term logic is resurfacing and prompting an important and very interesting debate among logicians, philosophers and linguists. By delving into the history and development of logic from its beginnings to the modern era, George Englebretsen rehabilitates term logic and demonstrates that an enhanced traditional logic remains a viab!e possibility. Taking inspiration from Fred Sommers' work, he creates an updated and fascinating version of term logic; one he believes to be just as legitimate as, anc;i in ways superior to, the currently predominant mathematical logic. Comprehensive, engaging and remarkably jargon-free, Something to Reckon With is liable to spark further discussions and developments on the fundamental concept of logic. "Englebretsen takes the reader through the history of logic, including our own times, when professional logicians succumbed to the temptation to use an unnecessarily complex technical symbolism. He tells how and why that happened and engagingly shows how we can p~t matters right again." FRED SOMMERS

George Englebretsen is a professor of philosophy at Bishop's University and is the author of Essays on the Philosophy of Fred Sommers: In Logical Terms. He is the editor of The New Syllogistic and has published numerous articles on logic, lan· guage and metaphysics.

UNIVERSITY OFOTIAWA PRESS ISBN 0-7766-0423-6 (paper)

SOMETHING TO RECKON WITH THE LOGIC OF TERMS

GEORGE ENGLEBRETSEN

With a foreword by FRED SOMMERS

UNIVERSITY OF OTIAWA PRESS

with perennial questions of The Ph1·tosophic:a series covers works dealing ---L- wor1cs wntten • • hi n wat . f hiJosophy. The series partieularlY ~ h d an astoryon PContinental and the analy tic: tradit1ons. · 1n confomuty · w1t ·h h e E uropea tthe . leo · · · 'th Press's editorial policy, the senes we mes manusc:npts wntten an ea er English or French.

canadian Cataloguing in PubUcation Data Englebretsen, George Something to Reckon With: The Logic of Terms (Philosophica; 48) Includes bibliographical references and index. ISBN 0.7766-0434-1 (bound) ISBN 0.7766-0423-6 (pbk.)

1. Language and logic. 2. Logic:, Symbolic: and mathematical. I. Title. D. Series: Collection Philosophic:a; 48. BC13S.E42 1996

160

C96-900379-X

This book has been published with the help of a grant from the Humanities and Social Sciences Federation of Canada, using funds provided by the Social Sciences and Humanities Research Councll of Canada. University of Ottawa Press gratefully acknowledges the support extended to its publishing programme by the Canada Council, the Department of Canadian Heritage, and the University of Ottawa.

Cover: Robert Dolbec Typesetting: Genevi~ve Boulet Distributed in the U.K. by Cardiff Academic Press Ltd., St. Pagans Road, Fairwater, Cardiff CFS 3AE. •All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording. or any information storage and retrieval system, without permission in writing from the publisher.·

University of Ottawa Press, 1996 Printed and bound in Canada

@

ISBN 0.7766-0434-1 (cloth) ISBN 0.7766-0423-6 (paper)

ForMorpn, definitely someone to reckon with

CONTENTS Foreword by Fred Sommers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . ix Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . xvii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . I Whatls a Logical Constant? . . . . . . . . . . . . . . . . . . . . . . . . . . I The Problem of Sentential Unity . . . . . . . . . . . . . . . . . . . . . . . 3 Notes ........................•................... 6 PART I Chapter One: The Good Old Days of the Bad Old Logic (or, Adam's Fall) .......................................... 9 Aristotle's Syllogistic . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . 9 Scholastic Additions . . . . . . . . . . . . • . . . . . • . . . . . . . . . . . . 16 Cartesian Interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Leibnizian Insights . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . 30 Nineteenth-Century Algebraists . . . . . • . . . . . . . . . . . . . . . . . 41 Notes ........................................... 49 Chapter Two: A Modem Success Story (or, Frege to the Rescue) .... 53 Frege ........................................... 53 Bradley and Ramsey Raise Some Doubts . . . . . . . . . . . . . . . 64 Russell and Wittgenstein . . . . . . . . • • . . . . • . . . . . . . . . . . . . 69 Strawson, Geach, and Quine . . . . . . . . . . • . . . . . . . . . . . . . . 78 Notes ....................•.•.................... 93 PART II Chapter Three: Coming to Terms with Sommers . . . . . . . . . . . . . . . . 99 The Calculus of Terms .............................. 99 The Logic ofNatural Language ...................... 122 The Truth .................•..................... 135 The Laws ofThought .............................. 142 Notes ...............••......................... 147 Chapter Four: It All Adds Up............................... 149 Plus/Minus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 The Simple System . . . . . . . . . . . . . . . . . . . . . . . . I 5 I Formulating English . . . . . . . . . . . . . . . . . . . . . . . 153 Relationals . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . I 54 Splitting Connectives . . . . . . . . . . . . . . . . . . . . . . . I 55 Wild Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 DONandEQ ...•.....................•... 164 Rules oflnference . . . • • . . . . • • . . . . . . . . . . . . . . 167 Names and Other Pronouns . . . . . . . . . . . . . . . . . . 174 Truth and What 'There' Is •......................... 185 A New System ofDiagrams .......•................ 188 Notes ...............................•.......... 237 Conclusion . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 239 Bibliography . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 243 Index ofNames . . . . . . . . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . . . . 269

FOREWORD by Fred Sommers I teach logic, but unless I want to be left strictly alone I've learned long ago not to tell that to anyone I've just met. Quite a few people are put off by logic (and logicians). Yet people reason all the time, and mostly they reason correctly. Why then is logic so alien a subject to so many? George Englebretsen's book tells why and introduces the reader to logic in a way that is altogether "friendly" and natural. Though it is not a formal text, it does what current logic texts do not do: it makes you recognize the processes of your own thinking; it shows them to you in ways that make them familiar again. It gives you one "aha" experience after another. To appreciate Englebretsen's novel approach to logic, it is necessary to have some idea of how the subject is currently taught and why it is so alienating. You hear someone say "I've petted a crocodile but I never petted an amphibian." You think to yourself: "That can't be right! Crocodiles are amphibians." Using your native logical abilities you intuitively saw that the following two sentences cannot both be true together: (PI) (1 *)every X is a Y; (2*) someone R'd an X but did not RaY.

The crocodile contradiction fits the pattern Pl, and so does this one: (1 ')every Norwegian is a Scandinavian.

(2') someone who cheated a Norwegian never cheated a Scandinavian. As our examples illustrate, we often recognize inconsistency when we come across it. Related to this is our ability to infer conclusions from premises. For example, given our feeling that sentences of pattern PI are inconsistent, we are confident that inferences ofform Ll are 'valid'. (Ll) (li) every X is a Y

ix

-

FOREWORD

(2i) /everyone who R's an X R's a Y (lbe little stroke sign that precedes the sentence in (2i) should be read as 'therefore', 'hence', or 'so'.) An example of an inference that fits the pattern (L I) is (Al)

every horse is a mammal /Everyone who rides a horse rides a mammal

Similarly, we easily recognize the validity of the inferences Every crocodile is an amphibian. So anyone who pets a crocodile pets an amphibian. Every Norwegian is a Scandinavian. So anyone who cheats a Norwegian cheats a Scandinavian. Here is a different pattern of inference that we all instinctively recognize as valid. (L2)

every X is a Y some X isaZ /some Y isaZ

An inference that fits this pattern is (A2)

every horse is a mammal some horses are white /some mammals are white

We intuitively see the conclusion of A2 really does follow from the premises. Our logical instincts are healthy, but when it comes to explaining why a given pattern of reasoning is valid or invalid, we must call on the professional logician. Just here, the modem teachers of logic have failed the public. For in most contemporary institutions of higher learning we practice and teach a style of logic that keeps the average intelligent layperson at arm's length. If you ask a well-trained logician to explain why a statement of the form 'every X is Y' is incompatible with a statement of the form 'someone R'd an X but didn't RaY', he will rightly tell you that taking them both to be true and applying rules of logic will lead you to an overt contradiction of the form 'some X is not an X'. But he will point out that before you can show this yourself, you will have to learn how to translate a sentence like 'some who petted a crocodile didn't pet an X

FOREWORD

amphibian' into the language of symbolic logic and learn the rules for manipulating its formulas. Now, learning the symbolic language and the rules usually takes a good half of a semester's college logic course. The translations use letters and symbols like the following: C for 'is a crocodile' A for 'is an amphibian' P for 'petted' (x) for 'for every x' (Ey) for 'there is an x' Pyx for 'y petted x' =>for 'if ... then' & for 'and' -for 'not' Using these symbolic abbreviations, the student learns to "translate" 'every crocodile is an amphibian' and 'someone who petted a crocodile did not pet an amphibian' as (It)

(2t)

(x)(Cx => Ax) (Ex)(Cx & (Ey)(Pyx & (z)(Az => -Pyz)))

Reading (It) as 'For every x, ifx is a crocodile, then xis an amphibian' and (2t) as 'There is an x such that x is a crocodile and there is a y such that y petted x and for every z, if z is an amphibian, then y did not pet z'. To prove the inconsistency of (It) and (2t), the student proceeds to apply rules to them to get new propositions that follow from these two, moving along until she or he deduces a formula that is of the form (Ex)(Px & -Px), which may be read as 'something is such that it is both a P-thing and a not-P-thing'. Now this is an overt contradiction and it shows that the original two propositions cannot both be true. I won't show you how the contradiction is actually derived. Even if you know how to translate the sentences of your arguments into the symbolic language, and know the rules for manipulating the formulas, the proofs take time and considerable ingenuity. Many students enjoy the challenge. Unfortunately, far more are put off by the whole process, feeling perhaps that something as obvious as good reasoning need not be approached in so complicated a fashion. That feeling happens to be right. We can say in one sentence why modern logic is so forbidding and arcane: Modern logic is unfriendly because, as it is currently presented, you cannot do logic unless you have learned an artificial symbolic language to be used in "translating" ordinary sentences into logical formulas. The symbolic translations that today's students of logic must master are alien to the average person because they contain phrases and constructions not found in the original sentences.

xi

FOREWORD

Consider the sentence 'every crocodile is an amphibian' and its translation as (xXCx =>Ax), which we may read as 'anything is such that if it is a crocodile then it is an amphibian'. That rendering is perhaps a bit closer to the English sentence than 'for every x, if x is a crocodile, then x is an amphibian'. Even so, both renderings are a far cry from 'every crocodile is an amphibian'. Both contain the construction 'if ... then' and a pronoun 'it' (in the form of the variable 'x'). The original English sentence has no pronouns and no connective phrase of the form 'if ... then'. So the student might well ask, Is this way of construing the sentence really necessary? Or take 'some horses are white', the second premise of A2. That is translated as a formula that uses (Ex) and &: (3t)

(ExXHx & Wx)

and is read as 'there is an x, such that x is a horse and x is white', or, more freely, as 'something is such that it is a horse and it is white'. But again, 'some horses are white' has no pronouns, nor does it contain the word 'and'. The translation of (3t) thus complicates the original sentence. And many a layperson gets the unhappy feeling that professional logicians may be making things unnecessarily complicated. The complexity shows up more dramatically in dealing with a sentence like 'someone who petted a crocodile never petted an amphibian'. The translation, (2t), introduces three pronouns and constructions using 'and' and 'if ... then', none of which are found in the original sentence. We are all logical. We all enjoy using our minds to make inferences, to detect false reasoning, to check on the reasoning of others. So it's really a pity that modern logic is so unfriendly. We should expect logic to be easy! It should be natural! It should be fun! And it can be. There is a way of doing logic that is natural and enjoyable. Something to Reckon With introduces you to it while giving the history of how logic took its unfriendly turn in the last century. Englebretsen begins where one must begin: with the analysis of the logical form of the simplest sentences we use in our everyday reasoning. That takes him immediately to the distinction between logical words such as 'some', 'and', 'ir, and 'not', which determine the form of the sentence, as opposed to words such as 'farmer', 'book', or 'runs', which contribute the content or matter of the sentence. The vehicle for rational thought is the sentence. For example, we may infer 'not every farmer is a non-citizen' from 'some farmers are citizens'. In 'some farmers are citizens'. 'citizens' and 'farmers' are the material elements. The words 'some' and 'are' determine the form of the sentence; they are called "formative" elements. Aristotle, who liked to place the formative elements between the two material elements, preferred to write 'some farmers are citizens' as 'citizen belongs to some farmer'.

xii

FOREWORD

The words 'citizen' and 'farmers', which now appear at each end of the sentence, are terms (as in 'terminals') and the expression 'belongs to some' then acts as the "term connective" that joins the terms to form the sentence. Another way of forming a sentence with these terms is 'citizen belongs to every farmer' (Aristotelian for 'every farmer is a citizen'). Since 'belongs to' is common to both connectives, we may dispense with it and write the two sentences as 'citizen some farmer' and 'citizen every farmer'. Englebretsen soon introduces the reader to an algebraic way of representing the two term connectives 'some' and 'every'. Using '+' for 'some' we write 'citizen some farmer' as 'C+F'. This allows us to represent the equivalence of'some farmer is a citizen' to 'some citizen is a farmer' as an equation: C+F=F+C citizen some farmer = farmer some citizen The equivalence shows that 'some' behaves like the addition sign. What about 'every'? To see how 'citizen every farmer' can be transcribed as an algebraic formula, we note that 'every farmer is a citizen' is equivalent to the denial of 'some farmer is a non-citizen' The Aristotelian form for the denial that some farmer is a non-citizen is 'it is not the case that non-citizen some farmer'. Using '-' for negative words like 'not' and for negative particles like 'un' or 'non-', 'not: non-citizen some farmer' transcribes as -((-C)+F) which is algebraically equivalent to 'C-F'. This suggests that in logic the word 'every' behaves like the subtraction in algebra. And indeed it does so behave. Note that 'not: non-citizen some farmer' (Aristotelian for 'it is not the case that some farmer is a non-citizen) is logically equivalent to 'citizen every farmer'. The equivalence is algebraic: C-F = -((-C)+F) citizen every farmer = not (non-citizen some farmer) Englebretsen's aim is to show the reader how we "reckon with sentences" in much the way we reckon with numbers or simple algebraic expressions. Using the plus/minus way of representing formatives, he soon introduces the reader to more natural ways of transcribing sentences into algebraic formulas. The following words or expressions are'+': 'some', 'and', 'is', 'it is the case that'. The following words or expressions are'-': 'not', 'non', 'every', 'ir, 'it is notthe case that'. Here are some examples of algebraic transcriptions.

xiii

FOREWORD

every fanner is a citizen some farmer is a non-citizen some fanner is a gentleman and a scholar some boy envies every astronaut no senator is a non-citizen everyone who cheats a Norwegian cheats a Scandinavian

-F+C +F+(-C) +F+ +B 1+(E 12 - A2> -(+S+(-C))

Note that a contradictory sentence such as 'some man is not a man' transcribes as a sentence of the form '+X+(-X)', which literally "says nothing." Note also that when we add sentences like 'someone who cheated a Norwegian didn't cheat a Scandinavian' to 'every Norwegian is a Scandinavian' we get a contradictory form: I. +(C 12+N 2)-(C,2+S2) 2. -N+S I +(C 12+S2)-(C,2+SJ

In contrast, anyone who uses the symbolic translations will fmd that it takes many steps and considerable ingenuity to show that the two sentences lead to a contradiction. It would, however, be a mistake to think that we can move easily from the vernacular sentence to the algebraic transcription. Some "regimentation" is required. An English sentence like 'every whale is a mammal' transcribes directly into algebraic notation as '- W+M'. Similarly, its equivalent, 'no non-mammals are whales', transcribes directly as '-(+(-M)+W)'. Sentences that come ready made for direct transcription are called "canonical." In real-life reasoning, however, canonical English sentences, all ready to be transcribed into algebraic formulas, are the exception rather than the rule. We are at least as likely to come across 'the whale is a mammal' and 'only mammals are whales' as 'every whale is a mammal' or 'no non-mammals are whales'. Regimenting sentences to make them suitable for reckoning is an important tool in practical reasoning. It is essential to expose the structure of argument by giving each sentence its proper form. The methods presented by Englebretsen may then be used to reckon with in order to arrive at a conclusion. Consider, for example, the following simple argument that is clearly valid. Some sport cars that have no automatic transmissions are convertibles. I Some convertibles that lack automatic transmissions are sports cars.

xiv

FOREWORD

Regimented and transcribed, the argument looks like this: ++C I ++S

Consider next a slightly complicated piece of reckoning that involves a relation with several subjects. 1. 2. 3. 4.

some sailor gave every child a lollipop Some children were orphans All lollipops are delicious Every sailor was an American

To see what conclusion we may draw, we transcribe the sentences algebraically: l. +S,+G,u-C2+LJ

2.+C+O 3. -L+D

4. -S+A Adding these premises we get the conclusion:

which is the transcription of 'some American gave some orphan something delicious'. As a final example of how logic is approached in this book, we look at a problem in a book on logic written by the author of Alice in Wonderland. Lewis Carroll presents three premises and asks the reader to supply a fourth statement as a conclusion. (1) No terriers wander among the signs of the Zodiac. (2) Nothing that does not wander among the signs of the Zodiac is a comet. (3) Nothing but a terrier has a curly tail. (4) I???

In solving problems of this kind we are better off relying not on our wits but on a mechanical procedure for drawing conclusions from premises. The tricky part is to regiment the sentences by paraphrasing each one in a way that permits us to transcribe it algebraically. The following transcriptions introduce letters that stand for the terms of the argument (viz., T =terrier, W =wanderer among the signs of the Zodiac, C =comet, S =curly-tailed). XV

FOREWORD

Using these tenn letters, we may represent the argument thus: (I) noT is a W

-(+T+W)

(2) no non-W is a C (3) no non-Tis an S

-(+(-W)+C) -(+(-T)+S)

Having represented the premises in algebraic transcription, we can add them up to derive the conclusion in a mechanical way. Driving minus signs inward gives us '-T-W-(-W)-C-(-n-s• which equals ·-c-s•. The conclusion ·-c-s• or '-(+C+S)' stands for 'every comet isn't curly tailed' or, equivalently, 'no comet is curly tailed'.

Well, a foreword should just briefly introduce; it may do no more than whet the reader's interest. Aristotle defined us as rational animals. To be rational is to think. Englebretsen's novel book brings you to a delightful landscape laid out with the pathways of good thinking. Along the way, he takes you quickly through the history of the subject, including our own times, when professional logicians succumbed to the temptation to use an unnecessarily complex technical symbolism. It is a symbolism that works well enough but alienates the lay public by straying too far from the language it actually uses in its everyday reasoning. Englebretsen tells how and why that happened and engagingly shows how we can put matters right again.

xvi

PREFACE Philosophy teaches us to talk with an appearance of truth about all things, and to make ourselves admired by the less learned. Descartes Through careful study of official answers to questions asked in Parliament one learns that the length and number of words used may e./foci a considerable economy in the quantity ofinformation conveyed J.A. Chadwick

Classical Greek rhetoric, in contrast to logic, allowed an argument to rest on examples and analogies, commonplaces and truisms. In inventing logic Aristotle recognized that, even as a tool for rhetorical science, something far more objective and rigorous is required of argumentation. Aristotle's formal logic was, in Leibniz' s judgment, one of the most beautiful inventions of the human mind. Beautiful or not, the fact is that for most of the past twentyfour centuries a relatively small number of people have taken a special interest in the discipline of logic. This is so partly because logic, like its close partners of old, grammar and rhetoric, has been considered trivial, not just in the literal sense of belonging to a trio, but in the sense of being unproductive. Yet, as Peter Geach has observed, "Logic is unproductive like book-keeping, but without sound accountancy a productive business may smash." Logic keeps the accounts of rational thought and discourse. And a good job that is. Formal logic may also have been considered trivial because it seems to care so much more for the little words than the big ones. Formal logicians care not a fig for words such as 'consideration', 'craftiness', or 'catalogue'. But they will go mad for words like 'if and 'and' and 'all', not to mention 'or' and 'no'. Still, if we are to have sound business practice as well as rational thought and discourse, it is well that we have accountants and logicians. To be logical is to think, speak, or write in a certain way. To reckon is (1) to consider, to heed, to include, to regard, (2) to count, to compute, to calculate, to sum. I use the word in both senses in this essay. Logic is something to be considered when engaged in any kind of rational endeavour, and it is, in itself, something worthy of special regard. Any attempt at an understanding of ourselves as rational beings must include an assay of our logical abilities. And in all of our rational discourse we must xvii

always heed the dictates of logic. This first sense of 'reckon' is used implicitly throughout this essay. What I do explicitly here is focus on logic as reckoning in the second sense. Thus to be logical, to reason properly and correctly (and consciously), is to count, compute, calculate, sum. Of course, logicians have always treated logic this way. In this respect, the logic that dominates today is no different from that of earlier times. But today's logic offers a quite different notion of what it means to compute logically. Today's logic is "mathematical logic," the standard version being the first-order predicate calculus with identity. It was developed just over a century ago, its authorship eventually, and rightly, credited to Gottlob Frege. In the 1880s, Frege outlined a system of logic very different from the traditional one (syllogistic), which had dominated the field since Aristotle. What made Frege's logic so different, so revvolutionary, was that it was grounded on a completely new theory oflogical syntax (the study of those little words). A theory of logical syntax is a systematic account of the logical forms of all sentences that enter into deductive inferences. Based on its theory of logical syntax, Frege's logic proved to be very effective. It allowed modem mathematical logicians to build a logical calculus adequate to the demand to account for logical reckoning involving all sorts of sentences: categoricals, compounds, singulars, and relationals. It is now used to do a very wide variety of logical and mathematical tasks and serves as a basis for much research in linguistics, psychology, and computer programming. It is definitely something to reckon with. The great power and beauty of the standard, Fregean system, not to mention its hegemony in the schools, notwithstanding, it is not a perfect tool for logical reckoning. Its rapid and complete ascendancy has been due to the weakness of its rival-the old syllogistic. That system was based on a theory of logical syntax very different from the one familiar to Fregeans today. The old theory construed all sentences as categorical. As such it had to make some unnatural concessions in order to accommodate singular sentences, and compound and relational expressions were virtually unaccounted for. This system, based on such a theory of logical syntax, was easily supplanted by its new, more powerful Fregean rival. Still, there are some things to be said in favour of the old theory. For one thing, its account of the logical form of categorical sentences at least (and these are, arguably, the most common of natural-discourse sentences) was far more natural than the one offered by Fregean syntax. Many linguists today continue to parse sentences of natural language in a way similar to the old theory rather than the new. Oversimplifying, we may say that modem mathematical logic is powerful but often complex and unnatural; old syllogistic is natural and relatively simple but weak. Mathematical logicians tend to dismiss the charge that their calculus is complex and unnatural by denigrating natural language as inherently illogical anyway. Mathematical logic, after all, was

xviii

PREFACE

developed by mathematicians and philosophers interested primarily in establishing the foundations of mathematics (thus accounting for mathematical reckoning). Their search for a theory of logical syntax, therefore, was guided by the model of mathematical expression rather than by natural language. As could be expected, once the system was built it fit poorly with inference made in the medium of natural language. Yet natural language, not mathematics, is the primary medium for thinking, speaking, and writing in the vast majority of contexts. Mathematics does indeed need a logic, as Frege saw. But whatever the logical needs of mathematics are, they are not necessarily the logical needs of those who seek to understand natural language. What is required is a logic of natural language that rests on a theory of logical syntax general enough to account for the logical forms of all kinds ofsentences entering into deductive inferences, powerful enough to account for the validity or invalidity of all such inferences, applicable to sentences expressed in the medium of natural language, and simple enough to be used as a tool of reckoning by all who must or care to do so. The latest version of a revitalized, revised, strengthened syllogistic logic, by Fred Sommers, is based on an exceedingly simple theory of logical syntax. Elements of such a theory can be traced back to the beginnings of logic itself-to Aristotle. Elements abound in the work of old syllogists from then on, but they are especially abundant in the logical remarks of Hobbes, the extensive writings of Leibniz, and the work of nineteenthcentury algebraists such as De Morgan. The algorithm for this new syllogistic logic reflects the simplicity of theory of simple logical syntax in that it borrows directly from simple algebra or arithmetic. Moreover, this single algorithm is adequate to the demands of analysing deductive inferences involving categoricals, singulars, compounds, and relationals. When contrasted with the first-order predicate calculus with identity, the differences are striking. It not only matches but surpasses the standard calculus by analysing with ease a variety of inferences beyond the scope of the entrenched system. What makes the system more natural than the standard system is that it formulates natural-language sentences with a symbolism whose own syntax closely matches that of natural language. (Few logicians today would dare to claim a high degree of naturalness, in this sense, for the standard logic.) The fact of its naturalness makes it easy to learn and teach (a second, foreign, "logical" grammar is not needed). It is also easy to learn and teach because no new symbolism is required. We are already familiar with the symbolism of simple algebra or arithmetic, with its unary/binary ambiguity for plus and minus signs. And this is how it should be. For centuries, logic was in the hands of a small group of priests and philosophers, useful but unused. For the past century, logic has been in the hands of a different small group of mathematicians and philosophers, useful but too complex to be used by others. The theory of logic presented here offers the hope of a system easily learned and easily used by anyone. And ease is important, for

xix

PREFACE

there is no gainsaying the fact that in an ever more complex (not to mention dangerous) age we can no longer afford the luxury of rearing generation after generation of citizens ill-equipped for the rigorous demands of clear, critical, logical thought and expression. A simple, natural, but powerful, system of logic, accessible to and usable by a majority, would be an important propaedeutic. It would, indeed, be something to reckon with. Since I refuse to take all the blame for this essay I must take this opportunity to thank several people. Those few readers who may be familiar with my work already know the tremendous debt I owe to Fred Sommers. I firSt read Sommers while a graduate student in the mid-1960s. At the time, I had fallen under the influence of O.K. Bouwsma and the latter Wittgenstinians. I was enrolled in a seminar on category mistakes led by the logician Charles Sayward. We began with Sommers's 1959 Mind paper, "The Ordinary Language Tree." But I was not fooled. In spite of the title, which should bring a glow of warm anticipation to any good Wittgenstinian ready to examine it, the piece was filled with symbols, diagrams, and neologisms. His next paper, "Types and Ontology," was even worse-more symbols and diagrams. Despite my background in mathematics, I had become wise to the false charms of formal methods, and had developed a keen sense of the absurdity of philosophy. So I gave Sommers my first reading in a skeptical, even hostile, frame of mind. Still, I kept reading. I began corresponding with him and eventually had a first meeting in 1967. By then, I had come around. He had won me over-not by his personality (charming as it is), not by easy words or facile arguments-but by the sheer power of his ideas. His early work on category theory is not always easy reading, but the effort required to come to a full understanding of it is paid off many times over by the acquisition of a theory about the nature of language as a whole, and ontology as a whole, and their connection, which is a potentially powerful tool for the analysis of a very wide range of philosophical problems and concepts. This theory, the "tree theory," became the subject of my doctoral thesis and much of my subsequent work over the next several years. Sommers's work on the tree theory eventually led him (with me, as usual, following more slowly) to the considerations of logic that resulted in his building of a new syllogistic, as found in his many papers after 1970 and his 1982 book, The Logic of Natural Language. The present essay is the latest of a large number of papers, books, and anthologies that I have produced exploring and exploiting his logical theories. I have been inspired by his thought, probed by his questions, humbled by his criticisms, provoked by his suggestions, warmed by his humour. Often I have done his ideas less than adequate justice and have even plagiarized him with hardly an effort to conceal, and, in spite of all this, he has yet to have me called to court. For all this (and so much more), I thank him. Others, logicians and non logicians alike, who must be tarred with my brush of thanks are Bill Shearson, Harvey White, Jamie Crooks, Dale XX

PREFACE Stout, David Seale, Mary Rhodes, John Woods, Greg Scott, Bert Halsal, Guyla Klima, Wallace Murphree, Bruce Thompson, Aris Noah, Francine Abeles, Thomas Coats, Martin Gardner, Thomas Hood, Bill Purdy, Charles Kelly, Anthony Lombardy, Hartley Slater, Michael BOttner, John Corcoran, Susan Haack, Philip Peterson, Stephan Theron, and my former student and colleague Graeme Hunter, from whom I have learned far more than he did from me. I have thanked Morgan by dedicating this essay to him (for he was with me for every page). But that dedication could be extended to the others whom I never fail to reckon with: Russell, Suzanne, Gael, and Genevieve. Finally, but importantly, this book has been published with the help of a grant from the Humanities and Social Sciences Federation of Canada, using funds provided by the Social Sciences and Humanities Research Council of Canada.

xxi

INTRODUCTION In no branch oflearning can an author disregard the results ofhonest reseorch with so much impunity as he can in Philosophy and Logic. Wittgenstein "One way ofviewing mathematics is in terms of number. I guess you lcnow what the other way is. I'll say the word in a more expressive language so there 'II be no doubt exactly what it is we 're talking about. " "I wish you wouldn't. " "Logilc. "softly said. Don DeLillo

What Is a Logical Constant? But the English... having such varieties of incertitudes, changes and Idioms, it cannot be in the compos ofhuman brain to compile an exact regular Syntaxis thereof James Howell (1662)

Is it possible to say, in a clear and precise way, just what constitutes the distinction between logical expressions (fonnatives, syncategoremata, particles, constants) and nonlogical expressions (material expressions, categoremata, tenns, variables)? Pessimism concerning this seems to be the rule among most modem logicians. For example, Tarski: "No objective grounds are known to me which pennit us to draw a sharp boundary between [logical and extralogical] tenns" ( 1956: 418-9); Mates: " ...unfortunately the question as to which words should be considered logical and which not involves a certain amount of arbitrariness" (1965: 14); Quine: "Each such word [i.e., particle] is in a class fairly nearly by itself; few words are interchangeable with it sa/va congruitate. . . . Instead of listing a construction applicable to such a word and to few if any others, we simply count the word an integral part of the construction itself. Such is the status of particles" (1970: 29); Allwood, Anderson, and Dahl: "In the last instance it is a matter of decision whether a word belongs to the logical vocabulary or not" (1977: 24); and, of course, Russell: "The logical constants themselves are to be defmed only by enumeration, for they are so fundamental that all the properties by which the class of them might be

INTR.Q!lli£TION

defmed presupposes some terms of the class. But practically, the method of discovering the logical constants is the analysis of symbolic logic" ( 1903: 37). In the face of this overwhelming pessimism, we must remind ourselves that there have been earlier times of optimism. For example, Leibniz: "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' " (1966: 3); De Morgan: "I think it reasonably probable that the advance of symbolic logic will lead to a calculus of opposite relations, for mere inference, as general as that of+ and- in algebra" (1966: 26); Sommers: "All formatives-including propositional 'constants'-are analogous to plus and minus signs of arithmetic" (1973: 249). If these optimists are correct, not only is it possible to draw a clear and precise distinction between the logical and extralogical expressions of a language, it is also possible to give a very simple characterization of the nature of logical expressions-they are all signs of opposition, analogous to the oppositional signs of mathematics. Such prospects are surely attractive. If so, however, why have most modem logicians turned pessimistic? The answer seems to lie in the shift from a traditional account of logical syntax to a Fregean account. We will examine that shift more closely below. But we should recognize now that there are important consequences of the traditionalists' optimism with regard to the logicaVextralogical distinction and the nature of logical expressions. Any substantiation of the attractive prospects offered by the traditionalist view must inevitably cast some doubt on the generally accepted modem, Fregean, view and its concomitant pessimism. The systematic ambiguity of plus and minus expressions in mathematical language (an ambiguity that we will explore below) is not only benign, it is a source of great expressive power for the mathematician. Leibniz, De Morgan, and Sommers have suggested that natural language has a logic that, like arithmetic and algebra, makes use of two kinds of basic expressions, the signs of opposition. In fact, their common position seems to be that all expressions of natural language that carry the responsibility for determining logical form are either positive signs or negative signs or signs defmable in terms of these. If this is so, it means, among other things, that one could build an artificial formal language that would model natural language by using the mathematician's opposition signs for all formatives. The result would be an algorithm for natural language that would model natural statements as arithmetical formulae and inference as arithmetical calculation. There is little doubt that this was Leibniz's goal throughout his logical studies, and Sommers has come very close to reaching that goal in his own logical work. It would seem, therefore, that the idea of using signs of opposition to model natural-language formatives is a good one, leading, as it seems to have done, to rich programmes of logical investigation and to viable systems for logical reckoning.

2

INTRODUCTION

One of the consequences of this idea has been great optimism among those who have felt that a clear and precise account of the nature of logical formatives, and of their distinction from nonlogical expressions, can be provided. In a sense, their account is quite simple: logical formatives, unlike other expressions, are oppositional in just the way that plus and minus are oppositional in mathematics. But to appreciate fully this kind of account we will look closely, in later chapters, at the oppositional character of formatives, their roles in inferences, and the kind of algorithm that could model those inferences. The Problem ofSentential Unity One ofthe 'insights ' ofmodern logic has been that nouns and verbs are basically the same sorts ofthings; both may be symbolized as predicates ofthe same sort. T. Parsons

The problem of sentential unity is simply the problem of determining what accounts for the fact that some strings of expressions form sentences while others do not. Just what is it that ties words of a sentence together to form a single linguistic unit, a sentence? The problem is an ancient one. The earliest clear attempt to solve it was Plato's. While Plato cannot properly be called a logician (who before Aristotle could be?) he did demonstrate in the Dialogues an ability to use very subtle argument forms, and he had an unschooled but intuitive sense of the form/content distinction. Yet he did exhibit a tendency to make elementary logical mistakes and seems to have believed that an accumulation of arguments, however weak, strengthened one's position.• The problem of sentential unity is one that he tried to solve in a careful and clear way. In the Sophist, especially, he sought an appropriate way to analyse simple statements that could serve as premises or conclusions of arguments. The view he took was that such statements have a binary logical structure. At 261 d, Plato asked "whether all names can be connected with one another, or none, or only some of them." He then concluded (262c) that a sentence is a string of words in which "verbs are mingled with nouns," for "the fJrst and smallest" phrase is a combination of a noun with a verb-that is, the binary analysis. According to this view, the minimal requirement for a string of terms to constitute a statement is that the string consist of two terms, and, furthermore, that one of the terms be an onoma (noun) and the other be a rhema (verb). So, for Plato, there were at least two kinds of terms, nouns and verbs, and a statement required one of each. One cannot form a statement from a pair of nouns, nor from a pair of verbs; one of each is required. One cannot form an axe by combining a pair of axe-heads, nor by combining a pair of axe-handles; one of each is required. According to

3

INTRODUCTION

Plato•s binary analysis, the two tenns that fonn a simple statement are logically heterogeneous. They are unfit for each other•s sentential roles. When Plato said that a simple sentence was properly constituted just by a noun and a verb, he did not demand that some other expression or sign be present to connect them. He believed that they could just naturally "mix,.. or "blend,, or "combine,.. or "mingle,.. all on their own. Moreover, his theory of logical syntax was tied to his theory of Fonns. Indeed, he wanted to hold that the blending of a noun and a verb to fonn a statement was a reflection of the blending of the two Fonns signified by those tenns. Naturally, his theory of syntax inherited many ofthe criticisms aimed at his metaphysics. 2 As we will see later, the idea of a binary analysis for logical syntax has had a strong and lasting appeal throughout the history of logic and is well entrenched in today•s standard system. Aristotle was the first to offer an alternative analysis, but only after an initial period of devotion to his teacher•s binary theory. In both the Categories and De lnterpretatione, Aristotle held that a sentence was a combination of a noun and a verb. 3 However, by the time he wrote Prior Analytics he had come to adopt a quite different view. He invented syllogistic in the Analytics. A key requirement for syllogistic inference is that at least one tenn occur as subject-tenn in one statement but as predicate-tenn in another. In other words, Aristotle•s syllogistic requires that tenns be logically homogeneous, tit for playing more than one role in different statements. Thus, he had to abandon the binary theory, with its distinction between nouns and verbs. The distinction may have been good grammar, but it had no place in (syllogistic) logic. In Prior Analytics, Aristotle introduced for the first time the idea that a simple sentence must consist not only of a pair oftenns (horos) but of something else as well. What is required is a logical copula. While the tenninology is Abelard•s, the idea is Aristotle•s. He says (24bl6), "I call term (horos) that into which the premise is resolved, viz., the predicate and that of which it is predicated, with be or not be added ... In effect, Aristotle had adopted a ternary theory of logical syntax. According to this view, simple statements consist of pairs of tenns connected by a third expression. Since the sole function of this third expression is to connect the two tenns to fonn a statement, it is appropriate to call it a copula, for it literally copulates (connects, joins, fuses, binds, links, unites) the two tenns. While Abelard is usually given credit for introducing the word 'copula• into logic, what he had in mind was primarily the (Latin equivalent of) 'is• in such statements as 'Socrates is wise•. But as we will see, 'is• here is no copula-it is a qualifier. Plato•s logical insight was supplied by his recognition of the grammatical noun/verb distinction. He took it to be a logical distinction as well-thus the binary theory. Aristotle, under the pressure of building a theory of fonnal deductive inference, came to see that the grammatical distinction was of no logical import. From a logical point of view, tenns are

4

INTRODUCTION

homogeneous. But this makes the question of sentential unity all the more urgent, for now grammar cannot be appealed to in order to unite pairs of terms (as it could be in order to unite a noun and a verb). Aristotle took the simple, commonsense view that any pair of items must be linked by some third thing in order to form a unit. As we saw above, early on in Prior Analytics Aristotle claimed that forms of 'to be' and its negation could be taken as the third item, the link or copula. But as the work progressed, he came to realize that this was too simple a view. What was required was a single expression that could be used to link the terms of any (statementmaking) sentence. There were four kinds of"Aristotelian" logical copulae. Their English versions are: 'belongs to every', 'belongs to no', 'belongs to some' and 'does not belong to some'. Thus 'Every man is rational' would be regimented so that the copula expression stood between the two terms (making the terms quite literally termini, endpoints): 'Rational belongs to every man'. The result of connecting a pair of terms by one of Aristotle's copulae was, of course, a categorical statement. Note that the copula determined both the quantity and quality of the categorical, for it should be kept in mind that the features of quantity and quality were properly applied to a statement as a whole-not to terms alone. The binary and ternary theories represent two quite different solutions to the problem of sentential unity. The former solves the problem by claiming that sentential unity is the result of two expressions naturally fitting together, mixing, blending, and so on, by virtue of the fact that they are formally different but nonetheless complementary to one another, so that together they form a unit. The second theory, having denigrated the grammatical differences between expressions (at least for logical purposes), solves the problem by positing a third expression, the sole duty of which is logical copulation. Unity is the result of the connection (via a connector) of a pair of expressions. Versions of each of these theories are still offered today. After a long detour, we shall return to the problems of logical constants and sentential unity. We hope that by then the best choice of solutions will be obvious.

5

Notes for Introduction See Patzig (1972) and Bochenski (1968), pp. 14-18. See Ackrill (1957), Kahn (1972), and Kneale and Kneale (1962). 3 Categories 1a16ffand De Interpretatione 16a1-17a37 (in Ackrill, 1963). 1

2

6

Part I

CHAPTER ONE

mE GOOD OLD DAYS OF THE BAD OLD LOGIC (or, Adam's Fall)

Logic, n. The Ol't ofthinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. Understanding, n. A cerebral secretion that enables one having it to know a house from a horse by the roofon the house. Its nature and laws have been exhaustively expounded by Locke, who rode a house, and Kant, who lived in a horse. Ambrose Bierce

Aristotle's Syllogistic L 'invention de Ia forme des syllogismes [est] l'une des plus belles de /'esprit humain, et meme des plus considerables. Leibniz Sweet Analytics, 'tis thou hast ravish 'd me. Marlowe Bachelors and Masters ofArt who do not follow Aristotle's philosophy are subject to a fine of5 shillings for each point ofdivergence. -fourteenth-century rule, Oxford University

Aristotle's syllogistic logic rests four-square on the theory of logical syntax worked out in Prior Analytics. As we have already seen, this theory is ternary, taking simple sentences to consist (logically) of a pair of terms connected by a logical copula. Aristotle had adopted the ternary theory of logical syntax only after abandoning the earlier binary theory, which he had shared with Plato. At least one prominent philosopher has likened Aristotle's shift here to Adam's fall. 1 According to Geach, the binary theory represents an earlier state of grace, which Aristotle foolishly abandoned after being

9

CHAPTER I

blinded by the false promises of syllogistic. The De /nterpretatione view that a noun and a verb are different in logically important ways (e.g., only the latter is tensed, only the latter can be negated) should never have been forsaken. But the development of syllogistic in Prior Analytics required the terms of a statement to be such that they could occur in either of the two term positions ("Aristotle's thesis of interchangeability"), and this thesis led directly to what Geach calls "the two-term theory,'' namely, the ternary analysis: statement-making sentences are logically formalizable as termcopula-term. The real sin of the two-term theory was its commitment to the logical homogeneity of terms, its failure to preserve in logic the grammatical noun/verb distinction. Geach concurs with most modem logicians in holding such a distinction to be foundational. Referring to nouns as "names" and verbs as "predicates,'' Geach claims (48), "It is logically impossible for a term to shift about between subject and predicate position without undergoing a change of sense as well as a change of role. Only a name can be a logical subject; and a name cannot retain the role of name if it becomes a logical predicate." Geach is defending the view most commonly held nowadays by logicians, according to which there are two quite distinct roles to be played by terms: referential and predicative. Names (viz., proper names and personal pronouns) are the only terms suited for the first role; only (verbalized) general nouns and adjectives and verbs are suited for the second.2 In Prior Analytics, the referentiaVpredicative distinction, a semantic distinction that even Aristotle could not deny, is not reflected in a syntactical distinction, and it is this failure that seems most to exercise Geach. According to Geach's reading of logical scripture-history, this failure, when coupled with the ternary theory (the two-term theory), unavoidably led to the ''two-name theory." The two-name theory was a transgression even more grave than the two-term theory. It was, says Geach, the theory of logical syntax held by the Scholastics, and later Mill, and even later Lesniewski. By Geach's description, the two-name theory parses statements as pairs of names (referential expressions) linked by a copula (viz., a form of 'to be'). And this copula is a mark of identity! Still, Adam's fall was not complete. It ended, for Geach, only with the coupling of the two-name theory with the view that the referent of a term is the class of individuals denoted by that term. When conjoined, these two theories led, finally, to the ultimate apostasy: "the two-class theory." This theory, with its doctrine of distribution based on a confusion concerning reference and denotation, is what is usually called ''Traditional Logic." And Geach's condemnation of the heresy of traditional logic is uncompromising: "Between such logic and genuine logic there can only be war'' (54). And to think this all started with Aristotle's innocent attempt to build a formal logic, and thus to shift from a binary to a ternary analysis. Did Aristotle really eat the forbidden fruit when he opted for the term-copula-term theory? One way to look for an answer is to look at 10

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

syllogistic, the logic made possible by the adoption of that theory. Those who believe that syllogistic logic (especially in its guise as Traditional Logic) is fit to be taught only in "Colleges of Unreason," that it is hopelessly inadequate to the demands of genuine logic, that its rival (modem mathematical logic) has shown itselfto be superior in every way, will, with Geach, happily reject the ternary analysis, syllogistic and, perhaps, Aristotle himself. 3 I do not share this belief. Aristotle's syllogistic is the first systematic, fully articulated formal togic. 4 It is not "traditional logic," but it does serve as the historical foundation for that logic. The crucial elements of Aristotle's syllogistic are its underlying ternary theory of logical syntax; its recognition of two kinds of negation; its use of the perfect fJrSt-figure forms as basic; its construal of proof as reduction to perfect forms; its explicit use of a small number of immediate inference patterns as rules of proof; and its implicit use of the dictum de omni et nullo as a rule of proof. There are certain important limitations on Aristotle's version of syllogistic: it ignores, for all practical purposes, the logic of compound statements; it deals with relational expressions only in a brief and superficial way; and it generally excludes singular statements from the domain of logical analysis. We have already seen how he abandoned the binary theory of logical syntax for the ternary theory. Aristotle never gave up the notion that statement-making sentences (statements), the kinds of expressions that enter into inferences as either premises or conclusions (as opposed to prayers, commands, etc.), are essentially copulated pairs of terms. Since he wanted to reject the grammatical noun/verb distinction for logic, thus allowing any term to occupy any term-place in a statement, he no longer had available to him the grammatical mechanism that permitted nouns and verbs to combine without an intermediary to form a linguistic unit. Once the two terms of a statement are seen as logically undifferentiated, no grammatical device can unite them. What Aristotle needed was a logical device for connecting the pairs of terms to form statements-the logical copula. In fact, Aristotle found four logical copulae. Throughout Prior Analytics he repeatedly refers to statements simply by listing their terms,-for example, 'AB', 'MN', and so on. I call these "protopropositions." In such cases a connecting logical copula is understood but left implicit for various purposes.5 For example, when determining syllogistic figure all one needs to pay attention to is the arrangement of terms. For such purposes, proto-propositions are sufficient. However, no statement consists ofjust a pair of terms; some expression, the sole duty of which is to connect, is required. At 24al6-23, Aristotle defines quantity in terms of his four copulae. From our post-traditional point of view, this seems to make no sense. We have learned to think of both quantity and quality as primitive logical concepts in traditional logic. And they are. But they were not in Aristotle's syllogistic. Statements, according to Aristotle in Book I, chapter I, of Prior Analytics, are either affirmative or negative

11

CH~I

and either universal, particular, or indefinite. But this is so not because, as the Scholastics would tell us, one term is quantified and the other qualified in certain ways. but because the tenn pairs are connected by one kind of copula or another, by virtue of which the entire statement is quantified and qualified. For Aristotle, every statement has the general logical form: A applies to (hyparche1) some/every B Here 'applies' is generic for either 'belongs' or 'does not belong'. So we have the four Aristotelian logical copulae: 'belongs to some', 'belongs to every', 'belongs to no'(= 'does not belong to some'- i.e., 'fails to belong to some'), and 'does not belong to every' (i.e., 'fails to belong to every'). The idea that a statement is the result of combining a quantified term and a qualified term is not Aristotle's. As we will see, it belongs to the traditional logic built first by the Scholastics.' The original syllogistic required that in each syllogistic inference at least one term occur in a statement as the first term and in another statement as the second term. This meant that term positions were undetermined with respect to either grammatical distinctions or semantic features. Any term was logically fit for any term position. The true logical work of a statement was determined by the expression connecting the two terms. I said above that Aristotle's recognition of two kinds of negation was a crucial element for his syllogistic. In the next chapter, we will see that modem logicians generally take all logical negation to apply to entire statements and never to terms alone. In other words, for modem logicians, generally speaking, all negation is sentential, never terminal. Yet Aristotle, and indeed virtually all logicians before Frege, recognized two kinds of negation. 7 Aristotle's two kinds of negation are tenn negation and term denial. The negation of a term, say P, results in a new term, nonP, which is (logically) contrary to the original term. Aristotle's discussion of contrariety (e.g., at Categories 12a26-12bS, De lnterpretatione 23b23-24 and 24b7-l 0 and Metaphysics lOSSa34) is far from clear. What is clear is that, at the very least, Aristotle wished to connect his view of logical contrariety with his notion that accidents but not substances have contraries. Two accidents are contrary whenever, with respect to a given primary substance, at most one can be truly applied. Thus, Socrates cannot be both white and red. Aristotle also holds that a substance can be privative with respect to some quality, in the sense that it is the sort of thing that could sensibly be said to have that quality but actually has some contrary quality. Thus, given that Socrates is white he is not also red. But he is the sort of thing which could sensibly be said to be red. So, Socrates is privative with respect to red. He is nonred. Of course, he is also nonblue, nongreen, nonorange, and so on. At Categories 12a26-12bS, Aristotle gives the example of a stone, which is not sighted but also not nonsighted (=blind) since it is not the sort of thing to be either sighted or privative with respect to sight

12

THE GOOD OLD DAY_!2E!HE BAD OLD LOGIC

Aristotle's second kind of negation is term denial. Two of his logical copulae are negative, used to deny rather than affirm one term of some or all of another. Let A and B be two terms. One way to form a statement from these is to connect them with the expression 'belongs to some': 'A belongs to some B.' Aristotle would say that this is a case of affmnation. 'A belongs to every B' would be another affirmation involving the same terms. Denial is achieved by replacing an affirmative copula with a negative one. Thus, 'A belongs to no 8' denies what 'A belongs to some B' affirms. The logical relation between such pairs of statements is contradiction. Two statements are contradictory only if exactly one is true and one is false. The use of 'belongs to some' and 'belongs to no' to connect the same pair of terms results in a contradictory pair of statements. The use of'belongs to every' and 'does not belong to every' to connect the same pair of terms results in a contradictory pair of statements as well. How are Aristotle's two kinds of negation related? Suppose that two terms are connected by the same copula in two statements except that the first term is negated in one, for example, 'A belongs to every B' and 'nonA belongs to every B.' It is clear that it is impossible for both statements to be true at the same time. Still, the two statements are not contradictory, since it is surely possible for both to be false. Consider 'Bald belongs to every logician' and 'Nonbald belongs to every logician,' neither of which happens to be true. Let's take another look at the negative copulae. We said that 'belongs to no' could be paraphrased by the expression 'does not belong to some.' Now, if A does not belong to B, it follows that B is nonA; thus, if 'bald' does not belong to some logicians, then 'nonbald' must belong to every logician. In other words, to deny a term of some/every X is to affirm its negation of every/some X. To summarize, a genuine "Aristotelian square of opposition" would look like this. A belongs to every B

A does not belong to some B (=nonA belongs to every B)

A belongs to some 8

A does not belong to every B (= nonA belongs to some B)

13

CHAPTER I

This is not familiar to us because it is not the traditional square of opposition that was developed by the Scholastic logicians. 8 Aristotle's favoured way of proving a valid syllogistic inference was reduction. In Book I, chapter I, of Prior Analytics, Aristotle makes an important distinction between perfect and imperfect syllogisms. A perfect syllogism is one such that the conclusion can immediately be seen to follow of necessity from the premises (cf. Prior Analytics 25b32-35); an imperfect syllogism requires one or more statements in addition to the premises before the implication of the conclusion can be seen. Perfect syllogisms need no proof of their validity, which is obvious, easily grasped by any rational being. But the validity of imperfect syllogisms must be demonstrated by a proof. 9 Aristotle had divided syllogisms into three "figures" (Book I, chapters 4-6). He made it clear there that he thought of a syllogism as a linear arrangement of terms, with the terms of the conclusion at the endpoints (viz., the "extremes") and the term occurring in each premise but not the conclusion in the middle (thus the "middle term"). When the terms of a syllogism are so arranged, and when the generalities of the terms are in descending order (e.g., 'animal', 'man', 'logician'), the syllogism is perfect-it is in the first figure. A valid syllogism in another, imperfect, figure can be proved by manipulating the positions of premises and of terms so as to produce a first-figure syllogism. Every syllogism is either perfect or can be reduced to one that is (29bl-2). However, sometimes this reduction is not actually to a first-figure syllogism but to one that premises the denial of the conclusion of a first-figure syllogism and concludes the denial of one of its premises. This kind of reduction is called by later logicians reductio per impossible or reductio ad absurdum. The rearrangement of premises or terms in imperfect syllogisms that results in a reduction to perfect syllogisms must be carried out systematically, according to rules. The rules were not always stated explicitly by Aristotle, but he did make use of such now-familiar rules as subaltemation, simple conversion, accidental conversion, contraposition, and obversion. Many modem logicians, following the lead of Lukasiewicz ( 1957), think of Aristotle's syllogistic as an axiomatic system modelled on Russell and Whitehead ( 191 0-13), even holding that the syllogistic itself is based upon the statement logic developed first by the Stoics and then again by modern mathematical logicians. 10 As will become increasingly obvious throughout this essay, I have little sympathy for such a view. I see Aristotle as using the rules of conversion as natural deduction rules governing immediate (single premise) inference. Mediate (syllogistic, two-premised) inference is governed by what the Scholastics called the dictum de omni et nullo. Proof-that is, reduction-is achieved, according to Aristotle (29a30ft), by applying the conversion rules to statements in imperfect syllogisms. The result will be a perfect syllogism. Perfect syllogisms are

14

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

valid by virtue of the dictum. Thus the dictum governs all valid syllogisms. Consider the four perfect syllogisms. I. A belongs to every B, B belongs to every C, so A belongs to every C 2. A belongs to no B, B belongs to every C, so A belongs to no C 3. A belongs to every B, B belongs to some C, so A belongs to some C 4. A belongs to no B, B belongs to some C, so A does not belong to every C (=nonA belongs to some C) A single principle can be formulated to describe each ofthese inferences: What does or does not belong to all or none ofsomething likewise does or does not belong to what that something belongs to-that is, the dictum. And since every valid syllogism is reducible to a perfect syllogism, the principle applies to all valid syllogisms. Aristotle himself came close to formulating this principle (Categories lb9-IS; Prior Analytics 24b27-31, 32b38-33a5). Needless to say, we will return to the dictum on several occasions. Aristotle thought of his syllogistic as a tool for the teaching of theoretical sciences. Unlike logic itself, the theoretical sciences were taken to be axiomatic. A syllogistic inference was used to deduce theorems from the universally and necessarily true principles of a given theoretical science. In Posterior Analytics, he set out to show just how this is so. Note, however, that this view of logic and its relation to science places an important limitation on the logic. Aristotle's logic is not universal; it is incomplete qua formal logic. One way in which the Aristotelian syllogistic is incomplete is in its failure to give any real account of inferences involving compound statements (the so-called hypotheticals). The Stoics invented just such a logic. In Prior Analytics, Aristotle was surely not unaware of such inferences (4Sbl5-20), but held them to be in some sense irreducible and thus inferior to categorical syllogisms (SOa 16-24). 11 Later Peripatetic logicians, such as Theophrastus, Alexander of Aphrodisias, and Boethius, did develop a syllogistic incorporating compound statements. But the ancients, including both Peripatetics and Stoics, were never clear about just what the relationship was between a logic of categoricals (a term logic) and a logic of compounds (a sentential logic). All of the textual evidence seems to suggest that the latter was patterned after the former. This, in tum, suggests that the logic of compounds may not be a part or derivative of the logic of categoricals, but a parallel, isomorphic, logic. A second limitation of Aristotle's syllogistic is its lack of a way of analysing inferences involving relational expressions (the so-called oblique

15

CHAPTER I

tenns, because they are tenns in oblique grammatical cases, cases other than nominal). Aristotle did say something about such inferences, 12 but he failed to incorporate them systematically into his syllogistic. Serious attempts to do so were eventually made by Leibniz and, still later, by De Morgan. But a satisfactory logic of relationals was not available until the advent of mathematical logic. My survey of Sommers's work in chapter three will show that a properly fonnulated logic of tenns can indeed incorporate relationals. It is the third limitation of Aristotle's syllogistic that is most directly the result of his view of the relationship between logic and theoretical science. Since the axioms of any theoretical science must be universal and necessary, they cannot make reference to individuals. Thus it would seem that no principle (axiom or theorem) of theoretical science can be stated in the fonn of a singular proposition. Various arguments to this effect can be found in the literature. 13 Nonetheless, Aristotle himself never actually bars singulars from syllogistic: indeed, he gives examples of syllogisms with singulars (e.g., Prior Analytics 43a35ff and Posterior Analytics 90a5-25). Still it must be admitted that Aristotle does not thoroughly and systematically show how singulars are incorporated into syllogistic. This, too, is be a topic that we will pursue more than once in this essay. Scholastic Additions There is another use of syllogistic, namely. that it enables one in a learned dispute to vanquish an uncautious adversary. But as this only belongs to the athletics ofthe learned, an art. however useful it may be otherwise, and does not contribute much to the advancement oftruth, I pass it over in silence. Kant

The Scholastic period in logic, especially from Abelard in the eleventh century to Ockham in the fourteenth, was a long, complex, often confusing series of attempts to recover and strengthen Aristotle's syllogistic. Of course, by that time much of what passed as Aristotelian contained material that belonged not only to later Peripatetics, but to Stoics and Megarians as well. Scholastic logic, a logic that still thrives in certain quarters, became what most philosophers now think of as Traditional Logic. While it was clearly based upon and inspired by Aristotle, its additions took it far beyond the Master. The Scholastic logicians amended and emended the original syllogistic in a number of ways, one of the most important of which was to add to it very elaborate semantic theories (usually under the heading of theories of"supposition"). These logicians tended to see logic as dealing

16

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

with what were later referred to as the "acts of the intellect." The first such act (simple apprehension) would deal with terms per se, the second (composition and division) would deal with statements (terms joined to form statements), and the third (reasoning) would deal with inferences (statements joined to form syllogisms). Each such branch of logic was supposedly inspired by a work of Aristotle's Organon in which he laid down the fundamental elements. Thus Categories was the inspiration for apprehension, De lnterpretatione for composition and division, and Prior Analytics for reasoning. It is in their discussions of apprehension that the Scholastics developed their semantic theories. 14 Medieval logicians generally took terms to have first signification and then supposition. The signification of a term was that by virtue of which it supposited (stood for) objects. The theory of supposition was intended to give a full account of just how a term could standfor objects. Some Scholastics distinguished between supposition and appellation, where the latter was the function of predicate-terms and the former the function of subject-terms. (I will discuss the notions of subject and predicate below.) Terms used in their normal sense (i.e., literally as opposed to metaphorically) were said to have proper (or sometimes natural) supposition. Here the term stood for, supposited, every object that does or could satisfy its signification. Proper supposition is what, after Mill, came to be called "extension" Gust as signification came to be called "intension"). A term with proper supposition can be either mentioned in a sentence or used in a sentence. When a term is mentioned it supposits itself (as 'red' does in 'Red is three-lettered') and is said to have material supposition. When a term is not mentioned but used it is said to have formal supposition. In the normal case, the terms of any statement all have formal supposition. Such terms are also said to have either simple or personal supposition. Terms have simple supposition, it seems, when they denote the properties, concepts, ideas, or universals determined by the term's signification. Thus, in 'Man is a species' and 'Wisdom is rare' the terms 'man' and 'wisdom' denote species or properties, and so are said to have simple supposition. A term has personal supposition when it is used to denote objects (Aristotelian primary substances) rather than universals (Aristotelian secondary substances). Terms used with personal supposition are either singular or general. A singular term used with personal supposition is said to have discrete supposition. A general term so used is said to have common supposition. There are two kinds of common supposition. The subject-term of a particular sentence is said to have determinate supposition; all other terms are said to have confused supposition. Terms with confused supposition consist of predicate-terms of aft"mnations, said to have pure supposition, and subject-terms of universals as well as the predicate-terms of negations, said to have distributive supposition. As it turns out, singular terms (those with discrete supposition) and terms with distributed supposition are logically distributed terms. The

17

CHAPTER I

rest, those with detenninate or pure supposition, are logically undistributed tenns. The above is, at best, a sketch of an enonnously large, complex topic. But what will be of most interest to us is the doctrine of distribution just alluded to above. Post-Scholastic logicians have tended to divide themselves between friends and foes of distribution. Soon, I will enlist on the side of distribution's friends. The Scholastics' theory of logical syntax is found in their discussions of composition and division. A statement is seen as either the composition or the division of pairs of tenns. Tenns are composed in affinnations, divided in denials. It would be a mistake, however, to think that these logicians simply bought wholesale Aristotle's theory of logical syntax, the ternary theory. As we have seen, an "Aristotelian" copula is a single (though many-worded) expression, namely, 'does/does not belong to some/every,' which, upon logical regimentation, comes between a pair of tenns and unites or binds them to fonn a sentence. But the technical tenn 'copula' (or its Greek equivalent) was never used by Aristotle. That innovation belongs to Abelard 15 (here I follow Kahn, 1972). At De lnterpretatione 21a2S, Aristotle raises the question of how we can attribute a property to what does not exist. For example, how can we say of Homer that he is a poet without thereby asserting as well that he is (i.e., is real, exists)? Aristotle's solution is to claim that in a statement such as 'Homer is a poet', 'poet' is predicated of 'Homer' but 'is' is only "accidentally predicated" of'Homer'. The notion of accidental predication, at least as used here, is far from clear. It was in his attempt to shed light on this notion that Abelard introduced the technical expression "copula." Abelard's theory of logical copulation is summarized by Kahn in the following theses. I. Every simple declarative sentence can be rewritten in the fonn X is Y and in particular every sentence of the fonn NV can be rewritten in the fonn N is Ving (where "N" stands for a noun fonn and "Jl'' for a verb). 2. In a sentence of the fonn X is Y, X and Yare tenns (in the sense of the tenns of a syllogistic premise), whereas is is a meaningful third part which is not a tenn. 3A. In such a sentence, the meaning of is is that of a sign of affinnation, signifying that the predicate Y is atranned of-said to belong to or to be true of-the subject X Similarly, is not is a sign of denial. 3B. (The same point otherwise expressed:) in X is Y, the verb is serves to link Y to X and thus to combine them in a complete sentence (or proposition), i.e. one which can be true or false. 4. In such a sentence is serves merely as a link or copula (in the sense of3A-B) and not also as a predicate which asserts the existence of the subject. S. In the ordinary NV sentence the verb fonn serves twice: first as predicate tenn (like Yin X is Y) and again as copu/ans or linking element. 18

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

The rewriting of NV sentences as N is Ving according to I. above (e.g. rewriting John runs as John is running) serves precisely to bring out this double role of the verb. (Kahn, 1972: 146) Notice first that Abelard has solved Aristotle's problem with accidental predication by denying that 'is' in a sentence such as 'Homer is a poet' is any kind of predicate at all. It is always, in such sentences, perfonning the logical task of linking. Also, in sentences with finite verbs other than 'is', the verb is forced to take on both the task of serving as one of the two tenns and the task of linking (Abelard called these "copulative verbs" [verbum copulativum]). Compare this with the Platonic theory of the Sophist. For Plato, the basic fonn of a statement is noun-verb (NV). Abelard held that in such sentences the verb is not only a tenn of the sentence but also, implicitly, a copula. In effect, such verbs contain the logical copula within themselves. So, for logicians after Abelard, the basic fonn of a statement is tenn-copula-tenn. Noun-verb statements must be regimented (the verb split into copula and tenn) to yield the logically basic fonn. This theory put an end to the Platonic binary theory of logical syntax. The idea that the basic fonn of elementary sentences could be viewed as noun-verb did not return to logic until the late nineteenth century. Where Abelard analysed, say, 'John runs' as 'John is running', these later logicians analyse 'John is running' as 'John runs'. Still, the theory proposed by Abelard is not quite Aristotle's ternary theory. Specifically, Abelard's copula is not Aristotle's. While Aristotle does, especially in De lnterpretatione, sometimes cite 'is' and 'is not' as logical copulae, his idea of such a copula is always that it is some kind of expression that serves solely to link or unite pairs of tenns to fonn sentences. For him, every expression used to link tenns can be rewritten as one of his four logical copulae. There is another reason that Abelard's copula is not Aristotle's, and this goes to the heart of the Scholastics' theory of logical syntax. Logicians of the Middle Ages could hardly have ignored, much less been ignorant of, Aristotle's ternary analysis of statements. In fact, they eventually came to adopt agreed-upon symbols for each of his four copulae, writing each of his four categorical fonns as, for example, •AaB', 'AeB', •AiB', and •AoB'. Here the 'a', 'e', 'i', and 'o' were seen literally to link the tenn pairs. But later Scholastic logicians also recognized that an Aristotelian copula always gives two kinds of infonnation about the sentences in which it operates, indicating both the quantity of the sentence and its quality. It indicates whether the sentence is universal or particular (indefinite sentences being implicitly one or the other), and whether the sentence is afflnnative or negative. Sentences fonned using Aristotelian copulae are as unnatural in our English as they were in the Latin of the Schoolmen, so they decided to cast Aristotle's regimented sentences into more natural Latin, and in so doing split the two roles of the copula. Aristotle's copulae became the

19

CHAPTER I

Scholastics' quantifier/qualifier pairs, the syncategoremata, which literally go 'with the categoremata (tenns)' to fonn a sentence. In Aristotle's tenninology, the tenn belonging or not belonging is the predicate; the other is the subject. This distinction is logical rather than grammatical (e.g., the noun/verb distinction). For the Scholastic logicians a sentence of the fonn •A belongs to some B' is parsed first as 'A belongs to/some B', with the copular expression now split, with one part, the quantifier, going with the subject-tenn and the rest, the qualifier, going with the predicate-tenn. Then, applying the restrictions of Latin grammar, the order of subject and predicate is reversed. Finally, the qualifier is replaced by Abelard's copula. Thus the logical fonn of any statement for the Scholastic becomes quantifier-tenn-qualifier-tenn (or quantifier-tenn[Abelard's]copula-tenn), the standard categorical fonn. But I said above that the basic sentence fonn for logicians after Abelard was tenn-copulatenn. Which is it? The difference here turns on the presence of singular subject-tenns. The example from Aristotle that Abelard first had in mind was 'Homer is a poet'. Such a sentence, having a singular subject-tenn and thus no explicit quantity, could be easily construed as a singular tenn, 'Homer', and a general tenn, '(a) poet' (no articles needed in Latin), linked by the copula, 'is'. Such a sentence was a model for Abelard's further analysis. But singular sentences are not logically typical (and certainly were not for Aristotle). General statements are more typical, and they always have a quantity (though its quantifier is sometimes implicit). Such sentences are analysed as having the logical fonn quantifier-tenn-copulatenn. We will see that most medieval logicians then fit singular sentences to this fonn by taking them to have implicit universal quantity (thus 'Homer is a poet' would be analyzed according to the pattern: [quantity]-tenncopula-tenn). But note now that the copula is no longer a logical copula; it is merely a qualifier. It serves not to link or unite the two tenns, but merely to indicate the quality of the sentence. In one sense we could say that the Scholastic analysis (quantifier-tenn-qualifier/copula-tenn), now the standard categorical analysis of Traditional Logic, is neither binary nor ternary but quaternary. But Traditional Logic has tended to view the analysis as binary. It has done this by taking the quantity-term expression to constitute a single logical unit, the Subject, and the quality/copula-tenn expression to be a second logical unit, the Predicate. As we shall see, more than a little mischief has been caused during the past century by the tendency to confuse a Subject-Predicate theory of logical syntax with a noun-verb theory. In a more proper sense, however, the Scholastic analysis is ternary. The two terms of a sentence are indeed connected by a third, nontenninal expression-a logical copula. But, unlike Aristotle's copula. the logical copula (not just the qualifier construed as copula) is simply split into two parts. The quantifier and the qualifier of a sentence are merely the two discontinuous parts of a single logical expression. The quantifier alone 20

THE GOOD OLD DAY..!QEIHE BAD OLD LOGIC

clearly does not link tenns into sentences. Nor does the qualifier (even if mistaken for a copula). The uniting of pairs oftenns to fonn sentences is accomplished jointly by the quantifier and the qualifier. Viewed in this way, the traditional theory of logical syntax is ternary. In an introductory philosophy text written by an anonymous Parisian master in 1245, one fmds this simple claim: "The art of disputation is logic."" For the men of the schools this was a commonplace. In the Middle Ages, schools were the product of the Church; naturally, the methods of the Church became the methods of the schools. In theological matters, disputes concerning the proper reading of a biblical passage were detennined, or settled, by rigorous argument. In the school setting, disputation became the means of leading a student to the truth. A master raised a quaestio concerning the reading of a given text (say, one by Aristotle). Students then presented arguments from reason or authority for various ways of reading and interpreting the text. The various arguments were, of course, incommensurate. The master then detennined, or settled, the question by producing a demonstration in the fonn of a rigorous argument establishing the proper, true interpretation. 17 This method of teaching, the practice of disputation, depended heavily on familiarity with a set canon of doctrines and methods concerning argumentation-a logic-shared by both master and students. The logic was, naturally, Scholastic. Since during this period the students entering the schools were boys generally aged twelve and thirteen, and since logic was seen as the essential tool for learning, methods were constantly being devised to aid the learning of logic. Keep in mind that logic was not a topic of disputation itself in the schools-it was the art, the very method, of disputation. Its tenninology, rules, and practice were already established. Students first learned them and then learned theology, medicine, or law by applying them. The teaching of logic to young adolescents relied especially on a variety of mnemonic devices. By the early thirteenth century, William of Sherwood had offered the verse containing the now-familiar names of the valid syllogistic moods in the four figures.•• These traditional names were a boon to students, since they constituted, in effect, recipes for the reduction of imperfect to perfect syllogism. In Prior Analytics (41 b6-36), Aristotle had already laid down some necessary conditions for syllogistic validity-for example, at least one premise must be universal, at least one premise must be affinnative. To these the Scholastic logicians added rules drawn from the doctrine of distribution to yield a set of necessary and sufficient conditions for validity. Such a set could then be used (e.g., by students) as a decision procedure to be applied to any inference under examination. Distribution was seen as a property that a tenn had relative to a sentence in which it was used. Aristotle seems to have had at least a primitive notion of such a property when he talked in De lnterpretatione (esp. 17b12-13 and l8al-2) oftenns being used in their "fullest extension." 21

CHAPTER I

The traditional laws of distribution demanded that in any valid syllogism the middle tenn must be distributed at least once, and that any tenn distributed in the conclusion must be distributed as well in the premise. Fonnulating these laws was easy. What was never easy, from then until the demise of Traditional Logic after the nineteenth century, was an adequate explication of the notion of distribution itself. Traditional logicians drew their doctrine of distribution in part from their semantics. Thus, any lack of clarity in their semantic theories was inherited by distribution theory. (I hope to show in the second part of this essay that a distribution theory based on syntax is viable and explicable, and thus preferable to the traditional theory.) The logic of the Schoolmen drew its texts and inspiration from Aristotle's Organon. Did it suffer from the same limitations that restricted the original syllogistic system? Could Scholastic logic offer adequate analyses of inferences involving singulars, relationals, and compounds? For Aristotle, as we have seen, syllogistic was a tool for the teaching of and research in the theoretical sciences, such as physics and theology. This seems also to have been the attitude of the Scholastic logicians. Syllogistic was a tool, an organon, for carrying out and teaching the theoretical sciences. But, of course, the first theoretical science, theology, was now something different. Aristotle could virtually ignore the role of singular tenns in his syllogistic, but the Scholastics could not. Teaching theology in Paris in 1250 or Oxford in 1450 meant dealing with such statements (and thus any inferences in which they might occur) as 'The Apostle Peter is a man', 'God is good', 'Christ was born of a virgin', 'Invisible God created the visible world', and so on. Singular statements can find a place in syllogistic only if they can be seen to have a logical fonn appropriate to the theory of logical syntax at the base of that logic. Since for the Scholastics statements involved in syllogisms must have the general logical fonn 'quantifier-tenn-qualifier/copula-tenn', singular statements having the grammatical fonn 'singular tenn-verb' must be reparsed. The verb, as usual now, was easily rewritten as 'qualifier/copula-tenn'. The singular tenn was then rewritten as a universal quantifier plus the tenn. The justification for this new move was provided by the doctrine of distribution, for a tenn was, generally, said to be used distributively in a statement when it was used to refer to every individual for which that tenn had personal supposition. Since a singular tenn such as 'Socrates' in 'Socrates is wise' is being used to refer to every individual for which 'Socrates' has personal supposition (viz., just Socrates), it is distributed. Universal quantity is the mark of any distributed subject-tenn, so singular subject-tenns were given an implicit universal quantity. This done, singulars could be incorporated into syllogistic on all fours with general tenns. By the thirteenth century it had become customary to include in texts and compendia of logic sections dealing with what had come to be called consequentia-the logic of propositions. 19 Scholastic thought

22

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

concerning propositional, or sentential, logic was deeply influenced by two ancient sources: the Stoic and Megarian theories of implication and Theophrastus's work on hypothetical syllogisms. Scholastic logicians included a wide variety of sentence forms under the heading "hypothetical." These included not only conditionals, conjunctions, and disjunctions, but non-truth-functionals such as causal sentences. Some logicians, such as Ockham, tended to keep the logic ofhypotheticals separate from the logic of categoricals (syllogistic), but later logicians tended to incorporate all logic into a general syllogistic theory. It seems that a fully articulated general theory of logic for compound sentences is not to be found among extant Scholastic literature, and this in spite of the enormous amount of attention paid to the subject, especially during the thirteenth and fourteenth centuries. This is partly due to the fact that these logicians were never fully clear about just what the relationship between a term logic and a sentential logic should be. As well, much of the discussion of so-called hypotheticals involved modal issues. Ivan Bob (1982) has suggested that the etymology of"consequentia" might provide another clue. The relationship of"following along" was commonly felt by medieval thinkers to hold between concepts, terms, propositions, and premise-conclusion pairs. Scholastic logicians seemed to take little care in distinguishing between saying, for instance, that the concept male is a consequence of the concept bachelor, the term 'male' is a consequence of the term 'bachelor', the consequent of the conditional 'If he is a bachelor then he is a male' is a consequence of the antecedent, the proposition 'He is a male' is a consequence of the proposition 'He is a bachelor'. and the conclusion 'He is a male' is a consequence of the premise 'He is a bachelor'. We will soon see that Leibniz intentionally tried to eliminate some of these distinctions, and later we will question whether such distinctions (now made almost instinctively by today's logicians) are the results of confusion or of keen logical insight. Propositions containing relational terms were said to be "oblique" because their object terms were in non-nominal-that is, obliquegrammatical cases. Scholastic logicians often tried desperately to account for such sentences. But their commitment to parsing statements as categoricals of the general form, quantifier-term-qualifier-term, made it difficult to formulate relational statements, which always had at least two quantified (or, at least, quantifiable) terms. 20

Cartes/lin Interlude Logick without Oratory is drye and unpleasing and Oratory without Logick is but empty babling. Richard Holdworth

By the sixteenth century, Scholastic logic was scarcely to be found as a

23

CHAPTER I

subject of inquiry or teaching in most European universities. 21 Indeed, Scholasticism itself(and with it Scholastic grammar, logic, theology, etc.) was in shambles. It would be a very long time before logic would again hold the pride of place in the academic curriculum that it enjoyed during the High Middle Ages. By then, both logic and the academy had changed beyond measure: most of the gains in logic made by the Scholastics were lost. Leibniz rediscovered or reinvented some, and modem mathematical logicians have built a cottage industry producing textual evidence that the Scholastics were precursors for many key elements of mathematical logic. While the rise of Humanism in the fifteenth and sixteenth centuries was not the direct cause of the decline in Scholastic logic, it did very rapidly replace the Scholastic curriculum with its own.22 And its view of logic was at odds in almost every way with that of its predecessor. Two names in particular are to be associated with the change in attitude toward logic-Peter Ramus and Rene Descartes. Humanism itself was the result of the rediscovery of large portions of the literature of ancient Greece and Rome. Cicero soon became a greater authority than Aristotle. Humanist grammar (with its emphasis on the linguistic customs found in Roman authors) replaced Scholastic grammar (with its careful attention to parsing and syntax). Rhetoric replaced logic. Eloquence replaced logic-chopping. Dialectic (i.e., logic) continued to hold a central place in the curricula of arts faculties in most European universities; however. the subject itself was gradually extended to cover topics that came to characterize Humanist logic. Ramus (IS I5-72), unlike many other Humanists, had been trained in the old Scholastic logic. But his contribution was not due to his logical abilities; he was the great popularizer of Humanist logic and its consequences for the universities. Having made a reputation for himself as a critic of Aristotle and Aristotelians, he went on to argue strenuously for the key theses of Humanist logic: that the logicians should primarily be interested in the conditions of good arguments-arguments that are persuasive and well presented-rather than valid arguments: that the logician/dialectician should view logic as a means for discovering new knowledge, rather than as a goal (i.e., the Scholastics' goal of valid inferences); and that the logician/dialectician should abandon the useless attempts to find a formal logic of language (viz., the artificial Latin of the Schoolmen), concentrating instead on ways to use effectively ordinary language, which is too slippery for the grip of formal syllogistic. Ramists saw the old school logic as unnecessarily difficult to teach and learn and as generally useless. The Scholastics' philosophical attitude toward logic was replaced by the Humanists' pragmatic attitude. Humanist logic was able to prevail in the university curricula during the fifteenth and sixteenth centuries partly because Aristotelian logic was no longer the area of research for significant numbers of first-rate thinkers. The best minds of the age were turned toward literature, Platonism, rhetoric, mathematics, and, eventually, the new science. The

24

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

only logic texts available were often poor and attenuated summaries, sometimes mixing Scholastic with Humanist theories. Nonetheless, by the end of this period certain elements of traditional logic had begun to reappear.2l At least two events account for this revitalization. First, new Greek editions of the Organon became available and replaced the Latin translations. Logicians already influenced by the Humanists were disposed to favour Greek over medieval Latin. Thus, Aristotle's logic in Greek gave a new respectability to the entire field of syllogistic. Second, logicians began to pay particular attention to problems of semantics and to recognize themes common to both Scholastic and Humanist philosophies of language. In particular, philosophers of language of both the medieval and Renaissance periods continued to hold that spoken language is purely conventional, and that spoken language corresponds to menta/language. The Aristotelian inspirations for these beliefs came from De lnterpretatione 17al-2 and 16a3-8, respectively. By the late sixteenth century, most debate centred on the second statement and concerned the question of whether words signifY concepts (in the mind) or things (in the world) or both, and the question of whether a mental proposition is a single unit or a complex of united parts, each corresponding to a mental term, combined by mental syncategorematic "acts." The interest of seventeenth-century philosophers of language in universal language, mental language, and artificial languages can be traced back in part to these earlier debates, which themselves harken back to the even older question of sentential unity. It was in this intellectual climate that Descartes opened the age of modem philosophy. Having rejected all that he had learned in school, including formal logic, Descartes made a typical Humanist turn. He decided that since logic could not provide knowledge (indeed, nothing taught in the schools could), a new method must be sought for discovery-an inventio, as Ramus had called it-a method for finding new knowledge, rather than iudicium, a tool for making judgments (viz., of truth and validity). Descartes's attack (especially in his Discourse on Method, Regulae ad Directionem In genii, and La Recherche de Ia Verite par Lumiere Nature/le) was aimed at the Scholastic institutions (such as the Jesuit College Henri IV de Ia Fleche, which he had attended) in general, and at their logic in particular. Knowledge of truth could be obtained by the proper application of a method (shades of Ramus). And this method, which Descartes devised for himself, required no master, no text, no school. The light of reason, common to all, is what allows one to see truth. The man who would seek knowledge need not ask entrance to the school. The rightthinking man (honnete homme) need merely follow the light of reason concerning things that are useful and least taxing to the memory (i.e., no need for Barbara, Celarent, Darii, Ferio, and the rest). Logic as taught in the schools, according to Descartes, is unnatural and forced upon the young before their natural reason has matured. Indeed, logic itself is unnecessary, because once natural reason has developed in the mind, all the useful truths

25

CHAPTER I can be seen or deduced by simple, self-evident chains of reasoning. No rules of logic are required. The logic of the schools is at best useless, at worst corrupting. Mais d'aucuns s'etonneront peut.atre que, cherchant ici les moyens de nous rendre plus aptes a deduire les verites les unes des autres, nous omettions tous les preceptes par lesquels les dialecticiens pensent gouverner Ia raison humaine, en lui prescrivant certaines formes de raisonnement qui aboutissent a une conclusion si necessaire que Ia raison qui s'y cootie, bien qu'elle ne se donne pas Ia peine de considerer d'une maniere evidente et attentive !'inference elle-mame, peut cependant quelquefois, par Ia vertue de Ia forme, aboutir a une conclusion certaine. C'est qu'en effet nous remarquons que souvent Ia verite echappe aces chaines, tandis que ceux-la memes qui s'en servent y demeurent engages. Cela n'arrive pas si frequemment aux autres hommes; et !'experience montre qu'ordinairement tous les sophismes les plus subtils ne trompent presque jamais celui qui se sert de Ia pure raison, mais les sophistes euxmemes. (1956: 71) In contrast with the obscurity of the old logic, the light of pure reason is nearly foolproof and, perhaps most importantly, so much easier. ... car comme il ne suit aucun maitre que le sens commun, et comme sa raison n'est gatee par aucun faux prejuge, il est presque impossible qu'il se trompe, ou du moins il s'en apercevra facilement et pourra etre ramene sans peine dans Ia bonne voie. (90 I)24 This Cartesian attitude toward logic dominated philosophy well into the nineteenth century. Locke's denigration offormallogic (especially in the fourth book of the Essa/5) is just one example of this attitude. Almost everywhere, the counsel was to relax and let the unschooled, natural tight of reason illuminate the truth; the tedious and difficult task of teaming the rules of syllogistic logic was unnecessary and unilluminating. Moreover, this new attitude was further encouraged by the developments taking place in mathematics during the sixteenth and seventeenth centuries. Given the current relationship between logic and mathematics, this may seem surprising. Nonetheless, during that period mathematicians began to abandon their traditional view of geometry (an axiomatic, logical system) as central to mathematics and began concentrating on algebra and analysis. In contrast to geometry, algebra and analysis were viewed as ways of discovering new knowledge. Thus mathematics was beginning to be seen

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THE GOOD OLD DAYS OF THE BAD OLD LOGIC

as a method of discovery rather than as a logical system. This shift, coupled with the new science of Galileo and Newton, helped to strengthen the Renaissance emphasis on method over reason, and contributed to the marginalization oftraditionallogic. Given Descartes's general animosity toward formal logic (and its role in the curriculum), it is ironic that one of the most influential logical works of the seventeenth and eighteenth centuries was produced by a group of Cartesians, the so-called Port Royallogicians. 26 The Port Royal logic is often best known for its revolution in semantics (its replacement of the Scholastic signification/supposition distinction by the comprehension/ extension distinction), but it is the Port Royal theory of grammar and logical syntax that is of most interest to us. While they were Cartesians, mixing theories of method and epistemology with logic and grammar, the Port Royal logicians were not Ramists. They took Aristotle as the chief authority on formal logic, and took logic to involve both the effective use of language and correct reasoning. The Port Royal theory is expounded in three important works, which were written separately and independently but can be treated "as one grammatico-logical work in three volumes" (Padley, 1976: 256). The three works are Nouvelle Methode pour apprendre facilement et en peu de temps Ia langue latine by C. Lancelot (Paris, 1644), Grammaire generale et raisonnee by A. Arnauld and Lancelot (Paris, 1660), and La Logique. ou /'art de penser by Arnauld and P. Nicole (Paris, 1662). 27 The theory of grammar and logical syntax found there formed part of a long tradition in linguistics, combining ideas from Aristotelian logic, ancient grammar, the grammaticae specu/ativae of the Scholastics, seventeenth-century philosophical grammars, and rationalism. That tradition was obviously fragmented and incoherent, including philosophers like Descartes and Leibniz as well as Bacon, Hobbes, and Locke. In contrast to the Humanists, the Port Royal logicians were sober, practical Jansenists, seeing logic, rather than rhetoric, as the source of syntactical insights. Indeed, they completely subordinated grammar to logic. The tradition to which they belonged posited a level oflanguage-a universal language-underlying and common to all of the various natural languages. The grammar of this universal language was a deep grammar, in that it was a grammar of concepts (which are shared by all persons) rather than of mere words (which are relative to each natural language). In fact, Lancelot's Latin grammar was meant to be a tool for teaching Latin by instruction not of Latin grammar rules but of universal rules. Thus it can be seen as an attempt to apply the grammaticological theory of the Logique and Grammaire. Cartesian linguists, such as those of Port Royal, recognized the frequent occurrence of natural-language sentences that differed markedly from their logical, deep forms. Most of the time, this is due to ellipsis. Certain terms or phrases essential to the deep sentence are omitted (for various reasons) from the surface, or "figurative," sentence. An important

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CHAPTER I

initial task of the logician, then, is to "resolve" surface sentences into deep sentences; they are natural in the sense that they carry the meaning of the surface sentence. Deep sentences have a logical syntax. The basic theses of this theory of logical syntax can be summarized as follows. 28 (i)

(ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

Every judgment or proposition (i.e., statement-making sentence) is a predication. Predication takes place in categorical sentences. Every categorical sentence consists of a subject and a predicate. Every subject is a universal or particular quantifier plus a term. Every predicate is a copula (i.e., qualifier) plus a term. Terms may be simple or complex. Predication takes place between the subject-term and the predicate-term. Predication is effected by the syncategoremata (i.e., the quantifiers and copulae). Predication is symmetric (he., the terms of a predication are logically homogeneous, fit for either a subject-term role or a predicate-term role).

It is obvious that these logicians were trying hard to preserve the generally

Scholastic view of logical syntax. The Port Royal logicians (Logique, II, 7) chastised logicians who taught merely that sentences consist of two parts-subject and predicate-without indicating anything more than that the subject is the farst part and the predicate the last part of a sentence. As the above theses show, they saw predication as a joining (or separating) of two concepts expressed by terms (categoremata). The job of syncategoremata is to do the joining (or separating). Indeed, where Aristotle and his followers marked the difference between categoremata and syncategoremata by saying that the former are independent and the latter dependent, the Port Royal logicians added that the former result from the farst act of the intellect (conceptualization, or simple apprehension), while the latter result from the second act (judgment, or composition and division-i.e., the formation of statements). Their theory of logical syntax is best seen in their account of verbs. Verbs are of two sorts: adjective and substantive. A "verb adjective" was analysed as an affirmation plus an attribute-that is, as a copula plus a term (this recalls Abelard's theory). A "verb substantive" was simply an affirmation-i.e., a copula. Consider the sentence 'Some man runs'. An alternative theory (viz., a binary one) might analyse this as a subject, 'some man', and a predicate, 'runs'. But, good Aristotelian temarists that they were, the Port Royal logicians demanded a connecting link between the two terms. Predication occurs not between a subject and a predicate, but

28

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

between a subject-term and a predicate-term. So, the verb 'runs' must be analysed into a copula (always in the qualifier sense) plus a term: 'Some man is running'. The two terms 'man' and 'running' are connected by the syncategorematic expression 'some ... is'. Where another analysis might take quantifiers and copulae (qualifiers) as optional, this analysis demands them. Sentences like the following all must be resolved into sentences containing syncategoremata. I. 2. 3. 4.

Socrates runs. Men reason. No men fly. Amamus.

Their corresponding deep sentences are: Ia. 2a. 3a. 4a.

Every Socrates is running. Every man is reasoning (or, is rational). It is not the case that some man is flying. Nos summus amantes (or, better, Omnes nostrum sunt amantes).

In 3a it must be noted that the phrase 'it is not the case' was seen as modifying the copula-changing it from one ofjoining to one of separating. Thus, 3a was taken as a sentence that negatively connects (separates) the concept of man and the concept of flying. It was not seen, as it is by modem logicians, as the negation of an entire sentence. An essential feature oflanguage, often recognized now by linguists, is its creative aspect. An infinite number of sentences can be generated from a fmite number of terms connected in a finite number of ways. A recursive formation rule, like (vi) above, permits this. To see how this is so, it is necessary to realize that the Port Royal logicians implicitly treated entire sentences as complex terms and resolved all complex terms as predications. A noun-adjective combination is an example of a complex term. Consider 'wise man'. This is resolved as 'man who is wise', where the phrase 'who is wise' is a predication between 'wise' and the "principal word," which is antecedent for the relative pronoun. In this case, the principal word is 'man'. In fact, they took all adjectives to be resolvable into predications. A substantive may be absent in the surface sentence but must occur in the deep sentence. For example, 'Some man is white' resolves into 'Some man is a thing which is white' (Logique, 1: 8 and II: 1). By admitting into their deep sentences predications that act as terms, the Port Royal logicians had to allow that not all predications are assertions. When I assert (to use one of their own well-known examples) 'Invisible God created the visible world,' I predicate 'invisible' of'God', 'visible' of 'the world' and 'creating the visible world' of 'God'. But only

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CHAPTER I

the latter predication is asserted. The unasserted predications are sentences "contained implicitly" in the surface sentence. The result of all this for logical syntax is simply that all logical complexity, for sentences and for complex terms, is accounted for in tenns of predication. Syntactically, complex terms are sentences-sentences are complex terms. The Port Royal logicians were acutely aware of the temptation to make logic a simple science of simple inferences. Consequently, they took care not to avoid complicated or difficult cases. But the kind of sentence that posed the greatest difficulty for their theory was one that they scarcely seemed bothered by-the relational sentence. Clues to how relationals might be incorporated into the theory are found in the Port Royalists' notion that all verbs connect subject-terms with attributes (predicate-terms), and the idea that complex terms are implicit predications. Consider 'Some boy loves some girl'. In the following section, we will see that Leibniz took this to be a conjunction of'Some boy is loving' and 'Some girl is eo ipso loved' in order to account for the fact that 'loves' somehow attaches to both 'some boy' and 'some girl'. However, by treating relational terms as complex, all complex terms as predications, and the predicate-term of a complex relational term as predicated as well of the main subject-term, one can incorporate relationals into the theory of logical syntax laid out by the Port Royal logicians without recourse to Leibniz's splitting procedure. Thus the Port Royal logicians, though they did not, could have analysed 'Some boy loves some girl' as a predication between two terms: a simple term, 'boy', and a complex tenn, 'loves some girl'. This latter term, being complex, can itself be analysed as a predication between two terms: 'loves' and 'girl'. Though some of the ideas concerning logical syntax put forward by the Port Royal logicians either are found in Leibniz or are similar to his, few, if any, have survived into our own day. In part 2 of this essay I will try to retrieve a few of them (along with other Scholastic and Leibnizian ideas).

Lelbnlt.lan Insights For Reason, in this sense, is nothing but reclconing, that is adding and subtracting, of the consequences ofgeneral names agreed upon for the marking and signifying ofour thoughts. Hobbes We do well to analyse matters most industriously and reduce everything to the simplest and most easily grasped inferences. so that even the most insignificant student cannot fail to see what follows and what does not. Leibniz

In the seventeenth century, there were a few who simply refused to abandon the logic of Aristotle and Abelard, of Sherwood and Ockham. Foremost among them was, of course, Leibniz. But there were many important figures

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THE GOOD OLD DAYS OF THE BAD OLD LOGIC

who influenced Leibniz's logical work. Naturally, Aristotle was a strong influence, with Leibniz going beyond Arnauld's claim in the Logique that all precepts of logic belong to Aristotle, and holding that Aristotle's syllogistic system was correct but incomplete. Plato was another influence, as he was on most philosophers after the Humanist revival of Greek. 29 But the most immediate influence on Leibniz's thinking concerning logic and language was Hobbes. That Leibniz, one of the greatest mathematicians and logicians of all time, should have been influenced by someone who was no logician and a bad mathematician may be surprising, but the fact is that Hobbes's work made a significant impression on the younger philosopher as early as the 1660s. Leibniz's famous letter to Hobbes of 1670 displays his deep respect for the ideas of the aged Englishman. But most important for our purposes is Leibniz's remark in "Of the Art of Combination": Thomas Hobbes, everywhere a profound examiner of principles, rightly stated that everything done by our mind is a computation, by which is to be understood either the addition of a sum or the subtraction of a difference (De Corpore, Part I, Chap. I, art. 2). 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. In that sense, then, 'is' is not properly a copula. but part of the predicate; there are two copulas, one of which, 'not', is named, whilst the other is unnamed, but is included in 'is' as long as 'not' is not added to it. This has been the cause of the fact that 'is' has been regarded as a copula. We could use as an auxiliary the word 'really'; e.g. 'Man is really an animal', 'Man is not a stone'. (1966: 2-4)

I will have more to say about Leibniz's view of the copula below. For now, I want to look at the two core logical ideas that Leibniz seems to be borrowing from Hobbes. The first is that all reasoning consists of computation-adding and subtracting; the second is that all statements consist of pairs of terms connected by a copula. Hobbes stated the first thesis on a number of occasions. In De Corpore, I, he wrote, By reasoning, however, I understand computation. And to compute is to collect the sum ofmany things added together at the same time, or to know the remainder when one thing has been taken from another. To reason therefore is the same as to add or to subtract, . . • Therefore, all reasoning reduces to these two questions of the mind, addition and subtraction. ( 1981 : 177)

And later in that work he defined a syllogism as "a collection of two

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CHAPTER I

propositions into a sum" (1981: 255). The same thesis is propounded in books IV and V of Hobbes's Leviathan (1904). Hobbes's second thesis is stated in his attempt to account for propositions (i.e. statement-making sentences) in De Corpore, 1: "Proposition is speech consisting of two copulated names by which the one who is speaking signifies that he conceives the name which occurs second to be the name of the same thing as the name which occurs first" ( 1981 : 225). He adds that the copula may be explicit (e.g., by the use of "is") or may be indicated by an inflection. And he continues, "Therefore, in every proposition three things occur that have to be considered, namely, two names, subject and predicate, and copulation" ( 1981 : 227). Leibniz did very clearly borrow the first idea from Hobbes. Throughout his logical studies, Leibniz persisted in his view of reasoning as computation. But Hobbes's second thesis is found only in a modified form in Leibniz. Hobbes viewed a proposition as a copulation of two names. However, he took names to be a fairly heterogeneous collection, including proper, common, and abstract terms as well as quantified terms. Though he made use of the Scholastic vocabulary of "subject," "predicate," and "copula," Hobbes had no great respect for the logic of the schools, so "subject" and "predicate" tended to be simply terms for the fJrst and second names of a proposition. There is little syntactical insight here. Leibniz, on the other hand, had great respect for the accomplishments of the Scholastic logicians, thinking only that they had not gone far enough. Like them, he believed that a proposition is best construed as a quantified term (subject) concatenated with a qualified term (predicate). That is why he said, in "Of the Art of Combination," that" 'is' is not properly a copula, but part of the predicate." Indeed, for Leibniz, 'is' is a qualifier. More generally, for Leibniz, a universal language would have a rational grammar. And the logical forms dictated by such a grammar would be revealed in naturallanguage sentences by particles-the copula being the foremost among these. 30 Before going on to look more closely at Leibniz, we cannot leave Hobbes without remarking on Geach's criticism of him. Geach accuses Hobbes of holding the "two-name theory" of predication (recall that, according to Geach's reading of history, the two-name theory followed the two-term theory and led to the two-class theory): "Hobbes, who held the two-name theory of predication, held also that the copula was superfluous; but we might very well object that on the contrary it is necessary, because a pair of names is not a proposition but the beginning of a list, and a redundant list at that if the two names do name the same thing" (1962: 35). It must be pointed out in response that (a) Hobbes did not hold the copula to be superfluous, for, as we saw above, he explicitly claimed that every proposition consists of two copulated names and that in analysing a proposition the copula is one of the three things to be considered; (b) Geach has misunderstood Hobbes's use of the word 'name'. Geach seems to think

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THE GOOD OLD DA ~HE BAD OLD LOGIC

that Hobbes's use of'name' is similar to his own, which construes a name as a referential expression (paradigmatically a proper name or a personal pronoun, but never a quantified expression). In Hobbes's Latin, the word usually translated as 'name' is nomen, but it could just as easily and accurately be translated as 'term', 'expression', 'linguistic item'. Furthermore, Hobbes's talk of such expressions "naming" cannot be understood simply as referring (as Geach and most moderns would have it). Rather, Hobbes held that when used in a given proposition a name (term) denotes an individual or individuals in virtue of its signification. Names do not signify thoughts, ideas, and so on. Names are significant by virtue of their uses in acts of signifying the thoughts or beliefs of speakers when intending to communicate to an audience. 31 Hobbes may not have been much of a mathematician or a logician, but he was a semanticist of the first rank. Leibniz, by contrast, was a mathematician and logician of the first rank. The idea that reasoning was a form of calculating, of adding and subtracting ideas, and the idea that statements are copulations of pairs of terms, may have been Hobbes's. However, only a logician of Leibniz's talents could develop these into a fully articulated theory of logical reckoning. Scholasticism hung on long enough in the German universities of the seventeenth century to touch Leibniz. His initial views of logic were clearly those of the later Schoolmen, yet he was soon surrounded by the antiScholasticism of the Humanist logicians-and was not completely uninfluenced by it. The Humanists had tended to dismiss logic in favour of a search for method (i.e., a means for discovering the truth). Having read Ramus, Descartes, and Hobbes, Leibniz, too, saw the search for method as a proper philosophical goal. What separated Leibniz from the others was his conviction that logic, far from being an impediment to the search for method, is a method for discovering truth. From his youth, Leibniz was convinced that it was possible to devise a symbolic calculus, the terms of which could be manipulated mechanically, according to simple laws, to yield truths. The terms of such a device would be supplied by a characteristica universalis, or alphabet of human thought. Moreover, Leibniz was always convinced that, given such an alphabet, all terms could be seen explicitly to be either simple or combinations of such simples. If an encyclopedia of established knowledge could be gathered, this, along with the universal language in which to express such knowledge and a logical algorithm for manipulating mechanically, according to logical laws, the terms of such a language, would place within the grasp of the human mind all possible knowledge. The task envisaged here was monumentally ambitious-and impossible for Leibniz to complete-but his enthusiasm for it never waned. He spent much of his life attempting to build a viable logical algorithm, and he made numerous attempts to enlist the aid of learned societies and other researchers in his project. Leibniz' s fertile mind produced a steady stream of insights into the

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CHAPTER I

nature of logic and logical algorithms. Of particular interest are those concerning logical syntax, of course, and his theses concerning the extensions and modifications of syllogistic. Russell made popular the charge that the failures of Leibniz's logical programme were due to his uncritical insistence on viewing all statements as subject-predicate in logical form. The fact is that, while Leibniz did insist on a subject-predicate analysis of statements, this attitude was well thought out, and the failures of his logic are due only to his inability to devise an appropriate system of notation for his logical algorithm. Indeed, Leibniz's insistence on a subjectpredicate analysis of statements (i.e., a Scholastic analysis) may have been suggested to him by his reading of Scholastic logicians and his respect for Aristotle, but his conviction that it was the correct analysis was the result of his attempts to modify and extend syllogistic so that it could be used as a general logic. We have seen that traditional syllogistic was unable to give an adequate account of inferences involving three kinds of statements: singulars, relationals ("oblique sentences"), and compounds ("hypothetical sentences"). Leibniz was convinced that the best course was to augment syllogistic so that it could be used to analyse inferences involving these kinds of statements. Since syllogistic, as he saw it, depends upon a categorical (subject-predicate) analysis of logical syntax, he saw his task to be to discover how to construe singular, relational, and compound sentences as categoricals.n For many years Leibniz's view of singulars was a typically Scholastic one. Singular terms in subject position were taken to have implicit universal quantity. Singular predicate-terms were simply treated like any other predicate-term.33 At one point, he did seem to suggest, however, that singular subjects could be construed as having an implicit particular quantity (Leibniz. 1966: 65). But he finally offered a considered view of the issue in a brief study written late in his career, "A Note on Some Logical Difficulties": Some logical difficulties worth solution have occurred 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 Apostle Peter is not a soldier'-since elsewhere a universal affirmative and a particular negative are opposed? Should we say that a singular proposition is equivalent to a particular and to a universal proposition? Yes, we should. So also when it is objected that a singular proposition is equivalent to a particular, 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'

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THE GOOD OLD DAY.!.QE_THE BAD OLD LOGIC

coincide, since the term is singular. (1966: liS) What Leibniz was claiming, in other words, was that for singular subjects we can arbitrarily take their implicit quantity to be either universal or particular, since the two "coincide." They coincide in the following sense. The reference of any subject expression (quantified term) is the result of the joint semantic work performed by both the quantifier and the denotation of the term. A universally quantified term makes reference to the entire denotation of the term (the Scholastics called this distributed supposition). Thus, 'every composer', when used in a statement, makes reference to all individuals denoted by the term 'composer': Mozart, Beethoven, Brahms, and so on. A particularly quantified term makes reference to an indeterminate part of the denotation of the term (for the Scholastics, such a term had undistributed supposition). Thus, 'some composer', when used in a statement, makes reference to some indeterminate part (though perhaps all) ofthe denotation of the term 'composer'-that is, to one or another of the individuals denoted by 'composer'. For example, while 'Every composer is a musician' uses a quantified term referring to Mozart and Beethoven and Brahms and ... , the statement 'Some composer is a musician' uses a quantified term referring to either Mozart or Beethoven or Brahms or ... (with 'or' inclusive). Now, suppose that the term involved is not general but singular, such as 'Bach'. The denotation of'Bach' is just Bach. So the universal quantification of'Bach' yields an expression that can be used to refer to the entire denotation of 'Bach', in other words, Bach. And the particular quantification of 'Bach' yields an expression that can be used to refer to a part of the denotation of'Bach', which, since it has but one part, is, again, just Bach. In summary, when the subject-term of a statement is singular we can arbitrarily take it to have either an implicit universal or an implicit particular quantity, since in either case the very same reference is made. In other words, as Leibniz said, the two "coincide." This idea, or one very close to it, has been advocated by others more recently, but as far as I know it belongs originally to Leibniz. Later we will see that Sommers has made much use of it under the title ''the wild quantity thesis."34 Finally, note in passing that in the quoted remarks above, Leibniz mentions without hesitation a sentence using a singular as predicate ('some writer is the Apostle Peter'). In the second part of this essay, we will see that Leibniz's wild quantity thesis, coupled with the admissibility of singulars as predicateterms, gives an advantage to syllogistic logic not enjoyed by mathematical logic (which must augment its calculus with an appended "theory of identity"). Modem logicians, following Russell in his criticism ofLeibniz's logic, generally have been most exercised by Leibniz's attempt to construe relational statements as categoricals. The problem with relationals is that they seem to have too many subjects. 'Some man is a lover' is clearly categorical, having one subject and one predicate. But 'Some man loves

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CHAPTER I

some woman' has two subjects, two quantified tenns, so how could such a sentence possibly be construed categorically? Leibniz's solution was to take such sentences to be at bottom conjunctions of categoricals. Thus, 'Some man loves some woman' would be parsed initially as 'Some man loves and some woman is loved'. But, of course, this leaves doubt about whether the loving man is the one who loves the beloved woman. So such a sentence is finally parsed as 'Some man loves and eo ipso some woman is loved'. In other words, 'A is R to B' would be analysed as 'A is an R' er and by that very fact B is R'ed'. Consider 'David is the father of Solomon'. A Leibnizian analysis renders this as 'David is a father and by that very fact Solomon is a son (i.e., is fathered)'. What Leibniz was aiming at here was an analysis that would construe relational tenns such as 'loves' and 'father of as being simultaneously predicated of two subjects, to be "Janus-faced," as Sommers has called it (1983: 188), turning each face toward a different subject.35 Russell quoted Leibniz concerning this very point: I do not believe that you will admit an accident that is in two subjects at the same time. My judgment about relations is that paternity in David is one thing, sonship in Solomon another, but the relation common to both is merely a mental thing whose basis is the modifications of the individuals. (Russell, 1900: 206)

It cannot be said that Leibniz was very successful in his attempt to give a categorical reading of relationals and thereby to incorporate inferences involving them into syllogistic. What he did do was introduce the notion that relational sentences should be analysed as triples (or n-tuples) of tenns, such that appropriate pairs of those tenns can be taken as fonning (by the presence of, possibly implicit, quantifiers and qualifiers) categorical phrases. Moreover, he saw that inferences involving such statements would be governed by the same rule of mediate inference that governs categorical syllogisms, namely, the dictum de omni et nullo.36 It was in "A Specimen of a Demonstrated Inference from the Direct to the Oblique" that Leibniz proposed a method for demonstrating the validity of inferences in which the tenns of the premise(s) are all nominative but some of the tenns of the conclusion are non-nominative (as is the object-tenn of a relational expression). Such arguments became of interest to Leibniz after reading Joachim Jungius's Logica Hamburgensis. One of Jungius's examples was: Omnis circulus est figura. Ergo, quicumque circulum describit figuram describit. Leibniz analysed a similar example: Painting is an art, therefore he who learns painting learns an art. In this case the tenn 'art' in the conclusion is not in the nominative case. Since his version of the dictum de omni et nullo (in tenns of subjects and predicates) requires that 'art' be nominative in the conclusion, Leibniz supposed an equivalence between any expression of the fonn 'thing that is X' (with 'X' nominative) and 'X'

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THE GOOD OLD DAY..!QE!HE BAD OLD LOGIC

(where 'X' is oblique). His proof then proceeds, in effect, as follows: 1. Painting is an art. 2. He who learns painting learns painting. 3. He who learns painting learns a thing that is painting. 4. He who learns painting learns a thing that is an art. S. He who learns painting learns an art.

premise assumption his equivalence the dictum his equivalence

Leibniz was on the right track here, but his system was still unsound. There are invalid inferences that it could prove, such as, 'Painting is an art, therefore he who learns an art learns painting.' What is required is a fonnulation of the dictum in terms of term distribution. Leibniz, like most traditional logicians, always took distribution in terms of quantity and quality. To break the hold of this dogma the logician must let go of the analysis of propositions into subjects and predicates. In part 2 we will see that Leibniz's failure here was due not to his inability to recognize that relational terms are always polyadic, never monadic.37 but to his inability to recognize the Aristotelian ternary form of relational predicates and to his failure to devise a suitable system of notation flexible enough to symbolize statements containing such terms. Thus far, we have seen that Leibniz was successful in his attempt to incorporate inferences involving singular statements into his syllogistic by construing singulars as categoricals. He was less successful in his attempt to do the same for relationals. His third task was to find a way of analysing compound sentences ("hypotheticals") as categoricals. He realized that by doing so the logic of compounds (what is now called propositional logic) would be then viewed as simply a part of the more general logic of categoricals-syllogistic. In "General Inquiries about the Analysis of Concepts and ofTruths" he wrote, "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" (1966: 66). While Leibniz was certain that entire propositions could be conceived of as terms, he was less than clear about just how this could be revealed logically. At any rate, he was clear about how to "treat all propositions universally." What he meant was that any proposition could be viewed as claiming that the concept expressed by the subject contains the concept expressed by the predicate. A proposition will be taken to be true whenever its claim holds-that is, whenever the subject concept does contain the predicate concept. If containment is the relation claimed to hold between a subject and a predicate, then, given that entire propositions can be taken to be terms, and that hypotheticals are to be taken as categoricals, the claimed relation between an antecedent and a consequent must be

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CHAPTER I

containment as well. And indeed this is just what Leibniz says: "Une proposition categorique est vraie quand le predicat est contenu dans le sujet; une proposition hypothetique est vraie quand le consequent est contenu dans !'antecedent" (quoted in Couturat, 1966: 423). Whether or not one accepts Leibniz's containment account of truth, the fact is that ifthere is to be a general logic incorporating both principles of inference for categorical statements and principles of inference for compound statements, it must be one that, as he saw, first takes entire statements to be terms and then bases its account of inference on a theory of logical syntax that sees terms as logically homogeneous (i.e., a non-binary theory of logical syntaxV8 Like De Morgan, Leibniz has become famous for a law-Leibniz' s Law. 39 That the law can even be found in the corpus ofLeibniz's work is questionable. At any rate, it is usually stated in the form of a biconditional or a conjunction of conditionals concerning the connection between identity and indiscemibility: two things are identical if they are indiscernible and only if they are indiscernible. Usually, indiscemibility is defmed as follows: Two things are indiscernible if and only if any predicate true/false of one is true/false of the other. Sommers (1982: 127-30) is among those who have denied that Leibniz formulated Leibniz's Law. Moreover, he has argued that what Leibniz did in fact formulate was an altogether different law, from which, when properly understood, Leibniz's Law can be derived. This other law is the Principle of Substitutability, and there is no doubt about its author. Leibniz states the principle as follows (quoted in Couturat, 1966: 259): "Eadem sunt quorum unum in alterius locum substitui potest, sa/va veritate" (Two things are the same that can be substituted for one another everywhere, without destroying truth). It is unclear here whether Leibniz is talking about objects or terms, but in a later work he makes it explicit. He is talking about the necessary and sufficient conditions for a pair of terms to be interchangeable: "Those terms are 'the same' or 'coincident' of which either can be substituted for the other wherever we please without loss of truth-for example, 'triangle' and 'trilateral' " (Leibniz 1966: 131 ). Earlier I mentioned that traditional logic, unlike modem mathematical logic, has no need for a special"theory of identity" (and I will say much more about this in later chapters). Suffice it to say for now that modem logicians take identity to be a relation that holds between an object and itself. Leibniz nowhere talks of such a relation. His Principle of Substitutability governs the terms of a sentence: it states the conditions in which a pair of terms are replaceable for one another in a given sentence without altering the truth-value of that sentence. It is only when seen in this way that the principle is regarded as an integral part ofLeibniz's general program for logic. For it is merely a special case of an even more general logical principle, one considered by Leibniz to be the most important principle governing logical inference-the dictum de omni et nullo. Look again at Leibniz's formulation of the dictum: "To be a predicate in a universal affirmative proposition is the same as to be capable of being

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THE GOOD OLD DAYS OF THE BAD OLD LOGIC

substituted without loss of truth for the subject in every other affinnative proposition where the subject plays the part of predicate" (1966: 88). In effect, the dictum is a very general principle of substitutability. It lays down the conditions governing when one tenn can be substituted salva veritate for another in a given sentence. Notice that there are pairs of tenns for which the dictum holds but the principle does not. In a Barbara syllogism, for example, the major tenn can be substituted for the middle term in the minor premise to yield the conclusion. But no law penn its the substitution of the middle tenn for the major tenn-they are not mutually interchangeable. Many pairs of tenns are such that one can be substituted for the other in a given sentence but the other cannot be substituted for the first. Other pairs of tenns are such that either can be substituted for the other in a given sentence. When is this so? According to the dictum, whenever each is truly afflrnled of the universalization of the other. In other words, 'A' and 'B' are intersubstitutable for one another in a given sentence salva veritate whenever 'every A is B' and 'every B is A' are both true. The principle of substitutability is merely a special case of the dictum. What, now, ofLeibniz's Law? Recall that the law states, in effect, that two things are identical if and only if they are such that whatever predicate is true of one is true of the other. This law can now be derived ftom the Leibniz's principle and his wild quantity thesis. Let 'a' and 'b' be two singulartenns. To say that a predicate, 'P', is true of'a' (given the wild quantity thesis) is to say 'every/some a is P', which, by the dictum, means that 'P' is substitutable salva veritate for 'a' in any sentence in which 'a' is affmned. Now, let 'P' be 'b'. To say, then, that 'b' is true of'a' is to say that 'b' can be substituted for 'a' in a given sentence salva veritate. The same holds for 'a'. So to say that a and bare identical is to say that 'a' and 'b' are intersubstitutable in a given sentence salva veritate. To say that Tully is identical to Cicero would be to say that every Tully is Cicero and every Cicero is Tully-'Cicero' and 'Tully' are intersubstitutable. Before leaving our discussion of Leibniz's insights, it would be instructive to say something more about the wild quantity thesis. It is possible to show that the wild quantity thesis can be derived ftom two other Leibnizian theses: the Conceptual Containment thesis and the Completeness thesis. 40 As we saw above, the first thesis holds that a statement is true if and only if the concept expressed, or "signified," by the subject-tenn contains the concept expressed by the predicate-term. The clearest mark of containment is the copula 'every ... is'. Thus, in 'Every man is rational' the concept signified by 'man' is claimed to contain the concept signified by 'rational'. The statement (made by the appropriate use of this sentence) is true just in case the claim is upheld. For Leibniz, every concept is either simple or complex. A complex concept is the result of a combination by addition or subtraction of less complex concepts, and any concept can be said to contain itself. Particular propositions are governed by the thesis only by taking them to be elliptical. Thus, Leibniz held that in a particular

39

CHAPTER I

proposition "something is added" to the subject-term. For example, 'Some man is musical' is true just in the case when the concept signified by 'musical' is contained in the concept signified by 'man' with an added term, 'musical', so that the real significant of the subject-term is the one signified by 'musical man'. Negative propositions simply deny their corresponding affmnations, and thus are taken as falsity rather than truth claims, or, equivalently, as truth claims depending on conceptual exclusion rather than containment. The Completeness thesis41 claims that the concept of an individual substance (Leibnizian monad) is complete. The content of an individual is a combination of concepts. That complex concept is complete in the sense that for any pair of logically incompatible concepts (e.g., rationaV nonrational, massive/massless, redlnonred) exactly one is contained in it. This thesis, along with the Conceptual Containment thesis, can be used to establish Leibniz's wild quantity thesis. Let 'X' name an individual (i.e., 'X' is a proper-name singular term). The wild quantity thesis holds that every X is Y if and only if some X is Y. Since subaltemation, a rule accepted by Leibniz, guarantees that every X is Y only if some X is Y, it is sufficient to show that every X is Y if some X is Y. Suppose it is asserted that some X is Y. According to the Containment thesis, this is true just in case 'some XZ is Y' is true (where Z is the concept added for particular statements and where every Z is Y and Z is contained in X). Next, suppose it is asserted that every X is Y. This is true, according to the Containment thesis. just in the case when the concept signified by 'X' contains the concept signified by 'Y'. Since X is an individual, according to the Completeness thesis, the concept of X is complete. No X could be a non Y. The concept of X contains the concept ofY. And, since X is an individual, its concept contains one of every other pair of incompatible concepts. No concept could be added to the concept of an individual that is not either already contained in it or inconsistent with it. So the concept of Z is already contained in that of X or it is inconsistent with it. If it is already contained in the concept of X, then 'Some X is Y' {='some XZ is Y') entails 'Every X is Y'. If it is not already contained in the concept of X, then 'Some X is Y' is a contradiction. So, if 'Some X is Y' is true, then 'Every X is Y' is true. The logical quantity of singular subjects is wild. Logicians of our own century are fond of pointing to Leibniz as one of the great precursors of modem mathematical logic (doomed to failure only by his excessive respect for tradition and the categorical analysis of statements). That Leibniz saw that mathematical techniques could be adapted to the needs of a logical algorithm, and that he viewed the establishment of a single general logic, fit for the analysis of all kinds of formal inferences, as a goal worth striving for, are indeed reasons for returning to his ideas and for claiming him as a source of inspiration. But the fact remains that Leibniz is best viewed not as the first mathematical logician but as the last (and best) Scholastic. 40

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

Nineteenth-Century Algebraists For whenever you think the two premises, you think and put together the conclusion. Aristotle In effect, we judge and reason with words, just as we calculate with numerals; and languages are for ordinary people what algebra is for geometricians. Condillac L 'expression simple sera a/gebrique ou e/le ne sera pas. Saussure

British philosophers from the Renaissance to the mid-nineteenth century were generally unsympathetic or critical toward the enterprise of formal logic. The great empiricists, especially Locke, were particularly harsh in stating their sentiments against logic. Ironically, however, it was in England, Scotland, and Ireland that the great revival (and the subsequent burst of creative activity) concerning formal logic began. The central figure in this revival was George Boote, but the stage for Boote's work had been set by a number of predecessors in logic, mathematics, and philosophy. As well, many of Boote's ideas were worked out in detail or set in proper order by those who followed his lead. These "algebraic" logicians-Boole, Hamilton, De Morgan, Jcvons, Venn, Peirce, Schroeder, J.N. Keynes, W.E. Johnson-dominated logic in the nineteenth century until its final few years, when Peano, Frege, and Russell revolutionized the subject completely. From the time ofLeibniz to the mid-nineteenth century the basic logic text in English schools had been Henry Aldrich's Artis Logicae Compendium. First published in I 691, it was a collection of aids, mnemonic verses, and so on. for traditional logic. Aldrich's text was replaced (finally) in English schools by Richard Whately's Elements ofLogic, which was published in 1826 and the standard text known to the British algebraists. In the 1830s Sir William Hamilton published a series of papers on logic that helped to initiate the renewed activities. In particular. Hamilton's work helped to establish the view (contrary to Kant's claim) that traditional syllogistic logic was far from complete, and that as yet unimagined alterations and additions could be made to the old logic. Mill's publication of A System ofLogic in 1843 was, in many ways. a negative instigator of the renewed interest in formal logic. Mill's logic was an attempt to carry out the older empiricist (and Hobbsean) tradition, which the algebraists generally abandoned. For these logicians, there was no question of going back to what they saw as Aristotelian syllogistic (as found in Whately). They were ready to reject the humanist (or, later, empiricist) notion that a logic that is not a tool of discovery is worthless (as represented by Mill). And they would never consider going 41

CHAPTER I

back to Scholasticism (as summarized in Aldrich); for one thing, they now knew far too much science and mathematics. The algebraists were convinced that it was possible to build an algebraic system for the manipulation of properly symbolized inferences in ordinary language. The most successful ofthese was, of course, Boote's. We will look briefly at his work below. But for my own purposes the work ofDe Morgan is far more interesting. We will look closely at some of De Morgan's original logical insights. George Boote built his famous logic as an example of how one could generalize algebra, which itself was a generalization of arithmetic. Like Leibniz, Boole saw certain analogies between numerical addition and multiplication and the logical conjunction and disjunction of terms. The idea that algebraic formulae could be used to express logical relationships followed naturally. Boole presented his logic in Mathematical Analysis of Logic, published in 1847, and (more fully) in An Investigation ofthe Laws ofThought on which are Founded the Mathematical Theories ofLogic and Probabilities, usually referred to as The Laws ofThought, in 1854. Here, Boole's algebra governs a language whose variable expressions ("elective symbols") are general enough that they can be interpreted in at least three ways. Such a term could be interpreted as a class name, a proposition, or a degree of probability. The first interpretation gives a generalization of the traditional logic of terms. now seen as a logic of classes. Boole's idea was that, once formalized, the drawing of syllogistic conclusions was simply a matter of mechanical manipulation of symbols according to the laws of his algebra. It is important to note that Boole, like Hamilton, De Morgan, and the other algebraic logicians, assumed that any algebra of logic must be equational. In other words, the general conviction was that the propositions of logic should be formulated as equations. Initially, this was taken to mean that the copula must be represented as a relation of equivalence,42 and that very idea led to the suggestion (made popular, if not originated, by Hamilton) that the predicate as well as the subject of a categorical proposition must be logically quantified. Boole, however, took the equivalence in question to hold between a specified class and either the universal class or an indefinite nonempty class. Boole symbolized the empty class by '0', the universal class (i.e., the class of entities constituting the "universe of discourse", a concept originally introduced into logic by De Morgan) by 'I'. The intersection (conjunction) of a pair of classes was indicated by juxtaposition. Thus, if 'x' stands for the class of men and 'y' stands for the class of unmarried adults, then 'xy' stands for the intersection of these two classes-namely, the class of bachelors. The union (disjunction) of two classes was indicated by the interposition of a '+'. The complement of a class was indicated by the subtraction of that class from the universe. For example, the complementofthe class of men (namely, nonmen), where 'x' stands for the

42

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

class of men, is symbolized as '1-x'. An E categorical, say 'No xis y', would be formulated as the claim that the intersection ofx andy is empty, equivalent to the empty set: xy=O. An A categorical, 'Every x is y', claims that the intersection ofx and the complement ofy is empty: x(l-y)=O. I and 0 categoricals could be taken as the negations of E and 0 (thus yielding inequations), or as claims that the intersections in question are equivalent to some indefinite nonempty class, which Boote symbolized by 'v'. So, I would be symbolized as: xy=v; 0 as: x(l-y)=v. Syllogistic inference was taken as the reduction of a pair of equations representing the premises to a single equation representing the conclusion, from which the middle term was then eliminated by means of the algebraic rules. These rules, by the way, tended to be those normally found in numerical algebra, with one important exception. Boote realized that his algebra could be interpreted in yet a fourth way, as an algebra of I and 0. In such a restricted algebra one finds the special rule: x~x. This allows the algebra to reflect the important fact about classes that any class is equivalent to its intersection with itself. Boote called the interpretation of variables as class names the ''primary" interpretation. By interpreting his terms as propositions (the "secondary" interpretation), Boote showed that his logic was also a propositional logic, a logic of compound statements. On such an interpretation 'I' was read as 'all cases', '0' was read as 'no case', and a propositional variable, 'x', was read as 'cases when x is true'. For example, a conditional, 'If x then y', was parsed as 'No case is a case when x is true andy is false': x(l-y)=O. In general, we have the following parallels: Formula

Term reading

x(l-y)=O xy=O xy=v x(l-y)=v

Every x isy Nox isy Somex isy Some x is not y

Propositional reading lfx then y lfx then noty xandy x andnoty

Boote noted ''the close and remarkable analogy which [the theory of Secondary Propositions] bears with the theory of Primary Propositions." He added, "It will appear, that the formal laws to which the operations of the mind are subject, are identical in expression in both cases. The mathematical processes which are founded on those laws are, therefore, identical also" ( 1854: 171 ). Notice that Boote did not claim that the logic of terms (classes), the algebra given the primary interpretation, was in any way more basic than the logic of propositions, the algebra given the secondary interpretation. Neither logic was taken as the foundation for the other. But the issue was not settled. Until the take-over by mathematical logic at the tum of the century, the question of which logic, term or propositional, was foundational for the other remained a topic of much dispute among algebraic logicians. By the first decade of the new century,

43

CH~I

the mathematical logicians and the algebraic logicians, such as MacColl, had established what is today generally held as the "default.. position: the logic of propositions is foundational for the logic of terms. While Boote, De Morgan, Pierce, and others were apparently neutral with respect to this position, I will not be. In the second half of this work I will reverse the now commonly held view. My claim will be that the logic of propositions is an important but small branch of the logic of terms, so that the logic of terms is foundational. 43 The great contribution made by Boote to formal logic was his production of a thoroughly general formal system. While those who followed him rejected various details of his logic, his vision of a fully formalized language, subject only to specified general rules, constituting an algorithm for modelling natural language inference, was shared by all of them. The traditional logic was never changed by Boote. He simply supplied it with a symbolic algorithm, which happened to admit alternative interpretations. The most ambitious attempt radically to change traditional syllogistic, without abandoning it, belonged to Boote's sponsor, friend, and fellow mathematician, Augustus De Morgan. The nineteenth-century algebraic logicians introduced three important ideas into logical investigations: the idea that there was at least a formal connection between the logic of terms and the logic of compound statements, the notion of a universe of discourse, and the realization that relational statements cannot be ignored by any adequate logic of terms. Boote was responsible for clarifying the first idea; De Morgan introduced the other two. De Morgan's work on logic extended from the early 1830s to the mid-1860s. His most original and important ideas are found in his series On the Syllogism ( 1846-62). It was well recognized by the early nineteenth century that traditional syllogistic could not easily model mathematical proofs (e.g., Euclid's proofs). One reason for this was the increasingly common view that mathematical statements are relational (and thus apparently not categorical). As a mathematician, De Morgan was committed to the view that mathematics is an instrument of sound reasoning. But the apparent relational character of mathematical statements means that either (i) mathematical reasoning is not syllogistic, or (ii) mathematical statements are in fact (and contrary to appearances) categorical. If the latter is the case, then the relational expressions of such statements must be construed as copulae. Initially De Morgan took the mathematician •s is and is equal to as copulae, trying to reduce relationals to categoricals. Eventually, however, he concluded that in effect these two copulae are actually abstract relational terms. In On the Syllogism, II, he wrote that an abstract copula is "a formal mode of joining two terms which carries no meaning, and obeys no law except such as is barely necessary to make the forms of inference follow .. (1966: S 1). This view soon led him to the more radical notion that all logic is simply the study of relations. Any relation (not just is and is equal to) can

44

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

be a copula. Moreover, only the formal features of a relation are important for logic. These formal features, for De Morgan, included especially transitivity and symmetry. In other words, De Morgan considered copulae/relations purely abstractly. Indeed, the attempt to look at logic as abstractly as possible (to achieve generality) was a hallmark of all of De Morgan's work in that area. In his Formal Logic of 1847, he wrote, "In the form of the proposition, the copula is made as abstract as the terms: or is considered as obeying only those conditions which are necessary to inference" (1966: ix). For De Morgan, then, inference in mathematics depended on the transitivity of the two copula. But to see this is to recognize-as De Morgan eventually did-that the copulae are themselves relations. Indeed, other mathematical relations (e.g., 'is greater than', 'is less than') are transitive and could easily be construed as copulae as well. 44 Since he took mathematical reasoning to be the discovery of new relations on the basis of known relations, it was natural for De Morgan to conclude eventually that (a) logic in general is the study of relations, and (b) any relation can be viewed as a logical copula. It was in On the Syllogism, II, that De Morgan began his attempt to build an algebra of logic. In doing so, he emphasized the important similarities between logical reasoning and mathematical reasoning. He held that all kinds of opposition (e.g., universaVparticular, affirmation/negation. blacklnonblack) are formally equivalent: "Every pair of opposite relations is indistinguishable from every other pair, in the instruments of operation which are required" (1966: 23). Thus, the mathematical+/- opposition could be used in an attempt to find an algebraic algorithm for reasoning involving any kind of relational opposition. "I think it reasonably probable that the advance of symbolic logic will lead to a calculus of opposite relations, for mere inference, as general as that of+ and - in algebra" ( 1966: 26). Moreover, the process of algebraic elimination could then serve to model logical inference in general: "Speaking instrumentally, what is called elimination in algebra is called inference in logic" (1966: 27). Note that in a valid syllogistic inference the middle term is eliminated. In building symbolic logic, De Morgan did not, unfortunately, take his own hint and make use of+ and - to symbolize any opposition. Instead, he introduced his "spicular" notation. In this notational scheme, parentheses are used to indicate the quantity (distribution) of a term, and negation is indicated by a dot or the use of a lowercase term letter. Distribution is indicated by a parenthesis facing the term; otherwise, the parenthesis faces away from the term. Thus the four standard categoricals would be formulated as follows. A: S))P, E: S).(P [or: S))p], 1: S()P, 0: S(.(P [or: S()p]. The system was cumbersome and far from perspicuous. A system of inference that takes any relation as a logical copula must account for inferences in which the premises do not share a common copula/relation. In such cases, there will be a question of which copula/

45

CHAPTER I

relation is to be found in the conclusion. De Morgan's response was to make use ofthe notion of the "composition of relations" (1966: SSff, 231, 253). An example of an inference whose two premises make use of different copulae/relations (a "bicopular syllogism") is John can persuade Thomas Thomas can command William So, John can control William Here, the two relations 'can persuade' and 'can command' are composed to yield the relation 'can control'. The relation 'can control' is seen as equivalent to 'can persuade what can command'. Logic is simply the study of such inferences-that is, the Logic of Relations; syllogistic is the part of logic that examines inferences in which the premise copulae/relations happen not to differ from one another. In On the Syllogism, 1/, De Morgan offered his famous challenge to traditional logicians. "I gave a challenge in my work on formal logic to deduce syllogistically from 'every man is an animal' that 'every head of a man is the head of an animal'" (1966: 29). He went on there to claim that this is not a syllogism but "the substitution, in a compound phrase, of the name of the genus for that of the species." What he meant was that, given the assumption 'every head of a man is the head of a man', one can substitute the name of the genus ('animal') for the name of the species ('man'), where the species-genus relation is given by the explicit premise, in at least one of its occurrences in that assumption. Substituting 'animal' for 'man' in its second occurrence in 'every head of a man is the head of a man' yields the conclusion 'every head of a man is the head of an animal'. De Morgan saw such "oblique inferences" as applications of the dictum de omni et nullo, and he took the dictum as a rule of substitution. For, just as in algebra ''we know that in x>y our right of substitution is, that we may for x write an equal or a greater, for y an equal or a less. In x>y, y is used after the manner of a universal term in logic, x after the manner of a particular" ( 1966: 28).45 Consider the following argument. Every person who loves some human is happy. Every American is human. So every person who loves some American is happy. Here the dictum requires 'human' to be universal in the first premise. But, as we will see later, this is merely the result of a confusion between the quantity of a term and its "distribution value." For traditional syllogisms, with simple terms, these are the same. But this is not so for complex cases, such as those containing relational terms. In the example above, 'human' is particularly quantified, but it has universal distribution. Once my own algorithm, making use of De Morgan's suggested+/- notation, is developed

46

THE GOOD OLD DAYS OF THE BAD OLD LOGIC

below it should be clear that the charge that "in the absence of some other criteria of particular and universal use, the dictum cannot be applied" (Merrill, 1990: 88) is best answered simply by providing just such a criterion. In general, De Morgan made the mistake of formulating the dictum in terms of species/genus. The fact is that often the universal premise is not a species/genus proposition at all. Thus in Formal Logic he wrote that "when X)Y, the relation of X toY is that of species to genus" (1926: 75). For De Morgan, all statements are relational (rather than categorical) in that they always relate (bring together, connect) pairs of terms, a view that at least reminds us of Aristotle's ternary analysis. While he was on the right track in taking relationals into a logic of terms, he was clearly mistaken in seeing relational terms as copulae rather than as (material) terms in their own right. In his recent study of De Morgan's logic of relations Merrill argues, "It is one thing to extend logic to include relations; it is quite another thing to think of subject-predicate propositions as relational propositions" (1990: 107). This suggests the stronger claim that it is also wrong to think of relational propositions as subject-predicate. I will reject that suggestion below. De Morgan, and the algebraists in general, was a "monist" in the sense that he held that there is just one general logical form for all categoricals and relationals. For most, this general logical form was categorical; for De Morgan, it was relational. Later logicians, especially Peirce and Frege, were "pluralists" in that they insisted that there are as many general logical forms as there are numbers of argument places of functional expressions {viz., 0 for unanalysed propositions, I for simple monadic predicates, 2 for binary relations, 3 for triadic, etc.). But De Morgan did not see such expressions as functions. Instead, he saw each proposition as consisting of a pair of terms bound together into a syntactic unit by use of a connecting element, a copula. All relational terms, in spite of the fact that they are prima facie material expressions, are copulae (thus formatives as well). But in at least one place in On the Syllogism, II. De Morgan suggests that it is possible to analyse a relational proposition as a categorical. Using "=" as the logical copula, he wrote, "The algebraic equation y=ifJx has the copula=, relatively toy and ifJx: but relatively toy and x the copula is =t/J. This is precisely the distinction between 'John can persuade Thomas' and 'John is {one who can persuade Thomas.}' " ( 1966: 56). Aside from the confusion due to using equality as the standard logical copula, this alternative analysis wants only a way to analyse expressions such as 'one who can persuade' as genuine nonformal terms connected by a logical copula to the next term. The result would be a logical syntax that takes, as Aristotle suggested, all propositions as pairs of terms connected by a logical copula. We will see how to achieve such a theory in the second part of this essay. Critics argued that the algebraic logicians, by making reasoning a

47

CHAPTER I

kind of computation (as Hobbes had seen it), subordinated logic to mathematics, particularly algebra. This charge may be appropriate for Boole, for example, who made important use of algebraic and geometric techniques in his system. But while De Morgan did want somehow to unite logic and mathematics, he did not actually make much use of mathematical techniques in his logic. He saw that traditional syllogistic was too weak to handle certain kinds of inferences (viz., those involving relationals) and developed (or tried to develop) the logic of relations as a general logic fit for the analysis of a very broad range of inferences. He saw syllogistic as being a proper part of that logic. What he did historically, in effect, was to usher in a period (along with Boole, Peirce, etc.) during which a number of attempts were made either to extend or to replace syllogistic, with the aim of obtaining the kind of universal logic envisaged by Leibniz, one fit especially for the analysis of proofs in mathematics. By the end of this period, logicians were convinced that syllogistic logic could no longer be considered seriously as anything more than a respectable but powerless alternative to a new, comprehensive, powerful "symbolic logic." Unfortunately, syllogistic logic was presumed identical to term logic. While De Morgan and other algebraists might have been willing to demote, or even reject, syllogistic logic, they were nonetheless term logicians. For better or worse, term logic was tarred with the anti-syllogistic brush, and symbolic logic (eventually, mathematical logic) soon replaced the old logic and quickly used its new success to bar the door to any term logic-even a nonsyllogistic one such as De Morgan had envisaged.

48

Notes for Chapter One Geach (1972a: 44-61 ). We will encounter this view (called "verbism") in chapter 2. For now, see Englebretsen (1982a, 1985a, and 1986a). 3 For Russell's attempt to reject Aristotle, see Russell (1945: 195-202). 4 I have offered a slightly different survey of Aristotle's logic in Englebretsen ( 1981 ). See, as well, Lukasiewicz ( 1957) and Lear ( 1980). Corcoran and Scanlan ( 1982) is an ideal place to start. See also the essays in Corcoran (1974a). 5 Geach claims on p. 53 of Logic Matters ( 1972) that Aristotle still held in Prior Analytics that a statement could consist of just a pair of unlinked terms, that such expressions as 'applies to' do "not supply a link between 'A' and 'B' "but were "meant only to give a sentence a lecturer can pronounce." 6 Cf. Englebretsen ( 1989, 1990b). 7 For a very thorough discussion of negation in all of its aspects see Hom (1989). See also Sommers (1982) and Englebretsen (1981b). 1 For more on squares of opposition see Sommers (1982, ch. 14); Sommers (1970); Englebretsen (1984a, 1984b). 9 For discussion concerning the Aristotelian notion of valid syllogism see Barnes (1969); Frede (1974); and Hadgopoulos (1979). 1°For an alternative view see Corcoran (1974b). 11 Cf. Englebretsen (1980a); Barnes (1983). 12 For a survey of Aristotle's attempt here see Englebretsen (1982b). See also Thorn (1977). 13 See especially Ross (1949: 289); Lukasiewicz (1957: 6-7); Bird (1964: 90); Patzig ( 1968: 5-7). A response to all of these claims is found in Englebretsen (1980b). 14 For a brief discussion of Scholastic semantics and its relation to modem logic see Henry (1972: 47-55). See as well A. de Libera (1982); Spade (1982); Nuchelman (1982). 15 See, especially, Abelard's Dia/ectica (1956). 16 Philosophica Disciplina in Lafleur ( 1988: 282). I owe this reference and the next, as well as other valuable information here, to Graeme Hunter. 17 See C. Bazan's "Les questions disputees principalement dans les facultes de theologie" in Bazan et al. (1985, esp. 40). Also see Angelelli (1970). 18 See, for example, Kneale and Kneale (1962: 232-3). 19 For a nice summary of this, see Kneale and Kneale (1962: 274-97). The classic extended discussion is Moody (1953). See also Durr (1951). 1

2

49

NOTES FOR CHAPTER I

A nice account of how Ockham might have used his supposition theory to account for oblique inferences is given in Sanchez ( 1987). 21 See Ashworth (1974). 22 For two excellent surveys see Jardine ( 1982, 1988). 23 See Ashworth ( 1988). 24 An excellent extended account of Descartes's views on logic is found in Gaukroger ( 1989) and in Clarke ( 1981 ). 25 For example, Locke ( 1924: 346-7) writes, "If syllogisms must be taken for the only proper instrument of reason and means of knowledge, it will follow that before Aristotle there was not one man that did or could know anything by reason; and that, since the invention of syllogisms, there is not one often thousand that doth." 26 A version of what follows has appeared in Englebretsen (1990c). 27 See especially Arnauld and Lancelot (1975); Arnauld and Nicole (1964). 21 A similar summary is given in Sommers (1983a). 29 See the excellent study by Castaneda (1982). 30 In the papers on grammatical analysis (Parkinson, 1966: 12-6), Leibniz shows how the number of natural-language particles can be reduced to a few, which correspond to logical particles. McRae ( 1988) has pointed out that in addition to this purely logical interest, Leibniz also had an interest (shared with Locke) in the project of using natural-language particles as indicators of the internal operations of the mind. 31 An excellent examination of Hobbes's theory is found in Hungerland and Vick's "Hobbes's Theory of Language, Speech, and Reasoning,'' in Hungerland and Vick (198la). See also Dascal (1987, ch.l and 2). 32 The account below closely follows those given in Englebretsen (198la, l982c). 33 According to Ashworth (1974: 247), "any proposition whose predicate was singular was treated as if it were of a standard form." 34 In addition to Sommers. two others have held something like the wild quantity thesis. See Czezowski (1955) and Copi (1982). All are discussed in Englebretsen (1986a, l986b, 1988b). 35 Compare Leibniz's remarks in Parkinson (1966: 14-5). For a brief defence of Leibniz's way with relationals against Russell's see Angelelli (1967: 19ft). 36 In "A Specimen of a Demonstrated Inference from the Direct to the Oblique" (in Parkinson, 1966: 88), Leibniz says that the following (i.e., the dictum) holds of such inferences: "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 proposition where that subject plays the part of predicate." 37 Parkinson (1966: xx) seems to be making such a criticism. 20

so

NOTES FOR CHAPTER I

For a useful, extended account of Leibniz's attempt to incorporate the logic of compound statements into syllogistic see Castaneda ( 1976, 1990). See also Ishiguro (1982, 1990). 39 More on what follows regarding Leibniz's Law is found in Englebretsen (1984c). 40 This has been shown in Englebretsen ( 1988b). 41 See, for example, section 9 of"Discourse on Metaphysics" (in Leibniz, 1989: 41ft) and section 40 of the "Monadology" (in Leibniz, 1965: 154). 42 As Patzig (1968: II) says, "I offer here the hesitant conjecture that the tendency which has continually reappeared in the history of logic, not least in more recent times, of conceiving a judgement as an equation, or even as an expression of identity, derives a good deal, if not all, of its force from the purely conventional wording of the schema'S is P'." 43 For more on the controversy among late-nineteenth-century logicians, see Shearman (1906). 44 For a time, De Morgan considered the idea that all relations could be reduced to a single one-identity. For more on this see the excellent discussion in Merrill ( 1990, ch. 2 and 3). 45 Merrill (1990) tries to show that De Morgan was unable to defend the use of his substitution dictum for cases where the substitution applies to parts of complex terms. De Morgan had formulated the dictum in terms of'some' and 'all' as signs of particularity or universality. But where these occur within complex terms, they are sometimes disconsonant with the true distributivity of the terms to which they apply.

31

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CHAPTER TWO

A MODERN SUCCESS STORY (or, Frege to the Reseue) Dear Senior Censor, In a desultory conversation on a point connected with the dinner at our high table you incidentally remarked to me that lobster sauce 'though a necessary a4iunct to turbot, was not entirely wholesome!' It is entirely unwholesome. I never ask for it without reluctance: I never take a second spoonful without a foeling of apprehension on the subject of a possible nightmare. This naturally brings me on to the subject ofMathematics . .. Lewis Carroll Far recent times have seen the development ofthe calculus oflogic, as it is called. or mathematical logic, a theory that has gone far beyond Aristotelian logic. It has been developed by mathematicians; profossional philosophers have taken very little interest in it, presumably because they found it too mathematical. On the other hand. most mathematicians, too, have taken very little interest in it, because they found it too philosophical. ThoralfSkolem We know that mathematicians care no more for logic than logicians for mathematics. The two eyes ofexact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye, each believing that it can see better with one eye than with two. De Morgan

Frege The logicians' conceit is due to their supposing that the ideas could only be learned or rendered clear by their method. their studies and their labours. They therefore interpreted a language in which they are weak

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and ofwhich their knowledge is imperfoct into another in which they are also weak and their knowledge is imperfect. This sort oftranslation they made into an art, and then declared that they have to do only with words, not with ideas. AbuHayyan Mathematicians are a species ofFrenchmen: ifyou say something to them they translate it into their own language and presto! it is something entirely different. Goethe

Generally speaking, the twentieth century has seen a fairly clear division of philosophy into two quite different branches. One, analytic philosophy, has been pursued mostly in English-speaking countries. It is (again, speaking quite generally) primarily interested in the investigation of relatively narrow problems, especially in epistemology and metaphysics, has had a fairly high regard for the natural sciences, and has tended to formulate its questions as concerning language. Most importantly, it has taken formal logic, in one guise or another, to be an essential tool in its investigations. The other branch of philosophy encompasses a much broader range of philosophical programmes, most of which have been pursued by philosophers on the European continent (thus it is often called "continental" philosophy). Continental philosophers have generally abjured recourse to the results of the natural sciences, and have tended to blur distinctions between philosophy and such disciplines as history, psychology, sociology, political theory, anthropology, literary theory, and so on. In particular, they have generally had little regard for logi~pecially formal logic. But the two very closely related fields of philosophy oflogic and philosophy oflanguage have come to dominate the work of analytic philosophers. These fields might well be said to be the core of twentieth-century analytic philosophy. To be sure, questions concerning the nature of logic and language are as old as Plato and Aristotle, but their modem versions go back only about a century or so. Indeed, many of the key questions in these areas were formulated by one man. Ironically, he was a German and a logician, and his ideas grew out of his attempts to provide a rational grounding for mathematics. Beginning in 1879, Gottlob Frege, a then relatively obscure mathematician at the University of Jena, developed a conception of logic and an adequate algorithm for it that were radically different from any earlier theory. What Frege started was a revolution in logic unmatched by any other change in logic since Aristotle.• The result of that revolution was the theory of logic that has virtually become canonical in most philosophical circles ever since-mathematical logic. From our perspective of more than a century later it is easy to look at Frege's revolution as representing a quick, clean break with the logic that had gone before (viz., that of the algebraists), but the fact is that there was a substantial period of transition (and even

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confusion2). Actually, the first attacks on the traditional logic of the algebraists came from non logician philosophers (especially idealists and pragmatists). Of course, the algebraists themselves often tarnished syllogistic's image by rejecting parts of it or by defending it inadequately. To understand Frege's revolution and the mathematical logic it initiated, one must farst understand something of what mathematical thinking was like during the period from the development of the Leibniz-Newton calculus to the mid-nineteenth century. From the time of Euclid to the nineteenth century, mathematicians, with few exceptions, looked at geometry as providing the foundation for the entire field of mathematics. To be sure, after the development of the calculus, and of analytic methods in general, there was a renewed interest in showing that mathematical reasoning was at least guided by physical observations. But even then, the model for mathematical proof was the derivation of a geometric theorem from Euclidean axioms. Physical science (especially astronomy, physics, cartography) merely supplied mathematicians with motivations and confirmations for their purely formal proofs. However, by the end of the eighteenth century, efforts to make mathematics, particularly the calculus, logically rigorous were becoming strained. An ever-expanding array of new functions, complex numbers, and even negative numbers were being formally manipulated by mathematicians who had little idea how one might construct convincing proofs of theorems in which these were involved. Most mathematicians during that period chose to ignore the problem of logical rigour and simply plunged ahead. As early as 1743, D' Alembert noted that "Up to the present ... more concern has been given to enlarging the building than to illuminating the entrance, to raising it higher than to giving proper strength to the foundations" (quoted in Kline, 1972: 619). Common sense, intuition, and conformity with nature were sufficient for eighteenth-century mathematicians. Their work, admittedly, was not modelled on Euclidean proof, but no one doubted that it could be. Copernicus, Kepler, Descartes, Newton, Leibniz, and their followers all were certain that mathematical truths were in harmony with, even reflections of, God's design of nature. There were exceptions, of course, the most famous being Laplace, who replied to Napoleon's remark that the mathematician's work contained no reference to God by saying that he had no need of such a hypothesis. But for most mathematicians, logical proof could wait. In spite of the confidence that mathematicians of the seventeenth and eighteenth centuries had in the possibility of grounding the truths of mathematics, a confidence due to the perceived objectivity and self-evidence of Euclidean geometry, little of that confidence survived the nineteenth century. The development of non-Euclidean geometries shattered the comforting old myth that Euclid could ultimately provide the foundations for all mathematics. Gauss, Bolyai, Lobatchevsky, and Riemann gradually convinced mathematicians that Euclidean axioms were merely empirical,

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CHAPTER2

based on everyday-but therefore limited-observations of mundane space. Any number of alternative geometries could be formulated that would be equally self-consistent and might very well describe space more generally. Eventually, the foundations of geometry, analysis, and, finally, set theory were all seriously challenged during the nineteenth century. By the middle of the century, the door was open to any number of formal systems unencumbered by the demands of self-evidence or empirical evidence. Thus Abel could charge, "There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner" (quoted in Kline, 1972: 47). Still, the hope that mathematics had a rigorous foundation was not completely lost. The search for mathematical rigour led to the investigation of logic as providing a sufficiently strong foundation for mathematics. But this led, in turn, to a search for rigour in logic itself, since traditional logic was seen as inadequate in too many ways. Thus mathematicians like Boole and De Morgan directed their efforts to algebracizing logic and thereby attaching it to mathematics (and separating it from philosophy and psychology, the perceived sources of its old weakness). Later, Hilbert and the so-called formalists sought to ground geometry on arithmetic and logic, which were purely formal systems bound only by the demand for consistency and designed to formally manipulate symbols. But by far the most influential and carefully articulated attempt to give a solid grounding to mathematics was that initiated by Frege. Working quite independently of his contemporaries, he axiomatized logic in his Begriffschrift (1879), and then in the Grundlagen (1884) and the two volumes of the Grundgesetze (1893-1903) he showed that mathematics is an extension of logic. This programme was eventually called "logicism." As it happened, however, Frege's revolutionary work was virtually ignored for more than a decade. It was rescued from oblivion only at the turn of the century, when Bertrand Russell gave the programme oflogicism and the new mathematical logic the prominent, international voice it needed. Frege's logic was revolutionary in several ways. Traditional logicians, with the exception ofLeibniz (who forged a connection between logic and algebra), saw no special relationship between logic and mathematics. The nineteenth-century algebraists' logical renaissance was partly due to their insistence that logic was merely a branch of mathematics. Frege went further still: arithmetic is the foundation of mathematics and arithmetic is logic. For Frege and his followers, all mathematics is merely an extension of logic. As a consequence of this view, Frege's logic was revolutionary in a second way. Logic was no longer seen as a special concern for either philosophers or psychologists. It does not describe reality, the knowable, or how one reasons. What it does is account for mathematical reckoning. Moreover, it has no close relationship with natural language, which, unlike mathematical language, is plagued by lack of clarity, ambiguity, vagueness, and inconsistency. The language of logic is,

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first and foremost, the underlying rational structure of mathematical language. Later logicians would go beyond Frege by holding that even natural language is, in a deep and hidden way, ultimately structured by this logic as well. The "look" of Frege's logic was also revolutionary. Earlier mathematizing logicians, such as Leibniz, Boole, and De Morgan, were content to use mathematical notation in their constructions of symbolic algorithms for logic. Frege, in order to avoid confusing mathematical expressions with underlying logical notations, chose an algorithm expressed in nonmathematical symbols. Though his own symbolism was not adopted by others, alternative nonmathematical notations were. As an ironic consequence, mathematical logic (often called "symbolic logic") is expressed in a symbolic vocabulary quite foreign to most mathematicians. But the most important way in which Frege's logic was revolutionary was in its theory of logical syntax-its account of sentential unity. Frege abandoned the traditional analysis of statements into subjects and predicates, substituting an analysis in terms of the mathematician's functions (i.e., function expressions) and arguments. 3 In his theory, complex statements are built up from less complex statements. The least complex of all are "atomic," consisting of a single function and an appropriate number of arguments. The distinction between a function and an argument is initially made in grammatical terms:4 arguments are grammatically complete, "saturated,'' contain no gaps; functions are incomplete, "unsaturated," contain one or more gaps. Frege recognized that two complete expressions alone could not form a sentence; nor could two incomplete expressions (again, two axe-heads do not make an axe-nor do two axe-handles). In the farst case especially, there must be a "binding agent." As he said, "An object-e.g., the number 2-cannot logically adhere to another object-e.g., Julius Caesar-without a binding agent [or cement: Bindemittle]. And this binding agent cannot be an object but must rather be unsaturated."5 For him, the "binding agent" is not a third expression (a logical copula). It is just an incomplete expression, with sufficient gaps to accommodate the complete expressions. Logical sentential unity is, according to Frege's account, not the result of expressions being connected by a connector but of one or more gapless expressions filling the gaps in a gapped expression (linkage as completion rather than connection). This account is supplemented by a semantic theory. Arguments are names that refer to objects; functions are not names and refer to concepts. No object is ever a concept; no concept is ever an object. No argument can be used as a function; no function can name an object-that is, be used as a name/argument.6 Consider the simple sentence 'Socrates is wise'. The function expression here is ' ... is wise', which contains a single gap (argument place), and the argument is 'Socrates'. 'Socrates' names an object (viz., Socrates), while the function refers to the concept of wisdom (which, in fact, is not being named by ' ... is wise'). Functions do not name,

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but they do apply to objects (viz., those objects which "fall under'' themi.e., those objects of which they are true). In 'Socrates is wise', 'Socrates' names an object and •... is wise' applies to that object. In effect, according to this theory, singular terms (proper names, personal pronouns, definite descriptions) are names; general terms (verbs, adjectives, etc.) are function expressions. Relational terms are also function expressions. In 'Socrates taught Plato'. the function is •... taught .. .', an expression with two argument places, two gaps. The logically simplest kinds of statements, atomic statements, consist of a single function expression (still often called a "predicate") and arguments (names) tilling each gap. Thus. atomic statements are always singular. Notice that in certain ways Frege's theory of logical syntax (at least for simple, atomic statements) is similar to Plato's binary theory. Each sees the nonformal terms of language to be divided into two exclusive sets: for Plato, nouns and verbs; for Frege, singular terms (names) and general terms (predicatesV In each case, logically simple statements contain no formal, logical expressions; they have zero degree of syntactical complexity. Complex ("molecular'') statements are built up from less complex ones by the use of "higher" functions. For example, pairs of statements can be combined by "truth-functional" expressions to form such compounds as conjunctions, disjunctions, and conditionals. The gaps of such expressions, like those of general terms, must be tilled by names. Grammatically, the only expressions that can till the gaps of a truth-function (e.g., •... or .. .') and result in a statement are statements themselves. For Frege, then, statements must be names, singular terms. Consequently, they are seen as naming objects. According to Frege, there are two possible objects that a statement can name: the True and the False. True statements name the former; false statements name the latter. Truth-functional expressions (now often called "statement connectives" or "propositional connectives") are formal terms that operate on one (in the case of negation) or two (in all other cases) statements to form logically more complex statements. This recursiveness guarantees that there is no limit to the degree of complexity that a statement can have. The logic of statements whose complexity is due only to the use of truth-functions is called "truth-functional logic," or "statement logic," or "propositional logic." Statement logic, according to Frege's theory, is taken to be basic logic. Complexity can be achieved in another way. Consider again 'Socrates taught Plato'. Suppose now that we replace each of the proper names here with a variable name, one which might name any object. Thus we might have •x taught y'. This is not itself a statement. The •x' and 'y' here (called "individual variables") are like the personal pronouns of a natural language (cf., 'She taught him'). And, as with pronouns, they can be used sensibly only with an antecedent expression that determines their reference. In the case of individual variables, their antecedents are quantifiers and are said to "bind" their subsequent individual variables. The

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expression 'x taught y' (called an "open formula..) might be thought of as a template for a statement. It can be turned into a genuine statement by binding each individual variable by a quantifier. There are two kinds of quantifiers, and, for our sample case, two variables to be quantified and two possible sequences of binding ('x' can be bound first or 'y' can be bound fllSt). So there are eight possible ways of turning 'x taught y' into a statement. The two quantifier expressions are 'something, call it v, is such that ... ' and 'each thing, call it v, is such that ... ', where 'v' is an individual variable such as 'x' or 'y' and the gap is filled by an open formula. In the present case, the eight possible statements are: I. Something, call it x, is such that something, call it y, is such that x taught y. 2. Something, call it y, is such that something, call it x, is such that x taught y. 3. Each thing, call it x, is such that each thing, call it y, is such that x taught y. 4. Each thing, call it y, is such that each thing, call it x, is such that x taught y. 5. Something, call it x, is such that each thing, call it y, is such that x taught y. 6. Something, call it y, is such that each thing, call it x, is such that x taught y. 7. Each thing, call it x, is such that something, call it y, is such that x taught y. 8. Each thing, call it y, is such that something, call it x, is such that x taught y. Frege is justifiably credited with being the fllSt logician to give an adequate account of the logical syntax ofsuch "multiply general sentences,.. sentences with more than one quantifier expression. According to Michael Dummett, His success here was due to his recognition of the fact that the logic of such sentences could be revealed by examining the history of their construction. Thus we begin with a sentence such as 'Peter envies John'. From this we form a one-place predicate 'Peter envies y' by removing the proper name 'John'-the letter 'y' here serving merely to indicate where the gap occurs that is left by the removal of the proper name. This predicate can then be combined with a sign of generality 'somebody' to yield the sentence 'Peter envies somebody'. The resulting sentence may now be subjected to the same process: by removing the proper name 'Peter' we obtain the predicate 'x envies somebody' and this may then be combined with the sign of generality 'everybody' to yield the sentence 'everybody envies somebody'. (Dummett, 1973: 10)

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Notice that beginning with the sentence 'Peter envies John' the choice of which proper name to replace by a variable name farst (and, in so choosing, the choice of which "sign of generality," quantifier expression, to use) will detennine a variety of different multiply general sentences (e.g., 'somebody envies everybody', or 'everybody envies everybody', etc.). Such "quantified" statements are genuine statements; thus, they can fill gaps in truth-functional expressions. (This is one of the reasons that, on a Fregean analysis, the logic of truth-functions is more basic than the logic of quantified statements.) Notice that quantifiers are higher functions. A system of logic that incorporates quantifiers as well as truth-functions is now called "quantificational logic," "the functional calculus," or "predicate calculus." One kind of statement seems to defy the neat syntactical theory worked out by Frege. Consider 'Shakespeare is Bacon'. In this sentence, 'Shakespeare' and 'Bacon' are clearly proper names and operating as arguments. But where is the function expression? In rejecting the traditional subject-predicate analysis of statements, Frege had banned the copular (i.e., qualifier) 'is' from logic.• For example, in 'Shakespeare is British' the 'is' plays no logical role. But if 'is' is ignored in 'Shakespeare is Bacon', there is no tenn to play the role of predicate (function expression). Frege's solution was to introduce the so-called "is of identity." The 'is' of' Shakespeare is Bacon' (contrary to appearances) is not at all like the 'is' of'Shakespeare is British'. In the case of'Shakespeare is Bacon' the 'is' is merely short for 'is identical to' (or 'is the same as'), and 'is identical to' is, without question, a function expression. incomplete, unsaturated, gapped (doubly). To ignore the logical distinction between the 'is' of identity and the 'is' of predication (i.e., the copular 'is', the old qualifier) would be to treat a name, say 'Bacon', as a predicate, as incomplete (and, as a consequence. to treat an object, say Bacon, as a concept). But, again, arguments (names, singular tenns) and functions (predicates, general tenns) are mutually exclusive from a Fregean logical point of view. A predicate logic that incorporates the identity function is called "predicate calculus with identity." Aristotle (and Plato) and all of his followers for centuries tended to see formal logic as applying to inferences made in the medium of a natural language (Greek, Latin, Gennan, English, etc.). Frege. by contrast, had little regard for the logical powers of natural language. He wrote, for example, If our language were logically more perfect, we would perhaps have no further need of logic, or we might read it off from the language. But we are far from being in such a position. Work in logic just is. to a large extent, a struggle with the logical defects of language, and yet language remains for us an indispensable

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A MODERN SUCCESS STORY tool. Only after our logical work has been completed shall we possess a more perfect instrument. (Frege, 1979: 252)9

For Frege, mathematics represented the paradigm case of rationality. His goal was not to build a system of logic for natural language; rather, he sought to construct a system of logic adequate to the needs of mathematics. The rigour that mathematics seemed to have lost in the nineteenth century was to be recovered by founding mathematics (especially arithmetic) on logic. The fact that the logical system he built required a theory of syntax remote from the forms of natural-language statements was of no concern to him. From Aristotle to the algebraists, nearly all logicians (and grammarians) recognized that both statements and terms have opposites. Thus, for the traditional logician, every statement that affirms a given predicate of a given subject corresponds to one that denies the same predicate of that subject. Such pairs of statements are contradictories. Every statement has exactly one contradictory. The denial is often said to be the negation of the affmnation. Terms were also taken to have opposites. Let 'T' be a term and 'nonT' be its negation. For every statement that affmns 'T' of a given subject there is one that affirms 'nonT' of that subject. Such pairs of statements are (logical) contraries. Each term has exactly one negation, one logical contrary. Term negation and statement negation were taken by traditional logicians to be clearly distinct, and logical contrariety between a pair of statements was accounted for by term contrariety. The logical contrary and the contradictory of a given statement were not to be equated. The former always entails the latter, but the latter does not always entail the former. So, to summarize the standard traditional view, statement negation, contradiction, is the result of denying the predicate. Statement contrariety is the result of negating the predicate-term. Thus, any statement can be negated and any term can be negated. There are two logically distinct modes of negation. 10 For Frege and his followers, there is but one kind of logical negation. In the Begriffschrift one finds only one sign (the short vertical stroke) for this: it is the sign for statement negation. It is applicable only to an entire statement and results in the contradictory of that statement. Frege recognized that grammatical negation occurs in various places within natural language sentences. But in his essay "Negation" (Geach and Black, 1952) he reminds the reader (as he did so often throughout his writings) that such "languages are unreliable on logical questions" and goes on to warn of the "pitfalls laid by language" ( 126). So, even though a grammatical sign of negation may appear to apply to a term or part of a sentence, in such cases the logical fact is "that we do [thereby] negate the content of the whole sentence" ( 131 ). According to Frege, there is no term negation-there is just sentential negation. However, as it happened, he did consider the possibility of a second mode of negation.

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Frege took pains to distinguish between what he called the "content of a thought.. and the "assertion of its truth.'' Suppose I say to you, 'Plato taught Aristotle'. In doing so in the appropriate context (with the appropriate tone ofvoice, etc.), I am claiming, at least implicitly, that what I say is true. We can think ofwhat-1-say as the content of my thought (the object expressed by my sentence). We can think of my implicit claim as the assertion of its truth. Some things one says are not asserted (as truths), such as questions, commands, and promises. Logically more important are sentences that are truth-functional parts of compounds, such as disjuncts or conditional antecedents. Suppose I say to you, 'If Plato taught Aristotle, then Plato knew syllogistic'. Here, I assert the entire conditional, but I do not assert its antecedent. With this distinction in mind Frege raised the possibility that in addition to sentential negation-the kind of negation that applies to the content of an entire statement and results in its contradictory-there could be a kind of negation that is the opposite corollary of assertion, a second way of "judging a thought... Are there two different ways of judging, of which one is used for the affinnative, and the other for the negative. answer to a question? Or is judging the same act in both cases? Does negation go along with judging? Or is negation part of the thought that underlies the act ofjudging? ( 129)

Frege's answer is that there is only one mode of judgment-affirmation (assertion): "it is a nuisance to distinguish between two ways of negating.. (128). 11 The result ofhavingjust one kind of negation, said Frege, is an "economy of logical primitives.. (130, cf. 48, 149ft). One of the prices paid for this economy of logical primitives is the loss ofterm negation, and thus, it seems, the notion of a statement having a logical contrary (as distinct from its contradictory). But, in fact. contrariety is preserved now in terms of sentential rather than term negation. Consider the two sentences 'All logicians are rational' and 'All logicians are irrational (nonrational)'. The logical contrariety between these two is accounted for by the traditional logician in terms of the logical contrariety between a tenn ('rational') and its negation ('irrational'). The logical contrariety between a term and its negation is primitive, and it is the basis for defining the logical contrariety between certain pairs of sentences. In contrast, for the Fregean logician, the only primitive negation is sentential. So the tenn negation in 'All logicians are irrational' is merely grammatical. Logically, the sentence is construed as a negation of another sentence. But what is that other sentence? It cannot be 'Not every logician is rational', since this is the contradictory of'Every logician is rational'(= 'All logicians are rational'), not its logical contrary. In such cases, the quantif.er expression is taken as

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a function whose argument is an entire sentence. That sentence is itself a compound (truth-function) of two sentences. The second of these is negated. Our two sentences. then, are construed as follows. Each thing, call it x, is such that: ifx is a logician then xis rational. and Each thing. call it x, is such that: ifx is a logician then it is not the case that x is rational. These are clearly not contradictories. It would not be possible for both to name the True-but they could both name the False. They are contraries (neither of which contains a negated term). So Frege, like traditional logicians, could preserve the contradictory/contrary distinction. And he could do so while economizing on the number of logical primitives (viz., forsaking the use of term negation). Nonetheless, in part 2 of this essay we will find reasons for judging this to be a false economy. Traditional logicians had little hope of building a truly unified logic, a single theory of logic (with an adequate algorithm for logical reckoning) that would accommodate inferences involving categoricals as well as those involving singular terms, compound sentences. and relationals. Leibniz had hope, and important insights, as we have seen. He recognized that unity could be achieved in part by reparsing all statements (including singulars, compounds, and relationals) as categoricals. But Boole and his followers saw that a logic of terms and a logic of sentences could be unifJed in the sense that a single algorithm could be used for either Primary or Secondary logic, as he called them. In other words, the logical syntax of categoricals and the logical syntax of compounds are isomorphic. De Morgan and Peirce then tried to fat relationals to this logic. Frege achieved a very high degree of unity for logic. In his criticism of Boole and the algebraists he pointed out that genuine unity is not achieved simply by letting term and sentential logic "run alongside one another, so that one is like a mirror image of the other, but for that very reason stands in no organic relation to it" (Geach and Black, 1952: 14). Frege reduced categorical sentences to compound sentences ( 17) because he took judgment (statementmaking) to be prior to conception (term use). 12 He was able to effect this reduction, and thus unify heretofore separate parts of logic, by replacing the subject/predicate distinction with the function/argument distinction. Every statement is a function of one or more arguments. The main function of compounds is truth-functional; relationals are low-level functions ("firstorder predicates") on more than one argument. All arguments (names) are singular. In the next chapter, we will see that this last thesis is what Sommers has called the "Fregean Dogma." I will say more about Frege in the remainder of this chapter. But before concluding these introductory remarks it would be useful to remind the reader, given the post-Fregean developments in philosophical logic, that

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Frege was a genuine, serious Realist. He believed that reality consists of objects and concepts, and that among those objects are not only material objects but Thoughts, functions, numbers, and the True and the False. This kind of Platonism has been vigorously rejected by most of the now nominalist heirs ofFrege's logic. As we have seen more than once, syllogistic has constantly been challenged by inferences involving three kinds of statements: singulars, compounds, and relationals. Solutions to these problems (such as those offered by Leibniz or De Morgan) were generally inadequate or unknown to late-nineteenth-century logicians. Frege's revolutionary logic offered a single system, the predicate calculus with identity, which could easily analyse all three kinds of statements. Syllogistic logic could then be incorporated into the new logic as just a small part of it-the part dealing with singly quantified statements containing a single nonrelational predicate. What Frege achieved was a system of formal logic that was far more powerful (in terms both of expressive power-the ability to formulate a broad range of kinds of statements-and of inference power-the ability to account for inferences in a perspicuous manner). After Frege, it was hard for any logician to look back to an earlier system of logic as having more than historical value.

Bradley and Ramsey Raise Some Doubts Logic sometimes creates monsters. Poincare

British empiricists in the nineteenth century, especially John Stuart Mill, had challenged the objectivity of mathematics, and of reasoning in general, by holding that in any judgment the constituent ideas can do nothing more than refer to subjectively held ideas in the mind of the one who makes the judgment. Moreover, the laws of logic were seen as mere generalizations of the natural associations among such ideas. Inference is not the active application of our rational faculty to propositions or judgments; rather, it is merely the passive recognition of the mind's passage from one idea to another in accordance with the laws of association of ideas. Logic is a matter of psychology and nothing more. This sort of view was known as psychologism. According to Frege, it was the view that posed the greatest danger to the possibility of objective, rationally grounded mathematics. In his view, the acceptance ofpsychologism in mathematics would render all mathematical truths subjective and relative. Mathematics would be pointless. In many ways, Frege's entire corpus in logic can be seen as a rebuttal of Mill's position. One of his most basic challenges was the argument that judgments (propositions) could not be analysed into ideas (terms). Propositions are more basic than terms (the "priority principle"),

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so that one can inquire into the meaning of a term only in the context of a proposition (the "contextual principle"). Propositions are not, then, built up out of terms, as traditional logicians had believed. Nevertheless, propositions can be analysed (into functions and arguments, as we have

seen). Logicians following Frege were eager to exhibit the powers of the new logic, now unencumbered by ties to either old syllogistic or more recent empiricism. They were particularly proud of the fact that they had successfully incorporated relationals into a unified system of formal logic. But celebration was barred almost from the beginning by a paradox usually attributed to Frege's British contemporary, F.H. Bradley. In his Appearance and Reality (1893), Bradley says, Relation presupposes quality, and quality relation. Each can be something neither together with, nor apart from the other; and the vicious circle in which they tum is not the truth about reality .... (21) But how the relation can stand to the qualities is ... unintelligible. If it is nothing to the qualities, then they are not related at all . . . . But if it is to be something to them, then clearly we now shall require a new connecting relation . . . And, being something itself, if it does not itself bear a relation to the terms, in what intelligible way will it succeed in being anything to them? ... we are forced to go on finding new relations without end. The links are united by a link. (27-8) ... If you take the connexion as a solid thing, you have got to show, and you cannot show, how the other solids are joined to it. And, if you take it as a kind of medius or unsubstantial atmosphere, it is a connexion no longer. (28)

Suppose I tell you, 'Alvin is to the left of Calvin'. How can we account for the truth of (the statement I make by appropriately using) this sentence? In one sense, the only things involved here are two people, Alvin and Calvin. If the truth of my statement is due to some thing (or things) having a (nonrelational, monadic) property, what could that property be? An inspection of Alvin and Calvin reveals no such property to account for the truth under consideration. So why not simply say that the truth here is due to a relation (of being to the left of) holding between the two things, Alvin and Calvin? Accordingly, one could say that it is true that Alvin is to the left of Calvin because Alvin and Calvin are related by the relation of being to the left of. But this would introduce a third thing (the relation) in addition to Alvin and Calvin. The question would then immediately arise: Are, say, Alvin and that relation related? And if they are, then it must be because of some other relation holding between them, and so on ad infmitum. One must not assume that Bradley's Paradox necessarily leads to a rejection of relationals in favour of categoricals, for the latter face the same sort of barrier (cf. Vander Veer, 1970: 40n). If it is true that A is B,

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then it must be that either A and B are identical or that they are not. If they are identical, then the statement is an empty tautology. If they are not, then there must be some connection between them (e.g., predication, exemplification, instantiation, etc.). But such a connection is a relation-and we have seen how relations lead to paradox. As Bradley says, "We wander among puzzles" (16). Bradley's Paradox raised a serious question about the possibility of giving a logical analysis of any statement, relational or otherwise. 13 Such a question is important for any Fregean logician since, from that perspective, while statements are not built up from constituents they are analysable into them. One way to avoid Bradley's Paradox is to deny that all the elements of a statement are on a logical par. What holds the elements together is never an additional element. Rather, the elements are united by virtue of one element's being logically fit for the others. This was Frege's (and, in a way, Plato's) solution. And how did Bradley respond to his own paradox? He argued that a judgment (proposition) has a unity that cannot be analysed. Judgment cannot be the mere union of ideas (and thus a proposition cannot be the mere union of terms). But a judgment cannot be a relation (or connection) of ideas because, the relation would itself be an idea, which in tum would stand in need of a relation to connect it to the other ideas of the judgment, and so on. So propositional analysis is impossible. Bradley actually had little regard for formal logic (which, after all, relies on such analysis), claiming that while it may be useful and worth preserving (he had already written Principles of Logic [1883]), it was metaphysically groundless. As an Absolute Idealist, he held that ultimately all statements are about the same thing-the Real. The unanalysed (indeed, unanalysable) statement expresses a single, unified idea, which is attributed to the Real. Relational thinking is merely a distortion of Reality. However, for finite beings, such as ourselves, to think in terms of "internal" relations, ones grounded in the natures of the relata, is not quite as distorting as to think in terms of"extemal'' relations, ones not so grounded (e.g., being to the left of) (1883, appendix B). Bradley's Paradox challenged logical orthodoxy while adhering to the standard traditional notion that singular and general terms are logically different. By the 1920s, even this was called into question. The logistic thesis advanced by Frege and Russell sought to secure the moorings of mathematics with the aid of logic (including set theory). In advancing their thesis, both Frege and Russell advanced as well a realist theory of universals. This version of realism holds that in addition to the existence of individual things (particulars), such as the Eiffel Tower, the moon, Socrates, my cat, and so on, there exist universals as well. Universals, unlike particulars, are not limited by space and time. A universal can occur in many places at a given time; examples are redness, blueness, wisdom, sphericality, and humanity. The moon can be in only one place at this time, but sphericality is where the moon is now and also where the sun is now,

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where Mars is now, where by old baseball is now, and so on. Particulars and universals are fundamentally different. Support for this view is often gleaned from grammar or logic. The grammarian's distinction between substantives, on the one hand, and adjectives and verbs, on the other, or the logician's distinction between subjects and predicates, is seen to be a revelation of the ontological distinction between particulars and universals. Even while rejecting the subject/predicate distinction for logic, Frege sought to preserve the particular/universal distinction in the guise of his distinction between objects and concepts. Again, objects are the referents of logically complete expressions (names); concepts are the referents of incomplete expressions (function expressions, predicates). Traditional logicians had taken particulars to be the referents of subjects, universals to be the referents of predicates. Frege preserved the ontological distinction while denying the traditional one by simply replacing the latter with his complete/incomplete (name/function) distinction. And that distinction is, in effect, a distinction between singular and general terms. 14 Russell (e.g., 1956) followed Frege here (but not everywhere, as we shall see). In 1925, young Frank Ramsey challenged both traditional logicians and his contemporaries by arguing that the theory of universals is a "great muddle.. and that there are no solid grounds for any of the asymmetries thus far alleged. 15 Both traditional logicians (represented for Ramsey by W.E. Johnson [I 921]) and Fregeans (represented by Russell) based the asymmetry of subjects (substantives, names, singular terms) and predicates (adjectives or verbs, functions, general terms) on the assumption that while the latter kinds of expressions can occupy either position in a proposition, the former can occupy only subject (or argument) positions-singular terms cannot be predicated. But, according to Ramsey, Both the disputed theories make an important assumption which to my mind, has only to be questioned to be doubted. They assume a fundamental antithesis between subject and predicate, that if a proposition consists of two terms copulated, these two terms must be functioning in different ways, one as subject, the other as predicate. (1925: 404)

Consider the two sentences 'Socrates is wise' and 'Wisdom is a characteristic of Socrates'. These may be different sentences, but they "assert the same fact and express the same proposition ... they have the same meaning.. (404). "Here,.. Ramsey concluded, "there is no essential distinction between the subject of a proposition and its predicate, and no fundamental classification of objects can be based on such a distinction.. (404). This is Ramsey's Symmetry Thesis. Russell had warned philosophers that the traditional notion that all propositions are subject-predicate in logical form ignores relationals. But, said Ramsey, ''Nearly all philosophers, including Mr. Russell himself, have

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been misled by language in a far more fundamental way than that ... the whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality, what is merely a characteristic of language" (1925: 405). One can avoid being misled by paying attention to "atomic" (noncompound) singular propositions (e.g., 'Socrates is wise'). Ramsey cited three possible theories to account for such propositions. The traditional theory, as found in Johnson, holds that the two terms here are linked by the copula, which is a "characterizing tie." The Fregean view, as represented by Russell, holds that the general term is incomplete, gapped, in such a way that it is completed, tilled, by the singular term, which is already complete in itself. The third theory is Wittgenstein's. According to Ramsey's account of this theory, there is neither a copula nor a privileged constituent (viz., an expression in need of completion). Atomic propositions depict atomic facts, and the objects that make up atomic facts simply "hang together like the links ofa chain." Ramsey said (408) that it is important to look only at the second theory. This theory recognizes the need for a verb in each atomic proposition (and verbs are incomplete). However, for Ramsey, the difficulty here is that all objects (Socrates and wisdom alike) are incomplete. What he seems to have had in mind is that an object is incomplete if it somehow depends upon other objects-and no object can occur in an atomic fact without depending upon some other object(s). Consequently, both 'Socrates' and 'wise' are incomplete as well. If all this is so, one naturally wonders how logicians could have insisted on singular/general (subject/predicate, name/function) asymmetry in their accounts of propositional unity. But what is this difference between individuals and functions due to? Again, simply to the fact that certain things do not interest the mathematician. Anyone who is interested not only in classes of things, but also in their qualities. would want to distinguish from among the others. those functions which were names .... So were it not for the mathematician's biassed interest he would invent a symbolism which was completely symmetrical as regards individuals and qualities; and it becomes clear that there is no sense in the words individual and quality; all we are talking about is two different types of objects. such that two objects, one of each type, would be sole constituents of an atomic fact. (Ramsey, 1925: 415-16)

But surely, it might be contended, there must still be a tie or relation (of characterizing) between these two "objects." Ramsey's response was, simply, "As regards the tie, I cannot understand what sort of thing it could be" (416). Consequently, Ramsey's own position regarding the problem of propositional unity is agnostic: "The truth is that we know and can know

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nothing whatever about the fonns of atomic propositions ... and there is no way of deciding any such question. We cannot even tell that there are not atomic facts consisting of two tenns of the same type" (417). Ramsey's Symmetry Thesis (that in a simple, nonrelational atomic proposition either tenn may be taken as the logical subject, or argument, and the other as the logical predicate, or function) implies that there is a logical symmetry between singular and general tenns. Fregean logicians assume that in such propositions the subject (argument) must be singular-the Fregean Dogma-while the predicate (function) must be a general tenn (recall Dummett's account of how multiply general sentences are built up from such singular atomic sentences). Those who wish to maintain the logical asymmetry of subjects and predicates have had to counter Ramsey by arguing for the logical asymmetry of singular and general tenns. The literature surrounding this "Asymmetry Thesis" has grown large in recent years. Many prominent philosophers and logicians, including especially Geach and Strawson, 16 have offered support for asymmetry. While many arguments have been advanced, most generally go something like this: 'Subjects and predicates are logically asymmetric because singular tenns and general tenns are logically asymmetric. Singular and general tenns are asymmetric because there are logical features which hold of the latter but not of the fonner.' Two such features are commonly cited: (i) General tenns can be negated; singular tenns cannot. Moreover, even if singular tenns could be negated (none denies that grammatically, if not logically, this is possible), the negation of the predicate (general tenn) would result in the negation of the entire sentence; the negation of the subject (singular tenn) does not result in the negation of the entire sentence. (ii) General tenns can be compounded (conjoined or disjoined); singular tenns cannot. Moreover, even if singular tenns could be compounded (again, none denies the grammatical, if not logical, possibility of this), the compounding of the predicate results in a compound sentence; the compounding of the subject does not. 17 In part two of this essay we will see why the Asymmetry Thesis, along with the Fregean Dogma, should be rejected. My challenge, then, will be to make logical sense of negated and compounded singular tenns.

Russell and Wlttgensteln Even in these semi-sophisticated times, we fall for the myth ofthe verb. J.L Austin They've a temper, some of them-particularly verbs: they're the proudest-adjectives you can do anything with, but not verbsLewis Carroll

While it was, ironically, Russell who in 1901 revealed to Frege a serious

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paradox (or "contradiction," as Russell called it) at the heart of the Grundgesetze, 18 Russell was the most articulate and best-known advocate of

the new logic (and the logicist theory). He was convinced some time before the turn of the century that traditional logic was fundamentally mistaken. His fJrSt target was the logician whom many contemporary logicians regard as the fJrStprecursorofmodern mathematical logic, Leibniz. Russell's main criticism ofLeibniz's logic (Russell, 1937) in particular, and pre-Fregean logic in general, was that it was dependent upon a false account of logical syntax (viz., the view that all propositions are logically analysable into subjects and predicates). Indeed, for Russell, this wrong logical view led to a wrong metaphysics of substances and attributes. But, while Russell's own logic was Fregean (or Peano-Fregean), his "philosophical" logic was non-Fregean in many ways. For both Frege and Russell the logic of functions and arguments was meant to serve (when supplemented by set theory) as the foundation of mathematics by virtue of its equivalence with arithmetic. Both viewed natural language as logically flawed. In his reply to Max Black, Russell wrote, "We ought [not], in our attempts at serious thinking, to be content with ordinary language, with its ambiguities and its abominable syntax. I remain convinced that obstinate addiction to ordinary language in our private thoughts is one of the main obstacles to progress in philosophy" (in Schilpp, 1944: 694). If anything, Russell had an even lower opinion of ordinary language than did Frege. Unlike Frege, however, Russell seems to have believed that natural language does in some sense have a logic. Natural-language expressions can be translated into expressions of the "logically perfect" language of mathematical logic (the syntax or grammar of which is presented in Principia Mathematica [Russell and Whitehead, 1910-13) and the vocabulary of which consists of "logically proper names" and simple predicates [Russell, 1918-19]). The logical form of a natural-language sentence is hidden by its prima facie grammatical form-thus the need for such a translation, for the logical form of a sentence in the logically perfect language is the same as its apparent, surface grammatical form. Russell came almost to identify logic with philosophy. And, unlike Frege, he took the study of logic and grammar to be a valuable tool for attacking philosophical problems. 19 Thus, for example, where Frege might account for the term 'ghost' by arguing that its referent was merely a mental image, Russell would analyse the term into a complex expression in his ideal language. 20 Finally, while both Frege and Russell could account for logical constants (which, in the long run, determine logical form) in no other way than simply to list them, they offered divergent accounts of propositional unity. As we saw above, Frege took the unity of a proposition to be the result of incomplete expressions being completed by complete expressions. The matter is purely one of logical syntax. Russell accepted this account. Yet it may not be too unfair to say that he often took his directions concerning logical syntax from his prior semantic theses (cf. Sainsbury,

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1979). For example, Russell held that the meaning of a complex expression was uniquely determined by the meaning of its component expressions (Frege's compositionality thesis). This, then, guided his account of logical form. The logical form of a sentence must exhibit the meaning of that sentence (by constituting a translation of the sentence into a sentence of the logically perfect language). Russell rejected the Fregean notion that an argument of a function (i.e., a name) could be a complex expression. Names cannot be complex. nor can they be empty. Frege took both ordinary proper names, such as 'Plato' and 'Kant', and definite descriptions, such as 'the teacher of Aristotle' and 'the man who broke the bank', to be eligible as arguments in first-order predicate expressions. Russell rejected Frege's sense/reference (or, for Russell, meaning/denotation) distinction. His semantics (and his epistemology') prevented him from allowing either kind of expression to play such a logical role (be a "logically proper name:• as he called it). According to Russell ( 1912), the denotation of a logically proper name must be a simple, existing object of acquaintance (such as a sense datum). As it turns out, ordinary proper names are merely abbreviations of definite descriptions. Indeed, as he says in Principia Mathematica, "In what we have in mind when we say 'Socrates is human' there is an apparent variable, (Russell and Whitehead. 1910-13: 50). In other words, we may say 'Socrates is human •, but what we "have in mind, (presumably the logical form) is a sentence/formula containing not the ordinary proper name 'Socrates' in a denoting role but a variable (say, 'he', or 'it', or 'x'). The contrary view-that definite descriptions are names-was the source ofthe Meinongian theory, which Russell (1904) had rejected on ontological grounds. Russell (1905) argued, then, that ordinary proper names could be analysed as disguised definite descriptions. And definite descriptions could, in tum, be analysed in terms of indefinite descriptions and identity. Finally, indefinite descriptions could be analysed in terms of predicates (functions) and individual variable arguments bound by existential quantifiers (this last eliminates the need for a predicate of existence in the ideal language). Suppose the ordinary proper name 'Socrates• can be replaced by the defmite description 'the teacher of Plato•. A sentence such as 'Socrates is wise' would then be analysed, according to Russell's lights, as 'There exists at least one thing that taught Plato and each thing that taught Plato is identical to it and it is wise' (a full logical analysis, of course, would then analyse away the name 'Plato'). The "apparent variable,, which we have in mind whenever we use such ordinary names, is simply the existentially quantified variable (expressed by 'it' above) of the logical analysis. One of Russell's reasons for denying Frege's amalgamation of proper names and defmite descriptions was a matter of logical syntax. For Russell, names, unlike descriptions, do not exhibit scope ambiguity (cf. Sainsbury, 1979: 66ft). Consider the following two sentences.

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(I) Socrates is wise.

(2) The man who broke the bank was French. Russell's claim was that any attempt to deny (I) would result in its (sentential) negation, i.e., ( 1.1) Not: Socrates is wise. (=It is not the case that Socrates is wise.) 'Socrates is not wise' and 'Socrates is unwise' would both be rendered as ( 1.1 ). Things are supposedly quite different for the denial of (2). This is because of what Russell ( 1918-19) called the "possibility of double denial." Sentence (2) can be denied in (at least) two nonequivalent ways: (2.1) Not: the man who broke the bank was French. (2.2) The man who broke the bank was not French.

Given Russell's analysis of definite descriptions, a sentence of the form 'The A is 8' will be false whenever there exists exactly one A and it is not 8, or there is more than one A, or there is no A. So (2.1) and (2.2) are not equivalent, say, when either more than one man broke the bank or no man broke the bank. In either of those cases (2.1) will be true but (2.2) will be false, and this difference is due solely to the difference in relative scopes of the descriptive expression ('the ... ') and the negative expression ('not'). In (2.1) the description lies within the scope of 'not'; in (2.2) 'not' lies within the scope of the description. To be absolutely fair to Russell, even 'Socrates' in (I) and (1.1) is not really a logically proper name. A sentence with such a name would be (3) This is red. Mathematical logicians like Russell would translate this into the ideal language as (3.1) Rx -that is, the function Ron the argument x. And no matter how we might deny (3) in natural language (e.g., 'This is nonred', 'This isn't red', 'This is not red', 'This is other than red', 'It is not the case that this is red', 'It is false that this is red', etc.) (3.1) can be denied in just one way-by the application of sentential negation: (3.2) -(Rx)

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In part two I will offer reasons for rejecting this way of distinguishing names and descriptions, as well as the contemporary way with negation. A central element of Russell's attack on traditional logic's subjectpredicate analysis of propositions (as represented by Leibniz) was his insistence that such an analysis must tum a blind eye to relations. 22 Russell's own view was that if relations were reducible to nonrelational properties (or classes), then relational propositions would reduce to categoricals (as Leibniz seems to have thought). But, according to Russell, such a reduction of relational propositions is not possible.21 And, indeed, Frege and Russell's formal language admits firSt-order functions with either one or more gaps (places for singular term arguments, i.e., relata expressions). The key to understanding Russell's rejection of Leibniz's view ofrelationals lies in Russell's Fregean assumption that all syntactical complexity is to be accounted for in terms of sentential complexity. The terms (function and argument[s], or predicate and name[s]) of an atomic proposition must always be simple-having no proper parts that are meaningful (i.e., that denote). Russell also assumed that, given that logical syntax is a guide to metaphysics, Leibniz's reduction of relationals to categoricals reflected his rejection of real relations (a view Russell found incompatible with the rest ofLeibniz's monadic metaphysics [cf. Ishiguro, 1972-76]). The Leibnizian theory that relational propositions can be logically reparsed in terms of subjects and predicates was only one oftwo accounts of relationals that Russell wanted to reject. The other theory was Bradley's.24 According to Russell, where the "monadistic" (i.e., Leibnizian) theory attempts to reduce a relational proposition, 'aRb', to a conjunction (of some sort) of subject-predicate propositions by replacing the relational expression by a pair of monadic predicates (i.e., 'aR1 & I\ b'), the "monistic" (Bradley's) theory analyses such a relational proposition as a subject-predicate proposition with a monadic predicate but a compound subject (i.e., '(ab)R1') (Russell, 1903: 221). For example, 'Paris loves Helen• is analysed by the first theory as 'Paris loves and by that very fact Helen is loved'. It is analysed by the second theory as 'Paris and Helen are lovers', where 'Paris and Helen• denotes a whole composed of Paris and Helen and 'are lovers• is a property of that whole. 2s Among Russell's reasons for rejecting the monistic theory was his observation that when the relation is asymmetric (as in 'David is the father of Solomon'), the proffered analysis would fail to preserve the inequivalence between the relational proposition and its converse. For example, 'David is the father of Solomon• is not equivalent to 'Solomon is the father of David', but, given that there is no difference between the whole composed of Solomon and David and the whole composed of David and Solomon, the analyses of the two propositions would be equivalent. When it came to Bradley's Paradox concerning relations, Russell insisted that the error that generated the paradox was the failure to recognize

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that certain elements of a proposition (viz., concepts-specifically, in this case, relational concepts as denoted by relational expressions) can occur in propositions without thereby being subjects of those propositions.26 The result of this failure is not only paradox, but inability to account for propositional unity. If each expression of a given sentence is viewed as simply naming a subject (thing, object), then the sentence will be nothing more than a list of names-not a unit. Russell directly addressed the problem of sentential, or propositional, unity in the Principles of Mathematics in 1903.27 The result of Russell's struggle is a theory of logical syntax that was different from Frege's in important ways simply because it resulted from Russell's attempt to solve the problem of sentential unity raised by his understanding of the source of Bradley's Paradox. In a way, this was never really a problem for Frege. In contrasting his new logic with that of Boote, Frege emphasized that he, unlike Boote, gave pride of place to the entire sentence as a unit, arriving only by analysis at the elements of the sentence. Boote had begun with terms and used them to build up sentences. For Frege, then, sentences are, by their very logical nature, unified (by being units!V' Russell accepted Frege's notions of functions and arguments, but, unlike Frege, he saw sentences as built up from such expressions (thereby abjuring Frege's priority principle). Russell held that sentences express propositions, the true objects of logical investigation. A (Russellian) proposition is a combination of"terms." A (Russellian) term is any object of thought, a unit, individual, entity; whatever can be mentioned (Russell, 1903: 43). Terms are either "things" (Fregean objects) or concepts. Concepts are expressed by "propositional functions" (or predicates). A propositional function is a proposition whose subjects (things) have been removed, leaving gaps. Singly gapped functions are class concepts; multiply gapped functions are relations. Relations are denoted by verbs; class concepts, by adjectives. Using "verb" in a sense wide enough to encompass all propositional function expressions, Russell claimed that, unlike terms that denote things (i.e., singular terms, names), a verb has a "twofold" nature. It can be used "as actual verb and as verbal name" (Russell, 1903: 49). Subjects (e.g., 'Socrates') are singular. They cannot be used as verbs because they do not have "that curious twofold use which is involved in human and humanity" (Russell, 1903: 45). 'Human' and 'humanity' are grammatically distinct but logically identical. Their grammatical differences merely reflect the two uses: as a verb ('human', as in 'Socrates is human') or as a verbal noun ('humanity', as in 'Humanity is the curse of this earth'). A verb used as a verbal noun denotes a thing. A verb used as a verb denotes a complex object---class or relation. The verb used as a verb is the source of propositional unity for Russell. A proposition in fact is essentially a unity, and when analysis has destroyed the unity, no enumeration of constituents will

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restore the proposition. The verb when used as a verb, embodies the unity of the proposition, and is thus distinguishable from the verb considered as a term [verbal noun], though I do not know how to give a clear account of the precise nature of the distinction. (Russell. 1903: 49-SO)

Russell said this with an eye on Bradley's Paradox. Consider again 'Paris loves Helen'. The proposition expressed here consists of three tenns: Paris, Helen, and the relation of loving. But when considering these three tenns, we are simply considering three things, three objects. The relating expression is merely a parataxis, not a unit. To unite them into a single propositional unit would require further relations, ad infinitum. To avoid this regress, Russell held that the verb ('loves'), unlike other kinds of expressions, has a twofold nature. It can be used "as a tenn" (thus naming an object, as 'humanity' does), but it can also be "used as a verb." When so used it does not denote another thing constituting the proposition; rather, it is what actually binds those other constituents into a unified whole-a proposition. As we said, Russell had one eye here on Bradley. His other eye was on Frege, who, as we have seen, in effect avoids the problem of sentential unity by means of his priority principle and his absolute and unqualified distinction between functions (which are incomplete) and names (which are complete). In "On Concept and Object" (in Geach and Black, 1952), he is clear about the price he must, and is willing to, pay for this; he had to admit that, for example, the concept of a horse is not a concept. This prima facie paradoxical claim is due, according to Frege, to a logical defect of language in general. "I mention an object, when what I intend is a concept" (54). One might very well wish to mention the concept of a horse, but to do so one must use an expression that is complete (i.e., a name) rather than incomplete (a function or concept expression). One is defeated by language. By insisting on the inviolate distinction between functions and objects (between incomplete predicates-relational and general tenns-and complete names), Frege refused to allow any expression the twofold nature of Russell's verbs. In effect, Frege embraced a radical dualism in logic (like Plato's dualism of onoma and rhema). Consequently, he was saddled with the 'horse' paradox, but he was able to avoid the problem of sentential unity. In contrast, Russell admitted a single category of"tenns," some of which were allowed a double use. In effect, he embraced (at least in 1903) a kind of Aristotelian (of the Analytics) monism in logic. Expressions are logically of one sort. Yet Aristotle had a solution to the problem of sentential unity. Russell's attempted solution was far from satisfactory. Why did Russell fail? What he lacked that Aristotle did not was a sentence-unifying expression-a logical copula. All of Russell's talk about verbs used as verbs embodying the unity of a proposition turned out to be an expression

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of hope in place of argument. Aristotle had the logical copula and Russell did not, having followed Frege in abandoning it. Recall that Ramsey had contrasted three theories concerning the logical structure of "atomic" propositions. These were the traditional (copular) theory, the Fregean (gap filling) theory, and Wittgenstein's (chain link) theory. However, in examining the view of logical form that Wittgenstein offered (especially in the Tractatus [1961]), it is absolutely imperative to keep in mind his distinction between what can be said and what can be shown (as found at remark 4.1212). In laying out his theory of logical form Wittgenstein kept in mind two principles-one from Frege, the other from Russell. Wittgenstein took from Russell the notion that the form of a proposition is determined by its predicate (verb); from Frege, he took the idea that what can be predicated can never be a subject. Subjects, for Frege and Wittgenstein, are names. But Wittgenstein meant by a "simple sign" (3.202) a name-not an ordinary name, but something like Russell's "logically proper" name. Names denote, or name, "objects." Predicates-general terms and verbs-denote concepts. Names say what they name; predicates do not. There is no "twofold" use of verbs, as in Russell of 1903. Thus predicates cannot name, or be used to say, what they denote. They can only show this by their use as predicates. It looks as if Wittgenstein held the Fregean view that a proposition consists of names and a predicate, where names and predicates are fundamentally different. But this is hard to reconcile with his "picture theory" of meaning, according to which an atomic proposition ("propositional sign") pictures an atomic fact (2.141 ). It does this because the fact is a combination of objects (2.0 I, 2.0272) and the proposition is a concatenation of names of those objects (4.22), names in "immediate combination" (4.221 ). Moreover, the names are combined in a defmite way (2.14, 3.14) corresponding to the configuration of objects in the state of affairs depicted (3 .21 ). As well, since every proposition has a defmite sense, there must be simple names (of simple objects) (3.23). Names have sense only in the context of a proposition-Frege's priority principle (3.3, 3.314). So, objects are always in the context of a fact. And the world is nothing more that the totality of such facts ( 1.1 ). Given the picture theory, a proposition is logically analysable as a combination of names (not names and predicates). And this is surely the view Ramsey was attributing to Wittgenstein when he claimed that for Wittgenstein the elements of a proposition are united like links in a chain (2.03, 3.14, 3.141, 3.142). In a chain, each link is like every other; they combine with one another not by means of an intermediary but by their very nature. Names, too, according to this view, are like one another (while names and predicates are not). Names simply link with one another in concatenation.29 Clearly, the kinds of propositions Wittgenstein had in mind were quite different from ordinary sentences (3.325, 4.002, 4.1213). Indeed,

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he admitted that it is not obvious that ordinary sentences have the kind of logical fonn that he attributed to them (4.002). This is especially so for

relational sentences. Concerning such sentences, he said (3.1432), "We must not say, "The complex sign 'aRb' says 'a stands in relation R to b' ..; but we must say, "That 'a' stands in a certain relation to 'b' says that aRb!' More specifically, we must not say this in a language of"adequate notation, (6.122; also 3.325, 5.533, 5.534). In other words, in a logically adequate, or correct, notation-a Begriffschrift (4.1273)-a notation in which sentences consist only of simple names naming simple objects-what would be expressed in, say, English by a sentence of the fonn 'aRb' would be expressed by a sentence in which only 'a' and 'b' appear. Such a sentence would, in tum, depict a state of affairs consisting only of the two objects a and b, standing in some relation to one another. The fact that a stands in relation R to b (if it is a fact-an existing state of affairs) is pictured by a sentence (of the logically correct language) in which the two names 'a' and 'b' are in some relation. The relation between 'a' and 'b' need not be the same as the relation between a and b (3.1431) (cf. Copi, 1958). In a logically adequate language, there are no predicates, no relational tenns, no Russellian verbs. In one stroke, Wittgenstein has accommodated Frege's demand that concepts are not objects (by not allowing predicates to be used as names-indeed, by allowing only names to occur in logical notation) and answered Bradley (by allowing relations among objects to be represented not by relational tenns but by relations among names). As a consequence of this theory, Wittgenstein was committed in the Tractatus (1961) to the view that all logically adequate propositions are relational, in that each consists only of names, which, themselves being objects, stand in relations, which, in tum, represent relations among the objects so named (2.0121). To be absolutely fair to Wittgenstein of the Tractatus, it must be noted that sentences of a logically correct language do have, in addition to names, logical constants as elements. However, these expressions, unlike names, are "not representatives, (4.0312, 4.441).30 Negation was the logical operation that most interested Wittgenstein (as it had Frege). Still, his remarks concerning negation hardly constitute a satisfactory theory. In the Tractatus, Wittgenstein claims that simple facts (the ultimate constituents of the world) are independent of one another ( 1.2, 1.21, 2.061 ). Because of this mutual independence, there can be no logical relations (implication, equivalence, incompatibility, etc.) between any two such facts, nor, presumably, between any two atomic propositions depicting such a pair of facts: "From the existence or nonexistence of one state of affairs it is impossible to infer the existence or nonexistence of another, (2.062). Suppose I know that the state of affairs depicted by a proposition of the fonn '-p' exists (i.e., is a fact). From this, I can infer the nonexistence of the state depicted by 'p' (5.512). Such inference would be legitimate, according to Wittgenstein, because at least one of these ('-p') is not atomic. But what of the seemingly obvious

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inference from 'This is red' to 'This is not blue' (cf. Ramsey, 1923: 18)? This inference depends upon the incompatibility of 'This is red' and 'This is blue' (given a common denotation for each token of 'this' here). As a Fregean, of course, Wittgenstein had no access to any kind of negation other than sentential (4.0641, 5.1241, 5.2341) to account for this. Consequently, he had to deny that any logical relation could hold between the two atomic propositions expressed by 'This is red' and 'This is blue'. Indeed, it appears that in the Tractatus view there can be no negative atomic facts at all. 31 Before leaving Wittgenstein, we must take note of a fascinating turn that he took in the Tractatus away from the logical path followed by virtually all other Fregeans. In "Sense and Reference," Frege had made a very clear distinction between the 'is' of predication and the 'is' of identity. A proposition such as 'Shakespeare is Bacon' must be analysed as a relation between two objects (viz., the relation of identity between Shakespeare and Bacon), rather than as a function on a single object (viz., the function of being Bacon applied to Shakespeare). This is because in the latter analysis a complete expression, a name ('Bacon'), would be used as an incomplete expression, a predicate. Wittgenstein did not offer an alternative to Frege's account of identity-he simply rejected the idea of identity altogether: "The identity sign, therefore, is not an essential constituent of conceptual notation" (5.533). In a logically adequate notation one cannot even express a proposition such as 'a is identical to b' (5.534): "Identity of object I express by identity of sign, and not by using a sign for identity" (5.53); "It is self-evident that identity is not a relation between objects" (5.5301). Either a and b are two things or they are not. "Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical to itself is to say nothing at all" (5.5303). It is safe to say that very few mathematical logicians have felt enough logical security, as Wittgenstein did, to throw out the oar of identity. Indeed, as we will see, some have even augmented it. Strawson, Geach, and Quine I am the most readily disposed person to do justice to the moderns, yet/ find that they have carried reform too far, among other things, by confusing natura/things with artificial things. Leibniz

As we shall see in the next chapter, Frege's revolution in logic has only recently begun to face serious challenges (both from those within the citadel of contemporary mathematical logic and from the few traditionalists beyond the moat). Still, now near the end of the twentieth century, the logic derived from Frege is well entrenched and powerful. Its hegemony is still intact. The sources of its pre-eminence are not in doubt. Modern mathematical logic was created by Frege and was soon pushed into a place of prominence

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in philosophy by Russell, Whitehead, and Wittgenstein. In comparison with what had thus far been offered by traditional logic, the new logic clearly deserved its rapid rise to power. It provided an algorithm for the analysis of mathematical reasoning far more perspicuous and effective than anything before. Even after the 1920s and 1930s, when the new logic's ability to generate mathematics (and the logicist programme in general) was forcibly abandoned, it continued to be logical orthodoxy-especially among analytic philosophers. Three of these philosophers, P.F. Strawson, Peter Geach, and W.V. Quine, are individually representative of certain aspects of late-twentiethcentury orthodoxy in philosophical logic. Of the three, Strawson is the least committed to orthodox mathematical logic. By the middle ofthe twentieth century, Fregeans, having generally abandoned logicism, had become more eager to claim that mathematical logic is actually the hidden, underlying logic of everyday discourse. This claim was not new, of course. Russell, for example, had held such a view early in the century. Indeed, Russell's Theory of Descriptions was generally regarded not only as a paradigm example of philosophical analysis, but as a way the new logic could be used to reveal the hidden, underlying logical forms of ordinary language expressions. Strawson first gained fame in 1950 with his "On Referring" (1950a), in which he challenged Russell's theory of descriptions. Shortly after, in Introduction to Logical Theory ( 1952), he extended his challenge to the standard logic in general. Strawson represented a brand of analytic philosophy called "Oxford analysis," or "ordinary language analysis." This kind of analysis was inspired primarily by Wittgenstein's work in the 1930s and 1940s (esp. 1953), when Wittgenstein abandoned the view that mathematical logic was the proper tool for language analysis (and that mathematico-scientific language was the proper language to be analysed). Ordinary-language analysts generally hold that the philosopher's central task is to solve (or at least dissolve) philosophical problems by showing how they are generated by a misuse or misunderstanding of expressions of ordinary language as they are used in everyday circumstances. Ordinary-language philosophers such as Strawson take mathematical logic to be in order as far as it goes, but they deny that it could shed much light on the workings of ordinary, nonscientific, nonmathematical language. In Introduction to Logical Theory, Strawson was at some pains to show that modern formal logic is insufficiently close to natural, ordinary language (e.g., 193-94). The logic of ordinary language is less elegant and less systematic than the artificial language of formal logic (232). In fact, "expressions of everyday speech ... have no exact and systematic logic" (57). In spite of Strawson' s early denial of a logic of ordinary language, he has spent much of the past forty-five years examining what can only be described as the logical form of simple statements made in the medium of ordinary language. 32 This concern is due in large measure to the fact that

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Strawson is one of the few contemporary philosophers who takes Ramsey's challenge to subject/predicate asymmetry seriously. Strawson is a staunch and persistent defender of the asymmetry thesis, and it is in this defence that one can discern how little he has actually moved away from the now entrenched view of logical syntax. As we have seen, Russell was willing to abandon (or at least temper) the radical asymmetry between names and function expressions (predicates) that Frege had advocated (an asymmetry essential in accounting for "the sharp distinction between concept and object" [ 1979: 177]). Ramsey went further and overtly challenged any such asymmetry. For Frege, predicates are never names (concepts are never objects). Since objects are, by definition, what are referred to by names (including definite descriptions), the concept of a horse is not a concept. Frege's distinction was primarily logico-linguistic-namely, the distinction between expressions which are complete, saturated (names) and those which are not (predicates). Strawson's asymmetry is neither as radical as Frege's nor primarily logicolinguistic. According to Strawson. the basic, logically simplest sentences consist of two expressions: a subject and a predicate. But his distinction is really one between the ways in which things are "introduced into discourse" (1957: 441 ). Things are either particulars (individuals) or universals (characteristics and kinds of individuals), and the distinction between particulars and universals is basic and ontological. Both particulars and universals can be introduced into discourse by expressions. Particulars are introduced by subjects (singular terms, such as names); universals are introduced by predicates (general terms). The differences between subjects and predicates are important because they reflect important differences between particulars and universals. Thus individuals, unlike universals, "cannot have instances" (1953-54: 31 ). For example, while man is a universal having as instances John, Peter, and Ralph, John himself is an individual, and nothing is an instance of John, according to Strawson. An individual is what can be counted as one. Consequently, "anything whatever is an individual" (1957: 442). Even universals can be counted as one and thus be individuals, which means that they can be introduced into discourse via subjects (i.e., by the use of abstract names). Indeed, for Strawson, we can ignore the distinction between abstract nouns and general terms (e.g., 'wisdom' and 'wise') (1987: 404). Individuals can never be introduced via general terms-that is, singular terms can never be predicated ( 1957: 446). Strawson, like an orthodox Fregean, presupposes that the singular/general distinction is reflected in (indeed, is identical to) the subject/predicate distinction, which in tum reflects the individual/universal distinction. 33 The ontological distinction between individuals and universals is reflected in the semantic distinction between singular and general terms. Whether one takes the logico-grammatical distinction between subjects and predicates to be reflective of such ontological and semantic distinctions depends upon how much of logical syntax one wants to be determined by ontology and

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semantics. In 1959, 1961, and 1974, Strawson continued his attempt to defend this notion that logic "must reflect fundamental features of our thought about the world. And at the core of logic lie the structures here in question, the 'basic combination' (as Quine once called it) of predication" (Strawson, 1974: 4; cf. 113). Strawson's most fully articulated defences of the asymmetry thesis are found in his "The Asymmetry of Subjects and Predicates" (1970) and Subjects and Predicates in Logic and Grammar (1974). According to Strawson subjects and predicates are logically asymmetric in at least two ways: regarding negation and regarding composition. With regard to negation, the argument is that general terms (the only kind that can play the role of predicate) "come in incompatibility groups" but singular terms do not (1970: 102-3; 1974: 19). Consequently, one can negate a subject-predicate sentence simply by negating its predicate, but one cannot negate a sentence by negating its subject (1970: 98). In fact. subjects (singular terms) cannot be negated at all. The negation of a general term, say 'red', can be thought of as the disjunction of all the terms in its incompatibility group ('blue', 'green', 'pink', etc.). So the negation of'red' ('nonred') is 'blue or green or pink or .. .'. The terms 'red' and 'nonred', and, generally, any nonsingular term and its negation, are logically incompatible, and it is this incompatibility that accounts for the incompatibility of sentences ( 1970: 104-5). But there is no term incompatible with a singular term. The asymmetry of subjects and predicates is due to a more fundamental ontological asymmetry, which "seem[s] to be obvious and (nearly) as fundamental as anything in philosophy can be" (1970: 102). As we have seen, subjects introduce particulars (individuals) into discourse, while predicates introduce universals (characteristics or kinds of individuals). It is the asymmetry of individuals and universals that is fundamental and obvious for Strawson. The negatibility of predicates is due to the fact that general terms come in incompatibility groups, which, in tum, is due to the fact that universals come in "incompatibility ranges" ( 1970: I02). The terms 'red' and 'nonred' are incompatible because the characteristic of red is incompatible with the characteristics of blue, green, pink, and so on. Singular terms have no negations because individuals do not come in incompatibility ranges. Nothing is incompatible with, say, Socrates. Now, the reason nothing (i.e., no individual) can be incompatible with Socrates is that any such individual would have to have all the properties Socrates lacks and lack all the properties he has. But any such purported individual would then have to possess incompatible properties, per impossibile (see 1970: 1lin, fmal paragraph). For example, a nonSocrates would have to be both French and Chinese, both over seven feet tall and under three feet tall. For Strawson, the fundamental ontological asymmetries between individuals and universals "explain and vindicate" (1970: 104) the logical asymmetries between subjects and predicates. The second of these asymmetries is that regarding composition. Strawson's claim here is that

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while predicates can be compositionally compounded (i.e., conjoined or disjoined) to form new predicates, "there are no such things as compound (conjunctive or disjunctive) subjects" (1970: 100; see also 1974: 4-9). Compound subjects have no place in logic; they are "pseudo-logical-subject terms" ( 1970: I0 I). Again, the difference between subjects and predicates (singular and general terms) is "explained and vindicated" by a fundamental ontological asymmetry. Universals (characteristics and kinds) can be conjoined or disjoined to one another to form new universals. We even have specific terms for some important such compounds (e.g., 'bachelor' for the conjunction of the characteristics male, adult, and unmarried). In contrast, there are no individuals that are composed of other individuals. Like negative individuals, such purported compound individuals would be impossible. Consider, says Strawson (1970: II In), Tom and William. Suppose there is an individual who is the conjunction of Tom and William, call him 'Tolliam'. Tolliam must have all and only those properties shared by Tom and William. Suppose Tom has property P and William is nonP. It follows that Tolliam, per impossibile, has neither P nor nonP. Suppose there is an individual who is the disjunction of Tom and William, call him 'Tilliam'. Tilliam must have all and only those properties that either Tom or William have. Again, suppose that Tom is P and William is nonP. Per impossibile, Tilliam must be both P and nonP. In summary, for Strawson, subjects and predicates are logically asymmetric because subjects (qua singular terms), unlike predicates (qua general terms), cannot be negated or composed, because subjects introduce individuals into discourse, while predicates introduce universals. No individual is incompatible with another individual, nor are any individuals composites of individuals. In contrast, universals come in incompatibility ranges and can be composites of other universals. We shall leave Strawson for now, but certain points concerning his view must be kept in mind: (i) he assumes that "basic" sentences must be singular (in other words, the Fregean Dogma); (ii) he believes basic sentences are, from the point of view of logical syntax, concatenations of two expressions, each having a unique semantic function; (iii) he assumes that singular terms can never be predicated; (iv) he believes that the negation of a singular term must be another singular term; and (v) he believes the composition of two or more singular terms must be a singular term.l4 As we saw in chapter one, Peter Geach has little love for traditional logic, a logic conceived in a brief Aristotelian Eden but lost in Prior Analytics. According to Geach, "Traditional 'Aristotelian' logic is full of mistakes and confusions" ( 1950: 461 ). It is ''the miserable mutilated torso that passed for the whole body oflogic from about 1550 to 1847" (l969a: 77). While most contemporary logicians have simply turned their backs on traditional logic, confident that the path first marked by Frege leads to logical perfection, Geach has taken up the challenge of showing just how the old logic is

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inadequate, misleading, or mistaken. Though he views the logic of Aristotle from a solidly Fregean point of view, and thus with a fair degree of hostility,35 his critique offers the kind of challenges that friends of traditional logic must meet if they are to have any hope of offering a viable alternative to the standard mathematical logic. Geach's Fregean outlook is not the result of a radical, unquestioning allegiance to Frege and his logic; it is tempered by some important departures from Frege.36 One of these is Geach's conviction that logicians must deal with natural (rather than artificial or mathematical) language. Throughout his work during the second half of the twentieth century, Geach has consistently urged logicians to take the new logic as the best (or even the only) tool for analysing natural language. Classical Fregeans ignored natural language or insisted that whatever can be viewed as logical in natural language can be seen as such only once expressions in that language have been translated into the formulae of the standard predicate calculus. The irony here is that Geach counsels the application to natural language of a logical tool designed on the understanding (by Frege) that natural language has no logic. Frege held that only after one has clarified the artificial, logically constructed language will one be in a position to apply it to natural languages. Geach feels ready now. A second difference between Geach's logical views and orthodox Fregeanism is his rejection of the contextualist thesis (that only in the context of a sentence can a word be said to have meaning): "The view put forward by Frege and Wittgenstein, that it is only in the context of a sentence that a name stands for something, seems to be certainly wrong" (1950: 462). This departure from strict Fregean doctrine is the result ofGeach's own theory of logical syntax, which is intended to extend Frege's theory. Accordingly, it insists on "an absolute" ( 1950: 464), a "fundamental distinction between names and predicates" (1950: 474, 476), and "predicables" (1962: 34), for it is only by insisting on such an inviolable distinction that Ramsey's challenge can properly be met (1950: 474). A predicable, for Geach, is a template expression that is turned into a sentence by tilling its blanks with names. Predicables are incomplete; names are complete. A predicable actually used in a sentence (i.e., one having its blanks appropriately tilled) is a predicate. Logically simple sentences are subject-predicate in form. Predicables have no logical role to play outside of their roles as predicates. In contrast, names can play the role of naming even outside of the context of a sentence (hence Geach' s rejection of the contextualist thesis). Used in a sentence, a name plays the role of subject. But it can also play the role of simply naming, "to acknowledge the presence of the thing named" ( 1962: 26). "An act of naming is of course not an assertion ... it does, however, 'express a complete thought' "(1950: 462). For Frege, a thought (actually, "Thought") is the sense of a sentence-never of a (nonsentential) name. At any rate, Geach's contention that names and predicables are fundamentally distinct is Fregean. Names

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can never be logical predicates, and a "predicate can never be used as a name" (1950: 463). Predicables can never be names because they never have a ''complete sense" ( 1962: 32). Any shifting about of logical positions (names to predicates; predicables to subjects) would require a shifting of sense. "It is logically impossible for a term to shift about between subject and predicate positions without undergoing a change of sense as well as a change of role" (1972a: 48). It is the licence for terms to play roles in either subjects or predicates without any semantic modifications that Geach finds most disgusting in syllogistic. Geach, like Plato and Frege, has conceived of terms (names and predicables) as logically heterogeneous (as sentence parts, i.e., subjects and predicates, undeniably are). Each kind of term has its own kind of sense-a sense that fits it to one logical role only (either subject or predicate). Moreover, he clearly takes naming as a logically primitive kind of sense, for he defines predicables as the results of removing names from sentences ( 1962: 22-25). An important consequence of Geach's theory of logical syntax (as ofFrege's) is that copulae have no place in logic, for "no link is needed to join subject and predicate; the incomplete sense of the predicate is completed when the subject is inserted in the empty place" (1950: 464). Failure to see this would lead to Bradley's Paradox. Geach points out that even "Aristotle had little interest in the copula" (1950: 465; 1962: 34). But what he has in mind here is the Abelardian copula (the qualifier) rather than the Aristotelian logical copula (consisting of the quantifier-qualifier complex). In his criticism of Strawson's theory of logical syntax, Geach ( 1980) accuses Strawson of preserving ''two bits of old logical lore" ( 179): (i) taking contrariety as logically more primitive than contradictoriness, and (ii) analysing (atomic) sentences as subject-copula-predicate rather than subject-predicate. His first charge is directed at Strawson's notion of "ranges of incompatibility." Geach argues that the idea of contradictoriness (between certain pairs of sentences) is a precondition for having the idea of incompatibility (between certain pairs of properties). (In part two I will offer reasons for holding neither idea to be a prerequisite for the other.) Concerning the second charge, Geach says, On the Fregean view, a predicative expression needs no glue or bond to make it adhere to a subject. At some point we must have an expression that simply adheres to others without needing some further linguistic device to make it adhere; why not ascribe this character to predicables themselves, rather than postulate a copula? Standum est in primo! If there were a genuine problem how subject and predicate can adhere together, then only by mere fiat could we avoid raising the problem how the copula can adhere to both. (1980: 182)

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Thus, for Geach, Bradley's Paradox is blocked only by Frege's syntax of complete-incomplete expressions or by fiat. Nonetheless, Strawson is no innocent here. He claims that the copula "indicates the mode of combination" (in van Straaten, 1980: 293) of the individual object specified by the subject with the concept, quality, or activity specified by the predicate, though he makes clear that the role of logical copulation and the role of predication are both played by the predicate. But, of course, Strawson's defence is ineffective. His analysis of the logical roles to be played in a sentence is tripartite. And while it makes sense to bind pairs of terms by a copula (as did Aristotle) or a quantified term with another term by a copula (as did Abelard) or subjects with predicates (sans copulae, as did Frege), short of taking subjects as nothing but subject-terms (as Frege and Geach do) and predicates as nothing but predicate-terms (which nobody does), binding subjects with predicates by means of a copula makes no sense. As I have already noted, Geach, like Strawson, has been a staunch defenderoftheAsymmetryThesis(e.g., 1962, 1969a, 1972a, 1975). So, for him, sentences can be conjoined, disjoined, and negated, and predicates can be conjoined, disjoined, and negated. In the latter cases, the result is a logical conjunction, disjunction, or negation of the sentences embedding the predicables. But names cannot be conjoined, disjoined, or negated. 37 One of the ways in which Geach uses the contrast between the negatibility of predicables (and sentences) and nonnegatibility of names is to enforce a distinction between genuine logical names and quantified general terms (cf. 1972b). Since he allows that names may be either proper or common, one might conclude that a noun phrase such as 'some man' is merely a complex name. But this would be a mistake, according to Geach. For him, as for Frege, a sentence is logically negated only by negating its main function expression (predicate). The latter may be a verb (as in 'Socrates walks') or a quantifier (as in 'Every man walks'V1 Geach holds that for each of these sentences negation is achieved in a manner that would not result in the negation of the other sentence if applied to it, because each sentence involves a different main function expression. 'Socrates walks' can be negated by negating '. . . walks'. its predicate. But the negation of '. . . walks' in 'Every man walks' results only in that sentence's contrary. The main function expression of'Every man walks' is 'Every' (a "second-order predicate"). It attaches to the first-order function, ' ... walks', to yield 'Every ... walks', which, in tum, is completed by the (general, common) name, 'man'. One of the features of traditional logic that has most exercised Geach is its doctrine of distribution. In chapter one, we saw how Geach has attacked Aristotle's position (as set out in Prior Analytics) that the two terms of a categorical are interchangeable (i.e., are logically homogeneous). Geach compared Aristotle's rejection of the Platonic binary theory (endorsed in De lnterpretatione) in favour of the ternary analysis of the

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Analytics to Adam's Fall. "Aristotle's going over to the two-term theory

was a disaster" (1972a: 47). Yet this "was only the beginning of a long degradation" (1972a: 51). The two-term theory was coupled with the view that since the subject-term could be taken as a name, the predicate-term could be viewed as a name as well. The two-name theory construed categoricals as pairs of names joined by a copula (now taken as a sign of identity). It was this theory, says Geach, that dominated the Middle Ages. 39 Finally, the two-name theory was coupled with the idea that what a general term names is the class of objects that it denotes. "By this slide the rake's progress of logic ... reaches its last and most degraded phase: the two-class theory of categoricals" (1972a: 53). The evils of the two-class theory are legion, according to Geach's reading of logic's history. Among them is the perverted notion that quantifier expressions somehow contribute to the reference of the terms to which they are attached, what Frege contemptuously called "quantificatious thinking." A second abomination was the doctrine of distribution. The Scholastics' doctrine of distribution was a result of their theory (or theories) of supposition-their semantics. According to the traditional account, since subjects are always syntactically complex, consisting of a quantifier and a subject-term, the semantic roles of subjects and subject-terms are different. The former refer while the latter denote. Subjects-terms are said to be distributed just in case their denotations and the references of the subjects in which they are embedded coincide; otherwise, they are undistributed. The doctrine was extended to singular subjects (taking them, usually, to be distributed), and then expanded into the well-known theory involving necessary conditions for syllogistic validity. As a Fregean, Geach rejects any claim that subjects might be syntactically complex. As a consequence, he rejects any distinction between reference and denotation, and with it the notion of distribution (cf. 1956; 1962, ch. 2; 1976). For Geach, reference/denotation is the role played by names. And any time a term is used to make reference it is in logical order to ask about what is being referred to. Even if we knew what 'referring' was, how could we say that 'some man' refers to some man? The question at once arises: Who can be the man referred to? ... which men will the subject refer to if a predication of this sort is false? No way suggests itself for specifying which men from among all men would then be referred to; so are we to say, when 'Some men are P' is false, all men without exception are referred to-and 'men' is distributed? ( 1962: 6-7)

Geach would reject the view that quantified terms (traditional subjects) refer at all. Thus 'some S' and 'every S' do not refer (i.e., do not denote, do not name). Much of the plausibility of his stand derives from the fact that he insists on including 'no S' among such expressions: "When 'nothing' or 'no

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man' stands as a grammatical subject, it is ridiculous to ask what it refers to" (1962: 12). But, of course, 'no' is no (logical) quantifier. One who, like Geach, takes all logical subjects to be names is right to require an answer to the question of what is being referred to, or named, by a given subject. Friends of distribution (among whom I will count myselt) must reject not only the Fregean Dogma (that all logical subjects are singular) but the weaker Geachian claim that all logical subjects are names.40 Atanyrate, Geach has not been content to reject just the semantic grounds of distribution.41 He has also sought to overturn the doctrine as it relates to syllogistic validity (e.g., 1962, ch. 1; 1972a, sections 2.1 and 2.2). The distribution of predicate-terms especially disturbs Geach. "If a predicateterm 'P' can indeed be understood to refer now to any and every P, and now only to some P, then it seems natural to mark this fact by attaching quantifiers to the predicate as well as the subject" (1962: 18). This is just what Hamilton tried, as Geach goes on to note, a century and a half before. But the fact is that no one defending distribution would say that predicateterms, or even predicates, refer. The doctrine holds only that a sentence in which the predicate-term is distributed implies a sentence in which that term is now a universally quantified term (and thus part of a referring expression). Even if we could make sense of the distribution of both subjects and predicates, does the doctrine actually work as applied to syllogisms? According to Geach, the syllogistic rules of distribution are inconsistent. Thus, the inversion of an A categorical to an 0 (viz., 'Every Sis P' to 'Some nonS is not P') has, contrary to the rules of distribution, a predicate term distributed in the conclusion but not in the premise. Suffice it to say for now that what needs to be doubted here is not distribution but inversion. A Keynesian sort of solution (seeing the inversion here as an enthymeme) is to be recommended. 42 Needless to say, we will return to Geach from time to time throughout the second part of this essay. A century after the initiation of the Fregean revolution in logic, no philosopher better represents the pervasive use of that logic as a philosophical tool than W.V. Quine. Through the period of accretions, changes, and challenges from the 1930s to the 1990s Quine has remained the heir to and defender of Frege and Russell's logicism. His contributions to logic, philosophy of language, epistemology, and metaphysics have been enormous and influential. Geach spoke for many when he wrote recently, "My intellectual debt to Quine is immeasurable" (in Lewis, 1991: 252). From his simplification in "New Foundations for Mathematical Logic" (Quine, 1937) of Russell and Whitehead's system to his most recent work in logic, Quine has been consistent in his advocacy of a small number of core theses. Among these are the following: ( 1) The proper logical task with respect to natural language is regimentation, the building of an artificial language free of the ambiguities, vagueness, and other qualities of natural language, and adequate for scientific discourse. The logic of natural

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CHAPTER2 language can be revealed only through a translation into the regimented language that eliminates the plaguing "quirks of usage" ( 1960a: I 58). The resulting constructed language-the standard first-order predicate calculus with identity-"is a paragon of clarity, elegance, and efficiency" ( 1970: 85). (2) In constructing an artificial logical language, the logician must seek to display the logical grammar of expressions (viz., sentences) as perspicuously as possible. Such forms must aim to reveal the truth-conditions of their sentences. Logic chases truth up the tree of grammar. . . The grammar that we logicians are tendentiously calling standard is a grammar designed with no other thought than to facilitate the tracing of truth conditions. And a very good thought this is. (1970: 3536)

Revision of grammar is an important part of the logician's activity... For the latter-day logician, logical regimentation of grammar is standard procedure.... what we call logical form is what grammatical form becomes when grammar is revised so as to make for efficient general methods of exploring the interdependence of sentences in respect of their truth values. (1980: 20-21)

(3) The distinction between statements true by virtue of their logical form alone (analytic statements) and those true by virtue of the way things are in fact (synthetic statements) is, at least, difficult to draw. An adequate understanding of the former would require an understanding of all members of a family of terms, including 'necessary', 'synonymous', 'meaning'. But none of the terms in this circle can be explicated clearly and independently. As a result, Quine has steadily maintained a conservative attitude toward all attempts to extend the standard system of first-order predicate logic to include modalities, propositional attitudes, and any kind of intensional object (such as propositions and meanings). (4) Logic is not ontologically neutral. While it cannot reveal what there is, it can reveal what a person or theory "says there is" (1960a: 253; see also 1953: 15). According to Quine, what is (i.e., what is assumed, presupposed, taken, counted to be) is just what a speaker (or theory) admits as a possible referent of his or her referring expressions, and in Quine's "canonical" regimented language these are bound individual variables-personal pronouns. As he so memorably put it, "To be is to be the value of a variable" (1953: 15; other, more accurate but less striking, versions are 1943: 25; 1953: 13, 14; 1960a: 242, 243). Quine's commitment to theses such as these has determined his ideas concerning logical syntax. In the canonical language all sentences are either atomic or functions of atomic sentences. Atomic sentences are formed by the "basic combination":

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The basic combination in which general and singular tenns find their contrasting roles is that of predication. . . . Predication joins a general tenn and a singular tenn to fonn a sentence that is true or false according as the general tenn is true or false of the object, if any, to which the singular tenn refers. ( 1960a: 96; see also 1970: 28)

This is a succinct but rich statement of Quine's central thesis concerning logical syntax and logical semantics. Predication is the joining of a singular and a general term. Natural-language singular terms (what he calls "ordinary" or "definite," as opposed to "indefinite" or "dummy," singular terms [1960a: 112-14]) are expressions such as names, definite descriptions, personal pronouns, and demonstratives (dummy singulars are quantified terms). The role of (definite) singular terms is reference. Since all such terms can be paraphrased in terms of just personal pronouns, existential quantifiers, and general terms, and since (by Quine's thesis 4 above) ontological commitment is revealed by those terms used to make objective reference, the singular terms of logically regimented sentences (i.e., individual variables) carry the entire burden of reference. Consequently, no other kind of expression, particularly general terms, can refer-the Fregean Dogma. In the basic combination, the singular term is referential. The general term is predicative. Indeed, it is only by this difference of"role that general and singular terms are properly distinguished" (1960a: 96).43 The predicative role is the role of being true ofwhat is referred to by the singular term (the subject). Only general terms can play the predicative role in a sentence. Like Frege, Strawson, and Geach, Quine subscribes to the Asymmetry Thesis. Singulars can never play the predicative role; general terms can never play the referential role.44 Sentences in the basic combination have no formatives. Sentential unity appears to be merely the result of the primitive relation of predication, which binds singulars to general terms when these are brought sufficiently close to one another. Most importantly, no logical copula is required to effect this binding. The grammatical copula (e.g., 'is') in some naturallanguage approximations of the basic combination is logically inert. Such expressions are nothing more than devices for converting "a general term from adjectival or substantival forms to verbal form for predicative position" (1960a: 97). Note that while general terms may have the grammatical forms of adjectives, substantives, or verbs, they are logically verbs, "for predication the verb may even be looked on as the fundamental form" (1960a: 96). Relative terms, likewise, are general terms that may be substantives (plus prepositions), such as 'brother or; adjectives (plus prepositions or conjunctions), such as 'part or, 'bigger than', 'same as'; transitive verbs, such as 'loves', 'killed'; or lone prepositions, such as 'in', 'under', 'like' (1960a:105-6). Fundamentally, then, a basic combination is

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a predication of a singular tenn and a verb. 45 The following represent ever closer approximations to logical purity. a. Socrates is a philosopher. b. Socrates is philosophic. c. Socrates philosophizes. According to Quine, indefinite singular tenns can be translated into the canonical notation via quantifiers and the individual variable that they bind. But, as mentioned above. definite singulars can be given this same sort of translation as well. Names can be paraphrased in tenns of identity (thus 'Socrates' becomes 'is identical to Socrates'), which, in tum, can be paraphrased in tenns of a general tenn (verb) designed to be true of just the thing named (thus: 'Socratizes') (1960a: 178ft). This procedure for translating names is generally called "Pegasizing", since it is meant to be applied primarily to names that fail to refer (e.g., 'Pegasus'). In such cases, it penn its the closing of truth-value gaps so that sentences like 'Pegasus runs'(= 'Something both Pegasizes and runs') are false. This way with names may seem artificial, but "All in all, who shall say whether English is more radically modified by a canonical notation in which names consort with the singular pronouns and indefinite singular tenns or by one in which they consort with the general tenns ( 1960a: 181 ). Singular definite descriptions. such as 'the queen of England' or 'the present king of France', are regimented according to Russell's theory (thus: 'exactly one thing is queen of England', 'exactly one thing is presently king of France'), again allowing truth-gap closure for cases such as the second example. Class, attribute, and relation abstractions (e.g., 'the class of red things', 'sanity', and 'superiority', respectively) are paraphrased as definite descriptions and then eliminated by Russell's theory in favour of bound variables and predicates. Thus evidently nothing stands in the way of our making a clean sweep of singular terms altogether. with the sole exception of the variables themselves. (1960a: 185) ... That variables alone remain as singular terms may be seen as testifying to the primacy of the pronoun. ( 186)

Quine's emphasis on the variable, in both logic and ontology, cannot be exaggerated. On one side it is the logical analogue of the naturallanguage pronoun;46 on the other it is the expression par excellence for reference to bare particulars (Locke's "unclothed substances," Wittgenstein' s "colourless objects"). As Quine says, "The pronoun is the tenable linguistic counterpart of the untenable old metaphysical notion of a bare particular" (1980: 165) and "The variable is the legitimate latter-day embodiment of the incoherent old idea of a bare particular" ( 1981 a: 25).

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Quine does not make it clear just what is "untenable" or "incoherent" in the idea of a bare particular-perhaps just that it is an old idea, for he appears to like it well enough in its "latter-day embodiment." At any rate, as he sees it, singular terms in natural language tend to work at two tasks simultaneously, though the two tasks-identification and reference--can be separated. In a regimented language, the identificatory work is assigned to general terms, leaving variables to do the purely referential work. Thus, what such variables refer to must be bare, unidentified, but merely enumerated. These bare particulars constitute the range of values, the domain of discourse, for the bound variables of a regimented sentence, "but the idea of [such sentences] being about certain things and not others seems dispensable" (Quine's reply to Strawson in Davidson and Hintikka, 1969: 321). Quine's emphasis on the variable notwithstanding, we will soon see that he has found ways of eliminating variables altogether from a regimented tangauge. But first, a few comments concerning Quine's notion of identity are in order. Modem mathematical logic has made much ofFrege's distinction between the 'is' of predication and the 'is' of identity. The distinction is seen as logically required, since otherwise some singular terms, when following 'is', would have to be construed as predicates. Thus, in 'Tully is Cicero' either 'is' is not the dispensable 'is' of predication or 'Cicero' is not singular. The 'is' of identity is seen as indicating a binary relation and is appropriately flanked by a pair of singulars. The identity statement is true if and only if the two singular terms are co-referential. Notice that Quine's Pegasizing procedure is a device not for predicating singular terms (in the case of names), but merely for converting such singulars to general terms (which can be predicated). Now, according to Quine, "Identity evidently invites confusion between sign and object" ( 1960a: 117). Oddly enough, he thinks that Aristotle was not subject to this widespread confusion (116n). This is odd because Aristotle and Quine hold quite different views of logical syntax-and thus of the logical form of identity statements. Quine sees his position to be similar to Aristotle's here only because he fails to see that Aristotle, unlike him, is willing to admit singular terms after the qualifier. Aristotle would take 'is' to be always predicational. Whether a term is used referentially or predication ally is, for Aristotle, independent of whether it is singular or general-it is just a matter of the term's position, or syntax. In contrast, Quine takes the role of' is' to depend on whether it is followed by a singular or general term.47 We saw that Leibniz was able to exploit this Aristotelian view in attempting to incorporate singular sentences into the categorical mould. In part two we shall see how Sommers has exploited it even further. 48 For Quine, all referential expressions-singular terms--can be paraphrased in terms of pronouns and appropriate predicates and formatives. Translated into the language of the canonical notation, these pronouns become individual variables. In effect, descriptive phrases, demonstratives,

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and names have been eliminated in favour of variables. Yet, as noted above, Quine allows for the possibility of eliminating even these. Given the enormous logical weight placed on variables. the natural question is: Why should they be eliminated, and how? The elimination of any device that does essential work in a given system must be accompanied by the substitution for that device of another device (or devices) that does the same job. From a theoretical point of view, the contemplation of such a substitution will reveal just what the work of the original device has been. The intention is to instil an appropriate level of understanding and appreciation of the old device. This is the motive for eliminating variables. But how can this be done? One of the jobs assigned to variables in the standard predicate calculus is that of indexing. Consider, for example, the sentence 'Every boy kissed every girl'. The canonical translation of this is usually '(x)(y)(Bx & Gy .::~ Kxy)'. Note that the (individual) variables here show which reference is bound by which quantifier and keep track of which is the subject and which the object of the relational ('kissed'). If one could reduce all polyadic (relational) predicates to monadic (nonrelational) predicates, such indexing-and thus the use of such variables-might be eliminated (see 1980a: 23-24). In a series of studies spread out over several years Quine developed a formal language, the Predicate Functor Algebra, "a drastic alternative to standard logical grammar'' (1970: 30), which makes no place for individual variables.49 As it turns out, the Predicate Functor Algebra, which makes use only of predicates and functors on them, has no need to distinguish between the singular and general terms of its lexicon. Singular terms-names, descriptions, demonstratives, and even pronouns-have been eliminated. Unanalysed sentences (as in the logic of truth-functions) have been taken as zero-place predicates. The result is a logic that is, in effect, equivalent to the first-order predicate calculus with identity. But, unlike the calculus, the algebra is a term logic, for, when all the terms of the lexicon are general terms, predicate terms, 'term' will do. Aristotle's syllogistic was a term logic. The universal characteristic sought by Leibniz was a term logic. The algebras of Boole, De Morgan, Jevons, and Venn were term logics. Quine's algebra of predicate functors demonstrates via its equivalence to the standard calculus that even the predicate calculus is a term logic. 50 In what remains of this essay, I will examine and develop a system of logic due to Fred Sommers. My contention is that it is a term logic par excellence .

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Notes for Chapter 2 1Indeed,

Kneale and Kneale (1962: 511) calll879 "the most important date in the history of [logic']," and Quine (preface to Clark 1952: v) says that "1879 did indeed usher in a renaissance [in logic]." 2 Some of this can be seen in the account of this transition period by one writing in the midst of it: Shearman (1906). 3 For example, see Frege ( 1879: 7, 12; 1892: 54; 1979: 141 ). 4 Prominent Fregeans, such as Geach and Dummett, disagree about the priority of the grammatical or the ontological distinction in Frege. Fora brief discussion ofthis debate, see Sommers (1982: 36-37). See also Wetzel (1990). For Frege's saturated/unsaturated distinction, see especially Frege (1891 : 31 ); see also Frege ( 1979: 177, 187). 5 Quoted in Furth (1968: 16). See also Frege (1892: 54-SS; 1979: 177). 6 See especially Frege (1891, 1892b). See also Furth (1968). 7 See Englebretsen (1986a). 1 For example, in Frege (1892a). See also Mendelsohn (1987) and Englebretsen (1990a). For an attempt to "repair" Frege by reintroducing the copula (as a predicate formative) see Wiggins (1984). 9 Also see 6f, 12ff, 142, 188, 190,266, 269ff. 1°For more on this contrast see, for example. Hom (1989); Sommers (1982, esp. ch. 13); Englebretsen (1981a); and Sanford (1966). 11 See Frege (1979: 185; also 198, 253). 12 See 46. This is the basis ofFrege's contextual thesis (only in the context of a sentence can a work be said to have meaning). 13 See Bradley (1914. 1935); Hunter (1985). 14 See Englebretsen ( 1986a, 1986c). 15 Ramsey ( 1925). See Russell's response ( 1931 ). 16 Geach (1962, ch. 2; 1969a; 1972a, esp. sections 3.5, 3.6, 8.1, 8.2; 1975); Strawson ( 1952, esp. 170ff; 1957; 1959, esp. part 2; 1970; 1974). See also Heintz (1984) (and the response by Linsky and KingFarlow [1984]). For criticism of the Asymmetry Thesis see, for example, Grimm (1966); Hale (1979, along with a brief comment by Geach, 146); Nemirow (1979); Clark (1983); and Bradley (1986). I have addressed the topic in Englebretsen (1985b, 1985c, 1987b, 1990d). 17 Traditionalists such as Johnson had held that general, but not singular, terms could be compounded. In response, Ramsey argued that even general terms could not be compounded (see 1925:405-07, 411). 11 See Russell's letter to Frege in van Heijenoort ( 1967a).

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NOTES F.Q!.£!:!APTER 2

This is especially so in Russell's Principles of Mathematics (1903) and "The Philosophy of Logical Atomism" (1918-19). zo This point is discussed by Kapian ( 1972: 239-41 ). 21 In addition to Sainsbury (1979), see White (1979). 22 Russell (1945: 13) claimed that Leibniz had actually made an "awkward discovery," relational logic (a logic that admits irreducible relationals), but that he chose to suppress his discovery, taking relations to be merely ideal. 23 See Korner (1979: 176ft). In fact, the reduction of relations to classes was achieved. See Wiener ( 1967). 24 See especially Russell (1903, ch. 26). Russell's critique is examined in Sprigge (1979). 25 Of course, as we saw above, Bradley went on to hold that both relational and non relational judgments are distortions of Reality. Nonrelational judgments are simply less distorting. 26 In fact, Russell had held that Bradley's Paradox was genuine but that, being "logically quite harmless," it posed no real threat to logic. See Russell (1903: 100). 27 The following remarks rely heavily on the excellent analyses of the problem in Palmer ( 1988) and Linsky ( 1992) (the brief appendix to the latter work is especially recommended). 28 See "Boote's Logical Calculus and the Concept-Script" in Frege (1979). Sluga (1987) calls Frege's insistence on giving priority to the sentence over its constituents the "priority principle." 29 As Linsky has said (1992: 26-7), "All of the constituents of the proposition in the Tractatus are incomplete." Names achieve propositional unity by completing each other. 30 But in a sense, claimed Wittgenstein, all of the logical constants occur in any atomic sentence; there is, in effect, just one such constant, which is what atomic sentences have in common-that is, their logical form (1961: 5.47). He had in mind "mutual rejection," the Sheffer stroke (5.1311, 6.001). 31 See remark 5.5151. Ramsey (1923: 17) made the same point. Wittgenstein (1929) tried to accommodate Ramsey's criticism concerning the incompatibility of 'This is red' and 'This is blue', claiming that such propositions are not contradictory but may "exclude" one another (35-36). 32 Since a statement is always a truth-claim, Strawson has also devoted much attention to the concept of truth (see, for example, 1950b, 1954, 1964, 1965). 33 For an extended discussion see Englebretsen (1986a). 34 For a more critical appraisal of Strawson concerning asymmetry see Englebretsen ( 1985b, 1987b). See also Strawson' s reply to Geach in van Straaten (1980: 293-94). 19

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Geach's "History of the Corruptions of Logic" (in Geach, 1972) is especially hostile. See Englebretsen ( 1981 c) for a response. 36 See Hintikka (1991) for a fuller survey ofGeach's hidden non-Fregean views. In his reply to Hintikka, Geach denies any departure from orthodox Fregean theory. 37 But Geach ( 1950: 463) does suggest that while the negation of a name is not a name, it isn't nonsense either. And in Logic Matters (1962a, ch. 7) he comes close to my own notion of composed names. For an alternative view of compound predicates see Stalnaker (1977). 38 For a more extensive discussion see Englebretsen ( 1985a). 39 Though Geach has not been reluctant to charge Scholastic logic in general with the sin of the two-name theory, he has exempted his favourite Scholastic, Aquinas. See especially Geach ( 1962: 22-46; 1969b: 42-64). Veatch (1974: 416-22) responds to Geach's views of Aquinas on logical syntax. 40 Recent discussions of distribution are: Toms ( 1965), Makinson ( 1969), Richards (1971), Williamson (1971), Sommers (1975), Katz and Martinich (1976), Friedman (1978ba), Rearden (1984), Englebretsen (1985d), Wilson (1987), Peterson (1995). 41 But see Geach (1950: 475), where his "total" and "partial" identity sound suspiciously like "distributed" and "undistributed." 42 See Englebretsen ( 1979: II 5-17). 43 For more on this topic see Englebretsen (1982a, 1986a, 1986c). See also Strawson (1969), along with Quine's reply in the same volume. 44 This last is meant to protect us from Platonism. 45 Recall Russell's notion of the verb "used as a verb," which embodies the unity of the proposition. 46 While many contemporary linguists have taken to heart this Quinean attitude toward the pronoun-variable parallel, not all have. See, for example, Wasow (1975) and Higgenbotham (1980). 47 For more, see Englebretsen ( 1985b). 48 In a sense, of course, Quine is willing to dispense with the identity relation altogether-at least, where the lexicon of the language at hand has a finite number of predicates. Given such a language, indiscernibility can do all the work of identity. For a brief statement ofthis see Quine (198la: 27-28). 49 See Quine (1936a, 1936b, 1937b, 1959, 1960b, 1971a, 1971b, 1972, 1981 c, 1981 d). Quine has often acknowledged his debt to the work ofSchOntinkel (1924) and Curry (see esp. Curry and Feys, 1958). 50 Though much of my knowledge ofthe Predicate Functor Algebra comes from a reading of Quine's work cited above, I owe most of my understanding and appreciation of the theoretical import of the algebra (particularly its status as a version of term logic) to the 35

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work of Aris Noah. No one should consider the study of Quine's algebra complete without having attended to Noah ( 1980, 1982, 1987).

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Part II

CHAPTER THREE

COMING TO TERMS WITH SOMMERS Classical logic was obviously in closer accord with natural language forms than its modern successor. The intriguing logical question remains, whether this modest rehabilitation might be pursued to create a "classical" theory ofinference. rivalling, say, Fregean predicate logic In breadth ofcoverage and elegance ofpresentation. J. van Bentham There are now two systems ofnotation, giving the same formal results, one ofwhich gives them with self-evident force and meaning, the other by dark and symbolic processes. The burden ofproofis shifted, and it must be for the author or supporter ofthe dark system to show that it is in some way superior to the evident system. Jevons

The Calculus of Terms Such is the advantage ofa well-constructed language that its simplified notation often becomes the source ofprofound theories. Laplace An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation carefully may be repaid by the time we save later by avoiding hesitation and confusion. G. Polya A good notation {is} .. . like a live teacher. Russell

According to contemporary theories of scientific evolution (first Popper's, but more particularly those of Kuhn and Lakatos), an established "normal" science, accepted as paradigmatic by those working within the field, eventually gives way under pressure to a new theory or science. A "paradigm shift" takes place. A new "research programme" is instituted.

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A revolution has taken place within the science. Such scientific revolutions are the result of several predictable factors. During the period of normal, pre-revolutionary science, the established theory is seen as accounting for a significant and large portion of relevant phenomena. Its ability to do so depends on the acceptance of certain assumptions and conjectures, which are accepted because of the explanatory power of the theory based upon them. Moreover, the established theory is "progressive" in that it opens up new areas of research and offers explanations, which, in tum, lead to even further new research programmes. Eventually, however, there comes a "crisis." Limitations are found in the main (and subsidiary) theory. Paradoxes are revealed that had been ignored or forgotten. Refutations of the explanatory theses and, eventually, of the ground assumptions and conjectures proliferate. Rival paradigms begin to demand attention. Defections of researchers from the old theory begin. This period of crisis, this revolution, ends only when one of these rival theories commands enough allegiance from researchers to serve as a new paradigm. Acceptance of the new paradigm is the result of the general agreement that the new theory can solve or dissolve the anomalies and paradoxes hidden within the old, and that it can match or exceed the old theory's power to solve problems and furnish explanations. A theory that may have been viewed as outlandish or radical or ultra-reactionary may, in time, tum out to be the new paradigm, the next normal science. "Theories, which looked counterintuitive or even perverted when first proposed, assume authority" (Lakatos, 1978,2: 41). From the point of view ofthe new science, the old theory is seen as riddled with paradox, nonprogressive, and dogmatic. This picture, or something fairly similar, is now generally accepted to be a good portrait of the rise and fall of scientific theories. Science is seen as continually progressing along this roller-coaster track. No theory can rest in the false security that it is irrefutable, that no revolutionary theory could ever capture the throne. A scientific theory that cannot even accept the possibility of its own replacement is no scientific theory at all. Kant told us in the first Critique that logic was just that kind of science-complete. For him, the possibility of the logic he knew being replaced by a different logic was unthinkable. But, of course, he was wrong. Unfortunately, even with the example of Kant as an object lesson, most contemporary logicians persist in believing that the system of logic now entrenched as the logic, normal logic, will never be replaced by any other theory. It is unfortunate that so many contemporary logicians have this attitude toward their field of study, since the fact is that logic is now in a period of crisis and revolution. And, as with any other science, this new revolution is just the latest in a string of such periods that have punctuated its long history. That most contemporary logicians fail to see this is due in part to the false security they derive from their numbers and in part to their lack of interest in the history of their subject. Researchers working in the midst of an established theory rarely look farther back in time than to the

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revolution that placed their own theory in power. All those who toiled the field before that time were simply wrong; all those who now try to till the field outside the castle walls are simply radicals. Neither deserves serious attention. Thus defenders oftoday's established logic see the real history of logic beginning with Frege. Still, Frege's revolution was not the firSt in logic, and today's logic no longer lacks serious challengers. The Stoic logicians may very well have thought of themselves as shifting from Aristotle's logical paradigm to their own. Thirteenth-century logicians certainly began a revolution, which eventually moved logic from the preservation of Peripatetic formulae to the exploration of the science of language. Humanist logicians of the Renaissance revolted against the established Scholastic logic and replaced it with a logic of invention and rhetoric. Nineteenth-century algebraists rejected the established mix of Scholastic, Humanist, and empiricist logic in favour of a logic modelled on the formal features of mathematics. The algebraist revolution never succeeded in establishing its own dominion (perhaps like many now-forgotten failed revolutions before it), for in the midst of the algebraic revolution Frege initiated a different revolutionary research programme that soon swamped all rivals and established itself as the normal logic of the day. Frege's revolution succeeded for the same reason that any successful scientific revolution succeeds. Frege rebelled not just against the algebraists {who, at any rate, had not yet accumulated much power), but against all previous logic-"traditional logic." The revolution succeeded because it was able to reveal a number of anomalies in traditional logic, because it provided a logical theory that had far more explanatory power than any previous theory, and because it was exceedingly progressive, leading to a very large number of new research programmes (in semantics, modal theory, mathematics. cybernetics, linguistics, etc.). The result of that revolution was the entrenchment of the now standard first-order predicate calculus as the established, correct theory of logic. Now, according to our Kuhnian picture, a new paradigm or research programme will maintain its hegemony as long as it is progressing. Eventually, however, it will begin to flounder and stagnate. Unresolved paradoxes and anomalies will re-emerge. Critics and rivals will proliferate. This period of crisis will ultimately give way to all-out revolution. Frege's logic has been fairly stagnant for the past two decades or so. Whatever progressive energy remains in its research programme is devoted to problems that are far more mathematical than logical (this is easily confirmed by reading the Journal ofSymbolic Logic). The established logic has become dogmatic and inflexible. It is taught to students as received truth rather than as a potent research tool. For the past quarter of a century, critics have ever more frequently pointed to unresolved problems and weaknesses in the logic. Rival theories (free logic, intensional logic, informal logic, paraconsistent logic, generalized quantifier theory, branching quantifier theory, fuzzy logic, relational algebra, and a variety of 101

CHAPTER3

other theories in formal semantics) have demanded hearings in larger numbers and louder voices. In spite of the continued hegemony of mathematical logic in the schools, that logic has been rejected in recent years by growing numbers of researchers. While there are sometimes trivial reasons for this rejection, there are some important reasons as well. Foremost among these are the following: (I) it is based on a theory oflogical syntax that is remote from natural language syntax; (2) while it is far more powerful in terms of its ability to model inferences than is traditional logic, there are, nonetheless, simple inferences beyond its scope; and (3) it is unnecessarily complex. Alternative, nonstandard versions of mathematical logic have been offered in response to (2); informal logic has been offered, in part, as a response to (I) and (3) (and perhaps [2]). Moreover, it is not only philosophers and mathematicians who are interested in the well-being of logic; the issue is of great import for the linguists who are seeking to show that there are language universals, syntactic and semantic characteristics common to all natural languages. Many of these researchers have been looking to formal logic for insights. Some cognitive psychologists now believe that the core of rational human thought is reflected in the kind of inferences that have always interested logicians. One of the most active areas of research in artificial intelligence involves attempts to develop a machine programme for translating from one language-say, English-into another-say, Spanish. This would make computers responsible for the job of translation, which is now done slowly (and expensively) by humans. All attempts to develop such a programme have so far failed, but many now believe that what is required is a "mediating language" into which any natural language could frrst be translated, and which would then be translated back into the second natural language. Such a mediating language could not be any given natural language, but an artificial language that exhibits all features common to all natural languages. Since systems of logic can be viewed as artificial languages, they are prime candidates for the role ofmediation. Finally, the old (but perhaps newly appreciated) crisis in education has led large numbers of education theorists to argue that the missing ingredient in education is the imparting of"critical thinking" skills. In each case, these theorists have come to look at logic (orthodox or otherwise) as the tool the students must acquire in order to be able to think critically. Yehoshua Bar-Hillel's challenge of a quarter century ago is still unmet by standard logicians. I challenge anybody here to show me a serious piece of argumentation in natural languages that has been successfully evaluated as to its validity with the help offormallogic. I regard this fact as one of the greatest scandals of human existence. Why has this happened? How did it come to be that logic, which at least in the views of some people 2300 years ago, was

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COMING TO TERMS WITH SOMMERS supposed to deal with evaluations of argumentations in natural languages. has done a lot of extremely interesting and important things, but not this? (Quoted in Staal, 1969: 256)

Keeping in mind that Bar-Hillel was referring to both traditional and mathematical versions of formal logic, my view is that what is needed in philosophy, linguistics, cognitive psychology, and other disciplines is a system of formal logic that is (I) natural (in that it is based on a theory of logical syntax close to natural-language syntax), (2) as powerful as (or more powerful than) the standard system now in place, and (3) provided with a symbolic algorithm that is easy to learn, simple to manipulate, and effective. Such a system of logic has been devised in recent years by Fred Sommers and his colleagues. Like most pre-Fregean systems, it is a logic of terms, a logic first begun by Aristotle and fully envisaged (but not completed) by Leibniz. Those who have abandoned formal logic have, perhaps unwittingly, closed the door not only to mathematical logic (which surely does not meet their needs) but to this revised traditional formal logic (which just might be found satisfactory). Philosophy, especially, cannot abjure formal logic completely. Sommers's logic is intimately connected with, and indeed grew out of, his work on language structure and ontology, which began in the late 1950s (see especially 1959, 1963a. 1965, 1971 ). He showed that between any pair of ordinary language categorematic terms is a semantic relation that either allows them to stand as subject- and predicate-terms of a meaningful categorical sentence or prevents them from so standing (the so-called U and N relations, respectively). Some terms share all of the same U and N relations (e.g.• 'blue' and 'red'); these are "categorially synonymous:• Any term and its logical contrary (its negation)-for instance, •red' and 'nonred'-are categorially synonymous. (Note that, while the contrast between 'blue' and 'red' is semantic, the 'red'/'nonred' contrast is syntactic.) Suppose one could write down all the (nonformative) terms of a language such as English, connecting pairs of terms (or pairs of categorially synonymous groups of terms) with straight lines. It turns out, according to Sommers, that the resulting diagram would be regular, subject to certain structural restrictions. It could be viewed as the "sense structure.. ofthat language. Moreover, whatever the language, the structure is always the same-an inverted finite binary tree (thus "The Ordinary Language Tree..). This way of looking at language provides a variety of insights into the nature of sense, ambiguity, and so on. But, from a philosophical point ofview, its greatest value lies in the fact (which Sommers came to establish as his main thesis) that ontology shares the structure of language-language and ontology are isomorphic. His arguments to support this thesis are grounded on his notion that, given any term and any individual object, either that term can be said to apply sensibly to that object (i.e., can be meaningfully affirmed or denied of it) or it caMot. A term that can so apply 103

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to an object is said to "span" it. All the objects spanned by a given tenn constitute a "category" of individuals (relative to that tenn). Any individual spanned by a given term is also spanned by any term categorially synonymous with that term. Thus a term and its negation span the same individuals. Categories of individuals stand, then, in a one-to-one correspondence to groups of categorially synonymous terms. This, and the fact that all relations between categories are governed by the same structural constraints as those for terms, guarantees that the structure of categories (the ontology, according to Sommers) is the same as the structure of language. Language, constituted by the senses of categorematic terms, and ontology, constituted by categories of individuals, are isomorphic.' The "tree theory" is a powerful analytic tool, applicable to a wide range of problems, and it has, in fact, been applied to issues in philosophical psychology,2 theology,3 philosophy ofscience,4 and philosophy oflanguage, among others. 5 At the heart of Sommers's theory are two theses of a particularly logical nature: (I) terms as well as sentences can be negated, and (2) all sentences can be logically construed as categoricals. Neither of these theses is compatible with the prevailing logical orthodoxy. Sommers was well aware ofthis. By the mid-1960s, he had come to formulate explicitly how his own logical view differed from those of Frege and his followers. In "On a Fregean Dogma" (1967), Sommers rejects outright a view held firmly by Frege and Russell, writing, "There is ... no good logical reason for saying that general and singular statements must differ in logical form" (47). The insistence that a singular (e.g., 'Socrates is mortal') and a general (e.g., 'Men are mortal') differ in logical form rests on the assumption that the general (but not the singular) predication involves quantifiers. 'Men are mortal' is true just in the case when 'mortal' is true of whatever 'man' is true of. In contrast, 'Socrates is mortal' is true just in the case when Socrates is mortal. The assumption made here is itself dependent on an even more fundamental assumption: predication cannot be to plural subjects. When we say that men are mortal, we are not predicating 'mortal' of men but of each individual of whom 'man' is true. The presumption is that all predication is logically singular. It is for 'Socrates is mortal'. It is construed so for 'Men are mortal'. This presumption is the Fregean Dogma. Note that what Sommers does not deny is the grammatical difference between singular and general predications. His aim is to preserve the traditional notion that, from a logical point of view, predication is indifferent to the singular/plural (general) distinction. His attack goes to the heart of Fregean logic, which, as we have seen, rests four-square on the notion that singular and general terms are fit for quite distinct logical roles. Exposing an assumption as a mere dogma is of no consequence in the absence of an acceptable alternative account that dispenses with that dogma. Sommers's task, then, is to show how one can construct a logic that does not formally differentiate between singular and general statements. Sommers proceeded independently of Quine in doing this. The first step is 104

COMING TO TERMS WITH SOMMERS

the "dequantification" of general statements (represented by the four standard categoricals). Each is treated as saying something about S's, differing only in what is being said about them. An A statement says ofS's that they are P; an 0 statement simply denies this, saying ofS's that they aren't P. An E statement says of S's that they are nonP; an I statement denies this, saying that S's aren't nonP. Thus the logical contrariety betWeen A and E is accounted for by the logical contrariety between a term and its negation. "One reason logicians have ignored the possibility of using plural predication for a dequantified interpretation of the four categoricals is their refusal to accord logical recognition to contrariety as a distinct logical relation between terms" (1967: 50). In offering this reading of the categoricals, Sommers insists on distinguishing what a statement says from its truth conditions. Such a distinction is either blurred or explicitly rejected by Fregean logicians, but they do so at their peril. After all, it is indeed a truth condition of'Men are mortal' that each man be mortal. But what 'Men are mortal' says is simply that men are mortal. One of the truth conditions of'Men are mortal' is that Socrates is mortal, but clearly the statement does not say this. The statement affirms mortality not of each man, nor of the class of men, but of men (49, 61-62). Predication is not a reflection of truth conditions: it is a mere linking of pairs of terms (recall that Plato saw no difference here). Sommers's next step is to account for formal inferences involving categoricals construed as dequantified. Two rules suffice. "Inversion" says that affirming/denying one term of another is equivalent to affirming/denying the second term's logical contrary (its negation) of the first's logical contrary. For example, 'Men are mortal' is equivalent to 'Nonmortals (immortals) are nonmen'. This, along with the innocuous assumption that the logical contrary of the logical contrary of a term is the term itself ('non-non-P'='P'), suffices to establish the usual relations of conversion and obversion (and contraposition). Mediate inference is accounted for in terms of inversion plus the rule of"transitivity," which says that affirmative predication is transitive. Thus, any term affirmed of a second can be afftrmed of any term of which that second term is affirmed. For example, ifS's are nonP and Q's areS, then Q's are nonP. Inversion and transitivity can be applied directly to syllogisms to determine validity. For example, consider Ferion: 'NoM are P, some MareS, so someS aren't P'. This is first paraphrased, via applications of inversion, as: 'M's are nonP, S's are M, so S's are nonP'. Next the inference is put in transitive form (again via inversion), thus: 'S' s are M, M's are nonP, so Ss are nonP'. Since transitivity guarantees that the conclusion follows from the premises, the syllogism is valid. The simple testing procedure outlined above suggested to Sommers an algorithm that allows for fast, direct testing of syllogistic validity by algebraic manipulation. Each statement is formulated as a fraction, with the subject-term as the numerator and the predicate-term as the denominator.

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Denial and term negation are both represented by a negative unitary exponent. The algorithm exploits the fact that an inference is valid if and only if the product of its premises algebraically equals its conclusion. The system can apply to syllogisms containing existential statements (construed now as predications with 'things' as subjects) and to syllogisms involving singular statements.6 As we have seen, Sommers's primary goal in "On a Fregean Dogma" ( 1967) is to establish the logical homogeneity for singular and general predications. However, in doing so he recognizes an important logical difference between singular and general terms-a difference first noticed by Aristotle. When the subject of a predication is singular, the contrast between affirming a given term of it and denying the logical contrary of that term of it collapses. For example, 'Socrates is poor' is logically equivalent to 'Socrates isn't nonpoor'. This special feature of singulars turns out to play a key role in the term logic that Sommers will subsequently develop. In "On a Fregean Dogma," we see the beginnings of a programme of logic. We also see most clearly here how that programme is tied to the earlier ontological-linguistic theory. 7 The crucial distinction between denial and term negation (i.e.• contradiction and contrariety) is again emphasized. And Sommers begins the process of displaying the fundamental presumptions ofFregean logic that he will reject: Frege's dismissal of the subject/predicate distinction, the absorption of term negation into statement negation, the analysis of general statements (and, after Quine, singular statements) in quantificational terms. Still, "On a Fregean Dogma" represents a programme, in the early state of development, for term logic, taking all statements as logically tying (predicating) one term to another. But the programme is not completed there. In the discussion that followed Sommers's presentation of"On a Fregean Dogma," several contemporary logicians (Kalmar. Quine, Dummett, Lejewski) expressed both incomprehension (especially regarding the denial/negation distinction) and reservation. In particular, they made it clear that the programme would have to be extended to handle relationals. Sommers admits that the scheme offered "is not meant to be a logical instrument of any generality." But he goes on to express his confidence that while extending such a system to relationals may present "formidable" technical difficulties, it is not impossible (78). In "Do We Need Identity?" (1969a), Sommers returned to the special feature of singular terms mentioned above. He had attributed to Aristotle the insight that while a logical contrast holds between the atfmnation/denial of one term of a second and the denial/affirmation of the logical contrary of the first term of the second when the second term is general, no such contrast holds when the second term is singular. For example, 'Men are mortal' is logically distinct from 'Men aren't immortal'. If this example is difficult to see because of the shared truth of both

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statements, consider instead 'Numbers are red' and 'Numbers aren't nonred'. These are clearly distinct. Indeed, the first is false while the second is true (as is 'Numbers aren't red' as well). But this logical contrast disappears when the subject is singular. For example, 'Socrates is mortal' and 'Socrates isn't immortal' are logically equivalent. Both are true (unless, again following Aristotle, there is no Socrates; see Sommers, 1967: 72-4). Sommers's intent in "Do We Need Identity?" is to show how a logic of terms need not distinguish a special sense of'is'-the so-called 'is' of identity-from the ordinary 'is' of predication. This will tum on the logical feature of singulars that divides them from general terms. Again, the identity sense of 'is' had been successfully urged on modem logicians by Frege (1892a). A statement such as 'Tully is Cicero' must have a predicate (function expression) that is incomplete and general, according to Fregean syntax. In 'Tully is verbose' this is the case:' ... is verbose' is incomplete and general. But, while we might be tempted to think that ' ... is Cicero' is incomplete, it is certainly not general. Frege insisted that singular terms can never be predicated. The only recourse for him and his followers was to raise the logical status of 'is' in 'Tully is Cicero' from playing no logical role (for that is his way with the 'is' of predication) to playing the role of predicate. In such cases, 'is' would be a relational predicate affirmed of the ordered pair (Tully, Cicero). Now any theory that would reject the distinction between the 'is' of identity and the 'is' of predication would have to allow for the predication of singular terms such as 'Cicero', and would thus treat "identity" statements as nonrelational. Such a theory is considered nonviable by Fregeans. Even Quine's Pegasizing procedure does not allow for singular predicates. 'Tully is Ciceronian' has a general, not a singular, predicate. For Sommers, not only can singular terms be predicated, they can (following the lead of the Scholastics and Leibniz) be the subject-terms of universal or particular statements-that is, they can be quantified ( 1969a: SO I). If singular terms can play all the same logical roles as general terms, there naturally arises the question of just what difference there is between them. Sommers's answer appeals to Aristotle's principle according to which the denial of a predicate of alVsome S is logically equivalent to the affirmation of its contrary to some/all S. For example, 'Some man isn't wise' is logically equivalent to 'Every man is unwise', and, generally: Every/some S is P

=

Some/every S isn't nonP

(keeping in mind that the 'n't' of denial and the 'non' of contrariety do not cancel one another). In the case where the subject is a singular term, as Sommers noted in "On a Fregean Dogma," the affirmation of a predicate is equivalent to the denial of its logical contrary. For example, 'Socrates is wise' is logically equivalent to 'Socrates isn't unwise'. By following the old course of universally quantifying such singular statements, and by applying

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Aristotle's principle, we arrive at the following kind of equivalence (where 's' is a singular term): Every s is P = Some s is P In other words, we are led to Leibniz's notion that singular subjects can be treated as (logically) indifferent with respect to their universal or particular quantification. Singular terms in subject position have "wild" quantity (Sommers, 1969a: 502). This, then, is the true nature of the special logical feature enjoyed by singular, but not general, terms. As it happens, since singular subjects have wild quantity, in such cases we are free to assign quantity as the (inferential) context requires in order to preserve validity. Thus, in the classic inference 'All men are mortal, Socrates is a man, so Socrates is mortal', formal syllogistic validity is guaranteed as long as the singular is given the same quantity in the premise and the conclusion (when the quantity is universal, the syllogism is a Barbara; when it is particular, the syllogism is a Darii). The syllogism 'Socrates is mortal, Socrates is human, so some human is mortal' is valid as long as the two singular subjects are assigned different quantities. This way of treating singular terms has the effect of eliminating identity as a logically primitive notion. In contrast, as we have seen, standard mathematical logic must treat the logic of identity as a special appendix (since identity as a relation is a special predicate) to the fii"St-order predicate calculus. While identity, for Sommers, is no longer primitive, it can be defined. We can define 'x is identical toy' ('x=y') as 'x is y' (where 'x' and 'y' are both singular). Such a definition conforms to the fact that identity is an equivalence relation. It is reflexive, since 'xis x' follows from the tautology 'Every xis x'. It is symmetric, since the equivalence of'Some xis y' and 'Some y is x' guarantees the equivalence of 'x is y' and 'y is x'. And it is transitive, since 'xis y, y is z, sox is z' is valid (a Barbara or a Darii) when the major premise is universal and the minor premise and the conclusion share a common quantity (1969: 502-3). The thesis that singular subjects have wild quantity can also be used to establish the so-called Leibniz's Laws: (i) indiscemibles are identical, and (ii) identicals are indiscernible. Regarding (i), given that y is indiscernible from x, what is true of x is true of y. Since every x is x it follows that y is x (i.e., by Sommers's definition, y=x). With regard to (ii), ifx=y, then every xis y. Given that xis P, it follows syllogistically that y is P when the major and conclusion are both particular. Sommers concludes "Do We Need Identity?" by listing the advantages of a theory that treats singulars and generals on a syntactic par, distinguished only by the wild quantity of the former when used as subjectterms. They are: (1) identity is derivable in traditional (i.e., syllogistic) logic, (2) a special, logically primitive 'is' of identity is unnecessary, (3) syllogistic can accommodate singular statements, (4) Frege's problems with

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identity statements like 'The Morning Star is the Evening Star' dissolve (504).1

Latter-day logicians, such as Frege, Russell, and Quine, have rightly criticized traditional logic for its failure to achieve Leibniz's goal of a universal characteristic-a system of fonnallogic adequate for analysing inferences involving not just simple categoricals but, especially, compound sentences and relationals. Russell offered the typical diagnosis of traditional logic's failure: it was tied to the subject-predicate fonn. The great advances made by mathematical logic are attributed in no small measure to Frege's rejection ofthis old theory of logical syntax. Compound sentences, such as conjunctions and conditionals, do not have (even implicitly) a subjectpredicate fonn. Relationals have more than two material constituents, so they cannot be construed categorically-that is, as subject-predicate in fonn. The same probably holds for singulars. The judgment is that traditional logicians simply ignored the logic of compound sentences and relational sentences because they could see no way to treat them categorically. The first of these failures is seen as especially important, since the logic of compound sentences ("truth-functional logic," "the propositional calculus") is taken to be primary logic. It is more basic than predicate logic (the logic oftenns). The categoricals themselves are analysed only with the aid of sentential "connectives" (e.g., 'and' and 'only ir). In one sense, of course, even a cursory look at the history of traditional logic shows this to be false. Traditional logicians did not ignore the logic of compound sentences, relationals, or singulars. Boole and De Morgan certainly did not; nor did Leibniz; nor did the Scholastics; nor even did Aristotle. Still, the fact is that the old tenn logic never succeeded in building a viable logic adequate to these demands. In "The Calculus of Terms" ( 1970) Sommers sets out to construct a tenn logic that (I) construes all statements as logically subject-predicate in form, and (2) reduces all logical inferences to syllogistic. In carrying out this programme, he devises a simple, perspicuous symbolism, which is, like those of the nineteenthcentury algebraists. arithmetic. Frege eschewed the use of mathematical symbols in logic, since he saw it as a source of confusion for logicians bent on the discovery of the foundations of mathematics itself. Neither the algebraists nor Sommers share Frege's goal for logic. As it turns out, they were right in following Leibniz's lead in mining arithmetic and algebra for familiar symbols and operations readily adapted to the needs of a logical algorithm. To repeat, the syntactic principle at the heart of Sommers's "new syllogistic," the calculus of terms, is ternary. This means that all statements can be logically analyzed as consisting of two expressions (simple or complex) such that the second (the predicate) is predicated-affirmed or denied-of the first (the subject). The difference between the two kinds of expressions, and thus the overall logical nature of entire statements, is fully accounted for in terms of two basic notions: quality and quantity. Quality 109

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is taken as positive or negative opposition. Sommers recognizes three types of quality. Terms themselves are positive or negative with respect to their logical contraries. Thus, for example, 'massive' and 'massless' are opposite in term quality; so are 'red' and 'nonred', 'happy' and 'unhappy'. Logically, term quality is symmetric...Which term of a pair we choose to call 'positive' and which 'negative' is a matter of logical indifference" (1970: 4). But it is important to recognize that we do not begin with neutral terms (e.g., 'red') and then define from them negatives ('nonred'). All terms come in logically charged-positive or negative-pairs. They have ..polarity.',.. The fact that natural languages rarely mark positive terms, leaving them looking uncharged, neutral, should no more mislead the logician than an unmarked numeral (say '2' in '2-3=-1') would mislead the mathematician into construing it as either neutral or absolute, rather than positive. Not only do terms per se have quality, so do entire predicates. Such quality (called ..predicative quality") is displayed by the presence of positive or negative ..term copulas" (Sommers. 1970: 5): •is' and 'isn't'. Let 'S' be any subject expression and '+P' and '-P' be a pair of logically contrary terms. Either of these terms could be predicated of'S', and in either case the resulting predicate could be either positive or negative. Thus: (I) (2) (3) (4)

Sis+P Sis -P S isn't +P S isn't -P

As it turns out, we can think of a negative copula as cancelling out a negative term quality. And, in general, if we mark the copulas as+ orappropriately, (1.1) S + +P

(2.1) S + -P (3.1) S- +P (4.1) S- -P

they reduce to (1.2, 4.2) (2.2, 3.2)

s+p s- p

In other words, predicative quality and term quality can normally be treated indifferently as a single kind of opposition. It is the opposition between logical contraries, and Sommers refers to it as ..C-opposition" (1970: 7ff). The third kind of quality is predicative, indicating whether the predicate (whatever its quality or the quality of its term) is affirmed or denied. Such opposition is rarely marked in natural languages, and was only

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dimly seen even by traditional logicians. It can be noted by the use of "predicative copulas" (Sommers, 1970: 5). Two sentences with the same subject and the same predicate, except that the predicate is affirmed in one and denied in the other, are contradictories. Predicative quality, then, affects the entire sentence. It is not to be confused with predicate quality, which applies to predicates alone and results in the opposition of logical contrariety-not contradictoriness. Predicative quality opposition is called "P-opposition" (8ft). A sentence like 'No A is B' displays negative predicative quality. Here, 'no' is a sign of P- (not C-) opposition. It is elliptical for 'Not: some A is B', indicating that the predicate, 'is B', is denied of the subject, 'some A'. As in the case of quality, quantity is oppositional ("Q-opposition," Sommers, 1970: 8ft). A sentence affirming a predicate of all S or denying a predicate of some S is universal in quantity; a sentence affirming a predicate of someS or denying a predicate of all Sis particular in quantity. We now have three kinds of opposition: C, P, and Q. The first has been symbolized using+ and - between the two terms of the sentence. We can also use + and - for P-opposition, displaying the sentence-wide scope of affirmation or denial by prefixing the sign to the entire sentence. This would give us the following schedule of the A, E, I, 0 categoricalsA E I

0

+(all +S + P) +(all +S - P) +(some +S + P) +(some +S- P)

-which is a QC schedule insofar as it makes no use of P-opposition, construing all four categoricals as affirmations. An alternative schedule would construe all four as having positive predicates: A E I

0

+(all +S + P) -(some +S + P) +(some +S + P) -(all +S + P)

This would, then, be a PQ schedule. A third kind of schedule would be PC, taking all quantity to be particular: A E I 0

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

Alternative QC, PQ, and PC schedules with all four categoricals taken as denials, or all predicates as negative, or all quantities as universal,

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respectively, are also possible. It would be of obvious benefit to be able to symbolize quantity in the way that quality has been symbolized. After all, quantity, like quality, is oppositional, which suggests a +/- notation. To determine which quantity is + and which - Sommers relies on the equivalences established by valid conversion. For example, I is convertible-that is, some S is P = some P is S +(some +S + P) = +(some +P + S) This equivalence will hold algebraically only if'some' is positive. E is also convertible-that is, noSisP = noPisS -(some +S + P) = -(some +P + S) +(all +S - P) = +(all +P - S) Such equivalences hold only when 'some' is positive and 'all' is negative. The general form of any sentence, then, will indicate all three kinds of opposition. +/-(+/-(+/-S)+/-(+1-P)) The fU'St plus-or-minus is the sign ofP-opposition, the second is the sign of quantity, the third and fifth mark term quality for'S' and 'P', and the fourth marks predicate quality. Since the predicate quality and predicate-term quality collapse, we can simplify this as +/-(+/-(+1-S)+/- P) Taking our subject-term, as usual, to be positive (and, as in arithmetic, suppressing positive quality signs), this reduces to +1-(+1-S+I- P)

with the three plus-or-minus signs representing P-, Q-, and C-opposition, respectively. Notice that what Sommers has done here is to show that the formatives of quantity and quality are uniformly alike in being oppositional in character. The so-called truth-functions can be treated as pairs of signs of quantity and quality. As a consequence, he is able to offer an answer to the question I raised in my introduction to this essay: What constitutes the distinction between formative and nonformative expressions? According to Sommers, ''the formatives-including proposition 'constants'-are analogous to plus and minus signs of arithmetic" (1973a: 249). Logical formatives are oppositional in nature. 10 Indeed, "the representation of all logical signs

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as signs of opposition is a consequence of the effort to realize Leibniz's program" (1976b: 41 ). An example of a symbolized QC schedule would be A E I

0

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

An example of a PC schedule would be A E I

0

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

The fact that any categorical can be symbolized using just two of the three kinds of opposition shows that each kind of opposition can be defined in terms of the other two. Since singular statements have no overt quantity, we can display them on a PC schedule, defining quantity as a consequence. Let 's' be a singular term. We have, then, the following PC schedule: A E I 0

-(s-P) -(s+P) +(s+P) +(s-P)

Using Aristotle's principle that affirming a predicate of a singular is equivalent to denying its contrary-that is, +(s+P) = -(s- P) we see that in the case of singular statements A=l (and E=O), which means, when the schedule is construed as QC, that singulars have wild quantity. Any denial can be converted into an affirmation by simply multiplying through (by -I). For example, 'No S is P' is symbolized initially as '-(+S+P)'. Multiplying through yields '-S-P' ('Every Sis nonP'). The equivalence is valid according to Sommers's law for immediate inference:

Iftwo statements L and M have the same logical quantity, then L entails Mifandon/yifL=M. (1970: 13)

Logicians in general tend to consider statements exclusively in their atfmnative forms. The following schedule (1970: 13) displays the standard

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classical immediate inferences algebraically. In doing so, it makes frequent use of the mathematicians• practice of dropping plus signs whenever no ambiguity results (signs of quantity can never be dropped). 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)

Subaltemation does not satisfy the law of immediate inference. For example, 'Every S is p• (-S+P) and 'SomeS is p• (+S+P) are neither algebraically equal nor similar in quantity. Nonetheless, a particular can be derived from the corresponding universal syllogistically once a tacit premise is supplied. In this case the missing premise is '+S+s•. Adding it to the premise '-S+P, yields a Darii syllogism. Sommers argues (1970: 20) that the option of such premises is benign and in no way forces unwanted existence claims. The classical term calculus is happily free of existential distinctions between universal and particular statements; nothing in the logic forces anyone to say that universal statements differ from particular statements in their existential import; we are perfectly free to consider all particular statements as having no existential import if we so wish. ( 1970: 20) 11

To account for syllogistic validity, Sommers formulates the following principle, which will also apply to immediate inferences (1970: 19).

(P.l)

An inference is valid if and only if (i) the sum of the premises equals the conclusion, and (ii) the number of particular conclusions equals the number of particular premises. 12

Consider Barbara: -M+P ~

-S+P This is valid, considering that the sum of the premises equals the conclusion (-M+P-S+M = -S+P) and there are no particular premises or conclusions. All of the classical valid syllogisms (including weakened moods) can be

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analysed as equating the sum of their premises with their conclusions and the number of particular premises with the number of particular conclusions. Thus far, Sommers has in hand a simple algorithm for determining the validity of inferences involving simple categoricals (including singulars). The challenge to the power of his calculus is to incorporate compound statements and relationals. To incorporate compound statements, he makes use of a notion first explored in "On Concepts of Truth in Natural Languages" (1969b). Since his calculus is a calculus of terms, it is imperative that entire statements be construed as terms, just as Leibniz had seen, in order to subsume the logic of compound statements under the logic of terms. The term form of a statement is achieved by "nominalizing" the sentence used to make it. Let 'p' be such a statement; then the term 'state of affairs in which p' (or, alternatively, 'case of p', 'p-state') will be its nominalization. The statement 'p and q' is compound, a conjunction of two sentences. The truth of this is preserved in the following nominalized form: 'Some case ofp is a case of q'. Nominalized statements are symbolized by placing them between square brackets. Our example becomes, then, '+(p]+[q]'. All the truth functions can be so construed, preserving Leibniz's insight and reversing the Fregeans' claim that the logic of compound sentences is more basic than the logic of terms. Here are some examples using nominalization. Not p = -[p] Ifp then q =All p-states are q-states = -[p)+[q] p or q =Not both not p and not q = -[+-[p)+-[q]] = --[p]--[q] If all men are fools, then some logician is unwise = -[-M+F]+[+L-W] Since terms, as well as sentences, can be compounded (by the use of operators such as 'and', 'or', 'only ir), the calculus can be extended as well to inferences involving compound terms (cf. Sommers,l970: 22-23). For example, 'Some men are fat and happy' can be seen as predicating the compound term 'fat and happy' of 'some men', and the compound term can be construed, on the model of the conjunction 'p and q', as '+F+H', using an alternative bracketing to indicate the compounding of terms rather than (nominalized) sentences. The entire statement is symbolized as '+M+(+F+H)'. Keep in mind that statements like this, and '-[p)+[+[q)+[r]]', are categorical. They consist of just two terms. It just happens that in such cases at least one of the terms is compoun~onstructed from two other terms. Relational terms are likewise construed as terms that have been constructed from other terms. For example, 'loves some girl' is taken as a (relational) term consisting of a relation ('loves') and an object ('some girls'). The objects of a relation are quantified; the relation itself is not. Though Sommers was far from clear about this in "The calculus of Terms"

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( 1970), relations are qualified. Consider the statements (a) (b) (c) (d)

Some boy loves every girl. Some boy fails to love some girl. Some boy is unloving (a nonlover) of some girl. Every boy sends some flowers to every girl.

These can be formulated as (a.l) (b. I) (c. I) (d.l)

+8+(+L-G) +8-(+L+G) = +8+(-L-G) +8+(-L+G) = +8-(+L-G) -8+(+S+F-G)

Notice that (b) is equivalent to 'Some boy is unloving (a nonlover) of every girl' and (c) is equivalent to 'Some boy fails to love every girl'. Contemporary logicians account for differences such as those between (b) and (c) in terms of differences of scope for quantifiers. Statements, compound terms, and relational terms are all syntactically complex. In using the calculus to analyse inferences involving such terms (in addition to elementary terms) Sommers specifies several principles that. in effect. allow for the iteration, association. composition, distribution, and so on. of the elements of such terms (1970: 23). For example. any complex term ofthe form '(+A+A)' is equivalent to '+A'; a particular subject can associate over a complex predicate (i.e., +A+(+8+C) =+(+A+8)+C, for example, 'Some man is fat and happy' = 'Some man who is fat is happy'); a universal subject can distribute over a complex predicate (i.e., -A+(+8+C) = +(- A+8)+(- A+C)). All butthe first are algebraically sound. In considering syllogistic inference in general, it is possible to formulate a principle that applies to all inferences, including those that involve complex (i.e., sentential, compound, or relational) terms. In effect, the principle is a description of the algebra governing (P.l) (1970: 26ffl. (P.2)

From a statement containing universal M (-M) and a second statement containing a positive M (+M) derive a statement exactly like the second except that +M is replaced by the entire first statement without - M. 13

A careful consideration of this principle shows that it amounts to saying that in any syllogistic inference middle terms are cancelled. And another way of looking at it is to see it as a version of the old dictum de omni et nullo: What is said ofall of something is said of what that something is said of

(or: What is true ofall M is true of any M). The 'something' here is the middle term, what is said of it is the first statement without - M, and what

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that something is said of is the second statement without +M. In "The Calculus of Terms," Sommers offers several examples of syllogistic inferences that illustrate the use of his system. A few simple examples will suffice for now. (i)

NoM isS Every Pis M So, noS is P

(Camenes)

This is first symbolized as (i.l)

-(+M+S)

.:£±M -(+S+P] Since we can convert any denial into an affirmation, this is simplified as (i.2)

-M-S

.:£±M -S-P Validity is guaranteed because the sum of the premises equals the conclusion and the number of particular premises equals the number of particular conclusions (zero). Or. by (P.2), '-S' in the first premise has replaced '+M' in the second to yield'- P-S', which is then simply converted to yield the conclusion. (ii)

Every circle is a figure. So, whoever draws a circle draws a figure.

(ii.l)

.:.QtE -(+D+C)+(+D+F)

Here, we require the analytic missing premise 'Whoever draws a circle draws a circle'. Thus: (ii.2)

-C+F - -(+D+C)+(+D+F)

Note that '+F' has replaced '+C' in the second premise to yield the conclusion according to the dictum de omni et nullo. (iii)

Tully is Cicero Tully is verbose

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So, Cicero is verbose Since 'Tully' and 'Cicero' are singular, they have wild quantity, allowing the assignment of quantity in each case as required for validity. (iii.l)

-T+C

±I±Y +C+V (iv)

Ifp then q lfq then r So, if p then r

(iv.l)

-[p]+(q] -[q)+[rl -[p]+(r]

(v)

Some horses are faster than some dogs. All dogs are faster than some men. So. some horses are faster than some men.

Adding the analytic missing premise that 'faster than' is transitive, we get (v.l)

+H+(+F+D) -D+(+F+M) -C+f±(+F+M)+(+F+Ml +H+(+F+M)

While application of the dictum to the first two premises yields '+H+(+F+(+F+M))', 'Some horses are faster than something faster than some men'. This, along with the missing premise, yields. again by the dictum, the conclusion. The algorithm presented in "The Calculus of Terms" (and in Sommers, 1973a, 1973b, 1975, 1976a, 1976b, 1976c, 1978a, 1982, 1983a) shows, at the very least, that it is possible to construct a term logic that has expressive powers to match those of the standard predicate calculus, and without requiring a special identity theory. Moreover, Sommers's algorithm is simple to use and rests on a theory of logical syntax that appears to be much closer to that of natural language. In building his own calculus Sommers rejects the following Fregean theses (see 1970: 38-39; 1976d: 589-90): (I)

Singular and general statements have essentially different logical forms. (2) The logic of statements is prior to the logic of terms.

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(3) Relational statements are not subject-predicate in form. (4) Compound statements are not subject-predicate in form. (5) Syllogistic is in no way a basic form of inference. One cannot but notice, when examining Sommers's account of term logic, that all statements, singular, general, relational, compound, and so on, contain some formative expressions, some signs of opposition. In effect, there are no atomic statements. Modem Fregean logicians accept as given a fundamental distinction between atomic statements (containing no logical signs) and molecular statements (containing one or more such signs). The distinction between the two logical views reveals a distinction between the Leibnizian and Fregean programmes in general. As we have seen, Leibniz took singulars to be on a logical par with general statements. Wild quantity and singular predication make this possible. In contrast, Frege took singular statements to be logically primitive, connecting a general term to a singular term in a primitive, undefined way, involving no logical formative. Such statements, then, are atomic. General statements are built up from these by means of logical formatives, among them the truth-functions, making the logic of (general) terms rest on the logic of statements. Sommers calls this "the key to Frege's revolution in logic" (1976d: 590; see also 1983a: 18182). For the traditionalist, subjects and predicates can be distinguished from each other syntactically. There is a prima facie syntactical difference between a quantified term and a qualified term. Quantifiers and qualifiers are distinct logical formatives. For the Fregean, on the other hand, the difference is that between singular and general terms, between saturated and unsaturated expressions, between complete and incomplete expressions. And this difference is nonlogical. It is a matter of semantics, not syntax. It is material-not formal} 4 The fact that all statements and all terms of a statement are charged, have positive or negative quality, and that quantity is likewise oppositional, guarantees that logically any statement can be viewed as a string of terms, each preceded by a (plus or minus) sign of logical opposition. In "Distribution Matters" (1975), Sommers shows that each such sign can be seen as a mark of the distribution value of the following term (once all external minus signs have been multiplied through). Thus, for logic in general, distribution does matter. Valid syllogistic inference requires that throughout a given inference extreme terms preserve their distribution values and middle-term pairs have opposite distribution values. In responding to Geach's attack on the doctrine of distribution, Sommers concludes that "there is a clear syntactical test for deciding the distribution value of a term in a proposition. and ... there is no serious question that distribution values play a role as necessary conditions for valid inferences" (1975: 39-40). 15 If it is granted that Sommers's calculus of terms is effective and that it matches the inference power of the standard logic now in place, one

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might still ask if it is natural. Does it come any closer in its syntax to the syntax of natural language? Perhaps one might even dare to ask (as we will in the chapter to follow): Does its account of inference reflect the ways in which we actually reason? Aristotle and all pre-Fregean logicians saw logic as aiming to lay bare the logic of natural language. While many traditional logicians took this to mean natural language in its broadest sense, some, including Aristotle, took it to mean natural language as used in theoretical science. In neither case did they take logic to be fit only for the description of an artificial language (viz., mathematics). Seventeenth-century logicians, such as the Port Royal logicians and Leibniz, often saw logic as concerned with the uncovering of a universal language, constituted by a grammar underlying and common to all natural languages. Any logician who claims that logic is concerned with natural language will tend to look for an algorithmic system that preserves salient logical features of natural syntax. In this sense, a logic is natural to the degree to which the theory of syntax on which it is grounded approximates the syntax of natural language. From this point of view, traditional term logic and modern mathematical logic differ markedly in their degrees of syntactic naturalness. Or, to put it another way, the two kinds of logic give radically different accounts of propositional unity. For the modem logician, the syntactic unity of a molecular statement is the result of operations on atomic statements, whose unity, in turn, is guaranteed by the predicational tie between a general term and an appropriate number of singular terms. In such basic cases, as we have seen, this tie is the result not of any binding agent (logical copula), but of the semantic nature of general and singular terms (the former being incomplete, the latter being fit to complete them). Indeed, in this Fregean view, atomic statements have no other logical syntax; they contain no formative expressions. In contrast, the traditional term logician holds that the unity of any statement ready for entry into logical inference is effected by the presence of formative expressions. In particular, pairs of terms are tied by a logical copula, resulting in a new complex term (which is either a sentence, a compound term, or a relational term). Sommers argues in "The Grammar ofThought" (1978) that each theory is consonant with a different theory of grammar. The modern theory of logical syntax "coincides with the constituent analysis into Noun Phrase and Verb Phrase which the modern linguist gives to sentences in natural language" ( 182). The traditional theory "coincides with the constituent analysis of Cartesian Linguistics" (184). By "Cartesian Linguistics." he means the account of natural-language grammar particularly associated with the Port Royalists. 16 Now, it can be argued that many of the ideas in the modern linguists' theory are inspired more by their reading of modern logic than by the nature of natural language. But whether or not modern logical theory coincides with modem linguistic theory, the question remains: How close is the syntax of modem logic to that of natural language? The fact is that it is not very close at all. And this should not be surprising; after all, Frege

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COMING TO TERMS WITH SOMMERS

explicitly abandoned natural language as a source of syntactic insights when he chose to set logic the task of establishing the logical foundations of mathematics. Jn doing so, he looked to mathematics (particularly analysis) for his logical insights. While the traditional term logician took as the paradigm sentence for logic one with a quantified general term and a qualified general term (e.g., 'Some men are fools'), the Fregean took as a model a sentence in which a general term (viz., a verb, as Russell noticed) is predicated (without the intervention of a formative) of a singular term (e.g., •Achilles runs'). Statements like the latter are hardly typical of ordinary language. Yet, for the mathematical logician, counselled by the intuitions of mathematics, a language adapted for the purposes of telling us what a thing does ('2 divides 14 evenly') is far more appropriate than one that facilitates saying how things are ('Men are mortal'). This difference is due in no small measure to the fact that mathematical discourse is carried out against a background consisting of an ontologically uniform universe of discourse-the set of mathematical objects (numbers, sets, etc.). Ordinary discourse, conducted in natural language, relies on a far less neat and homogeneous universe of discourse-the world or some part thereof. In rejecting the traditional subject-predicate analysis (i.e., the quaternary theory), Frege sought to abandon all earlier theories of logical syntax. The irony here is that in so doing he left intact Aristotle's ternary theory and, at the same time, took on Plato's binary theory. As we have seen, the binary theory offers an analysis of the simplest kinds of statements (atomic in modem terms) in terms of a pair of syntactically/semantically distinct expressions, joined without the aid of a connecting formative-a mark of predication. This is a subject-predicate analysis with a vengeance. Sommers offers a nice account of the contrast here in "Predication in the Logic of Terms" (1990), and in doing so he implicitly acknowledges his own move from (in 1970 and elsewhere) a commitment to a quaternary theory to a commitment to a ternary one. He argues for a theory of logical syntax that parses a statement as consisting of a pair of terms connected by a term functor: "The term/functor style of analysis may be said to go back to Aristotle, who formulated sentences in a way that placed the functor between the terms" (1990: 107). I conclude this section with an illustration of how each of the tbree theories of logical syntax would provide a distinct "constituent" analysis of sentences in general. 17 Binary theory (Plato, Frege):

Sentence

Subject (noun phrase)

Predicate (verb phrase)

name (complete expression)

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verb (incomplete expression)

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Quaternary theory (Scholastics, Sommers 1970s and 1980s):

•

~

A

quantifier

term

A

qualifier

term

Ternary theory (Aristotle, Sommers 1990s):

•

topcal copula

As we shall see in the following chapter, logical copulae can be "split," permitting a quaternary analysis to be derived from a ternary one.

The Logic of Natural Language It is not humanly possible to gather immediately from it what the logic of language is. Wittgenstein The ftxation on first-order logic as the proper vehicle for analysing the 'logical form ' or the 'meaning' of sentences in a natural language is mistaken. P. Suppes [The central idea in) my rejection offirst-order logic as the appropriate instrument for the analysis of natura/language ... is that the syntax of

first-order logic is too far removed from that ofany natura/language to use it in a sensitive analysis ofthe meaning ofordinary utterances. P. Suppes

122

COMING TO TERMS WITH SOMMERS For better or worse, most of the reasoning that is done in the world is done in natura/language. And correspondingly, most uses of natural language involve reasoning of some sort. Thus it should not be too surprising to find that the logical structure that is necessary for natural language to be used for a tool of reasoning should correspond in some deep way to the grammatical structure ofnatura/language. G. Lakoff

In 1982, Sommers published his long-awaited The Logic of Natural Language. It is a rich book, addressing a wide variety of logical topics. At its heart is a sustained attempt to offer, substantiate, and defend a logical programme contrary to contemporary Fregean orthodoxy. Both Strawson (1982) and Geach (1982) reviewed the book. Strawson called it a "rich, clever, and courageous book," and characterized Sommers as "an almost solitary champion, [who] raises the banner of traditional formal logic (TFL) and challenges the innumerable host of those who believe that quantificational or Fregean or modem predicate logic (MPL) has finally and rightly driven its ancient forerunner from the field" (786-87). In contrast, Geach hated the book, not least because he felt that Sommers had been either unclear or unfair to Geach's own published ideas. A bitter, but fascinating, exchange in the pages of the Times Literary Supplement ensued (Sommers, 1983b; Geach, 1983; Sommers, 1983c). In the remainderofthis section, I offer a brief overview of some of the main ideas found in Sommers's book. Sommers begins by laying out (in the introduction) the contrasts between TFL and MPL. TFL is "naturalistic" in that it seeks to preserve the syntax of natural language, finding itself only now and then forced mildly to regiment some sentences prior to analysing their roles in logical reckoning. MPL is "constructionistic," radically translating most statements into an artificial, nonnatural language in preparation for reckoning. Only TFL, therefore, can provide a model for how we actually talk and reason. Indeed, the contemporary logician no longer sees that if natural language is the typical medium for deductive reasoning then logic is, in an important sense, a part of cognitive psychology. Still, most logicians (and many philosophers, linguists, and psychologists) have abandoned TFL, recognizing in its traditional versions a system ill-equipped for a variety of logical tasks. "The negative verdict on the prospects for a viable traditional logic has had the effect of deflecting twentieth-century logic from its traditional task of characterizing the canonical fragment of natural language to the quite different task of constructing powerful but artificial logical languages" (1982: 11). One way of seeing the crucial difference between TFL and MPL is offered by the fact that only the latter recognizes a contrast between atomic and molecular sentences (chapter one). This contrast, as we have already seen, is the result of the Fregean Dogma, the idea that singular and

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general tenns are radically different for logic; sentences with general subject tenns can be analysed only in tenns of quantifiers and singular (pronominal) terms. General sentences are built up from more primitive singular sentences, logically primitive sentences, atomic sentences. Such a view assumes that singular tenns can never be either quantified or predicated. TFL abjures such assumptions and, consequently, has no need for an atomic/molecular distinction. While the binary theory of logical syntax on which MPL rests sees the contrast between subject and predicate as semantic, TFL' s quaternary theory of logical syntax accounts for the contrast in tenns of the fonnal distinction between quantifiers and qualifiers. TFL has looked to the fonnal features of certain expressions (quantifiers and qualifiers) to detennine the logical fonns of sentences in which they occur. MPL has, instead, sought to treat the logical fonn of any sentence as an expression of its truth conditions. Fregeans, without the traditionalists' recourse to the quantifier/ qualifier distinction, must seek other. nonsyntactic (i.e., semantic) grounds for the distinction between subjects and predicates. In "The Two Tenn Theory" (chapter two), Sommers critically examines attempts at this by Geach and by Dummett. An example of an argument used often by Geach holds that in a sentence, 'SP', 'S' is the subject and 'P' the predicate if and only if the negation of 'P' results in the contradictory of 'SP' but the negation of'S' does not. It follows, for Geach, that singular tenns, but not quantified general tenns, can be subjects. Thus, 'John' is the logical subject of'John is tall' (since its contradictory is 'John is not tall'), but 'Some man' is not the logical subject of'Some man is tall' (since its contradictory is not 'Some man is not tall'). Yet Geach's argument works only as long as singular subjects are not taken as quantified expressions. Sommers maintains the Leibniz-inspired wild quantity thesis, according to which singular subjects are indeed quantified. Moreover, in the traditional theory, the contradictory of a sentence is never achieved simply by negating the predicate-tenn. but results from denying the predicate of the subject. And this is equivalent (as tenn logicians from Aristotle to Sommers have seen) to simultaneously reversing the quantity of the subject and the quality of the predicate (i.e., negating the predicate). Since the implicit quantity on 'John' in 'John is tall' is wild, we can take it to be particular. In that case, the contradictories of 'John is tall' and of 'Some man is tall' are fonned in exactly the same way, leaving Geach with no reason to reject quantified general tenns like 'some man' as logical subjects. Dummett, taking seriously Frege's radical asymmetry between names and function expressions, holds all logical subjects to be proper names and offers three conditions for a tenn being a proper name. Yet Sommers shows that, given that singular subject-tenns (e.g., proper names) have implicit quality, Dummett's criteria hold equally well for both singular and general subject-tenns. The three criteria are, jointly, necessary conditions for an expression, E, being a proper name (=logical subject). 124

COMING TO TERMS WITH SOMMERS

They are (I) that from 'E is P' it follows 'Something is P' (i.e., existential generalization), (2) that from 'E is P' and 'E is Q' it follows 'Some P is Q', and (3) that from 'E is P or Q' it follows 'E is PorE is Q'. Consider (I): the Fregean must take existential generalization as an intuitively obvious rule of inference. But is it more obvious, more intuitive, than the singular/general distinction, which is partially explained in terms of it? Moreover, such an inference favours TFL since it is syllogistic, having the form (letting'*' mark wild quantity): *E+P *E+thin& +thing+ P Consider (2): this is just an expository syllogism: *E+P ~

+P+Q

Consider (3): the 'or' is distributed over particularly quantified subjects. Syllogistically, '*E + (--P--Q)' = '--[*E + P]--[*E + Q]'. Thus, while (I), (2), and (3) hold for singular terms, they also hold for subjects in general. Fregeans have worried so much about how to draw the subject/predicate distinction because they hold a theory of logical syntax that prohibits a simple syntactic account of the distinction. For the traditionalist, on the other hand, subjects are always quantified terms and predicates are always qualified terms. The quantifier/qualifier distinction is a clear, obvious, and purely syntactical one. Recourse to the singular/ general distinction is not required. 18 MPL is a pronoun-saturated language. Nonsingular statements are paraphrased canonically as sentences with quantifiers binding individual variable expressions. These expressions are the logical counterparts of natural-language pronouns, as Quine has so often reminded us. Consider a simple categorical, 'Every philosopher is wise'. In ordinary English, one fmds no pronoun here, and there is no suspicion that hidden (logical) pronouns lurk somewhere beneath the surface. Yet MPL formalizes such a sentence as 'Everything is such that if it is a philosopher then it is wise'. Reference is not only singular-it is pronominal. Now, the undisputed fact is that natural language does quite often make use of pronominal reference. But it does not require all reference to be pronominal. Sommers's calculus of terms (in the 1970s and early 1980s) remained to some degree unnatural 125

CHAPTER3

in that it failed to make any place for pronouns. But in chapters three, four, and five of The Logic ofNatural Language he set out to account for naturallanguage pronominal reference. For the modem logician, an expression such as 'someS' is not a referring expression at all. Recall Geach's argument that if it were a referring expression, one could legitimately ask which S is being referred to. According to these logicians, genuine referring expressions must be either definite descriptions or proper names (both of which can be cashed out in terms of pronouns). For term logicians, on the other hand, such an expression makes nonidentifying reference to an S. Sommers argues that "the fundamental form of reference is to be located in 'someS is P' and not in 'theSis P' or 'a is P' .... Expressions of the form 'someS' are the primary referring expressions of traditional formal logic" (59). According to the modem theory, the referent of a referring expression must be unique and must exist. Thus proper names and definite descriptions are paradigms of referring phrases. In the traditional theory, reference is the role played by quantified terms (logical subjects) and is to be distinguished from denotation, which is no kind of reference (being merely the extension of a term within the domain of discourse). Reference is determined by the quantity and the denotation of the subject-term. Thus 'someS is P' makes reference to an undetermined part of the denotation of'S'. While the modern theory holds that failure of reference on the part of a referring expression precludes meaning, or truth-value, for the atomic sentence, for the traditionalist, failure of reference by an indefinite referring expression (e.g., 'someS') merely results in falsity. Sommers contrasts this kind of reference with what he calls "epistemic reference" (57). Suppose I say 'A boy is in the garden' and there happens to be only a girl. I purport, but fail, to refer to a boy in the garden. However, I do succeed in making a weak reference to a girl in the garden. My reference to a girl in the garden is epistemic. I refer here to something that I take to be a boy. Epistemic reference, like purported reference and unlike actual reference, is incorrigible. I may fail to refer to a boy in such a case, but I surely succeed in referring to what I take to be a boy (even if it turns out to be a girl or even nothing at all}. Quine ( 1960a: 113} has argued that definite singular terms can have indefinite antecedents. Pronouns are definite singular terms with indefinite (quantifier) antecedents. Modem logicians are correct in recognizing Quine's ftrst point. However, the notion that pronominal reference (with a quantified antecedent) is primary reference is mistaken. Such logicians treat all pronouns as bound variables. Thus they require that the variable and the quantifier that binds it (its referential antecedent} occur within the same sentence. Yet pronouns in natural language do not always operate as bound variables, and, in fact, often do not occur in the same sentence as their antecedents.' 9 Indeed, Quine's Predicate Functor Algebra shows that pronouns, as bound variables, are completely dispensable. Pronouns are 126

COMING TO TE~ITH SOMMERS

used anaphorically or catephorically to make or repeat reference to what has been or will be referred to. When the antecedent of an anaphoric pronoun

is a universally quantified term, the subsequent pronoun merely repeats it. Thus, in 'Every man saluted the flag. They were all veterans', 'they' simply goes proxy for 'every man'. Referential pronouns have particularly quantified antecedents and make reference to what their antecedents referred to epistemically. Thus, in 'A boy is in the garden. He is trampling my peas', 'he' refers to what was taken to be a boy in the garden. If pronouns are referring expressions (logical subjects), then they must share the syntax of subjects. They must be analysable as quantified terms. Sommers's solution is to treat pronouns as implicitly quantified terms ("proterms"), whose denotations are the referents of their antecedents. Since they make definite reference (even when their antecedents do not), their implicit quantity is wild. The pronoun 'he' in the example of the above paragraph can be analysed, then, as syntactically complex, consisting of a wild quantity and a proterm that denotes what was referred to by 'a boy'. In his sixth chapter, Sommers reviews his account of how logic can dispense with identity as a relation. He shows how a nonrelational reading of identity statements is possible once singular subjects are permitted implicit wild quantity and singular terms are allowed to play the role of predicate-term. He also shows how the notion of wild quantity allows for the nonrelational, predicational construal of identity to be reflexive, symmetric, transitive, and consonant with the principles of indiscemibility of identicals and identity of indiscemibles. This last point raises an interesting question about the so-called Leibniz's Law. As I have said, there is no evidence that Leibniz himself ever formulated or even used the law as we now know it. 20 What he did formulate was a law of substitution, according to which any two terms that have the same denotation can be mutually substituted for one another salva veritate. Singular terms sharing a common denotation are just a special case of the general law. More importantly, the law is shown by Sommers to be nothing more than a Leibnizian version of the traditional dictum de omni et nullo (129-30). Modem logicians are remarkably tolerant of a wide range of difficulties that surround the relational version of identity. This fact probably indicates just how strongly the standard system is entrenched in the schools. An alternative, nonrelational version of identity, one that avoids such difficulties, is unimaginable for most modems. Sommers cites three examples, as follows. Frege took names as syntactically simple. Then, in order to account for how, for example, 'Tully is Cicero' is informative while 'Tully is Tully' is not, he had to introduce a degree of semantic complexity for names by giving them sense. Yet the wild quantity on singular subjects allows one to formulate these sentences as 'Every Tully is Cicero' and 'Every Tully is Tully'. The first has a nontautological form, 'Every A is B'; the second has a tautological form, 'Every A is A'. Tautologies are uninformative. Thus, the uninformativeness of 'Tully is Tully' is a matter

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of syntax rather than semantics. Geach' s relative identity thesis (cf. Geach, 1962) is fatally coupled with a claim that a sentence such as 'x is P' is analysable as 'x is the same Pas x'. But this would mean that, for example, 'Socrates is a man' and 'Socrates is Socrates' are logically equivalent, since they are both analysable as 'Socrates is the same man as Socrates'. Finally, there are worries about Kripke's (1972/80) thesis that a true identity is necessarily true. Sommers suggests that the notion of requiring de re here leads to the view that there are sentences of the form 'Every A is necessarily A' that are self-evident. a doubtful thesis at best. In chapter seven, Sommers argues for his method of incorporating relationals into term logic. Traditional logic's failure to do so was due not to its syntax, but to (i) the absence of an adequate system of formal notation, and (ii) the failure to make explicit the logical forms of pronouns. Since both (i) and (ii) can be rectified by Sommers, his claim is that the advantages of the new system of logic over the traditional one are merely practical, rather than theoretical (140). For example. the equivalence between a relational sentence and any of its converses must be taken as an analytic truth (semantic rather than syntactic) for the traditional theory. The modem theory must also take such equivalences to be nonformal for atomic sentences (e.g., 'Plato taught Aristotle' I' Aristotle was taught by Plato'), but for nonatomic sentences the equivalences are always formal and are seen in the order of the variables following a polyadic predicate. Yet even the practical advantage that the new theory has over the old in dealing with relationals is merely apparent. Sommers suggests that once the traditional theory is provided with pronouns it can simply be used to map the forms of modem predicate logic. The practical differences between the two systems would then disappear. Whether a term logic chooses to map modem forms or to introduce analytic converse theorems, the fact remains that term logic (contrary to the charge made by nearly all of its critics) can account for inferences involving relationals. Such inferences are syllogistic (often enthymemes with innocuous missing premises), and the general rule of syllogistic is always the dictum de omni et nullo, which simply allows a term affirmed of a universally quantified term to replace that term in any sentence in which that replaced term is predicated. Sommers concludes the chapter by showing that the usual charge that the traditional logic failed with relationals because of its basis on the subject-predicate form is groundless. Indeed, the logical forms of the modem predicate logic are themselves covertly subject-predicate. Sommers then offers an ingenious method for mapping modem forms onto categoricals. It is a method that ought to be welcomed, even by his critics, as offering a pedagogically effective way to translate natural-language sentences into the canonical forms of the standard calculus.21 Kant held out for an absolute difference between categoricals and

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compound sentences, while Leibniz argued that all assertions, including compounds, could be formulated as categoricals. Frege, on the other hand reduced categoricals to compounds. The fact that reduction is possible i~ either direction suggested to Sommers (chapter eight) a thesis once held by Peirce. 22 The thesis is that categoricals and compounds are "analytically autonomous and structurally isomorphic" ( 159). Thus, conjunctions share the syntactic structure of I or 0 categoricals; conditionals share the structure of A or E categoricals. This isomorphism opens up the possibility of a uniform system of symbolic representation, where formatives are read as either term functions or sentence functions. Such a system stands in marked contrast to the modem standard amalgam of a sentential calculus and a predicate calculus (plus identity theory). Sommers also advances in this chapter his claim that statements in I and 0 forms are "elementary" (160ft). Elementary statements are denotative of states of affairs and are not negative (i.e., denials). A and E forms can be defined from I and 0 forms. For example, 'Every Sis P' =df 'No S is nonP' (= 'Not: an S is nonP'. the denial of 'An S is nonP'). Furthermore, A and E forms are not state-denoting. Sommers argues for the last claim both in this chapter and in appendix 8, but his arguments, as he admits, are not conclusive. At any rate, the notion of elementary sentences permits an easy account of truth, since the truth of any elementary sentence depends upon the existence of the state which it denotes. Sommers uses '[p]' for 'the state denoted by p'. So we can say that 'p' is true if and only if[p] exists. We will see later that by the 1990s he had come to an improved and clearer account of the logic of propositions and of truth. In chapter nine. Sommers presents the formal algebra for his term logic. This is, in effect, the system derived from "The Calculus of Terms," supplemented with devices for incorporating pronouns. Pronoun antecedents are given indices. which in tum are used as the proterms of the subsequent pronouns. Consider, for example, 'A boy kissed every girl who loved him'. This has the form (where redundant positive signs of quality are suppressed and wild quantity is indicated by Sommers's '*'): +B; +(K -(G + (L *i)})

Since all formulae in this system are concatenations of positive and negative terms. rules of transformation and derivation are virtually rules of algebraic equivalence, addition, and subtraction. The key derivation rule, of course, is the dictum de omni et nullo. which is now formulated as

-X+Y ~

... Y... This is a general form for all syllogisms, and it reveals how conclusions

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result simply from the addition of the premises. The distributed X and the undistributed X cancel out and are replaced by Y. Several familiar inference patterns are seen to be merely instances of this general pattern (including modus ponens and Leibniz's Law [205]). Sommers's system here is simple and effective, and it leads him to make a far-reaching claim: "It is far more likely that the actual procedures we use in getting from the premises to the conclusion are closer to the model of cancellation than to the model of instantiation and generalization familiar to the practitioner ofMPL" (206). In chapter ten, "Truth and Logical Grammar," Sommers claims that while in modem predicate logic "we put our syntax where our semantics is" (207), the traditional formal logic "has no apparatus for regimenting sentences in a manner that makes truth conditions perspicuous" (208). For example, Russell's theory of descriptions would formulate sentences with defmite description subjects so as to exhibit, among other things, the truth condition that the referent of the subject exist. Strawson accepted Russell's view that if the existence of the referent was materially implied by the sentence, then the sentence must be logically existential. He denied the existential form of sentences with definite description subjects, and, consequently, rejected the view that the relation between the sentence and the atfarmation of existence to the referent was one of material implication. Sommers argues that Russell was correct in claiming that the relation here was one of material implication, but that Strawson was correct in denying that the implying sentence was logically existential. Sommers's solution is to take the inference from a sentence of the form 'The A is B' to 'The A exists' to be an enthymeme. The missing premise is 'Every B exists'. In general, any sentence with a singular or particular subject has existential import whenever it is appropriate to supply a missing premise affirming existence of whatever satisfies the predicate. Of course, it is not always appropriate simply to add such a premise, as when we wish to produce sentences about fictitious or other nonexistent entities. Sommers's claim is that the terms of a sentence are always used with a specific "amplitude" (212-13), where the amplitude ofa term in a sentence is determined by the domain of application of that term. Thus we might, say, use 'horse' with amplitude in the domain of Greek mythology or in the domain of the actual world. A validity condition on inferences will then be that all the terms of the inference share a common amplitude. Sommers uses the notion of amplitude to argue against Donnellan's ( 1977) claim that referential uses of definite descriptions have no existential import, to suggest that Kripke's (1972/80) theory of rigid designation is preferable to David Lewis's (1971, 1972, 1973) counterpart theory, and to eliminate standard puzzles about opaque contexts. In connection with this last, he argues that the terms of an embedded sentence need not share the amplitude of the terms of the embedding sentence. For example, consider (1) Tom believes that some survivors reached the island.

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(2) All survivors were logical positivists. (3) Therefore, Tom believes that some logical positivists reached the island. The reason this is invalid is not that it breaks anything like "Leibniz's Law," but that the middle term has different amplitudes in its two instances. In the next two chapters, eleven and twelve, Sommers returns to the consideration of pronominalization, arguing for the thesis that "proper names are pronouns" (230). They are "special duty" pronouns. Moreover, since pronouns are always rigid designators, names are, as Kripke ( 1972/80) argues rigid designators as well. The proterm theory accounts for the rigidity of pronouns by pointing to the fact that the proterm of a pronoun is specifically designed to denote 'the thing in question', a fact that explains why new tokens of the pronoun continue to designate that thing or those things in every context of use, including modal contexts. (Sommers. 1982: 229)

Nonetheless, Sommers's theory of names is not Kripke's. In place of Kripke's causal chain theory, Sommers offers a "pronominal chain theory." In this theory, the reference of a name is fixed by the epistemic reference of an antecedent indefinite description. Nominal reference may be preceded by pronominal and descriptive reference. For example, in 'A boy is in the garden. He is trampling my peas. He must be Little Sherman', the two pronominal tokens refer to what was taken to be a boy in the garden, and the proper name is a special-duty pronoun introduced to do the job of the pronoun and to pick up its reference. Since proper-name reference is fixed by an epistemic reference by means of an antecedent indefinite description, a proper name could never be the initial link in a referential chain. Sommers had earlier distinguished between pronouns whose references are fixed by definite descriptions used in nonepistemic contexts so that they are corrigible, and those whose references are fixed by antecedent ascriptions in epistemic contexts so that they are incorrigible. Thus, 'it' in 'Socrates owned a dog and it bit him' is a descriptive pronoun whose reference is fixed by 'the dog owned by Socrates'. In 'A dog is on the rock ... well, it may not be a dog', the 'it' is fixed by 'a thing taken to be a dog on the rock'. It is an ascriptive pronoun. In pronominal chains, each pronominal link makes the same reference as, but is more comprehensive than, its predecessor in that it accumulates preceding descriptions (for descriptive pronouns) or ascriptions (for ascriptive pronouns). One of the questions Sommers wants to answer here is: Do proper names accumulate ascriptions throughout chains? His answer is that they do, since they are "modal free" terms (261 ). A term is modal free whenever whatever it is true of in the actual world is such that it is true of

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CHAPTER3 that thing in any other world in which that thing is said to exist. Thus, modal-free terms, including proper names, accumulate background ascriptions. Consider 'Hesperus is Phosphorus'. Each of these names, being a special-duty pronoun, has an accumulated ascriptive background. For example, 'Hesperus' might be 'a thing taken to be a star and called Hesperus and said to be the brightest superlunary body in the evening sky'. When we learn that 'Hesperus is Phosphorus' is true we simply add to the ascriptive background of'Hesperus' whatever information might be found in the ascriptive background of' Phosphorus'. 'Hesperus is Phosphorus' is necessary because the denotation of names is conventionally fixed (for all possible worlds). But it is informative (unlike 'Hesperus is Hesperus') because it provides new information about the ascriptive backgrounds of the names. 23 Sommers distinguishes between ascriptive properties, which belong to a thing "in virtue of the way the thing is spoken or thought about,'' and descriptive properties, which belong to a thing "in virtue of what that thing is in itself apart from the manner in which anyone thinks or speaks of it" (268). For example, when I learn that Felix is a cat, I learn not how Felix is thought or spoken of, but what Felix is. When I learn that Hesperus is Phosphorus, I learn not what Hesperus is (e.g., self-identical), but how Hesperus is spoken of (viz., as Phosphorus). This ascriptiveness of identities is often mistakenly accounted for by saying that identities are about names rather than things. In Sommers's account, then, identities are not about names-they are about things-not in virtue of what those things are, but in virtue of how they are spoken of or thought about. Sommers's early work on language and ontology led him to the distinction between the contradictory of a given sentence and the logical contrary of that sentence. Recognition of this distinction, not found in MPL, then led to a distinction between category mistakes and vacuous sentences.24 In chapters thirteen and fourteen, he returns to this cluster of distinctions in order to review his theory of category structure and to show how it affects TFL. The modem logician takes, for instance, '4 is nonred' as 'It is not the case that 4 is red'. Accepting that '4 is red' is not true, the logician must then conclude, by the law of noncontradiction, that '4 is nonred' is either true (Quine) or without truth-value (Strawson, Ryle). In contrast, traditional logic, recognizing the difference between two kinds of negation (denial and term negation), can admit, without danger to the law of noncontradiction, both '4 is red' and '4 is nonred' as false. A crucial result of the failure to mark the logical distinction that Sommers has made is a complete confusion concerning the law of excluded middle. Sommers contrasts the standard law of excluded middle (called the sentential law of excluded middle-SLEM) with two other laws. SLEM demands that a sentence or its contradictory be true. The predicative law of excluded middle (PLEM) demands that a sentence or its logical contrary be true (e.g., 'xis P or xis nonP'). And the categorical law of excluded middle (CLEM) demands that a term be

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predicable of a subject (e.g., 'x is P or nonP'). The Fregean equates all three of these. Yet Aristotle has shown, in chapter nine of De lnterpretatione, that when the subject is undetennined with respect to the predicate-tenn, while SLEM and CLEM hold, PLEM does not. Now, PLEM entails CLEM, which in tum entails SLEM. SLEM always holds. But PLEM does not always hold; nor does CLEM. Statements for which PLEM fails are vacuous. A sentence will be vacuous whenever (i) it is a category mistake (e.g., '4 is red'), in which case CLEM will also fail, or (ii) its subject is empty (e.g., 'The present King of France is bald'), or (iii) its subject is undetennined with respect to the predicate-tenn (e.g., 'Hillary will be president in 2004'). Sommers has defined a primitive proposition as one containing only fonnatives that are intuitively plus or minus (286). All nonprimitives can be defmed in tenns of primitives. Thus, 'Every Sis P' is defmed as 'No Sis nonP' (='Not an Sis nonP'). And 'lfp then q' is defined as 'Not both p and not q'. The distinction between various laws excluding the middle (and thus between category mistakes, other vacuous sentences, and nonvacuous sentences) guarantees that on some occasions nonprimitive sentences will be undefined. We can summarize this with the aid of a square of opposition on which primitives are displayed. Nonprimitives can then be attached in a way which shows that they are defined. 25

NoS iaP

NoS ianonP

Every S ia P

Every S ia nonP

Some S is nonP

SomeS iaP

Sommers's claim is that a and e are defined (as A and E, respectively) only as long as the sentences are nonvacuous (PLEM holds). When PLEM fails, both I and 0 will be false, A and E (as contradictories ofO and I) will be true, and both a and e will be undefined. The traditional square makes use of what Sommers calls the diagonal law of excluded middle (DILEM), which requires that either I or e be true and either 0 or a be true. But DILEM will fail for vacuous sentences. Sommers's theory of opposition accounts in a unifonn way for the paradoxes of existential import for universals and the paradoxes of material implication. For the modem logician, when there are noS's, 'Every S is P' and 'Every S is nonP' are taken to be true. Thus the modem logician must

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CHAPTERJ say, paradoxically, that every unicorn is blue and that every unicorn is not blue. According to Sommers's account, when there are noS's, PLEM fails, so that 'Every S is P' and 'Every S is nonP' are undefined. Again, for modern logicians, when 'p' is false, 'lfp then q' and 'lfp then not q' are both true. Thus the modem logician must say, paradoxically, that if Earth is larger than Saturn it is larger than Mars and that if Earth is larger than Saturn it is not larger than Mars. According to Sommers's account, conditionals with false antecedents are undefined. This last point is easily seen once one recalls his argument that the logical forms of categoricals are shared by compounds-that categoricals and compounds are syntactically isomorphic. In his final chapter, Sommers returns briefly to argue that terms such as 'exist' may (contra Kant) be predicated. Such terms do not characterize subjects, but they do locate them in some domain of discourse, and that is why they have unrestricted amplitude. He also argues here against Geach's claim that quantified phrases are not referring phrases. Geach consistently equates referring with naming. Quantified phrases, his argument goes, never name. This argument rests on the senseless contention that since (i) either all quantified phrases name or no quantified phrases name, and (ii) phrases of the form 'noS' cannot possibly name, it follows that no quantified phrases refer. The point, of course, is that phrases of the form 'no S' are not quantified phrases. 'No' is no quantifier. It is not primitive, but defined (in terms of denial and particular quantity, viz., 'no' = df 'not some'). It is important to understand that, in spite of its size, The Logic of Natural Language was not intended to constitute either the final stage or the culmination of Sommers's work in logic, nor is it merely a summary of his preceding work. In developing his calculus of terms. he had progressively refined and clarified his notion that natural language, contrary to Frege's claim, has a logic. In his book, Sommers's perspective has shifted from an earlier one, which took the formation of a formal calculus as prior to the attempt to fit it for application to natural language, to one that looks directly to natural language as a guide to the construction of the calculus. As well, an important lacuna in that calculus is filled by Sommers's account in the book of the logic of pronouns. Still, work on the logic of terms was not complete in 1982. In particular, Sommers had accounted for the logic of unanalysed statements in terms of a semantic theory that calls for statements to express states of affairs in the world. A statement is true just in the case when the state that it expresses is a fact-that is, exists in the world. In referring to this view in his preface to the softcover edition of The Logic of Natural Language, Sommers wrote, "I should now wish to shift some of the semantic positions taken in the book. On the other hand, the basic syntactical thesis seems to be firmly in place" (vii}. Much of his attention since 1982 has been given over to the attempt to defend a correspondence theory of truth that is not burdened with the problems of ontology and 134

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semantics implicit in his earlier account of truth. The Truth

You will not find in semantics any remedy for decayed teeth or illusions ofgrandeur or class conflicts. Nor is semantics a device for establishing that everyone except the spealrer and his friends is speaking nonsense. Tarski To say of what is that it is is to speak the truth. Aristotle I never let the facts interfore with the truth. Farley Mowat There is only one world, the "real" world Russell

Sommers's farst attempt to defend the correspondence theory of truth was in "On Concepts of Truth in Natural Languages" (1969b). It was in this work

that he offered his nominalizing procedure, which forms a term ("sentential term") from a sentence. Recall that this had been one of Leibniz's goals. A (sentence used to make a) statement both is about certain things and specifies a state of affairs.26 Consider the statement made by 'Politicians are liars'. It is about politicians and liars. It specifies the state of affairs in which politicians are liars. Generally, if 'p' is a statement, then '[p]' is the state specified by 'p'. Since states are themselves things, one sentence can be used to say something about the state specified by another. Consider 'It is disturbing that politicians are liars'. This statement is about disturbing things and a state of affairs (the one in which politicians are liars). It does not, however, specify that state. It specifies the state in which it is disturbing that politicians are liars. Let us symbolize 'Politicians are liars' as 'p' and 'It is disturbing that politicians are liars' by 'q'. What 'p' specifies is [p]; what 'q' specifies is [q]; what 'q' says is that [p] is disturbing (i.e., [q] = [[p] is disturbing]). Here one statement is embedded in another (thus one state is embedded in another). Leibniz had sought a way of turning entire sentences into terms in order to paraphrase so-called hypotheticals (compound sentences) as categoricals. According to Sommers, such statements are about states of affairs (more simply, states). A statement of the form 'lfp then q' is about [p] and [q]. What it says is that all [p] are [q] (symbolically: '-(p)+[q]'). What it specifies is [-[p)+(q]]. What it is aboutare [p] and [q]. We have seen above that this process allows Sommers to incorporate the logic of statements into the logic of terms by giving a uniform formulation of compound statements and categoricals.

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Not all statements that embed other statements are compounds. Consider 'Nixon believes that all politicians are liars'. It is obvious that the object ofNixon's belief is not a sentence. Nor is it a state of affairs. What Nixon believes (according to the above sentence) is that a certain state is a fact. Here we are approaching the theory of truth. "The relations between sentences, states, statements, and facts are at the centre of any theory of truth" (Sommers, 1969b: 281 ). Different sentences can be used to make a given statement. Thus, 1can say 'Morgan is beautiful' and my wife can say 'Notre fils est beau', each of us producing different sentences but making the same statement. A statement specifies a state. But two different statements can express a given fact. Suppose it is a fact that on 30 March 1992 Morgan was born. I can express this fact by stating on 30 March that Morgan was born today. I can also express this fact by making the different statement on 31 March that Morgan was born yesterday. According to Sommers, "Only states of affairs will do for a healthy correspondence theory" ( 1969b: 279). Truth is defined in terms of correspondence as follows. i) A sentence is said to correspond to the state of affairs it specifies. If that state exists the sentence is said to correspond to reality. In that case it is true. ii) A statement corresponds to the state of affairs specified by any sentence that may be used to make that statement. If that state exists, the statement as said to correspond to reality. In that case it is a true statement. ( 1969b: 282) After 1982, Sommers's account of a "healthy correspondence theory" was altered in very important ways. This was due primarily to the fact that he had come to reject the notion that states of affairs and facts are things in the world. Strawson ( 1950a) had rejected any version of the correspondence theory because such a theory would require the presence in the world of such things as facts. states, situations, and so on. Yet even untutored intuition tells us that what makes 'Some logicians are fools' true is not the presence of some fact (e.g., that some logicians are fools), but the presence in the world offoolish logicians. One might find foolish logicians, red stars, happy hookers, or green apples in the world, but one searches in vain for facts as the objective correlates of true statements. Having failed to locate facts in the world, Strawson rejected any correspondence theory, since, according to him, any such theory would require facts in the world to make statements true. One ofSommers's aims in recent years (1987, 1990, 1993, 1994) has been to defend a version of the correspondence theory of truth that, nonetheless, takes seriously Strawson's observation that facts and the like cannot be located in the world. A second aim is to provide a plausible 136

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account of how the logic of statements is, contrary to Fregean orthodoxy, merely a special part of the logic of terms. As it turns out, a proper understanding of what is special about statements is exactly what is required to account for the inclusion of statement logic within term logic. We will briefly survey here Sommers's attempts to achieve each of these goals. Recall that in chapter eight of The Logic of Natural Language Sommers had characterized "elementary" statements as those having I or 0 forms, statements that say either that some such-and-such is so-and-so or that some such-and-such is not so-and-so. Such statements are easily reparsed as statements to the effect that something exists or does not exist. For example, 'Some logicians are fools' can be paraphrased as 'There are foolish logicians', a form favoured by modem predicate logic. Quine's dictum that to be is to be the value of a variable commits him to the view that, logically, we speak not of a thing's existence but of the existence of a thing which is so-and-so. Tarski ( 1935/36) had argued for the view that the truth of any statement is always relative to the domain of discourse under consideration. "We may put Tarski's insight in Quine's idiom: to be is to be denoted in the domain of interpretation" (Sommers, 1987: 300). To state that something is P is to claim that the domain relative to which the statement is being made can be characterized as having in it a P-thing. To deny that something is P-that is, to state that nothing is P-is to claim that the relevant domain cannot be characterized as having in it a P-thing. Such characterizations are "constitutive" (300). Consider the soup I made last night. It had lots of broth, carrots, peas, spices, garlic, chicken, and onions. It had no salt. I could characterize the soup as tasty and cheap. Such characteristics apply to the soup in virtue of what it is like as a whole. I could also characterize the soup as oniony but not salty. Such characteristics apply to the soup as a whole but in virtue of what does or does not constitute it. These are constitutive characteristics. A constitutive characteristic applies to a totality in virtue of what does or does not constitute it. Sommers holds that existence and nonexistence are not characteristics of things (say, the moon and Atlantis). "Existence and nonexistence are constitutive characteristics of domains" (300). Sommers's use of "domain" here is closely allied with his use of "amplitude" in The Logic ofNatural Language. There are any number of possible domains of discourse. Which domain any given statement is made relative to is determined by context and the speaker's intentions. In normal conversation the domain is simply the actual world (as when I say 'Some politicians are honest'). Sometimes the domain is just a part of the actual world (as when I say to my daughter 'Everything is a mess' with the understanding that the domain is not the entire world but just her room). Occasionally, the domain at hand is a special totality (as when I say 'Some prime is even' taking the domain to be understood as just the set of natural numbers). Now, true statements signify states of affairs (or facts). Since a state of affairs is the existence or nonexistence of something that is so-and-

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so, statements constitutively characterize the domain. Just as 'wise' nonconstitutively characterizes Socrates as being wise, 'Socrates is wise' constitutively characterizes the world as being wise-Socrates-ish. The statement is true just in the case when the state that it signifies obtains-that is, the world is so characterized. A state of affairs that obtains is a fact, so a statement is true just in the case when it signifies a fact. Facts are the objective correlates that make true statements true. Facts are what true statements correspond to. The term 'red' signifies a nonconstitutive characteristic of things. It denotes whatever has that characteristic (say, apples, Mars, and fire trucks). The statement 'Something is red' signifies a constitutive characteristic of. say, the world. If the world is reddish (i.e., contains a red thing), then the statement is true. Just as 'red' denotes whatever has the characteristic signified by 'red', 'Something is red' denotes whatever has the characteristic signified by 'Something is red'. What 'Something is red' denotes is the appropriate domain-the world, say. Now, some expressions denote many things; others denote just one thing; still others denote nothing. For example, 'red' denotes many things; 'Socrates' denotes just one thing; 'the present King of France' and 'Atlantis' denote nothing. What, if anything, an expression denotes is dependent on what has the characteristic that that expression signifies. If nothing is so characterized, the expression is denotationally vacuous. A statement is an expression that signifies a (constitutive) characteristic. If a domain has that characteristic it is denoted by that statement. If no domain has that characteristic the statement is denotationally vacuous. A true statement denotes its domain of discourse; a false statement denotes nothing. For example. 'French' can be used to characterize Quine (falsely), but Quine is not in the denotation of'French'. Likewise, 'Some logician is a professional basketball star' can be used to characterize the world (falsely), but the world is not denoted thereby. The important point to keep in mind is that. while facts are what correspond to true statements and make them true (here. Strawson was mistaken), facts are not themselves constituents of the domain; they are not in the world (Strawson was right about this). Facts are characteristics ofthe world. It is a point that is easy to miss. As Sommers confesses, The mistake of looking for a notion of fact as something that exists in the world was made by Russell. Wittgenstein, Austin (in the famous debate with Strawson) and by most correspondence theorists. It was made by me in [The Logic of Natural Language] but the august company of those who are drawn to the correspondence theory but who fall into the error of looking for facts and states in the world mitigates the recanting of it. ( 1987: 304)

I will have more to say about truth in my final chapter, but, before turning to Sommers's account of how to incorporate statement logic into term logic,

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we would do well to note some important formal distinctions involved in his truth theory. We have seen that any term is either positively or negatively charged. So is any statement. It is important to distinguish between the state signified by a statement that claims the existence of a nonP thing and the state signified by a statement that claims the nonexistence of a P thing. (Again, remembering that existence and nonexistence properly characterize (constitutively] domains rather than things, and that what we actually mean here is presence or absence rather than existence or nonexistence.) Consider, for example, the following. (I) Some snakes are pink.

(2) Some snakes are nonpink. (3) No snakes are pink. (I) characterizes the domain as pink-snake-ish; (2) characterizes it as

nonpink-snake-ish; (3) characterizes it as un(pink-snake-ish). The first two statements positively characterize the domain; the third negatively characterizes it. The first two claim that the domain is characterized by the presence in it of certain kinds of snakes. The third characterizes the domain as failing to have in it certain kinds of snakes; the domain is characterized by the absence of such snakes.27 Let us tum now to Sommers's account of how statement logic is a proper part of term logic (as found especially in Sommers, 1993). We will see that it is closely tied to his theory of truth. Modem Fregeans have been in no doubt about the fact that the propositional calculus, the logic of unanalysed statements, precedes the predicate calculus, the logic of terms. In the schools, propositional calculus is taught before predicate calculus. And, indeed, the latter cannot be understood without acquaintance with the former. From the point of view of modem mathematical logic, the logic of unanalysed statements is primary logic. Term logicians, such as Leibniz and Sommers, reverse this order, taking the logic of terms to be primary logic and viewing the logic of statements as secondary. Historically, the term logic of Aristotle came before the statement logic of the Stoics. As both Leibniz and Sommers have seen, the first step in incorporating statement logic into term logic is to construe entire statements as terms. Leibniz was less than perfectly clear about just how to do this. Sommers, as we have seen, accomplishes it by taking statements to be complex terms signifying states of affairs. Sommers has taken conjunctions and conditionals to share the logical form of I and A categoricals, respectively. For example, 'p and q' (where 'p' and 'q' represent unanalysed statements) is first paraphrased as 'Some state of affairs in which p is the case is a state of affairs in which q is the case'. This is then formulated as '+(p)+[q]' ('Some [p] is [q]'), an I form. 'lfp then q' is paraphrased as 'Every state of affairs in which pis the

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case is a state of affairs in which q is the case'-that is, '-[p)+[q]', an A form. Here the unanalysed statements have been nominalized to create "sentential terms." Now, sentential terms are terms, but they are special in important ways. One consequence of this is that in certain specifiable ways the logic of statements does not appear to conform to the more general logic of terms. There are theorems that appear to hold for one but not the other. For example, 'Ifp then q' is immediately derived from 'p and q'. In other words, from '+(p)+[q]' we should be able to derive '-[p]+[q]'. However, in general term logic, an I sentence does not immediately yield the corresponding A sentence. Moreover, while an I and its corresponding 0 sentence are logically compatible (e.g., 'Some men are married'/'Some men are unmarried'), their nominal versions are not (i.e., '+(p]+[q]' ('p and q') and '+(p]-[q]' ('p and not q') are incompatible). Sommers's solution to these kinds of disanalogies leads to his account of how term logic is primary and properly includes statement logic. Again, true statements, like any other term, signify properties, or characteristics. A term such as 'wise' signifies the property of wisdom; a term such as 'unmarried' signifies the property of being unmarried. True statements signify constitutive characteristics of the domains relative to which they are made. To make a statement is to use a sentence to make a truth claim. Suppose I state that some logicians are fools by using (appropriately) the sentence 'There are foolish logicians'. In so doing, I implicitly make a truth claim to the effect that the domain (presumably the actual world) has among its constituents foolish logicians. My statement signifies a (positive, in this case) constitutive characteristic of the world. ('There are no foolish logicians' would signify a negative characteristic.) The term 'wise' denotes whatever has the property of wisdom. The statement 'There are foolish logicians' denotes whatever has the constitutive characteristic that it signifies. Thus, if it is true it denotes the world. The statement that there are no foolish logicians is false; it denotes nothing. False statements are denotationally vacuous. Any term that denotes nothing is denotationally vacuous. Such terms express a concept, since they are meaningful. But, as there are no unhad properties, they do not signify any property, and are both denotationally and significantly vacuous. However, no meaningful term is conceptually vacuous. The term 'wise' is meaningful; when used, it expresses the concept of wisdom or the sense of 'wise', signifies the property of wisdom, and denotes all wise things. The term 'king of France in 1985' is meaningful; when used, it expresses the concept of being a king of France in 1985. But, as nothing has the property of being a king of France in 1985, it is denotationally and significantly vacuous. Statements follow the way of terms. 'There are no foolish logicians' is meaningful. It expresses the concept (thought, proposition) that no logicians are fools. But this concept does not characterize the world. It is denotationally and significantly vacuous. On the other hand, 'There are foolish logicians' not

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only expresses a proposition but signifies a property that characterizes the world. It signifies a fact. It is true. Recall that an elementary statement has the form 'Something is P' or 'Nothing is P'. Elementary statements are the components of compound statements. Elementary statements signify elementary states. So, compound statements signify compounds (conjunctions, disjunctions) of elementary states. Thus, a conjunctive statement, 'p and q', signifies the state of the world that is both (p] and [q]. Let 'x-world' be a term that denotes a world that is [x], a world constitutively characterized by the characteristic signified by the statement 'x'. For example, since the statement 'There are foolish logicians'(= 'Something is a foolish logician') is true, it denotes a foolishlogician-world. Since what a statement denotes, if anything, is whatever has the characteristic it signifies, we are free to say that what nominalized statements (the term forms of statements) denote are worlds. Let 'W' be read as 'world'. We can read any (unanalysed) statement, elementary or compound, as a categorical concerning worlds. For example (letting 'x' stand for 'W that is [x]'): 'p' is read as 'some W is (p]' 'not p' is read as 'some W is non[p]' 'p and q' is read as 'some W that is [p] is a W that is [q]' (briefly, 'some pis q') 'if p then q' is read as 'every W that is [p] is a W that is [q]' ('every pis q') We can take the statements of statement logic to be about the world. This contrasts with analysed statements of term logic-for example. 'There are foolish logicians' -that make truth claims about the world but are about things in the world (say. fools and logicians). This contrast between what a statement is about and what it claims is inert for unanalysed statements. Such statements, then, can be construed as being about what they denote-that is, the world. And, indeed, according to Sommers's account, such statements are uniquely denoting-they denote one thing, the world. Talk about some world that is [p] being a world that is [q], and the like, is merely a logical regimentation assigning an explicit quantity to what is actually a singular term (viz., 'world that is [p]'). It is the fact that (nominalized) statements are always singular. denoting just one thing, and that this one thing is always the same thing-the world-that Sommers uses to account for the disanalogies between statement and term logic that we saw above. 'Some A is B' (+A+B) does not generally entail 'Every A is B' (- A+B), but 'p and q' (+p+q) does entail 'lfpthen q' (-p+q), because the logical subject of'p and q' is singular and thus has wild quantity (i.e., its particular quantification entails its universal quantification). Likewise, while 'Some A is B' (+A+B) and 'Some A is nonB' ('+A-B') are logically compatible, 'p and q' and 'p and not q' are

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not, since, there being but one world, it cannot be characterized as both [q] and non[q]. The doctrine that all true statements denote one and the same domain (though signifying different facts) is the key to understanding why all of the "general categorical" statements of a terminized propositional logic are semantically singular. That all nonvacuous propositional terms denote the world fully accounts for the unique features of propositional logic as a special branch of term logic. (Sommers. 1993: 181)

Implicit in Sommers's theory of truth and the primacy of term logic is an uncompromising commitment to actual ism: the only world is the actual world; there are no possible, fictitious, conceptual, or other worlds beyond the actual world. This is an ontology that takes what there is. the actual world, to consist of individual things and the properties that characterize them. It is a theory that contrasts sharply with the type most popular today among analytic philosophers, possibilism, which opts for a plurality of possible worlds, one of which is actual. Often, such theories result from the mistaken assumption that since some given property does not characterize any actual individual it must characterize some nonactual thing. Here one sees the result of ignoring the distinction made by Sommers between what a term expresses and what it signifies. Conflating these two results is taking the concept expressed by any meaningful term to be the property it signifies. From this it follows that, since there are no unhad signified properties, such terms are never denotationally vacuous. Given that they denote no actual things, nonactuals, possible things, are posited to serve as their denotata. Sommers's distinction between the concept expressed by any meaningful term and a property that may or may not be signified by that term allows a simple way to avoid the ontological complexities of a nonactualist ontology. The Laws of Thoug/11 Logic itself is a science that describes reasoning and does not merely provide formal materials for it. Sommers Syllogistic and propositional logic [appear to) express, at some level, a common structure ofreasoning. Colwyn Williamson "Contrariwise. " contin11ed Tweedledee, "if it was so, it might be: and if it were so, it would be: but as it isn 't, it ain 't. That's logic. " Lewis Carroll And don 'I mess with Mr. In-Between Harold Arlen

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COMING TO TE~ITH SOMMERS

There is little doubt that logic is a scientia sermocinalis, as the Scholastics said, a systematic study of what we say. In this sense, logic is concerned with language, terms, statements, arguments. But the Scholastics also called logic a scientia rationalis, a systematic study of how we think when we think rationally. In this sense logic is concerned with rational thought. Aristotle seems to have believed that thought comes before language, so perhaps logic's primary concern is rational thought, while language, merely reflecting thought, is simply what logic deals with immediately. In this sense, logic's concern with language is justified by the fact that language reflects thought, logic's ultimate concern. Most Scholastic logicians appeared to believe that logic was concerned with thought and language at the same time; no priority was given. This seems to have been Leibniz's view as well, though he did make remarks that suggest the more Aristotelian view. Whether thought and language were seen as co-ordinate or given an order of priority, the fact is that traditional logicians quite generally believed that there was some essential connection between the two and that, consequently, logic had a legitimate concern with our rational thinking. Recall that Boote called his great work of logic The Laws of Thought. Part of Frege' s revolution in logic was the rejection of the idea that logic is in any way concerned with actual thinking: "Perhaps the expression 'laws of thought' is interpreted by analogy with 'laws of nature' and the generalization of thinking as a mental occurrence is meant by it. A law of thought in this sense would be a psychological law" (Frege, 1979: 17). The idea was that thinking is merely a succession of images (dim copies of sensations). To reason is to proceed through such a succession. Logic (laws of reason, laws of thought) can be nothing more than an investigation of the laws describing this sort of succession of images-that is, a matter of observation. Thus logic is simply a part of psychology. Such a view, which Frege associated most immediately with Mill, was known as psychologism, and Frege was intent upon excluding it from logic. "The laws in accordance with which we actually draw inferences are not to be identified with laws of valid inference; otherwise we could never draw a wrong inference" (Frege, 1979: 4). "The so-called deepening of logic by psychology is nothing but a falsification of logic by psychology. In the form in which thinking naturally develops the logical and psychological are bound up together. The task in hand is precisely that of isolating what is logical" (5). Thus, for Frege, the logician must "conduct an unceasing struggle against psychology" (6). For him, and then for his legions of followers, logic, the foundation of mathematics, can never involve the contingent. Logical truth is necessary and universal. In contrast, thinking, even when rational, is subjective, contingent. It takes place in the minds of individuals. Thus it is dependent on the nature of minds-contingent affairs. "If ... by the laws of thought we understand psychological laws, then we cannot rule out in advance the possibility that they should contain mention of something that varies with

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time and place and, accordingly, that the process of thinking is different nowadays from what it was 3000 years ago" (5). While logic, then, could not be concerned with thoughts. it might be concerned with Thoughts, the objective, independent senses of statements. While our thoughts are the proper objects of study for psychology, Thoughts are not (253): "Thoughts are by no means unreal but their reality is of quite a different kind from that of things. And their effect is brought about by an act of the thinker without which they would be ineffective ... and yet the thinker does not create them but takes them as they are" (Frege, 1967: 38). Frege's Thoughts, in contrast with thoughts, are not psychological. There is no room for psychology, the systematic study of thinking, in logic. Psychologism remains anathema to most logicians today. Sommers, true to form. has gone against today's current in logic to defend a version ofpsychologism, one weaker than the Millian version. If logic is taken to be psychologically real (as Sommers claims), then one must first determine whether this is because it can be used to describe how persons actually perform when they think rationally or because it can be used to describe how persons would ideally think rationally-that is, possess the competence to think rationally. The former, descriptive, claim certainly seems stronger than the latter, prescriptive, claim. Consider the weaker claim. One way to relate logic to our competence to reason is to argue that such reasoning must be reflected in our natural language. In particular, the claim is that our competence to reason is nothing more than our competence to use language-our grammatical competence. When linguists in the seventeenth and eighteenth centuries (including Leibniz) and in the twentieth century (especially Chomsky) sought a universal grammar, what they were looking for was a set of universal, necessary constraints on any natural language. The idea here is that if all ideal-language users are constrained by a common set of grammatical rules, rules independent of whatever natural language is considered, then such rules must be necessary features of any natural language and somehow be intimately related to the very nature of the one species that uses such languages. If the universal grammar is constituted by the universal and necessary grammatical (syntactic) rules common to natural languages, then there can be no gainsaying the key position held by logical rules in such a grammar, for logical rules are themselves universal, necessary constraints on language. Logic must be a part of universal grammar (cf. Sommers, 1973b: 170). Most modem logicians, and the linguists who follow them, have adopted the Fregean fear of tainting logic with psychology. Frege's own reluctance to allow logic to be guided by a psychology that in his day was poorly developed is understandable. But, in the case oftoday's logicians and linguists, it precludes even the possibility of considering psychologically real versions oflogic. Even the contemporary logician tempted to construct a "cognitive logic," a logic that could account for rational competence, must hesitate at the prospect of abandoning the standard predicate calculus in the 144

COMING TO TERMS WITH SOMMERS

absence of a viable alternative. Nonetheless, the plus-minus calculus of tenns is a viable alternative. And nothing in it bars one from taking it to be a cognitive logic. The logic takes its cue, in part, from suggestions by philosophers such as Hobbes and Leibniz (Dascal, 1976) that both rational thought and language could be viewed as calculating-reckoning. "For words are wise men's counters. They do but reckon by them" (Hobbes, 1904: 25). This reckoning, Sommers has argued, takes the simple fonn of adding or subtracting tenns (i.e., cancelling the middle tenns of syllogisms). Confidence that such a logic reflects our competence to use natural language rests on the fact that the syntax of the logic is closer to the syntax of natural language than is the syntax of any other logic. Indeed, this logical syntax has been drawn from the syntax of natural language. So, for Sommers, "the suggestion that we reason by cancellation of elements that have opposed signs is a plausible candidate for a theoretical description of the deductive process" (1976d: 6 I4). A logic of natural language, rather than a logic of an artificial, constructed language, "will illuminate the actual process of reasoning" ( 1978a: 42). Indeed, "any logician who is interested in a cognitive logic must adhere closely to natural syntax" ( 1983d: 40). Even if the case has been made for considering logic to be descriptive of our competence to reason, the question remains whether logic is descriptive of our actual reasoning perfonnance. Sommers's remarks seem to suggest that he is actually defending this stronger thesis. But the simple fact is that studies by cognitive psychologists tend to continn the suspicion that our unschooled rational perfonnances rarely match the level of our ideal competence.28 At any rate, the intimate relationship (whatever its exact nature turns out to be) between language and thought cannot be denied. To keep one's language in order must surely be an aid to one's ability to think clearly. A threat to language, as Orwell saw, is a threat to thought. Viewed from the narrower perspective of the twentieth century, Sommers's contributions to logic are clearly outside the mainstream. But this appearance of unorthodoxy obscures the fact that, from a broader perspective, his work represents the latest stage of a long development that began with Aristotle. This long tradition sees the logic oftenns as primary logic. It also takes the syntax of canonical statements to be independent of any semantic singular/general distinction among tenns, allowing any kind of tenn to play any categorematic role in a sentence. But old truths bear repetition, and Sommers has recovered from a past buried beneath a recent Fregean superstructure a number of tine old truths. Yet his mission has been more than just one of rediscovery. The old logic, in spite of its simplicity and naturalness, was simply no match for the Fregean alternative. The latter may .lack the old logic's degree of simplicity and may contort ordinary statements into quite unnatural fonns, but it enjoys an expressive power and breadth that most traditionalists hardly envisaged and others (Leibniz in 145

CH~3

particular) could only dream of. Even though he has carried out his work in an intellectual context saturated by Fregean assumptions, Sommers has achieved what term logicians before Frege could not. He has built a uniform, simple, natural, powerful algorithm for term logic. It is a universal characteristic. And while Aristotle began it, the Scholastic logicians refined it, Leibniz envisaged its simplification and generalization, and the nineteenth-century algebraists had a relatively clear vision of its algebraic nature, only Sommers has done it.

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Notes for Chapter 3 1 For

a detailed discussion and summary see Englebretsen (1985c; 1990a, ch. 1) the literature surrounding the Tree Theory is fairly extensive. See the references cited in Englebretsen (1990a). 2 E.g., Sommers (1965), Englebretsen (1972b, 1975). 3 E.g., Sommers (1966), Englebretsen (1971a, 1973). 4 E.g., Sommers (1970-71), Englebretsen (1971b). 5 E.g., Sommers (1963b), Englebretsen (1972a). 6 For more on Sommers's fraction algorithm see Friedman (1978a). 7 For example, Sommers argues here that any two terms predicatively tied must be U-related and that the universe of discourse for any statement (or inference) is confined to the intersection of the categories determined by the terms involved. 1 For more on wild quantity see Englebretsen (1986a, 1986b, 1987a, 1988b). On wild quantity and Leibniz's Law see Englebretsen (1984c). For more on identity, see Englebretsen (1981d). On syllogistic with singulars see Englebretsen (1980b). 9 For more on logical polarity see Englebretsen ( 1987a, ch. 14). 10 Sosa (1973) tries to argue that Sommers's account offormatives would lead to the claim that, for example, 'hot' and 'cold', being opposed, are formatives. But this is simply to miss the important distinction between logical and nonlogical contrariety. 'Hot' and 'cold' are nonlogical contraries. They are not "opposed." 'Hot' and 'nonhot' are logical contraries, and their opposition can be accounted for in terms of term quality. Sosa also argues that Sommers's calculus is ineffective since "there are valid inferences that [it] would pronounce invalid" (256). Yet his counter-example (254) mistakes a syllogistic (mediate) inference for an immediate one. 11 For a discussion of Sommers's use of premises ofthe form '+S+S' see Englebretsen (1979, 1981e). 12 This is my much simplified version of Sommers's principle. In subsequent writings, he offered successively simpler and more concise formulations of the conditions for syllogistic validity. 13 Again, this is a much more concise, simplified version of the one Sommers (1970) formulated. 14 For more on this see Englebretsen (1986c). 15 On the importance of viewing syllogistic as a logic of distribution values see the excellent study by Williamson (1971); see also Englebretsen (1979, 1985d). 16 See Englebretsen's "Cartesian Syntax" (1990c). 17 See Englebretsen ( 1989).

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See Englebretsen (1986a). For more on this point see Vendler (1967, ch. 2), Paduceva (1970), and Chastain (1975). A theory similar to Sommers's is found in Heim (1982, ch. 3). 20 See Englebretsen (1984b, 1985e). 21 See Sommers (1982, app. A) and Englebretsen (1982d). 22 A fine account of this is offered in Dipert ( 1981 ). 23 For a similar account of proper names see Lockwood (1971). 24 See Englebretsen ( 1972c) for a clarification of Sommers's account of this distinction. 25 A much more extensive examination of this device is found in Englebretsen (1984b). 26 Other logicians, including especially Boole and Frege, had held similar views. Frege took every statement to refer to either the True (what makes the statement true) or the False (what makes the statement false). 27 In ch.l4 ofEnglebretsen (1987), I try to show that while, with respect to any term 'P', a given thing may be either P or nonP, with respect to any constitutive characteristic [p], every domain is either [p] or un[p]. In other words, the polarity of nonsentential terms is reversible but the polarity for sentential terms is not. This fact is the true basis for the contrary/contradictory distinction. 28 For recent work on this question see, for example, Braine ( 1978), Evans (1982), Henle (1962), Johnson-Laird (1983), Johnson-Laird and Byrne (1991), Osherson (1975), Wetherick (1989), and Rips (1994). 18

19

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CHAPTER FOUR

IT ALL ADDS UP We cannot go back to the prison that would confine all logic to the Aristotelian syllogism, but it is possible to defond (a) something like the view that the form "Every X is Y" is more fundamental than either "For all x, f(x)" or "Ifp then q" and (b) the traditional ignoring (in inforence by suba/ternation, etc.) ofterms that have no application. A.N. Prior

Plus/Minus Certainement ca/culer c 'est raisonner, et raisonner c 'est ca/culer .... Lorsque je dis que les quantites sont ajoutees ou soustraites, et que consequemment je /es distingue en quantites en plus et en quantites en moins, je ne /es confonds pas avec /'operation qui les ajoute ou qui les soustrait; et on voit comment, etant /es memes en algebre que dans toutes /es /angues, il n y a de diffirences que dans Ia maniere de s 'exprimer: mais quand on nomme quantiti positive /'addition d 'une quantile, et quantile negative Ia soustraction d 'une quantile, on confond /'expression des quanti/is avec /'expression de /'operation qui les ajoute ou qui les soustrait, et un pareillangage n 'est pas fait pour repandre Ia lumiere. Aussi les quantites negatives ont-elles ete un ecueil pour tous ceux qui ont entrepris de les expliquer. Condillac The concepts ofaddition and subtraction. The rudiments oflogic. Don De Lillo

In chapter three I offered a brief summary of the many contributions to term logic made over the past several years by Sommers. Such a summary cannot in any measure serve as a substitute for Sommers's own work, but I hope that it will kindle a degree of interest in it. As well, it is meant to show to some extent just how Sommers's logical ideas are actually the latest stage of a very long historical development that did not, contrary to the view of many contemporary logicians, end with Frege or retreat to a few "Colleges

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CHAPTER4 of Unreason." My aim in the present chapter is to give a simple, consolidated picture of the logical algorithm for term logic (the plus/minus system envisaged by Hobbes, Leibniz. and De Morgan and built by Sommers). Along the way I will offer a few modest amendments and additions. According to the Scholastic logicians, as we have seen, the proper concern of logic is at once both speech and thought (scientia sermocinalis and scientia rationalis). As a science of thought, it aims at an account of what were commonly known as the three "acts of the intellect": (I) understanding/comprehension, (2) composition and division, (3) reasoning/ syllogistic. The first deals with the meaning of terms; the second with the formation of sentences from terms; the third with the formation of arguments (syllogisms) from sentences. The nature and order of the three parts was claimed to have been inspired by Aristotle's Categories, De lnterpretatione, and Prior Analytics. In modem terms, it would be fair to describe the content of the three studies as (I) semantics, (2) syntax, (3) deduction. Serious debates have erupted from time to time among post-Fregean logicians, but traditional logicians were in general agreement about this ordering. One cannot understand syllogisms without first accounting for the sentences that constitute their matter. And these, in tum, require a prior account of the terms that constitute sentential matter. Post-Fregeans have been particularly worried about the relative order of semantics and syntax. Some give priority to the former; others to the latter; and still others take semantics and syntax to be interdependent. As we saw in chapter two above, whatever view is taken. there is a sense in which the standard system of mathematical logic now must give pride of place to at least a certain measure of semantics. In particular, the syntactic theory of the standard logic rests on the prior division of the elements of the lexicon into absolutely distinct categories: general terms (predicates) and singular terms (names, pronouns, individual variables). For all the talk of those who would give priority to syntax, the fact remains that such lexical classification must rest on semantic distinctions. Ironically perhaps, in spite of the traditional priority of semantics, pre-Fregean logicians avoided the temptation to allow logical syntax to be determined by semantics. The first act of the intellect was directed toward understanding the meanings of terms. In studying this act, the logician is concerned with accounting for the various semantic roles that any term can play in a sentence. The theory of supposition and comprehension (in all of its many guises) was meant to provide a complete account of the extensional and intensional meanings of any (used) term. There was little interest at this stage in classifying kinds of terms. All terms-singular, general, relational, sentential, compound, and so on-were given the same semantic treatment and were passed on to the second, syntactic, stage undivided. This traditional way, approaching syntax with an undivided lexicon, was followed by Sommers. I shall follow it as well.

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IT ALL ADDS UP

There is a systematic ambiguity of plus and minus expressions in mathematical language. This ambiguity is not only benign, it is a source of great expressive power for the mathematician. Leibniz, De Morgan, and Sommers have suggested that natural language has a logic that, like arithmetic and algebra, makes use of two kinds of basic formal expressions, the signs of opposition. In fact, their common position seems to be that all the expressions of natural language that carry the responsibility for determining logical form are either positive signs or negative signs or signs defmable in terms of these. If this is so, it means, among other things, that one could build an artificial formal language that would model natural language by using the mathematicians' opposition signs for all formatives. The result would be an algorithm for natural-language reckoning that would model natural statements as arithmetical, indeed algebraic, formulae and inference as algebraic calculation. There is little doubt that this was Leibniz's goal throughout his logical studies, and Sommers has effectively reached that goal in his own logical work. It would seem, therefore, that the idea of using signs of opposition to model natural-language formatives is a good one, leading, as it seems to have done, to rich programmes of logical investigation and to viable systems for logical reckoning. One of the consequences of this idea has been, as we saw, great optimism among those who have shared it that a clear and precise account of the nature of logical formatives, and their distinction from nonlogical expressions, can be provided. In a sense, their account is quite simple: logical formatives, unlike other expressions, are oppositional in just the way that plus and minus are oppositional in mathematics. But to appreciate fully this kind of account, we need to look more closely at the oppositional character of formatives, their roles in inferences, and the kind of algorithm that could model those inferences. For if natural language has a logic (something assumed by all traditional logicians but denied by many modem logicians), then it ought to be possible to devise a formal language that models all kinds of statement-making sentences, as well as inference patterns among them. In other words, it ought to be possible for the logician to construct a formal system that closely matches the expressive and inferential powers of a language such as English. The Simple System We begin with a simple, abstract formal system consisting of the following: upper-case letters, a plus sign, and a minus sign (as well as any parentheses, brackets, etc., that we need for punctuation). The letters are the system's variables, its lexicon or vocabulary; the plus and minus signs are its formatives. The plus is a binary formative; the minus is a unary formative. The formation rules are: (i) Every letter is a term.

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(ii) If X is a tenn, so is -X. If X andY are tenns, so is X+Y (called a phrase).

(iii)

Our binary coMective, +, is symmetric; thus the tenns of a phrase like A+B can be commuted to give us B+A (for convenience, we will usually allow fonnulae to be their own quotations). As well,+ is associative. Thus a tenn like A+(B+C) is equivalent to (A+B)+C. So far the system is extremely simple-and weak. We can apply commutative and associative rules to phrases to yield new (and equivalent) tenns. But we could not, for example, derive a new phrase from a pair of phrases, neither of which is equivalent to the new phrase. What is needed, of course, is a binary connective that is transitive. Suppose we had such a coMective, perhaps •. Then we could fonnulate a deduction rule such that, for instance, A•c followed from A*B and B*C. As yet, we have no transitive binary connective, but we can define one in tenns of our unary minus and binary plus. Notice that if X+Y is a tenn, so is -(X+Y). A system consisting ofjust a unary minus and a binary plus is already familiar to us. We can all do arithmetic usingjust negative and nonnegative numbers and addition. Subtraction is a binary operation defined in tenns of negativity and addition. Thus: 3-2 = 3+( -2), where the minus sign in 3-2 is a binary operator defined by the binary plus (addition) and the unary minus (negativity) of3+(-2). In like manner, we will define a binary minus in tenns of our unary minus and binary plus as follows: D.l

X-Y =df -(-X+Y)

(Compare: 5-4=-(-5+4).) This new binary connective is reflexive and transitive, and, unlike our binary plus, is nonsymmetric. We now have a binary plus, a binary minus, and a unary minus. As well, we have implicitly defined a unary plus. Thus: 0.2

+X = df -(-X)

And then, as in arithmetic and algebra, we suppress unary plus signs when convenient, taking all unmarked tenns as implicitly positive. The addition of these defmed fonnatives simply amounts to a conservative extension of the original system. The system of four fonnatives (two binary, two unary) is still relatively simple, but it has far greater expressive powers than does our original system of a binary plus and a unary minus. And the introduction of our transitive binary connective yields increased inference power. Let us say that the expressive power of any fonnallogic is a function of the extent to which it can fonnulate natural-language expressions. The greater the number ofkinds of natural-language expressions that can be fonnulated, the greater the expressive power of the system. The inference power of a logic

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is a function of the extent to which it can model inferences made in the mode of a natural language. Ideally, the logician wants not only a system exhibiting power (expressive and inferential), but, as well, a system that is simple (relative to alternative logics of comparable power). An additional criterion of adequacy might be naturalness. One logic is more natural than another in the sense that the first formulates natural-language expressions in ways that are closer (syntactically) to the original than does the second. The criterion of naturalness has not been accepted by all logicians (recall Frege's views about this), and even among those who do accept it, naturalness has often been honoured more in word than in deed. We will apply the criterion.

Formulating English The simple system we have in hand thus far can be used to formulate many natural-language (e.g., English) expressions. Let our variables stand for natural-language terms. In natural languages such as English, terms come in charged (positive/negative) pairs (e.g., 'massive' /'massless', 'wed' I 'unwed', 'confidence'/'nonconfidence', 'painful'/'painless'). These charges are clearly reflected by our unary formatives. And, just as a sign of positive charge on terms is rarely explicit in English, our positive unary sign is generally left tacit. Our binary plus has the formal features of nonreflexivity, symmetry, and nontransitivity. In English there are several formative expressions with just these formal features. The most obvious of these is 'and', as in 'wealthy and happy', where 'and' connects a pair of terms to form a compound term, and in 'It is raining and it is cold', where 'and' connects a pair of sentences to form a compound sentence. Let us place a phrase in angular brackets when it formulates a conjunctive term, and in square brackets when it formulates a conjunctive sentence. We might formulate our two samples here as (W+H) and [r+c], respectively (adopting in the latter case the additional convention of symbolizing logically unanalysed sentences by lower-case versions of our variables). In the Ana/ytics, Aristotle tended to paraphrase categorical sentences by using a single formative expression between pairs of terms. For example, he would write (the Greek version of) •A belongs to some B' and •A belongs to every B', rather than 'Some B is A' and 'Every B is A'. The first of these, 'belongs to some', is a binary connective that forms a sentence ftom a pair of terms. It is the Scholastics' 1-connective-as when they formulated the conclusion of Darii as 'SiP'. Such a connective is nonreflexive, symmetric. and nontransitive. So, like 'and', it can be formulated using our binary plus (for we can think of our binary plus as having been created with just those formal features and no others-recall De Morgan's notion of the purely formal copula,=). We could then formulate an I categorical, paraphrased first as •p belongs to some S', as P+S. All of the other three categorical forms can be expressed in our 153

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fonnallanguage. An 0 categorical makes use of our binary plus and unary minus. 'NonP belongs to some S' would be fonnulated as - P+S. A and E categoricals are the contradictory negations of 0 and I, respectively. Thus we could fonnulate 'P belongs to noS' as -(P+S). And, given D. I, this can be expressed in tenns of our defined binary minus as -P-S, where the second minus is now our defined binary connective and can be read as 'belongs to every'. Similarly, we could fonnulate 'P belongs to every S' as -(- P+S), which, by D.l and 0.2, is equivalent toP-S. The new connective, read here as 'belongs to every', is reflexive, nonsymmetric, and transitive. The English connective 'if has the fonnal features of reflexivity, nonsymmetry, and transitivity. This suggests that we can fonnulate it using our defined binary minus, and a bit of calculation shows that we can indeed do so. A sentence of the fonn 'p if q' is the contradictory of '(both) not p and q', which is symbolized as [-p+q]; its negation would be -[-p+q]. By applying D.l and 0.2, we get [p-q] for 'p if q'. In effect, we have defined 'if in tenns of 'not' and 'and'. It might be objected at this stage that we are treating 'and' and 'belongs to some' as equivalent expressions (likewise for 'if and 'belongs to every'). But from a purely fonnal point of view they are equivalent; both 'and' and 'belongs to some' are nonreflexive, symmetric, and nontransitive (and both 'if and 'belongs to every' are reflexive, nonsymmetric, and transitive). They share the same fonnal features. (This fact is the one De Morgan sought to express when he tried to reduce, for example, 'only if to 'is'.) Moreover, the following kinds of equivalences ought to convince one of the fonnal parallels we have drawn. Some A is B and C = B and C belongs to some A to some C and A = B+(C+A) = (B+C)+A

= B belongs

Here the associativity of our binary connective reflects the fonnal equivalence in English between 'and' and 'belongs to some'. Similar equivalences show that 'if and 'belongs to every' are fonnally equal. Every A is B if C = B if C belongs to every A = B belongs to every C and A = B-(C+A) = (B-C}-A Relationals Thus far, we have seen that a system of two unary and two binary fonnatives can be used to express a wide variety of kinds of natural-language expressions: conjunctive and conditional tenns, conjunctive and conditional sentences, and categoricals. But it can be used to express still more. Consider a sentence like 'Some boy loves every girl'. Its Aristotelian paraphrase is 'Loves every girl belongs to some boy', which we might begin to fonnulate as (L every G)+B. We are now tempted to

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symbolize 'every' by our binary minus since we have already used it for 'belongs to every'. Yet, 'every' and 'belongs to every' appear to be different expressions; are they formally equivalent-do they share the same formal features of reflexivity, nonsymmetry, and transitivity? The answer is yes. One way to see why this is so is found in Leibniz's idea that relational terms are "Janus-faced," facing in two directions at once. Consider 'Paris loves Helen'. Leibniz would analyse this initially as 'Paris loves and, eo ipso, Helen is loved'. The relational term 'loves' applies to 'Paris' as subject and to 'Helen' as object. In 'Some boy loves every girl', the expression 'loves every girl' has 'girl' as the object of the relation-that is, 'loves' (with its passive face, 'is loved') is said to belong to every girl (by virtue, eo ipso, of 'loves every girl' being said to belong to some boy). Leibniz was correct that relational terms are Janus-faced, but he was wrong to conclude that relational sentences must therefore be conjunctions. Relative terms, such as 'loves', 'killed', and 'gave ... to', can be predicated of more than one term at a time. We will indicate this by subscribing to each subject or object of a relation a unique numeral, which, in tum, will be subscribed to the relational term, along with other such numerals, in the appropriate order, to indicate how the relational is to be read. Thus, in formulating 'Some boy loves every girl' we symbolize 'boy' as '8 1', 'girl' as 'G2 ',and 'loves' as 'L 12 ': (L 12 -G2)+B 1• Our letter variables can be used to symbolize any kind of simple (noncompound, nonsentential, nonrelational) term. So far, we have used them to formulate general count nouns (e.g., 'logician') and adjectives (e.g., 'happy'). But they can be used as well to symbolize general mass nouns (e.g., 'wine', as in 'Some wine is sour'-i.e., 'Sour belongs to some wine', or, S+W). And, most importantly, they can be used to symbolize singular terms, especially proper names. Thus, to assert 'Socrates is wise' is to say of Socrates that wise (=wisdom) belongs to him, in other words, 'Wise belongs to Socrates': W+S. Notice that we have used our binary plus here. This is justified if a sentence like 'Socrates is wise' can be commuted (since our binary plus is symmetric). And, indeed, 'Socrates is wise' is logically equivalent to 'Some (one who is) wise is Socrates'-that is, 'Socrates belongs to some (one who is) wise', or S+W. But suppose I add 'Some Greek is Socrates'-that is, S+G. Now, from 'Socrates is wise' and 'Some Greek is Socrates' one can intuitively derive 'Some Greek is wise', W+G. Yet this is contrary to our observation that the binary plus is nontransitive. The solution here is to see that a sentence like 'Socrates is wise' ('Wise belongs to some Socrates') is logically equivalent to 'Wise belongs to every Socrates'. In other words, when Sis singular we can say W+S = W-S. To see how this is so, we will make an important change to our formal system. Splitting Connectives We began with a formal system consisting ofjust a unary minus and a binary

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plus. Recognition of the fonnal features of these operators revealed a surprising expressive power, which was then greatly increased by adding to the system defined unary plus and binary minus operators. However, in spite of this greatly increased power of expression, we have yet to achieve very much naturalness. For example, Aristotelian paraphrases of categorical sentences are less than perfectly natural. In English, for example, the expressions fonnulated so far as binary fonnatives in fact often (indeed, in some cases, usually) come not as single expressions (e.g., 'belongs to some') but as a pair of expressions-split (e.g., 'some ... is ... '). What is clearly wanted, then, is a way to split our binary connectives, so that they more closely match their natural-language counterparts, while simultaneously keeping their fonnal features (symmetry, transitivity, etc.). Let us begin by looking at some fonnatives in English for combining pairs of sentences to fonn compound sentences. For example, we say 'Sue will go to the party only if Ed is not there', or 'Ed will not be there if Sue goes to the party', where the binary connective, or fonnative, is a single expression occurring between the two constituent subsentences. But we just as readily express the same proposition by splitting the connective: 'If Sue goes to the party then Ed will not be there', with the single connective, ' ... if .. .' (or ' ... only if .. .'), now split into 'if ... then ... '. So, in English, at least, we fonn conditionals with split as well as unsplit binary fonnatives. The same is true for compounds such as conjunctions and disjunctions. Thus, ' ... and ... ' can be split into 'both ... and ... ', and ' ... or .. .' can be split into 'either ... or .. .'. When it comes to compound sentences and phrases, both split and unsplit fonnatives are common. What now of fonnatives used to bind tenns into categoricals? We have seen that the Aristotelian fonnatives 'belongs to some' and 'belongs to every' are not always natural. What is natural is split versions of these. We do not ordinarily say 'Happy belongs to some bachelor', but rather 'Some bachelors are happy', where the single binary fonnative is now split into 'some ... are ... '. In the same way, we nonnally split ' ... belongs to every ... ' into 'every ... is ... '. When it comes to relational expressions, however, we have seen that the unsplit version is the nonn. Thus we say 'Some boy is kissing every girl', with 'every' as an unsplit binary fonnative connecting the tenns 'kissing' and 'girl' to fonn the relational expression 'kissing every girl', but the split binary connective 'some ... is' connecting the tenns 'boy' and the relational phrase to fonn the categorical sentence. Traditional logicians were of two minds (perhaps appropriately) when it came to the categorical fonnatives. On the one hand, they developed an algorithm that took such connectives as unsplit. Thus they used the a, e, I, and o signs as symbols for the unsplit tenn connectives 'belongs to every', 'belongs to no', 'belongs to some', and 'does not belong to some'-that is, 'PaS', 'PeS', 'PiS', and 'PoS', for the four standard categorical fonns. On the other hand, they recognized the natural split

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versions of such fonnatives, and, indeed, elaborated various semantic theories based on the analyses of categoricals fonned with split connectives. Each half of a split Aristotelian fonnative can be treated as if it were an independent logical expression. Yet it must always be kept in mind (though this has not always been so in the tradition) that the two parts of any split connective are just that-parts of a whole. They are not two formatives; they are two parts of a single formative. The two parts of these formatives are called, respectively, the quantifier and the qualifier. In what follows, I shall restrict the use of the term logical copula (or simply, copula) to any unsplit Aristotelian formative. The part of a sentence consisting of a quantifier and a term is the subject; the part consisting of a qualifier and the other term is the predicate. A subject, then, is a quantified term. Thus. from a formal point of view, object expressions in relationals are logical subjects, since they are quantified terms. Predicates are qualified tenns. The fact that traditionally qualifiers were often called copulae indicates the longstanding ambivalence over split and unsplit fonnatives for categoricals. Quantifiers are of two kinds: particular (expressed in English usually by expressions such as 'some', 'one', 'a(n)', 'at least one') and universal (e.g., 'all', 'every', 'each'). It should be kept in mind that in ordinary uses of English particular quantifiers can often play the logical role of universal quantifiers, though this can be detennined by context. Moreover, in the ordinary use of a natural language, contextual clues often allow the explicit use of any quantifier to be suppressed. Qualifiers are of two kinds as well: affinnative and negative. English examples of affinnative qualifiers are such words as 'is', 'are', 'was', 'were'; negative qualifiers include 'is not', 'isn't', 'ain't', 'was not', 'wasn't'. Given any pair of tenns, it is easy to see that using pairs of a quantifier and a qualifier one could fonn four possible kinds of sentences (viz., the four classical categorical fonns): universal affinnations, universal negations, particular affinnations, and particular negations. Each fonn is the result, we recall, of splitting the single fonnative, the copula, which connected the tenns to fonn a categorical. Thus •p belongs to every S' is now fonned as 'Every Sis P'; •p belongs to someS' is now 'SomeS is P'; 'P belongs to noS' is now 'NoS is P'; 'NonP belongs to someS' is now 'SomeS is nonP'. We have symbolized our unsplit fonnatives (copulae) by + and - signs. How shall we symbolize our fonnatives now that we have split them into quantifiers and qualifiers? We are going to continue to use plus and minus signs as our only fonnative symbols. This means that our original unsplit copulae signs must now be rendered as pairs of signs (quantifiers and qualifiers). In the case of qualifiers, the choice of signs is natural. Affinnative quality can be symbolized by +; negative quality can be symbolized by -. Our choice of signs for the two quantifiers is now detennined algebraically by the requirement to preserve commutativity for I categoricals and reflexivity and

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CHAPTER4 transitivity for A categoricals. Consider 'Some logicians are fools\ which had been symbolized (with an unsplit formative) as F+L. Splitting the connective, and using + for the affirmative qualifier, we have the following first approximation: some L +F. Now, given that such a sentence can be commuted (i.e., its two terms can exchange places to yield a formally equivalent sentence), we have F+L = L+F. So, our particular quantifier must be symbolized in a way that will guarantee some L + F = some F + L. And clearly, only a+ will do here: +L+F = +F+L ('Some logicians are fools'= 'Some fools are logicians'). Symbolizing the particular quantifier by + suggests that the universal quantifier be symbolized by -. And, indeed, the proper algebraic equivalence is preserved for transitivity by doing so. Consider 'Every logician is a philosopher, and every philosopher is wise, so every logician is wise'. The transitivity that 'every ... is' inherits from 'belongs to every', and that guarantees this logical truth, can be preserved only by symbolizing the universal quantifier as a minus. Thus, 'Every logician is wise' is formulated as -L+W. It is easy to see that 'every ... is' is nonsymmetric. There is a second way to show the proper symbolism for universal quantifiers. Recall that our original system consisted of a binary plus and a unary minus. We then defined binary minus in terms of our binary plus and unary minus, and we defined a unary plus in terms of our unary minus. Our binary plus and unary minus were elementary formatives. Let us take the split version of our binary plus, along with our unary minus, as elementary as well, defining other split formatives in terms of them. Thus an I categorical and an 0 categorical can be formulated using only elementary formatives: Some S is P (+S+P), Some S is nonP (+S+(- P)). A and E forms are the contradictory negations of 0 and I forms, respectively. They can be formed by applying the unary minus to the entire sentence. So we can negate our I sentence, 'Some S is P'. to give us 'Not: some S is P' (- (+S+P]), an E form. Likewise, we can negate our 0 sentence, 'Some S is nonP', to give us 'Not: someS is nonP' (-(+S+(-P)]), an A form. These defined forms are not natural, however. What is required is a method that will allow us to distribute these external minus signs into parenthetical expressions. And this is just what we have in arithmetic and algebra (e.g., -(2+3)=-2-3). Consequently. we will adopt 0.3

-X-Y =df -(+X+Y)

Our E form can now be simplified, using 0.3, to give us -S-P. Applying 0.3 to our A form yields, first, -S-(- P), which, after then applying 0.2, gives us -S+P. In effect, then, we have symbolized the universal quantifier by a minus, defined in terms of our elementary formatives. Our new formulations for the categoricals closely match their natural-language counterparts. Each consists of a sign for quantity, a subject term, a sign for quality, and a predicate term. 158

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A Every Sis P -S+P

E NoS is P -S-P

SomeS is P +S+P

0 Some S is nonP +S+ -P

It is important to keep in mind that a pair consisting of a quantifier and a qualifier is simply a split version of a logical copula, an unsplit formative. Just as the formatives connecting pairs of terms for forming sentences have been split, we can split those formatives when they are used to form compound terms and sentences. Thus, just as we split 'belongs to some' to give us 'some ... is', we can split 'and' to give us 'both ... and'. And just as we defined 'every ... is' in terms of'some ... is' and our unary minus, we can define 'if. .. then' in terms of'both ... and' and minus. So, in summary, we have the following formulations (with which we will take the liberty of using the traditional labels for forms having the same formal features). A

Every Sis P -S+P lfpthenq -p+q

E NoS isP -S-P If p then not q -p-q

SomeS is P +S+P Both pand q +p+q

0 Some S is nonP +S+ -p Both p and not q +p+ -q

The parallelism between the categorical and compound forms here is striking, and hints at the possibility of a single algorithm for analysing inferences involving either kind of statement. For convenience, I shall continue to refer to the first part of any split connective as a quantifier and the second part as a qualifier. Thus, for example, we will talk of 'if as a quantifier and 'then' as a qualifier. So far, we have seen that our formatives can be used to express not only statements formed from simple terms (e.g., the categoricals), but also conjunctive and conditional compound terms and conjunctive and conditional compound statements. But if we are to build a system of logic that is as natural as possible, we cannot ignore disjunctive compounds. Consider a simple disjunctive statement of the form 'Either p or q'. This is the negation of 'Both not p and not q'. So we could initially formulate it as - (+(- p]+(- q]]. This formula, after we have algebraically distributed the external minus sign, yields -[-p]-[-q], or, more briefly, --p--q. (Keep in mind that the first and third minuses here constitute a split connective; the second and fourth are unary.) We will use the defined'-- ... -- .. .' notation as a formulation of'either ... or'. For example, 'Some senator is a liberal or a democrat' could be formulated as +S+(-- L-- D). It is important to note that in introducing our splitting procedure we are not reducing 'every ... is' to 'if ... then' (nor 'some ... are' to 'both ... and') or vice versa. We are simply symbolizing each of these by an

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expression,- ... +(and+ ... +), that has just those formal features shared by each formative. Compare this to the practice of others who tried to reduce compounds to categoricals. For example, De Morgan wrote, "In the forms of propositions, the copula is made as abstract as the terms: or is considered as obeying only those conditions which are necessary to inference" ( 1926: ix). Logicians such as Leibniz and De Morgan tried to place all of the burden of form on the "copula" (seen as equality,=) rather than on the logical copulae (e.g., Aristotle's or my split formatives). Wild Quantity I am now in a position to explain my earlier claim that 'Wise belongs to some Socrates' and 'Wise belongs to every Socrates' are equivalent. These sentences are in their unsplit, Aristotelian formats. Let us split the connectives here to give us 'Some Socrates is wise' and 'Every Socrates is wise'. My claim, then, is that these are, in effect, logically equivalent (the Leibniz-Sommers wild quantity thesis). When the subject-term of a sentence is singular, there is no logical difference between taking the quantifier to be particular and taking it to be universal. (Singular statements are simply particulars that semantically, nonformally, entail their corresponding universals. One might say that their "default" quantity is particular.) In natural language, this logical indifference to quantity for singulars is reflected in the fact that no quantifier at all is attached to singular subjects. Nonetheless, sentences with singular subjects enter into all kinds of logical relations with other sentences, and, as we will see, from a logical point of view they can be thought of as having whichever quantifier we want them to have. Singular sentences are indifferent to quantity because they happen to share the formal features of both particulars and universals. For example, 'Socrates is wise' is like 'Some philosopher is wise' in that it is commutable. Just as 'Some philosopher is wise' is logically equivalent to 'Some wise (person) is (a) philosopher', 'Socrates is wise' is logically equivalent to 'Some wise (person) is Socrates'. This suggests that singular sentences are implicitly particular in quantity; thus '(Some) Socrates is wise' is, formally, +S+W. But, unlike particulars. singulars are both reflexive and transitive (like universals). Thus, just as 'Every human is human' is tautologous, so is 'Socrates is Socrates' tautologous. Moreover, just as 'Every logician is wise' follows from 'Every logician is a philosopher' and 'Every philosopher is wise', so does 'Socrates is wise' follow from 'Socrates is a philosopher' and 'Every philosopher is wise'. This suggest that singular sentences are implicitly universal in quantity. Thus: '(Every) Socrates is wise', formally: -S+W. We have seen that singular terms can be subject-terms, which requires that they be logically quantified. We have seen as well that such quantity is not overt in natural language, but that logic requires that such

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subjects have (as any logical subject must) some (at least tacit) quantity. And we have seen that there are reasons for taking the quantity of singular subjects to be indifferently either particular or universal. There is one further consideration that should strengthen our resolve to so treat singulars (and that will, in the long run, actually contribute to the simplicity of our system). We have, so far, made no semantic distinctions among the terms fit for formulation by our symbolic algorithm. Any term, singular or general, mass or count, abstract or concrete, or whatever, can be symbolized by one of our letter variables for terms. And any term can be either quantified or qualified, so that any term can be a logical subject and can also be a logical predicate. In other words, our variables are semantically opaque. One might object at this stage that we have, after all, introduced the semantic singular/general distinction into our discussion. Yet it should be noted that we have recognized this distinction not by a distinction of variables but by a formal distinction, for, while the singular/general distinction is indeed semantic, its only effect on formal inference (i.e., its only logical effect) is syntactic (i.e., due to the indifference to quantity of singular terms when in subject positions). But this still leaves us with the fact that any kind of term can also be qualified, and thus become a logical predicate. When singular terms are in predicate positions, the subjects of those sentences are usually singular as well. Sentences whose terms are both singular are no different, formally, from other sentences. They consist of a pair of terms connected by a formative. When the formative is split, one part of it is the quantifier and the other is the qualifier. Now, modem mathematical logicians have assumed what was not generally assumed before the late nineteenth century: that all predicate terms must be general (the other side of the Fregean Dogma). This assumption presents logicians with a problem when it comes to accounting for sentences in a natural language. Consider, for example, 'Shakespeare is Bacon'. Ignoring for now the problem of whether the subject is quantified, what is the logical form of the predicate? If predicate terms cannot be singular, then 'is Bacon' cannot be construed as a qualified term. Since 'Bacon' is undeniably a term (even though it is singular), the only option appears to be to deny that the 'is' in such a case is a qualifier. And this is exactly what the modem logician does. He or she claims that 'is' (and such words) is systematically ambiguous. Sometimes it is a genuine qualifier and other times it is itself a general term. It is a qualifier whenever it accompanies a general term, but when it accompanies a singular term, since that term cannot be the predicate term (and there must be a predicate term), 'is' itself must do duty as the predicate term. Thus 'is' in such cases must, contrary to all appearances, be a general term. In these cases, the 'is' is taken to be the " 'is' of identity" and generally read as a contraction of the expression 'is identical to', which is, in tum, without question a general term. 'Shakespeare is Bacon' is taken to be 'Shakespeare is identical to Bacon', and the general term here is

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appropriately symbolized by '=', which, as in mathematics, indicates the equivalence relation par excellence. Needless to say, this logic, like traditional logics, makes no assumption about the semantics of predicate terms. Singular terms, like general terms, can be qualified and thus predicated. 'Shakespeare is Bacon' is construed, then, as a pair of terms connected by a formative. The formative in such a case has been split and the quantifier part suppressed. But such an analysis now presents a problem for us. So-called identity sentences are transitive, reflexive, and symmetric, all of which are guaranteed in a formal system that analyses them as pairs of singular terms connected by a general term that is itself an identity (equivalence) relation. How can our system preserve these features for such sentences? Since they have singular subjects, we are free to assign to them whichever quantifier we choose in any given context. By taking our tacit quantity to be particular, we guarantee that sentences like this are commutable (for alii categoricals are commutable-'belongs to some'. 'some ... is' are symmetric). By taking our tacit quantity to be universal. we guarantee (by the reflexivity and transitivity of' belongs to every', •every ... is') the reflexivity and transitivity of sentences whose terms are both singular. In summary: Symmetry: Reflexivity: Transitivity:

(Some) Shakespeare is Bacon. So (some) Bacon is Shakespeare. (Every) Bacon is Bacon. (Every) Shakespeare is Bacon. (Every) Bacon is Johnson. So (every) Shakespeare is Johnson.

We have symbolized the particular quantity by a plus and the universal quantity by a minus. Whenever the logical quantity of a formula is indifferently either particular or universal we will follow Sommers and indicate its quantity by* (e.g., 'Shakespeare is Bacon' becomes *S+B). Before leaving the topic of symbolization, we can take an additional step toward the naturalization of our symbolic language. Sentences like *S-P +S-P -S-P (with split connectives) can be read as having the predicate 'isn't P'. But this is a contraction of 'is not P'. Is this 'not' binary or unary? Were we using a language like Latin, our question would be: Is the predicate to be read as 'non est P' or as 'est non P'? In other words, is the predicate to be parsed as - (+P) or as +(- P)? As it turns out, the two are logically equivalent. The rule of obversion was the traditional logician's recognition of this equivalence. English, unlike Latin, allows us to suppress even the

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appearance of a difference by contracting the 'is' and the 'not' to give 'isn't'. Consider: 'No human is immortal' is equivalent to 'Not a human is immortal', which equals 'Every human isn't (fails to be) immortal', which, in tum, is equivalent to 'Every human is mortal'. Symbolically: -H-(-M) = -[+H+(-M)] = -H-(-M) = -H+M Our system is further simplified and naturalized by two additional notational conventions. Consider the relational sentence 'A man gave a rose to a woman'. Here, the subject is 'a man' and the (complex) relational predicate is 'gave a rose to a woman'. The phrase 'gave a rose to a woman' consists of a logical predicate, the (complex) relational term 'gave a rose', and a subject-term, 'woman'. These are connected by the unsplit binary plus. This relational term is itself composed of a (simple) relational term, 'gave', and a subject-term, 'rose', and they are connected also by the unsplit binary plus. Fully symbolized, the sentence is:

This formula is cluttered with several numerical subscripts. It can be simplified by adopting a pair of conventions. First, the final 1 is unnecessary Gust as both subscripts are in +F 1+8 1 (='Some flowers are blue'] to yield +F+B). Such subscripts merely indicate which pair of terms constitutes a phrase. We will assume that for any well-formed phrase the two terms that constitute it must share at least one numerical subscript, which may be suppressed if unnecessary (as when there are no other subscripts or when one of the terms is relational). So our formula now becomes:

The subscribed 1 and 3 indicate that the complex relation of having given

a rose holds between a man and a woman. Thus, in effect, we are taking all relations to be binary, two-place. However, we can adopt an additional convention of amalgamating the subscripts of complex relations to yield relations of higher degrees. In other words, relational terms nested within relational terms are amalgamated, fusing their subscribed numerals so as to preserve order. Thus:

Now we see 'gave' as a three-place relation. The convention gives us a new formula that is simpler and more natural. One further note before leaving this section: We assume that every well-formed phrase consists of a pair of (possibly complex) terms sharing a common (but sometimes implicit) subscript. We will soon specify a deduction rule that will permit 163

CHAPTER4 simplification from complex formulae. For example. from our formula above we can derive any of the following: I. +M,+(+G,2l+R2) 2. +M 1+(+G, 23 +W 3)

3. +G 123 +R 2 4. +G 123+W 3 5. +M 1+GI2)

(where apparently extraneous subscripts, such as the 2 and 3 of the fifth formula, are ignored). But we cannot derive: 6. +M,+R 2 7. -t-M 1+W 3

The derivable formulae can be read as: 1.1 2.1 3.1 4.1 5.1

A man gave a rose. A man gave to a woman. Something given was a rose. Someone given to was a woman. A man gave.

DON and EQ

Our system of split connectives provides us with a formal logic that is relatively natural, simple, and expressively powerful. Its ability to model in a perspicuous manner a wide variety of kinds of inferences will be a measure of its deductive power. A natural deduction system consists primarily of a small set of rules for deducing the conclusion from the premises of valid arguments. If the arguments can be formulated in a single, abstract notational system, then the deduction system amounts to an algorithm for manipulating the symbolic representations of the premises m order to arnve at the symbolic representation ofthe conclusion. The notational system of variable letters and plus and minus signs was motivated by our desire to use an algebra-like algorithm for deduction. Indeed, I have already made use of a part of such an algorithm when, for example. I showed the equivalence of phrases with symmetric formatives after commutation of their terms (+S+P = +P+S). The fundamental principle of this algorithm recalls the ancient law known as the dictum de omni et nullo. As we have seen, this has often been claimed (though not without challenge) as the underlying principle of traditional syllogistic reasoning. In effect, the law says that whatever is said of all or none of something is likewise said of what that something is said of. In other words, any term predicated (affirmed or denied) of a universal

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subject (quantified term) is predicated in the same manner (affirmatively or negatively) of any subject of which the universally quantified term is predicated (viz.• affirmed). Consider. for example, the inference Every A is B Every B is C So every A is C This inference satisfies the law and is valid. The term C, affirmed of the universally quantified term B (as in the second premise), is affirmed (in the conclusion) of the subject, 'Every A'. of which that term, B. was affirmed of (in the first premise). Consider next the inference Some boy kissed a girl. Every girl is a female. So some boy kissed a female. Here, what is affirmed of a universally quantified term ('female' in the second premise) is affirmed in the conclusion of the subject of which that quantified term ('girl') has been affirmed ('some boy kissed' in the first premise). The dictum de omni et nullo applies directly to classical valid syllogisms and. as we saw, can be extended to apply to inferences involving relationals. But. as stated. it is hard to see how it can be extended to apply to all kinds of inferences. Nevertheless. a close inspection of the law and my examples reveals what Sommers saw, that the law really amounts to a rule of substitution. It says, in effect. and most generally now, that, given a sentence with a universally quantified term (subject or object) and another sentence in which that term is positively qualified, we can deduce a third sentence that is exactly like the second sentence except that the given term has been replaced by the first sentence minus the given term. In classical syllogisms, the given term is the middle term; the first sentence is the major premise; the second sentence is the minor premise; the second sentence minus the middle term is the minor term; and the first sentence minus the middle term is the major term. In effect, we substitute the major term for the middle term in the minor premise to get the conclusion. Thus we substituted C forB in 'Every A is B' to get the conclusion of our first example above, and 'female' for 'girl' in 'Some boy kissed a girl' to get the conclusion of our second example. This rule. allowing the substitution of one term for another in certain circumstances, always results in the cancellation of a term (viz., the middle term). The cancellation of terms suggests algebraic addition, as when we add 'a+b' and 'c-a' to get 'b+c'. where pairs of oppositely charged terms of an addition are cancelled. This is exactly what happens in this logical algorithm. Middle-term pairs are oppositely charged

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and, so, can be cancelled. We symbolize the two inferences above as I.

-A+B ~

-A+C

Notice here that the middle term, B, is positive in the minor and negative in the major. We cancel these two, in effect, adding the premises to get the sum-the conclusion. For a valid inference, it all adds up. 2.

+B 1+(+K 12+G2) -G+F

In this case, as in I, the application of the substitution law amounts to adding the premises algebraically (i.e., cancelling middle terms) to get the conclusion as sum. Consider now a slightly more difficult inference. No A is B Some B isC So some C is not A Before applying the dictum directly, we would have to commute both the major and minor premises. Once we recognize the law as merely a rule of algebraic addition, however. we merely need to symbolize and add: 3.

-A-B ~

+C-A But in carrying out this addition we see the need for a logical restriction. Notice that we could have added our two premises to get -A+C ('Every A is C'), which does not follow from our premises. +C-A and -A+C are algebraically equal, but they are definitely not logically equal. What we require is a further restriction on premise addition to guarantee validity. The equivalence of the conclusion with the algebraic sum of the premises is a necessary but not sufficient condition for validity. A second requirement, which is also a necessary condition for validity, will, conjointly with the other restriction, be a sufficient condition for validity. Keep in mind that our split binary connectives will be said to consist of two parts: the quantifier and the qualifier, and we will continue to use this terminology even for split binary connectives that combine with phrases and sentences as well as with simple terms. We can now say that our second necessary condition for the validity of any inference is this: the 166

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number of conclusions with particular quantity must be the same as the number of premises with particular quantity. It follows that in the case of logical equivalence the two statements must not only be algebraically equal but must have the same quantity as well. From now on, we will refer to the necessary and sufficient conditions for validity as EQ. E reminds us of the algebraic equivalence condition; Q reminds us of the quantity condition. EQ: A conclusion follows validly from a set of premises if and only if(l) the sum of the premises is algebraically equal to the conclusion and (2) the number of conclusions with particular quantity (viz., zero or one) is the same as the number of premises with particular quantity.

The principle EQ amounts to a definition of 'validity' and accounts for classical conversion, obversion, contraposition, and all valid categorical and hypothetical syllogisms. We can use it, in effect, as a decision procedure for determining validity. Thus, with our example 3 inference above we now know that +C- A, but not - A+C, follows from the premises. Both formulae satisfy the equality condition, but only the former also satisfies the quantity condition. Rules of Inference Once an inference has been determined to be valid, what is next required is a proofof that validity. As we have said, the fundamental principle of our algorithm for proving validity is the dictum de omni et nullo. We will see what role it plays in proving validity shortly. Let us say that a proof is a finite sequence offormulae such that the first n (n~O) formulae in the sequence are the premises, the last formula is the conclusion, and every formula is justified by as least one rule of inference. In the algorithm offered here, there are two kinds of inference rules. Rules that permit the creation of a new formula on the basis of a single previous formula are Rules of Immediate Inference; rules that permit the creation of a new formula on the basis of a pair of previous formulae are Rules of Mediate Inference. The main rule of mediate inference will be the dictum de omni et nullo, also often called the Rule of Syllogism. As is normally the case for systems of proof, each of these rules is nothing more than a simple and obvious pattern of valid inferences. And, as is usual, there is no restriction on the number of rules in this system. But too many rules, while making proofs short. will fail to model perspicuously the ways we ordinarily deduce conclusions from premises; too few rules, while making each step in a proof perspicuous, render proofs too long to be practical. I, like others, seek an optimal number of rules; as we shall see, the number can be kept relatively low because, unlike the standard algorithm now in general use, this one recognizes the common formal features of statements

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composed of simple tenns as well as those composed of subsentences. Any subsentence is a sentence; any sentence is a phrase; any phrase is a (complex) tenn. This is a tenn logic. A single set of rules will suffice for reckoning inferences involving all kinds of statements. A further consideration in choosing these rules is my intention to preserve as many of the correct logical insights of traditional logic as possible. Before presenting the rules of inference, I will define 'tautology' in my system and show how tautologies can occur in proofs. I define a tautology as follows: any universally quantified formula that is algebraically equal to zero is a tautology. Note that we could think of a tautology as the conclusion of a valid zero-premise argument. Such an inference satisfies EQ; the algebraic sum of the premises is zero and so is the conclusion, and the number of particular conclusions (0) is the same as the number of particular premises (0). In general, statements of the universal aftinnation fonn -X+X will be tautologies. Obviously, the negation of any tautology will be a contradiction. A contradiction is a particularly quantified fonnula that is algebraically equal to zero. I begin with the Rules of Immediate Inference. Such rules allow the creation of a new fonnula on the basis of a single previous fonnula in the proof sequence. Rules of Immediate Inference Premise (P): Any premise or tautology can be entered as a line in proof. (Tautologies that repeat the corresponding conditional of the inference are excluded. The corresponding conditional of an inference is simply a conditional sentence whose antecedent is the conjunction of the premises and whose consequent is the conclusion.) Double Negation (DN): Pairs of unary minuses can be added or deleted from a fonnula (i.e., recalling 0.2, -- X=X). External Negation (EN): An external unary minus can be distributed into or out ofany phrase (i.e., recalling 0.3, -(±X±Y)=•X•Y). Internal Negation (IN): A negative qualifier can be distributed into or out of any predicate-tenn (i.e.,±X-(±Y)=±X+(•Y)). Notice that we need different rules here, because external negation is tenn negation while internal negation is actually negative quality. External negation is unary; internal negation is part of a split binary fonnative. The first minus of -(+S+P] is a (sentential-)tenn negation. So we have: (+S+P] = -S-P. But the minus of +S-P is a qualifier. Thus: +S-P = +S+(- P) = +S-(+P). Consequently, we have a rule for distributing tenn

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negation and a rule for "amalgamating" a qualifier and the unary formative of the following term-namely: ...+(- ... ) = ... - .. ... -(+... ) = ... - .. .

... +(+ ... ) = ... +.. . ... -(- ...) = ... +.. . Commutation (Com): The binary plus (split:+... +) is symmetric (i.e., +X+Y=+Y+X). Association (Assoc): The binary plus (split:+ ...+) is associative (i.e., +X+(+Y+Z)=+(+X+Y)+Z). Contraposition (Contrap): The subject- and predicate-terms of a universal affirmation can be negated and can exchange places (i.e., -X+Y=-(-Y)+(-X)). Predicate Distribution (PO): A universal subject can be distributed into or out of a conjunctive predicate (i.e.• -X + (+Y + Z) = +[-X + Y] + [-X + Z]) and a particular subject can be distributed into or out of a disjunctive predicate (i.e., +X+(-(-Y)- (-Z)) =--[+X+ Y]-[+X +Z]). Iteration (It): The conjunction of any term with itself is equivalent to that term (i.e., +X+X=X). Notice that the traditional rules of conversion are preserved in the rules above. I tum now to the rules of mediate inference. Rules of Mediate Inference Dictum de Omni et Nullo (DON): If a term, M, occurs universally quantified in a formula and either M occurs not universally quantified or its logical contrary occurs universally quantified in another formula, deduce a new formula that is exactly like the second except that M has been replaced at least once by the first formula minus its universally quantified M.

Simplification (Simp): Either conjunct can be deduced from a conjunctive formula; from a particularly quantified formula with a conjunctive subject-term, deduce either the statement form of the subject-term or a new statement just like the original but without one of the conjuncts of the subject-term (i.e., from +(+X+Y)±Z deduce any of the following: +X+Y, +X±Z, or+Y±Z), and from a universally

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CHAPTER4 quantified formula with a conjunctive predicate-term deduce a new statement just like the original but without one of the conjuncts of the predicate-term (i.e., from - X±(+Y+Z) deduce either- X±Y or -X±Z). Addition (Add): Any two previous formulae in a sequence can be conjoined to yield a new formula, and from any pair of previous formulae that are both universal affirmations and share a common subject-term a new formula can be derived that is a universal affirmation, has the subject-term of the previous formulae, and has the conjunction of the predicate-terms of the previous formulae as its predicateterm (i.e., from -X+Y and -X+Z deduce -X+(+Y+Z)). Note that Add incorporates, in part, It. For example, from- X+Y and- X+Z one can deduce -X+(+Y+Z), which, by It, equals -(+X+X)+(+Y+Z). It should be noted that DON accounts for all valid first-figure syllogisms (and much more besides). As I said earlier, DON amounts to a rule of substitution (the predicate-term of a universal statement can be substituted for the subject-term of that statement in any other statement in which that subject-term occurs positively). The subject-term just mentioned is the middle term of traditional syllogistic. In effect, DON permits the cancellation of middle terms. The same rule accounts not only for syllogistic inference, but for such an inference as modus ponens and for instances of Leibniz's Law as well. That modem logic must make use of three different rules for these three kinds of inferences while this logic needs only one is due to the fact that I, unlike most modem logicians, take categoricals (including relationals and singulars), compound statements, and "identity statements" as all sharing a common logical syntax--each is viewed as a pair of (possibly complex) terms connected by a binary connective/formative (split or unsplit). I will now show how DON operates in these three kinds of cases and then give some sample proofs to illustrate all of the rules spelled out above. First, consider Cesare. Its premises are symbolized as -M-P, -S+M. The conclusion, -S-P, follows directly by DON since -P replaces M in the minor premise (a statement in which M occurs positively, not universally quantified). Consider next a slightly more complex inference. 'Every animal runs from a bear. All bears are carnivores. Hence, every animal runs from some carnivore.' The premises are formulated as -A 1+(+R12+B2), -B+C. The middle term here is B, which occurs universally quantified in the second premise and positively in the first. Thus, by DON, C can replace B in the first premise to yield the conclusion - A1+(+R 12+C2). Before going on, it should be noted that the usual proof for this simple inference using the standard predicate calculus requires twelve steps beyond the premises, makes use of such rules as conditional proof, addition, simplification, modus 170

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ponens, universal instantiation, existential instantiation, universal generalization, and existential generalization. Now, the original argument is extremely simple, and virtually any rational person can draw the appropriate conclusion from the given premises quickly and with very little effort. A system of logic making use of a rule like DON can at least claim some degree of psychological reality. The standard system now in place cannot, does not, and would not. A general modus ponens inference can be formulated as -p+q, +p, therefore +q. Again, the conclusion can be seen to follow directly by DON. Here, the middle term is p, which occurs quantified universally in the first premise (recall that we have agreed to call the first part of any split binary connective a quantifier) and positively in the second premise. DON simply allows us in such cases to substitute the predicate of the first premise for p in the second premise. An alternative account of modus ponens (following Sommers's method of incorporating statement logic into term logic) would treat the second premise and conclusion as having as their subjects the singular term 'the (actual) world', *W. Thus: -p+q, *W+p, therefore *W+q, a Barbara or Darii syllogism. Recall that a formula like *W+p can be read as 'the (actual) world is a p-world'. Notice that not only modus ponens but rules of the sentential calculus such as modus to/lens and chain argument are also merely instances of DON. Finally, consider Leibniz's Law. This rule is explicitly a rule of substitution. It says that from premises of the general form 'a is identical to b' and 'b is so-and-so' one can derive the conclusion 'a is so-and-so'. The rule embodies the notion that when two terms stand in an identity relation one can be substituted for the other in any (nonintensional) statement in which the other is used. In other words, one can derive 'Pa' from 'a=b' and 'Pb'. Nonetheless, this rule is merely another instance of DON. We formulate the premises as *a+b, *b+P. The middle term here is b, which (given that it is singular) is indifferent with respect to quantity in the second premise, and, so, allows us to regard it as universally quantified. It occurs positively in the first premise. So, by DON, the conclusion, *a+P, follows directly, since we can simply substitute the term predicated of universal b forb in the other premise. I offer now a few sample proofs using the system outlined above. From 'No Pis M' and 'Every Sis M' derive 'NoS is P': I. -P-M

2. 3. 4. S. 6.

-S+M -[+P+M] -[+M+P] -M-P -S-P

p p

I, EN 3,Com 4,EN 2,5,DON

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From 'Every circle is a figure' derive 'Every drawer of a circle is a drawer of a figure': I. -C+F 2. -(D+C)+(D+C) 3. -(D+C)+(D+F)

p

P (tautologous assumption) 1,2, DON

From 'Every boy loves some girl', 'Every girl adores some cat',' All cats are mangy', and 'Whatever adores something mangy is a fool' derive 'Every boy loves a fool': I. -81+(+L12+G2)

2. 3. 4. 5. 6. 7.

-G2+(+A 23 +C]) -C+M -(+A23 +M 3)+F -G2+(+A 23+M3) -G+F -81+(+L12+F2)

p p p p

2,3,DON 4,5,DON 1,6,DON

The traditional term logic failed to hold the field against the new logic introduced by Frege. For the most part, this was due to its inability to offer adequate analyses for three kinds of inferences-those involving singulars, relationals, and compound sentences. We have seen that this disadvantage in inference power was not inherent in term logic. Sommers, following suggestive hints from his pre-Fregean predecessors. has built a new version of the old logic of terms. As we have also seen, it enjoys the same advantages of expressive and inference power as does the Fregean logic. Indeed, its powers here actually exceed those of the Fregean system, for just as the old logic was faced with three kinds of inferences beyond its capacity, the new Fregean logic is faced with three kinds of inferences beyond its scope. Consider the simple inference 'Plato taught Aristotle. So Aristotle was taught by Plato'. The standard system formulates this inference as Tpa I Tpa Unschooled intuition, as well as grammar, sees the conclusion as semantically equivalent but syntactically distinct from the premise. We naturally take a sentence in the active voice and its "passive transform" to be different sentences even though they share the same truth conditions. Taking logical form as nothing but the revelation of truth conditions, Quine has said. "The grammar that we logicians are tendentiously calling standard is a grammar designed with no other thought than to facilitate the tracing of truth conditions. And a very good thought this is" (Quine, 1970: 35-36). Frege had held the same view, and on the basis of it he dismissed the

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active/passive distinction. His claim was that since they "express the same thought" the grammatical difference between them "is of no concern to logic" (Frege, 1979: 141 ). The present system formulates the arguments as

It turns out that though the premise and the conclusion entail one another, the two are formally distinct. The transformation from active to passive (or vice versa) is accomplished in this case by an application of Com (twice) and Assoc. A second type of inference beyond the scope of standard predicate calculus is represented by the argument 'Socrates taught a teacher of Aristotle. So one whom Socrates taught taught Aristotle'. The best that mathematical logic can do in terms of formulation is (3x)(Tsx & Txa) I (3x)(Tsx & Txa) Again, the standard system is powerless to exhibit the formal difference between the premise and the conclusion. In the present system, the inference has the form

Modem grammarians call this a case of"associative shift." While modem logicians see it as a trivial reiteration, this logic recognizes the formal distinction between premise and conclusion. The conclusion is derived by an application of Assoc. The third kind of inference that challenges the Fregean logician is represented by the following example: 'Plato taught Aristotle with a dialogue. So Plato taught Aristotle.' The standard formulation is (3x)(Dx & Tpax) I Tpa The two relational predicates are distinct; one is a three-place function (' .. . taught ... with ... '); the other is a two-place function (' ... taught ... '). For the inference to be valid there must be a hidden assumption of an analytic, semantic tie between these two predicates (like the one between, say, 'bachelor' and 'unmarried'). The formalization below retains a more natural syntax and preserves the common-sense view that 'taught' is univocal throughout its two uses here.

(The subscribed numerals here are not to be confused with bound variables of the predicate calculus. The latter simultaneously keep track of reference

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and the order of subjects and objects with respect to interpretations of relational predicates. The subscribed numerals perform only the second of these tasks.) The inference proceeds by the application of Assoc and Simp. The standard logic is essentially mute in the face of inferences involving passive transformations, associative shifts, or simplifications with polyadic predicates (relationals). I conclude this section with an example of a simple inference that is beyond the scope of the standard system in all three ways: 'A man loves a woman. So some lover is a man.' My proof is I. 2. 3. 4.

+M 1+(+L 12+W2) +(+M 1+L 12)+W2 +W2+(+M 1+L 12) +W2+(+L 12+M 1) 5. +L 12+M 1

p

I, Assoc 2,Com 3,Com 4,Simp

Notice that each of the intermediate lines in the proof "makes sense" in natural language. Thus: 2. What some man loves is a woman. 3. A woman is what some man loves. 4. A woman is loved by some man. Names and Other Pronouns First, it is offundamental importance to grasp that the properness of proper names is a feature-in Saussurean terms-of 'parole ', not of 'langue'. L.J. Cohen Augustus, meeting an ass with a lucky name, foretold himself good fortune: I meet many asses but none ofthem have lucky names. Swift

Singular terms, especially names and personal pronouns, are a prominent feature of natural-language discourse. Pronouns, in the guise of individual variables, play an important role in the formal language of the predicate calculus. Even Quine's Predicate Functor Algebra, which eliminates such variables, is meant to reveal just what their roles are in the calculus. I have followed Leibniz and Sommers in giving them wild quantity when used as subject-terms. Whatever grounds exist for distinguishing names and pronouns from general terms are semantic ones. Although my path has been, in contrast to the Fregeans. to degrade this distinction in the building of a formal language, the semantics of such terms demands our special attention. There is both a classical and a modem semantic theory for the standard predicate logic. The former holds that all singular expressions 174

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refer, and that reference is determined by the sense of those expressions. Since general terms are terms with sense, pronouns and names are replaceable by appropriate definite descriptions, whose senses are determined by the senses of the general terms that occur in them. The classical semantics is most closely associated with Frege and Russell. The more modem theory, associated particularly with Kripke and Putnam, holds that a distinction must be made between singular expressions that are "rigid" and those that are not. Pronouns and names are rigid; definite descriptions are flaccid. The former, unlike the latter, do not have their references determined by their senses. Rigid designators refer directly to their referents without detour through senses or meanings. Thus a rigid designator refers to or designates the same object in all of its referential uses (cf. Salmon, 1986). Since the new theory is usually part of a possible-worlds semantics, it is said that a rigid designator refers to the same object in every possible world in which that object exists. A logic of terms requires something like the rigid/nonrigid distinction. But it need not accept the entire modem theory of semantics. It especially eschews possible worlds. My semantic theory recognizes at least two levels of reference for terms: denotation and reference (proper). In the normal, nonvacuous case, every term used in a statement, whether the term is charged positively or negatively, has a denotation determined by its signification and the domain (of discourse) relative to which the statement is made. What a term signifies is a property. For example, 'red' signifies the property redness, 'wise' signifies the property of wisdom, 'pious' signifies piety, 'nonsquare' signifies nonsquareness. Whatever is denoted by a term has the property signified by that term. The converse need not hold. Every used term is used in a used (statement-making) sentence, or statement. Every such sentence is used relative to some specifiable domain of discourse. A domain is a nonempty totality of compossible objects. Ordinarily our domain of discourse is simply the actual world. But any world, any part of the actual world, or any set of objects could serve as a domain. When we say 'Some man held a horse on his shoulders' our domain is, ordinarily, the actual world. When we say 'Some man held the Earth on his shoulders' our domain is, presumably, the world of Greek mythology. A term used in a sentence denotes the objects that have the property signified by that term and that are in the domain relative to which the sentence is used. We can think of denotation, then, as the intersection of a domain and the set of objects having a given property. We have seen that normally each term used in a sentence has a denotation. Now, each term used in a sentence is either quantified (i.e., it is a logical subject) or qualified (a logical predicate). Subjects have a mode of"reference," however, not shared by predicates. Every term used in a sentence, whether a subject-term or a predicate-term, has a denotation, but only a subject (quantified term) refers (in the proper sense). The referent of a subject is determined by its quantity and the denotation of its term. A universally quantified subject refers to the

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entire denotation of its subject-term; a particularly quantified subject refers to some (possibly specifiable) part (perhaps the whole) of the denotation of its subject-term. So. while predicate-terms may denote, they do not refer. Thus far. no distinction has been made between singular and general terms. In the standard predicate calculus the distinction is allimportant. A general term used in a statement makes reference (denotatively) to one or more individuals. A term whose denotation is either unique or not unique to individuals is not a general term. A term whose denotation is not individual is a mass term (e.g., 'wood', 'water', 'wool', 'wine'). Mass terms. those with nonindividual denotation, and singular terms, those with unique individual denotation, contrast with general terms, those with nonunique individual denotation, in having tacit quantity when used as subjects. When a mass term is explicitly quantified it is because the term is being used with an understood, thus implicit, phrase such as 'sample(s) or, 'piece(s) or, 'drop(s) or. 'chunk(s) or, and so on. For example, 'Water is dripping in the sink' is understood as 'Some drops of water are dripping in the sink', and 'Wood is combustible' is understood as 'All samples ofwood are combustible'. Terms like 'drops ofwater' and 'samples of wood' are not mass but general. Singular terms denote uniquely; they denote just one individual object. A singular subject (a logical subject whose term is singular) refers to the object denoted by its term. The implicit logical quantity of a singular subject is always understood to be particular. The canonical form of 'Socrates is wise' is '(Some) Socrates is wise'. Particularly quantified subjects make undistributed reference to the denotations of their terms. Universally quantified subjects refer distributively; they refer to the entire denotations of their terms. Since we know the subject of a given sentence to be singular (with tacit particular quantity}, we can infer a corresponding universal sentence from it. This, again, is the wild quantity thesis. Such inferences are informal. depending as they do on our extra-logical knowledge of the denotations of the subject-terms. We saw above that the denotation of a term is in part determined by its signification. What a term signifies is a property. The term 'red' signifies the property of redness and 'wise' signifies wisdom. Let [T] be the property signified by the term 'T'. Thus, 'red' signifies [red] (= redness) and 'wise' signifies [wise](= wisdom). A singular term such as 'Socrates' signifies a property as well. The term 'Socrates' signifies [Socrates](= the property of being Socrates, Quine's Socratizer). Indeed, [Socrates], [wise], [Greek], [teacher of Plato], and [philosopher] are some of the properties that Socrates has. Many things have the property [Greek]; not so many things have the property [wise]; a small number of things have the property [teacher of Plato]; just one thing has the property [Socrates]. As long as Socrates belongs to our domain of discourse we will denote Socrates by 'Socrates', since he is what has the property [Socrates]. Indeed, in any domain in which Socrates is located, the use of'Socrates' will denote (and, when in subject position, refer to) Socrates, for the signification of

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'Socrates' is invariable from one domain to another. Names are rigid. As subject expressions, their references are immutable. Definite descriptions have the form 'the x', where 'x' is a (usually complex) general term. While 'x', as used in a sentence relative to a specifiable domain, denotes all the objects in the domain that have [x], if there is but one such object the term 'the x', similarly used, will denote it. As subject expressions, definite descriptions are not rigid in reference because their constitutive general terms have denotations that may vary from one domain to another. Thus, for example, 'man who fought a duel', when used relative to the domain of Hamlet, denotes, among other things, Hamlet but not Hamilton. The same term. used relative to the actual world, denotes, among other things, Hamilton but not Hamlet. Likewise, the use of 'the man who fought a duel' in subject position may refer on one occasion to Hamlet and on another to Hamilton. In summary, since the denotation of a used term is determined in part by its signification (which is invariable) and in part by the domain relative to which the sentence in which it is being used is used (which is variable), its denotation is variable from domain to domain. Definite descriptions are variable in denotation, as the denotations of their constituent general terms are variable. Names are invariable (i.e., rigid) from domain to domain only because the properties that they signify cannot (as those signified by general terms can) be possessed by different objects in different domains. Consider again the term 'man who fought a duel'. Here, it is possible to have two different domains relative to which the denotation of the term is nonempty, yet which are such that the two sets of objects constituting the denotation are disjoint. In contrast, consider 'London'. Here, given any two domains, ifthe denotation of'London' relative to each is nonempty it can only be because London is in both (compare the actual world and the world of Sherlock Holmes). So names, but not definite descriptions, are rigid. What of pronouns? Here, roughly, is one version (Kripke, 1972/80) of how names are introduced into discourse (the so-called causal theory). Names are introduced in an initial "baptismal ceremony" ('I name this child "Socrates",' 'I dub thee "Sir Lancelot".' 'Let's call this place "London",' 'Henceforth this bridge will be called "Pons Asinorum" '). A subsequent user of the name. in the normal course of events, will use that name to designate (denote) the same object only if such use can, in theory, be traced back through intermediary uses, ultimately ending in the initial baptism. As long as the intention of each user was to denote just the object that the preceding user intended to denote, the denotation from the baptism to the present use is constant. Constancy is preserved by a uniformity of shared intentions (to denote). Uses of a name to refer to the same object on different occasions are links in a chain of intentions. Our theory, derived from Sommers, holds that nominal subjects are simply links in anaphoric chains. Indeed, names, so used, are just special duty pronouns, "pro-pronouns." And generally, most links in an anaphoric

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chain are pronominal. But how do chains get started? And how are their links bound together? On this theory, initial reference to an object must always be by the (sometimes implicit) use of an indefinite description. We say (or assume) such statements as 'A child is born', 'Here is a newly discovered mountain', 'There is a star'. These sentences have the overall form 'Some X is Y'. where 'some X' is indefinite but makes specific reference. (Compare: 'Some man is at the door' -specific-and 'Some man will be the first to walk on Pluto'-nonspecific.) Subsequent reference to the object is pronominal: 'A child is born. She is beautiful.' According to Sommers's theory, pronominal subjects are, like any subject, logically analysable as quantified terms. As singulars they enjoy wild quantity. The denotation of the term of a pronominal subject, the "pro-term," is determined in part by the reference of its antecedent. In our example the antecedent of'she' is 'a child'; the denotation of the pro-term is also partly determined by the ascription of its antecedent sentence. The antecedent sentence ascribes the property of being born to the referent of its subject. The pro-term of the subsequent anaphoric pronoun. then, denotes a child who is born. The reference of the pronoun is determined by its implicit quantity and the denotation of its pro-term. Since it denotes all of what the antecedent refers to, it has universal quantity, but it also inherits the quantity of its antecedent. So, in effect, a pronoun whose antecedent is particular (the normal case) has wild quantity. As Quine has said, "[Pronouns] may have indefinite singular terms as antecedents but they can be supplanted only by definite singular terms" ( 1976: 46). In our example, 'she' refers to the child who is born. In fact, we could make this explicit by using the definite description 'the child who is born'. where the definite description is anaphoric. Thus: 'A child is born. She (or: the child who is born) is beautiful.' And we could go on: 'She has red hair.' In this last statement, 'she' refers to the child who is born and is beautiful, as ascriptions are accumulated from link to link in an anaphoric chain. We have yet to introduce names into these chains. Thus far. the initial link in a chain is an (often implicit) indefinite description having the logical form 'some X'. Subsequent links are anaphoric, picking up the referents and ascriptions of their antecedents. These links are pronominal or, when the ascriptions are explicit, definite descriptions. Sometimes, though far from always, we make such frequent reference to an object or have such interest in it that, by fiat or custom, we create an expression whose job is to refer specifically to just one object. These expressions are names. We name an object in what Kripke calls a baptismal ceremony. But our introduction of the name there is not the initial link in an anaphoric chain containing uses of that name. Names are (nonvariable) pronouns. Nominal reference is a special kind of pronominal reference. Names are usually introduced after (or perhaps at the same time as) pronominal reference has been made to the object named. Thus: 'A child is born. She is beautiful. She has red hair. Let's call her Lucy.' Even

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where no intervening pronominal links occur between the initial link and the nominal one, we cannot say that the naming of an object is the initial link in an anaphoric chain. Suppose upon the birth of a child I say, 'Let's call her (this, that, it) Lucy'. On such an occasion, the object being named has been picked out by 'her'; and this pronominal reference must be accompanied by an indefmite reference, such as 'a child', in an implicit statement, such as 'A child is born', or 'Here is a child'. Names, then, are like any other pronouns in that when used as subjects their denotations are determined by the referents of their antecedents and the accumulated ascriptions, and, as singular subjects, they have wild quantity. Names differ from other pronouns in that they are introduced only on special occasions to make anaphoric reference to just one specified object. We could do without names (as we usually do in the case of cows and crows). Names are specialduty pronouns--they are pro-pronouns. Names, then, are rigid because they are referentially pronominal; and pronouns are always rigid, in the sense that they are always used to refer to the same object on each occasion of their use in a given anaphoric chain. This is guaranteed by the fact that each pronominal pro-term in a chain is determined by its antecedent's reference and the accumulated ascriptions. These are what bind pairs of links in order to form a cohesive chain. This theory of rigidity contrasts favourably with the causal theory, which binds links in a referential chain by means of common intentions on the part of users of those links. We have seen that for the Fregean the difference between singular terms and general terms is that the former can never be used as functions (or predicates, as they are still called). According to the Asymmetry Thesis, this difference is of the greatest logical import, so let us briefly review this thesis. According to defenders of Asymmetry: (i) a sentence can be negated by negating its predicate but not by negating any of its arguments (singular terms), (ii) a pair of sentences can be conjoined or disjoined by conjoining or disjoining their predicates but not by conjoining or disjoining their arguments. Thus, according to the thesis, 'Socrates drinks' is negated by 'Socrates is not drinking' (or 'Socrates does not drink'), but not by 'NonSocrates drinks'; 'Kant is serious' is negated by 'Kant is nonserious' (or 'Kant is not serious'), but not by 'NonKant is serious'; 'Socrates eats' and 'Socrates drinks' can be conjoined as 'Socrates eats and drinks', but 'Plato and Aristotle carried the piano' is not a way of conjoining 'Plato carried the piano' and 'Aristotle carried the piano'. A corollary of(i) is that singular terms do not have negations. A corollary of (ii) is that there are no logically compound singular terms. Finally, there is the basic, underlying claim of the Asymmetry Thesis: (iii) singular terms are never predicated. Many twentieth-century analysts have offered a variety of arguments intended to support the various tenets of the Asymmetry Thesis (we saw in chapter two how Geach and Strawson were prominent among these philosophers). Here are the outlines of some of the more important or common arguments. 179

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( 1)

Names cannot be negated, because if they could be there would have to be negative objects that negative names enote. But there cannot be such objects.

(2)

Names cannot be compounded, because if they could be there would have to be compound objects that such names denote. But there cannot be such objects.

(3)

All negation is logically sentential. Any colloquial form of negation that cannot be construed as sentential is not genuine (logical) negation. Name negation cannot be construed as sentential, it is not genuine.

(4)

All compounding is logically sentential. Any colloquial form of compounding that cannot be so construed is not genuine. Name compounding is not sentential, so it is not genuine.

(5)

All logical predicates are general terms (adjectives, verbs, common nouns, etc.). Names are not general terms, so they are never logical predicates.

Each of these arguments rests on the conclusion of an unstated argument: (6)

All logical arguments (subjects) are singular. Negated and compound names are either singular or general. If they are singular, then they denote impossible objects. If they are general, they are not singular. Therefore, such terms cannot be logical arguments. No logical subjects are negated or compound.

The first premise of (6) is the Fregean Dogma. It is essential to the Asymmetry Thesis. Some of the claims made in the above arguments cry out for examination. Defenders of the Asymmetry Thesis hold that a name like 'Socrates' cannot be negated because the result, 'nonSocrates', would have to name an impossible object. The reasoning here is that any object denoted by 'nonSocrates' would have to lack all of the properties that Socrates has and have all of the properties that Socrates lacks. But, since the properties that Socrates lacks are not compossible, it is impossible for an object to have all such properties. Thus, such an object is impossible. Suppose Socrates is white, Greek, and male. By the above reasoning, nonSocrates must be nonwhite, nonGreek, and female. So far so good. However, the argument further requires that nonSocrates have all the properties that Socrates lacks. Well, Socrates lacks the properties of being green, red, black, Roman, 180

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French, Japanese, and female (and many other properties besides). NonSocrates may be a black Japanese female, or a yellow French female, or a green Roman female, and so on, but it/she cannot possibly be all of these. NonSocrates, so the argument goes, is impossible-so 'nonSocrates' is not a genuine name. The Asymmetrist's argument rests on a simple logical fallacy involving De Morgan's Laws. The negation of a conjunction is a disjunction of negations, not, as per the argument above, a conjunction of negations. Consider: Socrates is white. Thus he lacks the properties of being green, red, and black, and so on. In other words, Socrates is nongreen, nonred, and nonblack. What follows with regard to nonSocrates, then, is not that it/she is (per impossible) green, red, and black, but rather that nonSocrates is not nongreen, nonred, and nonblack-that is, nonSocrates is either green or red or black, etc. There are at least two sources of the confusion over negative names. One is lack of clarity concerning the distinction between properties that an object lacks and properties that do not apply to that object. The number 2 is even and lacks the property of being odd, but it neither has nor lacks the property of being green. The term 'green' simply does not apply to (span, in Sommers's terminology) the number 2. Likewise, Socrates may be white and nonRoman, but he neither has nor lacks the property of being Marxist. Consequently, whatever properties an object named by 'N' has, nonN can neither have nor lack any properties that N neither has nor lacks. 'Marxist' does not apply to Socrates-so it does not apply to nonSocrates either. Whatever nonSocrates might be, it/she is the same sort of thing (i.e., has exactly the same terms applicable to it, is spanned by all the same terms, belongs to the same categories) as Socrates is. The second, and perhaps more fundamental, source of confusion is the refusal of standard mathematical logicians to recognize the distinction between term negation and sentential negation. We have seen that the logical contrary of a term is semantically equivalent to the disjunction of all terms incompatible with that term. For example, the nonlogical contraries of'white' are 'red', 'green', 'yellow', 'pink', 'black', and so on. 'Male' has 'female' (normally). 'Greek' has 'Roman', 'French'. 'Japanese'. and so on. 'Six feet tall' would have infinitely many nonlogical contraries (e.g., 'four feet tall', 'six feet one inch tall', 'six feet two inches tall'). Sentence pairs that differ only in that their predicate-terms are nonlogical contraries are themselves nonlogical contraries. Sentence pairs that differ only in that their predicate-terms are logical contraries are themselves logical contraries. The nonlogical contrary of a given sentence entails the logical contrary of that sentence. The converse need not hold. The logical contrary of a given sentence entails the contradictory of that sentence. The converse need not hold. Fregean logicians today do not recognize the notion of logical term negation. As a result, this way of drawing the contrary/contradictory distinction is not to be found in standard mathematical logic. Now, if terms

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cannot be negated, then clearly names cannot be negated. Nonetheless, we are free to abjure the Fregean Dogma and the Fregean refusal to recognize term negation. Ordinary language practice (not to mention the counsel of many contemporary linguists and all traditional logicians) suggests such a departure from contemporary logical cant. There are negative terms, and among them are negated names. We have already seen that the argument that negative names must name impossible objects is logically (thus fatally) flawed. But there is something else plaguing such an argument. The simple fact is that the negation ofa singular term, such as a name, is not itself singular-the negation of a singular term is a general term.

Consider 'Socrates is not Roman'. As the logical contrary of 'Socrates is Roman', we read this as 'Socrates is nonRoman', which is semantically equivalent to 'Socrates is Greek or French or Japanese or ... '. What, now, of 'Socrates taught someone other than Plato'? What could 'other than Plato' (logically, 'nonPiato') refer to? Suppose I tell you that someone other than me taught my logic course last year. Logically, I may be construed as saying 'Someone who is not me (other than me, nonme) taught my logic course last year'. To whom am I referring? Well, at least not to myself. In fact, I refer to some member of the set of professors excluding me. But in all probability only one of these other professors taught the logic course. So how could the term 'nonme' be general? One more example: 'Olivier read every Shakespeare play but Hamlet' (logically: 'Olivier read every nonHam/et Shakespeare play'). My contention is that the negation of a singular term (name, pronoun, etc.) is a general term. Thus, 'nonPiato', 'nonme', 'nonHam/etShakespeareplay' are general terms. They are not used to make reference to any particular (possible or impossible) objects. To see how this is so, we need to recall key elements of my semantic theory. My view is that every used term, in the normal case, denotes objects, and the denotation is the intersection of the domain relative to which the term is used and the extension of the term. When quantified, such terms refer. The reference of a quantified term (logical subject) is determined in part by its denotation and in part by its quantity. Universal subjects make distributed reference to their entire denotations; particular subjects make undistributed reference to a part of their denotations. When I state that Olivier read every nonHamlet Shakespeare play, the term 'nonHamlet Shakespeare play' denotes all the plays in the Shakespeare corpus-minus Hamlet, and the quantifier 'every' shows that my reference is to all of these. The term is primafacie general, denoting not a mysterious, purported play by the Bard called "NonHamlet," but the entire body of plays, excluding just one. Now, recall the example 'Some nonme taught my logic course last year'. The term 'nonme' denotes all the people in the domain (say, the faculty) who are not me. But I do not refer to all of them. My use of a particular quantifier shows that reference here is made to one

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or more of the faculty members, but not necessarily all. Again, 'nonme' is not a singular term. Finally, consider once more 'Socrates taught some nonPlato'. To what (or whom) does 'nonPiato' refer? Suppose our statement is made relative to the domain of Athenian philosophy students in the fifth century B.C. The expression 'some nonPlato' must refer undistributively to those-minus Plato. Again, 'nonPlato' is clearly not a singular term naming a strange object, but a general term denoting a large number of people (in this case, the fifth-century-B. C. Athenian philosophy students, minus Plato). 'NonPlato' is not an impossible or even strange name-it is no name at all. Again, the fact is that the negation of a singular term is not singular. Recognition of this point would diminish much of the enthusiasm for asymmetry. Recognition of another, but closely related, point would also contribute to this general loss of enthusiasm: the conjunction or disjunction ofsingular terms is not singular. We saw in chapter two that Strawson has argued that if, say, 'Tom and William' did make a reference, it would have to be to an individual (again, the Fregean Dogma) that possessed all and only those properties that Tom and William both have. But if Tom were short and William tall, then this third individual would be neither short nor tall. If Tom were two-legged and William one-legged, then this third individual would be neither. Such an individual is impossible. So a conjunction of names cannot refer-thus cannot be a name. Similarly, a disjunction of names cannot name. 'Tom or William' would have to name an individual that had all and only those properties that either Tom or William possess. But this would require, for example, that such an individual be both short and tall. In order to preserve the Fregean Dogma, a statement like 'Tom and William played squash (together)' cannot be construed by the asymmetrist as having 'Tom and William' as its subject. Terms are either singular or general. If this term is general, it cannot be the subject; if it is singular, it names an impossible object. Consequently, the statement must be paraphrased so that the 'and' is seen as a sentential rather than a term connective. The statement is logically paraphrased as 'Tom played squash and William played squash'. Now, the Fregean Dogma is just that-a dogma, and so are such well-entrenched beliefs as the one that demands that all connectives (negation, conjunction, etc.) are logically sentential. Like Euclid's parallel postulate, they can be rejected with impunity. We have already seen that a logic that permits negative names is natural, and so is one that countenances compound names. The key is, again, the realization that the compound of a pair of singular terms is not itself singular. 'Tom and William', for example, is not the name of anyone or anything-even an impossible thing. From my logical point of view, 'Tom and William' and 'Tom or William' are quantified terms-logical subjects. They refer distributively and undistributively, respectively, to the denotations of their terms. Let us say that 'a and bare C' has as its subject 'a and b', and that

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the subject-term is ' {a,b} '. This latter term we will call a term of explicit denotation. All used terms have a denotation-but it is almost always implicit, unstated. Consider the statement 'Every logician is wise' (made relative to the actual world). The term 'logician' here denotes Aristotle, Chrysippus, Abelard, and so on. The number of denoted items is quite large in this case. In 'Every Canadian province has a park', the term 'Canadian province' denotes British Columbia, Alberta, Ontario, Quebec, and so on. Here the number of objects denoted is only ten. Notice that we could use terms of explicit denotation in place of' logician' or 'Canadian province'. We could say 'Aristotle, Chrysippus, Abelard, Ockham, Leibniz, Frege, Russell ... and Quine are wise', for example. But clearly there is much practical advantage in using terms of implicit denotation-and there is no alternative in cases where the denotation is infinite (e.g., 'Every prime greater than 2 is odd'). The advantage virtually disappears, however, when the number of denoted objects is very small. 'Every author of Principia Mathematica was British' has little to recommend it over 'Russell and Whitehead were British'. Now, the denotation of 'author of Principia Mathematica' is Russell and Whitehead and no one else. Let '{Russell, Whitehead}' be a term denotatively equivalent to 'author of Principia Mathematica'. The farst is denotatively explicit; the second is denotatively implicit. A logical subject is a quantified term. The reference of a subject is determined by the denotation of the term and the quantifier. Given that they are made relative to the same domain, we know that 'Every author of Principia Mathematica was British' and 'Russell and Whitehead were British' are equivalent. They share a common predicate and their subject-terms are denotatively equivalent. The quantifier in the first case is explicit ('every'), indicating that the subject refers to the entire denotation of the subject-term. In the second sentence there is apparently no quantifier. But, since the two sentences are logically equivalent, the logical quantity of the second must, like the first, be universal. The universal quantity here is indicated by 'and'. The two sentences have the general forms (I) Every author of Principia Mathematica was British. (2) (Every) {Russell, Whitehead} was British. But in (I) the term 'author of Principia Mathematica' is simply a convenient shorthand for '{Russell, Whitehead}', just as 'logician' is a (very) convenient shorthand for ' {Aristotle, Chrysippus, Abelard, ... Quine}'. (2) is the logical version of the colloquial 'Russell and Whitehead were British'. lf'and', as in the example above, indicates universal quantity, then it is reasonable to expect its dual, 'or', to indicate particular quantity. Consider 'Some (one of the) author(s) of Principia Mathematica was an earl'. Replacing the term of implicit denotation with a denotatively equivalent one of explicit denotation yields 'Some {Russell, Whitehead}

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was an earl'. This is rendered more naturally as 'Russell or Whitehead was an earl'. Terms like 'Tom and William', 'Russell or Whitehead', and so on, are not singular terms. They are general terms-indeed, quantified terms (subjects). Of course, it is possible to use a term of explicit denotation that happens to denote just one object. Consider 'Socrates is wise'. Let us say that '{Socrates}' is a term whose denotation is Socrates. It is a term of explicit denotation. A subject of the form 'every {Socrates}' refers to the entire denotation of '{Socrates}', and a subject of the form 'some {Socrates}' refers to a part of the denotation of ' {Socrates}'. Now, the entire denotation of ' {Socrates}' is Socrates, and the only part of the denotation of '{Socrates}' is Socrates. So, whether universally or particularly quantified, the reference in each case is the same object, Socrates. This is a further explanation of why there is no need for singular subjects to be explicitly quantified and no logical need to replace a singular term with its denotatively explicit equivalent. Singular subjects have wild quantity. every {Socrates} =some {Socrates} = {Socrates} =Socrates

Truth and What 'There' Is

If what is is what is said. then the more we talk. the more being there is. UmbertoEco "There is something better than logic." "'Indeed? What is it?" "Fact." Mark Twain As scarce as truth is. the supply has always been in excess ofthe demand Josh Billings

Every sentence is a (complex) term. In the normal case, every used term signifies a property and denotes those objects (if any) that both have that property and are also in the domain of discourse relative to which the term is used. Thus, every sentence (used to make a true statement) both signifies and denotes. What a true statement (i.e., sentence used to make a statement) signifies is a constitutive characteristic of the domain relative to which it is used. If the sentence is (used to make) a statement, then it is used with the implicit accompanying (truth) claim that the domain is so characterized. A true statement denotes its domain. A false statement denotes nothing; nor does it signify anything; it is doubly vacuous. Compare the term 'red' and the term 'present king of France'. The first signifies the property of being red and denotes whatever has that property in the domain at hand (say, the actual world). The second term expresses the concept of being a present king of France. But since nothing corresponds to that concept in the actual

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world, it is vacuous; it denotes and signifies nothing. A true statement denotes whatever has the property it signifies. So if the domain at hand has that property (that constitutive characteristic), then the statement denotes the domain. Otherwise it denotes nothing. "It is just true statements that have a corresponding entity" (Davidson, 1969: 74). The actual world is Mars-ish and red-ish (in this case, and many others, we actually have the word 'reddish') and logician-ish and shy-ish because it contains such things as Mars, firetrucks, Quine, and me as constituents. It is un-ghost-ish and un-unicorn-ish because it does not contain such things as ghosts and unicorns. The actual world is red-ish and nonred-ish and un-ghost-ish because it contains firetrucks and lemons but no ghosts. In general, to say that a domain, D, is P-ish is to say that some (at least one) P-thing belongs to, constitutes (in part at least), D. To say that D is un-P-ish is to say that no P-thing is a constituent of D. Note that for any D and any P, D is P-ish or D is un-P-ish, but it need not be the case that Dis either P-ish or nonP-ish. The domain of natural numbers is either redish or un-red-ish because either it has a red constituent or it has no red constituent; but it is neither red-ish nor nonred-ish, since it has no red constituent and also no nonred (blue or green or pink or ...) constituent. To make a truth claim is implicitly to characterize the relevant domain constitutively. When I state, relative to the actual world, 'Mars is red', I implicitly characterize the actual world as being in part constituted by red Mars, as being red-Mars-ish, as being [red Mars]. When I state 'Some logician is shy' I characterize the domain, normally the actual world, as shy-logician-ish [shy logician]. Notice that for 'Mars is red' to be true, all that is required is that something that is both Mars and red be in the world; there is no need for states or facts also to be in the world. Suppose I state 'Mars is male'. Whether my implicit claim here is true or false will depend on the domain with respect to which I make my statement. If that domain is the actual world, then the claim is not true. For the actual world has no constituent that is both Mars and male. But if the domain is the world of Greek mythology, my claim is true, since that domain is constituted by, among other things, a male Mars. To say of an object that it exists is, in effect, to say that it is a constituent of the domain at hand. Existence, as Kant and so many others have argued, is not a (real) property. What existence is is a constitutive property-not of objects. not of concepts (as Frege thought), but of domains, totalities of objects. Recognition of this leads to an interesting thesis concerning the term 'there'. Indeed, we can now say what 'there' is. Our thesis can be stated in simple terms: the English word 'there', as used in such statements as 'There is an X', 'There are X's', 'There is no X', 'There are no X's', is nothing more than the simple locative adverb, equivalent to 'in/at that place'. The received view among logicians and philosophers has generally been that in addition to the locative use of 'there' (as in 'There is the book I was looking for', 'The lighthouse is there, on the

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other side of the cape', 'My car is not there, it's been stolen') there is an existential use of'there'. To use 'there' in this second way (e.g., 'There is a Kantian joke', 'There is no god>, 'There are no unicorns', 'There are honest politicians•) is to affinn or deny existence of some thing or things. Thus, to say 'There is a Kantianjoke• is just to say 'A Kantianjoke exists•, the phrase 'there is' being reduced to 'exists'. In like manner, one can generate 'No god exists•, 'No unicorn exists', and 'Honest politicians exist•. There is no denying that this way with 'there' leads to the growth ofPiato•s Beard, taking, as it does, the use of 'exists' in these paraphrases as the predication of a property (existence/nonexistence). But there have been good arguments advanced against treating 'exist' as a predicate. Hume gave one in the Treatise (I.ii.6 and I. iii. 7), and Kant gave an even more famous version in the "Transcendental Dialectic, of the Critique (II.iii.S). Others have argued this since. Modem logicians are hardly prepared to deny the Hume-Kant position. Existence is not a property that any thing has or lacks. Still, there is an existential use of 'there'. Their way with 'there• is to treat it as a "higher, function-namely, a quantifier. To say 'There is an X' is to say 'There is at least one thing such that it is X'. The phrase 'there is at least one thing such that' is standardly treated as a Fregean "second-level function••. To say 'There is an X' is just to say that something falls under the concept referred to by 'X'. Accordingly, existence may not be a property of objects but it is surely a property of concepts. My claim is that there is no existential use of'there•-au (nonnal) uses are locative. Admittedly, such a thesis cannot stand alone. If'there' is always locative in statements such as 'There is/are an/no X(s)', then the question immediately and naturally arises: Where? Using 'there• locatively, 'There is an x• is convertible into 'An X is there•. Thus: 'A Kantian joke is there•, 'A god is not there', 'Unicorns are not there•, 'Honest politicians are there'. So, where is the Kantianjoke, the honest politician; from where is god missing; from where are unicorns missing? Here I need my second thesis: In statements of the fonn 'There is an X', 'There' is always used to locate objects in the same place-the world. The locative sense of 'there' can always be expressed by the wordier 'in/at that place', where the demonstrative 'that' is interpreted differently according to the context of its use. The same paraphrase applies to 'there• in 'There is an X', and so on. Thus: 'In that place is an X'. But in such cases 'that place• is always used to designate the same locale, the world. Thus: 'There is no god' ('No god is in the world', 'No god is a constituent of the world', 'The actual world is un-god-ish•); 'There are honest politicians• ('Some honest politicians are in the world,, 'Some honest politicians are constituents of the world,, 'The world is honest-politician-ish'). The use of 'there• is always locative, and the "existential, use of 'there• locates objects in the world.

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A New System of Diagrams• To find a lucid geometric representation for your ... problem could be an important step toward the solution. G. Polya Be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will-a straight Line he looks and nothing else. E.A. Abbot Novelty, by itself, is no drawback to a scheme; in some cases (as with milk. eggs, and jokes) it is a positive advantage. Lewis Carroll

Euler and Venn diagrams are simple and effective devices for illustrating syllogistic validity. Their potential is limited, however, since they cannot apply to arguments with more than four terms. Attempts at extending the scope of plane figure diagrams (e.g., by Carroll, 1958, 1977) have been only marginally successful. 2 Aristotle probably used some sort of diagram method in teaching the syllogistic, and many ancient commentaries made use of linear diagrams, though our understanding of just how they worked is sketchy. 3 If the ancient syllogists used linear diagrams rather than planar ones, and if they diagrammed not only simple syllogisms but sorites, polysyllogisms, and compound syllogisms, then it is likely that there is a satisfactory linear method of logical diagrams that can readily go beyond the virtual four-term limit on plane diagrams. Leibniz tried to use line diagrams in analysing syllogisms,4 and a century later Lambert attempted linear diagrams for syllogistic. 5 In what follows I will describe such a linear diagram method, illustrate some of its uses, and extend the method to relationals and compounds statements.6 Rather than follow the nineteenth-century practice of representing each term of an inference as a set of points constituting a closed plane figure, let us follow the ancient suggestion of representing such terms as points of a straight line segment. (Topologically, we might think of the line as a covering space on a Venn circle.) We can think of the place in which a given diagram lies as constituting the relevant domain of discourse. A term such as 'animal' (symbolized by 'A') will be represented as a straight line segment, the extent of which is undetermined. More precisely, the line represents the denotation of the term. Each such line segment will be labelled at its right terminus. ---------------------------------•A

Terms may be negated or unnegated (i.e., implicitly positive). In either case, their diagrammatic representation is a straight line (segment). Thus 'nonA'

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will be diagrammed as --------•nonA

Since nothing could ever satisfy both a term and its negation, their linear representations can have no point in common. In other words, the two lines representing such terms must be parallel.

--------•A ---------•nonA

This diagrams the logical truth that no A is nonA. A limiting case of a line segment is a single point. Singular terms will be represented quite naturally by such lines (point-lines). For example, a term such as 'Fido' will be diagrammed as a single point. • Fido If Fido is a dog, then we want the point-line representing Fido to be one of the points constituting the line representing dogs. If

--------•D represents the term 'dog', we will place the point representing 'Fido' at the left terminus of this line (since we have agreed to label each line at its right terminus and a point-line has no other point to its left). Fido •

-----------•0

I will now show how categorical sentences in general are represented by linear diagrams. But first a preliminary condition: a line consisting of no points is no line, so no terms are empty. Every term is represented as a line of one or more points. We have seen how to diagram a sentence such as 'Fido is a dog'. Suppose, however, that we want to diagram 'Some pet is a dog'. Here, what needs to be illustrated is the claim that there is at least one thing common to both pets and dogs. The lines for 'pet' and 'dog' must have at least one common point-they must intersect.

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• pet

Notice that we represent 'pet• and 'dog• as having a single common point. Yet, for all we know, they may have many points in common. Nonetheless, from a logical point of view, the truth claim made by 'Some pet is a dog• is just that at least one thing is both a pet and a dog. This is what we have diagrammed. Generally, then, an I categorical ('Some S is p•) will be diagrammed as I

If two lines do not intersect, then they must have no common point; they must be parallel. The contradictory of an I categorical, therefore, must be represented by parallel lines. An E categorical ('No S is p•) will be diagrammed as E

----------------•8 ----------------•P

Universal affirmations claim that whatever is denoted by the subject-term is denoted by the predicate-term. So the subject-term line must be represented as a (possibly proper) part of the predicate-term line. An A categorical ('Every Sis p•) will be represented, then, as A

s

----- • -----•P

Notice that if every S is P and every P is S, then the number of points between the right terminus ofS and the right terminus ofP will be zero. To be very clear, then, our diagrams for universal and particular 190

IT ALL ADDS UP

affinnatives are stipulated to be understood in such a way that

pennits interpretation (or reading) wherein more than oneS is P (more than one point shared by lineS and line P, and possibly all points on S are on P, and even vice versa, and

s

---•P

permits interpretation wherein all the points on Pare points on S as well. So, one line crossing another at a single point is to be interpreted to mean that at least one point is shared by the two lines. (This parallels exactly a single 'x' on a Venn diagram interpreted as 'at least one'.) And one line, sayS, partially coinciding with another, P, is to be interpreted to mean that all points on S are points on P and possibly no points on P are left over. The contradictory of an A categorical claims that at least one thing satisfies the subject-tenn but not the predicate-tenn. So, an 0 categorical ('SomeS is not P'} must be diagrammed as an S-point outside the P-line.

0

•S

-----------•P

Note that 'S' is represented as a point-line here. But, for all we know, there may be more than one S, and the line representing them may or may not be parallel to the P-line. Indeed, to say that some S is not P is to say that some Sis nonP, which can be diagrammed as

------------•P 191

CHAPTER4 The simpler diagram for 0 consists just of the P-line and the point of intersection of the S-and nonP-lines. Line diagrams represent inferences in the usual way. First, the premises are diagrammed. Then, either the conclusion has already been diagrammed or it has not. If it has already been diagrammed, then the inference is valid; if it has not, then the inference is invalid. Equivalences are immediate inferences in which each of a pair of propositions can be validly drawn from the other. For example, simple conversion equates 'Some S is P' and 'Some Pis S'. Our diagram method illustrates this by representing both sentences by a pair of intersecting s- and P-lines.

Universal negatives are likewise simply convertible. Both 'No S is P' and 'No P is S' are diagrammed by parallel S- and P-lines.

------------------------------•P

----------------------------•8 Subalternation is an example of an immediate inference between two nonequivalent statements. Any universal statement will validly entail its corresponding particular just because no term is empty. Diagrammatically, whenever there is a line there must be at least one point in that line. For example, we can derive 'SomeS is P', from 'Every S is P', where the premise is diagrammed as

s

---•P

Since every part of a line intersects (at least once) with that same line, it follows that at least one point in S is in P. Also, from 'No S is P' we can validly derive 'Some S is not P', since if lines S and Pare parallel (given the premise)

------------------------------•8

------------------------•P 192

IT ALL ADDS UP

and since every tennis nonempty (every line consists ofat least one point), there must be at least one point in lineS outside of line P. Obversion is an example of immediate inference relying on the fact that a term and its negation satisfy nothing in common (so that their line representations must be parallel). Consider, for example, 'Every Sis P'. It is diagrammed as

s

---•

•P

But we also know that 'nonP' is logically contrary to 'P', so that a nonP-line is parallel to the P-line. By adding this we have

s

- - - ---•P - - - - - - - •nonP

Since every point on P is outside of nonP, and since every point on S is on P, it follows that every point on Sis outside ofnonP. In other words, lines S and nonP are parallel (i.e., 'No S is nonP', the obverse of our premise, 'Every Sis P'). However, the true importance of obversion is seen when applied to E and 0 fonns. 'No S is P' has been diagrammed thus far as

----------•8 --------•P By obversion, 'No S is P' is equivalent to 'Every S is nonP', thus:

s

---•nonP

Likewise, 'SomeS is not P' is equivalent to 'SomeS is nonP'. Thus, both

•S

--------•P and

~

~ 193

..

•nonP

CHAPTER4

can be used to diagram an 0 categorical. Now, an obverted A statement can be converted. The resulting statement can then be obverted to yield the contrapositive of the original. The contrapositive of 'Every S is P' is 'Every nonP is nonS'. Diagrammatically, then, the full representation of 'Every S is P' must be

A (full)

s

---•---•P ---• noDP

•nonS

This represents such equivalent statements as 'Every Sis P', 'Every nonP is nonS', 'NoS is nonP', 'No nonP isS', 'No nonS is P', 'No Pis nonS', 'No S is nonS', 'No nonS is S', 'No P is nonP', and 'No nonP is P'. These last four are tautological and are instances of the law of noncontradiction. A full representation of any statement will necessarily represent the law of noncontradiction as well. Consider the I statement 'Some S is P'. Conversion and obversion on I demand that a full representation must exhibit such equivalent statements as 'Some PisS', 'SomeS is not nonP', and 'Some P is not nonS'. Thus:

I (full)

-------------•nonP

~.·P ~ S

•nonS

Universal and particular negations can also be given full representations in order to exhibit logical equivalences.

194

IT ALL ADDS UP E (full)

s

- - - - - • - - - - - •nonP - - - - - • - - - - - •nonS p

0 (full)

-----------•P

~-"' •S

"'-·nonS

Note that the law of noncontradiction is also represented (twice) by eachfu// diagram. The full representation of a categorical will always be a diagram consisting of two pairs of parallel lines. However, for most purposes of logical reckoning, the simple A, E, I, and 0 diagrams are sufficient. These are the results of"minimizing" (Gardner, 1982: 72) the full diagrams, which will usually represent far more information than we need. Here are our minimized, simple diagrams for the four categoricals. A

s

•P

E

·S •p

s•

or

•nonP

I

0

•S_ _,.p __

or

=:........, ~ • nonP •S 195

CHAPTER4

The classical simple syllogisms are easily diagrammed by my linear method-using just the simple (minimized) diagrams. The premise diagrams for the ftrSt figure are as follows (in each case the conclusion can be seen to be already diagrammed-the mark of validity): Barbara: Every M is P Every Sis M So every S is P Here, the major is diagrammed first as

M - - - - - • - - - - - - •P

..

The minor is then added to get

s •

----·

- - - - .P

From this the conclusion can be read directly. Darii:

Every M is P SomeS isM So someS is P

We diagram the major as M

- - - - - • - - - - - •P

The minor is then added:

~:----•P The conclusion is already diagrammed, since the S- and P-lines intersect. Celarent: No M is P Every Sis M SonoSisP The major is diagrammed as M - - - - - • - - - - - •nonP

196

IT ALL ADDS UP

Adding the minor, we get 8 M •nonP -----·-----·-----

Ferio:

No M is P SomeS isM So some S is not P

The major is M - - - - - - - • - - - - - - - - •nonP

Adding the minor:

··~ ~------- •nonP ~•S Similar diagrams can easily be constructed for all24 valid classical syllogisms from AAA-1 to EI0-4. The method is complete. It is sound as well. An exhaustive (and exhausting) check of each of the 232 classical invalid forms shows that none is valid by my method. The diagram method (whether planar or linear) is most effective in determining validity and in discovering missing premises or conclusions. Consider the premise pair 'No P is M' and 'Some M is S'. These are diagrammed linearly simply as

------------•P

We can readily see that the inference of 'Every PisS' from this is invalid. But we can also see what the missing conclusion is-'Some Sis nonP'. Enthymemes with missing premises are most easily resolved by

197

CHAPTER4

diagramming the explicit premise along with the contradictory of the conclusion. What follows, then, will be the contradictory of the missing premise. For example, let the explicit premise be 'Every A is 8' and the conclusion be 'Some C is not A'. We first diagram the premise as A - - - - - • - - - - - •B

Next, we add the contradictory of the conclusion (viz., 'Every Cis A'). C

A

-----•----•----•B From this we conclude 'Every C is 8', which is the contradictory of the tacit premise 'Some C is not 8'. Thus far, we have seen that linear diagrams can do virtually what planar diagrams can do. One minor advantage they may enjoy is that they are faster to construct (since lines and points are easier to draw than circles, squares, ellipses, etc.). But their major advantage is their ability to represent inferences involving relatively large numbers of terms (viz., more than four) without destroying the original simplicity of the diagrams. Here is an example of a relatively elementary valid argument that Venn diagrams are powerless to represent in a simple, perspicuous manner. Every A is 8 Every 8 is C NoCisD Some Dis E So some E is not A Diagramming the first premise gives us A - - - - - • - - - - - •B

Adding the second: A

8

-----•---• ---•C

Adding the third:

A

B

---•---•---•C

----------•0 198

IT ALL ADDS UP

Finally, we add the fourth: A

B

-----•-----•-----•C

That's all-three lines, five labels. The conclusion is already diagrammed. Sorites of any number of terms can be diagrammed using the linear method. The geometric restrictions on closed plane figures, which prevent perspicuous representations involving more than four terms using simple continuous figures, do not apply to the simpler linear figures. Identity statements are easily diagrammed by my method. A statement of the form' A is (identical to) 8' claims that the A-point is on the 8-line. Since the 8-line is a point-line, this means that 'A' and '8' label the same point: An argument such as the following: Tully is Cicero Cicero is Roman So Tully is Roman would be diagrammed as

c T

-------------------•R

where, by the first premise, 'C' and 'T' label the same point-line. The simple diagrams outlined above can be extended to represent relational propositions. I shall make some tentative suggestions as to how this can be done. The key idea here is to see relational expressions as terms. Consider a simple relational proposition such as 'Paris loves Helen'. The claim here is that Paris is among the things that love Helen. Let us diagram these by a line labelled 'loves Helen'.

- - - - - - - - - · l o v e s Helen

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Paris is one of them. So: Paris • ------•loves Helen

'Some person loves Helen' would be diagrammed as an I categorical.

•Penon

(Notice that the converse here is 'Something that loves Helen is a person'.) So far, 'loves Helen' is treated just like any other term. It is represented by a single line segment representing things that love Helen. Often in inferences, however, it is necessary to analyse such relational terms, abstracting from them one or more of their constituents for independent treatment. Consider the argument 'Someone loves Helen, but Helen is a vamp. So someone loves a vamp'. The first premise can be diagrammed as

But now, in order to diagram the second premise, we need a way to represent the relative term, 'loves', and the object term, 'Helen', separately. Let us represent relative terms by arrows (indicating the "direction" of the relation) connecting their relata. Then 'loves', in this case, would be represented by an arrow from lovers of Helen to Helen.

~---.=....::"'~------•loves •Penon

Helen

+lovu •Helen

'Helen' is now extracted from the relational'loves Helen' and we are free

200

IT ALL ADDS UP

to diagram our second premise, 'Helen is a vamp'.

~-__::~-----•loves """"•Person

Helen

~loves

•-------•Vamp Helen

Given that from this we can see that every lover of Helen loves Helen, the conclusion, 'Someone loves a vamp', can be read directly. Look once more at our diagram for 'Paris loves Helen'. Paris • - - - - - - •loves Helen

We know that whatever loves Helen loves Helen. In fact, we can say, generally: Whatever is r to some/every X is r to some/every X, i.e.:

F.,.,,,

- - - •r to some x

--'·'-r-•x

----•x

It is this tautology that permits the tautological

-------------•loves Helen

+loves 0

Helen so that 'Paris loves Helen' can be rendered as Paris• --------------•loves Helen

+loves

•

Helen

201

CHAPTER4

Thisfo// representation, however, diagrams more information than we often need in computing logical inferences. But it can be simplified (by suppressing tautological information, as with categoricals in general) to the more natural looking

or simply

If we agree to read arrows in reverse direction as representing the converses of the relations they represent when read in their indicated direction, we can take the preceding diagram to represent both 'Paris loves Helen' and 'Helen is loved by Paris'. This same process can be used to simplify diagrams for statements such as 'Some man loves some woman'. ltsfo/1 representation is

Simplifying, by suppressing tautological information, we get

Notice that the locations of the end points of a relational arrow indicate the quantities of the relata. The quantity is universal when the relational arrow meets the term-line at the line's right terminus; otherwise the term is particular in quantity. For example, 'Some A is r to some 8' is diagrammed as ------•A

~·

------•8 202

IT ALL ADDS UP

'Every A is r to some B' is diagrammed as

___t. .

·A_r_ _ •B

'Every A is r to every B' is diagrammed as ------•A

f

------•B And 'Some A is r to every B' is diagrammed as:

------·•B In spite of these simplifications, it is important to remember that in the context of a given logical inference it may be necessary to restore some or all of the full relational expressions. This is especially so when those relational expressions occur as logical subjects in subsequent premises or conclusions. For example, a proposition such as 'Some senator gives away some money' could be diagrammed in one of three ways: (i)

--~--·S ~. .M

__l..____

(ii)

(iii)

~•gaomeM ~•S -""--.,..,.__ _ _ _ •g some M

""'·___ S _,J.,_ Wg __ 203

•M

CHAPTER4

We can use (i) if the relative tenn 'gives away' occurs elsewhere without the logical object 'money'. Thus, suppose the second premise is 'All money is tainted'. Our premises are diagrammed, then, as (i.l)

--,----•8

____t~·-·--------•T M from which we might conclude 'Some senator gives away something tainted'. We can use (ii), however, whenever the analysis of the relational expression 'gives away some money' is not demanded by any subsequent premise or conclusion. Suppose our second premise is 'Whoever gives away some money is generous'. Then we can diagram our premises as (ii.l)

And from this we could read the conclusion 'Some senator is generous'. Finally, we are in need of the .full representation, (iii), when the relational expression occurs subsequently both analysed and unanalysed. For example, suppose our second premise is 'All money is tainted', our third premise is 'Whoever gives away some money is generous', and our fourth premise is 'Whoever is generous loses some money'. The conclusion, 'Some senator loses something tainted', is diagrammed by diagramming the premises thus:

(iii.t-'-)_ _ _ _c ______ .p

f. -

204

----T

M

IT ALL ADDS UP

Consider next the famous inference 'Every circle is a figure. So whoever draws a circle draws a figure'. The premise is easy enough:

c

--------·

- - - - - - - - •F

We also know that whoever draws a circle draws a circle. So: - - - - - • d some C

___i"--d--•C Together, these give us

_f~-e-C----•F c

And here we can see 'Whoever draws a circle draws a figure'. We can develop a useful general rule out of the preceding example, call it Rule R, based on four kinds of cases. These cases are argument forms that each have the premise 'Every X is Y'. Then, two kinds of conclusions occur with four variations in the remaining premise, as follows: Case 1: Every X is Y Some S is R to some X I So some S is R to some Y Case 2: Every X is Y Some S is R to every X I So some S is R to some Y Case 3: Every X is Y Every S is R to some X I So every S is R to some Y Case 4: Every X is Y Every S is R to every X I So every S is R to some Y So, in general, the conclusion has one and the same grammatical predicate, ' ... is R to some Y', even when the same predicate in the second premise

205

CHAPTER4

has an 'every' in it. All of these cases seem valid intuitively. Each is confirmed to be valid when placed on linear diagrams: Case 1:

~""s f~_x___ .y X

Case 2:

':.., """'·S

r, ....,-X to

t ____

.y

X Case 3:

s

----•----~rmaomeX

---~-•

Case4:

X

----•s ----i:

•Y

to every X

'-----•Y X

It is important to note that as obviously valid as cases 2 and 4 appear in our diagrams, they are not valid in the predicate calculus. To prove them valid in the predicate calculus would require adding a premise tantamount to the existential import embedded in our diagram method (viz., every line has at least one point). The premises required for 2 and 4 are 'There are X's' or 'There are Y's'. Now, Rule R is simply the generalization from these cases-to wit: Rule R: If every X is Y, then whatever is R to some/every X is R to some Y. Diagrammatically, we state it as follows: 206

IT ALL ADDS UP

RuleR:

r to 1ome/every X

r to some Y

---------·---------· .............. ..···· r ···...

Jf/

···~-•Y X

Thus, RuleR penn its extending the line for 'is R to some/every X' so that it is a sub-portion of'is r to some Y', so that the top line above can be read 'Everything that is R to some/every X is R to some Y'. Now let us diagram the proposition 'Paris gave a rose to Helen' as p., _____________ • g some R to H

I -------1------•R to

I have arrived at this representation in the following way. First, diagram the proposition simply as p., ________ • g some

R to H

Next, analyse the relational expression 'gave a rose to Helen' as a relative tenn, 'gave a rose to', and its logical object, 'Helen':

P• - - - - - - - •g some R to H

~gsomcltto •H

207

CHAPTER4

Now, the relative tenn 'gave a rose to' is itself a relational expression. We analyse it as a relative tenn, 'gave ... to', and its logical object, 'a rose'. Thus: P• - - - - - - - . • g some R. to H g ... to

----+----

•R.

•H

Here, the arrow labelled 'gave a rose to' has been replaced by

I· .. to -----'---•R

Since the vertical line is now only a part of the arrow from 'gave a rose to Helen' to 'Helen', we can re-label the arrow segments to give us: P• - - - - - - • g some R to H

g

---+---•R to

As before, we can often, unless the context demands otherwise, simplify this as:

•P g

•R.

to •H

208

ITAL~SUP

And this diagrams as well all th~ equivalent converse relationals, passive transforms, such as 'Helen was g1ven a rose by Paris', 'A rose was given to Helen by Paris', 'Helen was given by Paris a rose', and 'A rose was given by Paris to Helen'. In diagramming relational expressions, it must be kept in mind that ultimately the entire arrow represents the relation. This is especially important for relations that are not usually expressed in natural language by multi-word terms. Consider. for example. 'Paris is between a rock and a hard place'. This can be simply diagrammed as

•P

But there is no way to label independently the arrow segments. The entire arrow represents the 'between' relation. Perhaps the most perspicuous diagram would thus be one that labels all such arrow segments by 'between':

•P b

-------4--------•R b

------~---------·H Our analysis of relationals with more than one object is useful for diagramming such inferences as

Some boy gave a rose to a girl. Every rose is a flower. Every girl is a child. So some boy gave a child a flower. The first premise is diagrammed (simply) as ------~---------·8 g

------~~-------•R to

------~~-------·0

209

CHAPTER4

Adding the next two premises gives us

---.------•8 8 R

-----•F

to G -~----•------•C

The conclusion is read directly. Of course, when entire relational expressions occur more than once in an argument, we are often required to make use of their unanalysed representations. Consider: Every boy loves some girl. Every girl adores every cat. Whoever adores every cat is a fool. So every boy loves a fool. Here 'loves some girl' requires analysis, while 'adores every cat' does not. So we diagram the first premise as

--------•B

_____.Y'---_·o Adding the second premise yields

-----•B

tl

____.f.____ G•----· a every C (We could add the analysis of 'adores every cat', viz.:

~·

-------•C 210

IT ALL ADDS UP

but it is unnecessary in this context.) Finally, we add the last premise:

---•B

_1_?--aeveryC

•---·F

Let us consider one last example: Every boy loves some girl. Every girl adores every cat. Every cat is mangy. Whoever adores something mangy is a fool. So every boy loves some fool. In this case, unlike the preceding one. both relational expressions must be analysed. But, as we will see, 'adores every cat' must have an unanalysed representation as well. The first premise is diagrammed as

-------------·8

t.

----------~t_____ •G Adding the second premise, we have ---•B

_l_?_~ t~c

____

The third premise gives us ---•B

_l_?~ J_•_ev_e_ry__C---•M -------c211

CHAPTER4

Now, the fourth premise is diagrammed as asomeM

----------•----------•F Clearly what is missing to connect the two diagrams is a representation of 'Whatever adores a cat adores something mangy'. And, in fact, this does hold, given the third premise and our Rule R, for Rule R explicitly sanctions changing a diagram like

into aeveryC - - - - - - - • -------·• some M

~·

---------·~----•M

c

So, instead of trying to add an F-line to theM-line (a mistake, since 'Every M is an F' is not a premise), we change the last diagram for this argument into the following (which does not add more information and is merely a rearrangement): -----•B

j_~___•_e_v~

__c___

~·

•aoomeM

- - - - • ----•M

c

For then we can add a representation of 'Every adorer of some M (mangything) is an F (fool)' correctly and, thereby, show how the conclusion, 212

IT ALL ADDS UP

'Every B (boy) loves some F (fool)' is already represented:

---•B

j_?_ ~·

a every C a some M •---• •F

---c---·M

And notice that this final diagram also contains a very clear representation of the intermediate conclusion that many might think is the most important one for the argument-namely, 'Every girl adores something mangy'. We have adopted the convention that any arrow that touches the right terminus of a line touches every point on that line; an arrow that touches a line only at a point left of the right terminus touches that line at no other point. Yet the presence of repeated references raises the possibility of counter-examples to the second part of our convention. Consider the statement 'Some barber shaves some barber'. If we diagram this using 'Some boy loves some girl' as a model-that is, ---.,----•B

!.

_ _ _t.......__ _ _ •G

we get this: ---~---•B,

---L-1

1

_ • 82

where 8 1 represents shaving barbers and 8 2 represents shaved barbers. But clearly, 8 1 and Bz must denote the same objects. After all, if even just one barber shaves himself then the statement is true. In fact, 8 1 and 8 2 must be identical lines. Let us represent barbers, then, by a single line (as we have always done). Then our statement can be diagrammed by drawing an arrow from one nonterminal point to another. Thus: s

213

CHAPTER4

Here the distance between the base and the head of the arrow may be equal to or greater than zero; equal to zero if some barber shaves himself. This last suggests that we have a preliminary way of diagramming the statement 'Some barber shaves himsetr: s

and the statement 'Every barber shaves himsetr: s

0 •8

But how, given our right terminus convention, do we diagram 'Every barber shaves every barber'? What is required for us to distinguish diagrammatically between statements of the form 'Every X is R to every X' and 'Every X is R to itsetr is an additional convention. Our new convention will involve the representation of pronominal expressions. Consider the inference 'Some girl is loved by every boy. She is lucky. So some boy loves something that is lucky'. Here, specific reference has been made to some girl (i.e., to a certain girl rather than to some girl or other). Then a pronoun is used to make subsequent reference to that very girl. We don't know her name, but we can give her an arbitrary, variable one ('she'). Let us agree to label every unnamed individual to which specific reference is made with a small roman numeral. That label can then represent the pronoun in subsequent, anaphoric pronominal references. For our sample inference, we can diagram the first premise initially as

But, since the reference to some girl is specific, we will label it: 214

IT ALL ADDS UP

r

----•----·0 i The second premise says that she (1, the girl loved by every boy) is lucky. Adding this yields

•G

Consider next the argument 'Some boy loves a girl. She hates him. So he loves a hater of him'. We diagram the first premise, adding the pronoun labels for subsequent use:

-------·----------·8

t·

---------·----------•0 ii Adding the second premise, we get

ii

The conclusion is diagrammed here once we recognize that 'hates him'-that is, i

•

215

C~4

is a simplification of'is a hater ofhim'-that is, i

•

___t: This same simplification allows us to diagram 'Some boy loves every girl. They hate him. So some boy loves a hater of him':

Next, consider the statement 'Every owner of a donkey beats it' .7 Analytically, every owner of a donkey owns a donkey: - - - - - . • o some D

L ------~t~-----•D

It, the donkey so owned, is beaten by its owner. So

We are adopting the convention of labelling individuals to which specific reference is made by small roman numerals, which are then used to represent subsequent pronominal references. This convention permits us to diagram, now, 'Every barber shaves every barber' and 'Every barber shaves 216

IT ALL ADDS UP

himself as follows: Every barber shaves every barber: s

Q Every barber shaves himself:

s

Q i

Now, since 'Every barber shaves every barber' entails 'Every barber shaves himself, we might justify this by a rule (called "i-insertion" by an anonymous reader) analogous to existential instantiation in the standard predicate calculus. Such a rule allows us to pronominalize at will by marking any point on a given diagram line with a roman numeral. It must be noted that once a pronoun is so marked it cannot subsequently be ignored (otherwise one might derive, e.g., 'Every barber shaves every barber' from 'Every barber shaves himself). We would diagram the valid argument 'Some barber shaves every man. Every barber is a man. So some barber shaves himself as follows. Second premise: B

-----------•·----------•M

Adding the first premise: s

(i.e., 'Some barber shaves every man'). Given our pronominal convention (i-insertion) and our right terminus convention, we can read the conclusion directly from

217

CHAPTER4

s

_f.~ B

In particular, the latter convention permits movement at the arrowhead to the left (with the i-point as its left limit). In other words,

0

----•----------- • M

i

B

Next, consider the statement 'lago is a hater of a lover of Desdemona'. Leaving 'hater of a lover of Desdemona' unanalysed, we have

I• - - - - - - •h some I D Here lago (I) is an individual member of the set of things that hate (h) some lover of (I) Desdemona (D). We could analyse the relation here first as

1·--------•hsomelD

_L•ID where

t. -----*L----•10 represents 'hates a lover of Desdemona'. Further analysis yields

218

IT ALL ADDS UP I•

-------~:oome I D

____.t___ •l D

~·

•D

By suppressing some of the analytic content here, we could simplify this as I•

~b

____.f____ •ID

~·

.D

Finally, consider the argument 'Every lover of an adorer of a cat is a fool. Every boy loves some girl. Every girl adores some cat. So every boy is a fool'. We diagram the second premise simply as ----•B

_1_.0 There is no need to analyse 'adores some cat', so we can add the representation of the third premise to give us

-----•B

I. .

_ L 0 - - - - - • a some C Now, by Rule R, we can add 219

CHAPTER4

B

- - - - - • - - - - - - •I some a some C

~•-----•asomeC G

Finally, we add the first premise to get

8 ~

laome a some C

l.

•------•F

----•asomeC

G

from which we can read the conclusion directly. Let us note (under the prompting of our anonymous but friendly reader) that negative relationals can be diagrammed by our method. First, we must recognize that, as with nonrelational terms, relational terms have corresponding negatives. Consider 'Some boy does not like every vegetable'. Given no contextual clues, the sentence is ambiguous (in at least two ways), between (I) 'Some boy dislikes every vegetable' and (2) 'Some boy fails to like every vegetable'. The difference here is due to the scope of the negation. In (1) only the relational term 'likes' is negated (i.e., 'Some boy does not-like every vegetable'). In (2) the entire relational predicate 'like every vegetable' is negated (i.e., 'Some boy does not-(like every vegetable)'), an 0 form. We can think of'dislike' as the logical contrary of 'like'. Such relationals are diagrammed just as nonnegative relationals are diagrammed. Thus, (I) can be diagrammed as

-p•B _ _ _ ____;.y

(2) can be diagrammed as a simple 0 categorical:

- - - - - - - - - • 1 every V 220

IT ALL ADDS UP

The real value of recognizing negative relationals is seen when we approach inferences in which such expressions play logically effective roles. Consider the argument 'Paris loves Helen. Every Greek fails to love every Trojan. Helen is a Trojan. So Paris is not a Greek'. The third premise is diagrammed as (i)

H·------•T

Now (without accounting for the relations between contrary and contradictory negative relations) we can diagram the second premise (along with the third) as (ii)

-------·0

k•l

H• _ _ _ _ _ _ _

Adding the first premise. we have

(iii)

1_ f·' 1

__

At this point, the question naturally arises concerning how we know that the P-point is not on the G-line. For if it is not, then our conclusion is diagrammed. First, note that the I-line and the non I-line are (indeed, must be) parallel. Now, we could not diagram the P-point on the G-line without contradicting the first premise. So, the only way to diagram all three premises consistently is to keep the P-point off the G-line. Diagram (iii) represents the simplest and most perspicuous way of doing this, and from it the conclusion is easily read. Let us tum now to the diagramming of compound statements. Compound

221

CHAPTER4

statements offonns 'p and q', 'p and not q', 'ifp then q' and 'neither p nor q' could be exhibited using Venn-type diagrams, where the circles are labelled 'p' and 'q' and each is taken to represent all states of the world for which the labelling statement is true. Truth tables and truth trees have generally supplanted the use of diagrams for analysing inferences involving compound statements. Nonetheless, my linear diagram system can extend to compounds, pennitting an easy geometric representation for inferences involving such statements. Besides, the initial idea that there ought to be a single diagram system for all sorts of inferences is a sound one and mirrors my Sommersian claim that a single logic oftenns suffices for both analysed and unanalysed statements. We begin with some preliminaries. Noncompound statements (symbolized by appropriate lower-case letters) will be diagrammed by line segments labelled at their right tennini by statement symbols. Thus, for example, a statement symbolized by 'p' will be diagrammed as

----------------•P Each point on the line could be taken as a p-state. Again, to assert or state 'p' is to claim that the state(s) that make it true (the p-states) are among the states that obtain-that is, characterize the domain of discourse-the world. Thus, to assert both 'p' and 'q'-that is, to assert 'p&q'-is to assert that at least one of the states that obtains is both a p-state and a q-state. In other words, 'p&q' is true whenever the p- and q-lines intersect:

What is wanted, of course, is a way to represent states that obtain. Let us imagine that all line segments are in a single plane. For practical purposes, we can assume that the plane is finite in area so that it has a central point (it is easiest to think of the plane as having a circular perimeter). Assume as well that the plane is bisected along its vertical axis, resulting in a positive (right) sub-plane and a negative (left) sub-plane. Let us label the central point 'T', and think of lines extending from T to the right at any angle except ninety degrees vertically. We will call the right sub-plane of our plane the 'T-field'. Let 'F' also label the central point. Lines from F extend to the left in just the way that lines extend to the right from T. We will call the left sub-plane the 'F-field'. Twill represent those states that obtain. To make a truth claim by the use of a statement is to claim that the

222

IT ALL ADDS UP

state of affairs expressed (signified) by that statement obtains (i.e., that the statement is true, that the line representing it lies in the T-field). Tis truth; F, naturally, is falsehood. In the left, or F-field, lines will be labelled at their left rather than right terminus. A very rough picture of our plane would look like this:

The important vertical line here is the left limit of the T-field and the right limit of the F-field. We can, for convenience, think of it as a single line that is the T-line or the F-line. To make a statement using 'p' is to claim that 'p' is true. Thus:

(Here we take the mid-point of the vertical to be the T-point.) To assert that p and q is to claim that 'p&q' is true, that Tis on each line, and that both lines intersect at T:

To deny a conjunction (e.g., -(p&q)) is to claim that no p-state is a q-state, that the p-line and q-line do not intersect, that they are parallel:

223

CH~4

_ _ _ _ _ _ _ •p

_ _ _ _ _ _ _ •q

Since no statement and its negation are ever both true together, the law of noncontradiction guarantees that for any 'p' _ _ _ _ _ _ _ •p

-------·-q Excluded Middle is guaranteed by the bifurcation of our plane, so that any line segment must lie in either the T-field or the F-field, either to the left or to the right of the vertical. To say that two line segments are parallel is not to suggest that they lie in the same field. Since every line segment in the T-field must have T as its left terminus (and F as its right terminus if in the left field), no two line segments in a given field are ever parallel. Any two parallel line segments must, in fact, be colinear but not coplanar; they must lie in opposite fields. Nonetheless, we will adopt the convenient convention of representing parallel lines in the usual way unless their truth-values are known. Thus, in general we know that

_______ .,

--------·-q But, knowing that 'p' is true-that is,

we likewise know that '-p' is false-that is,

224

IT ALL ADDS UP

This suggests that we can represent the negation of a statement, say 'p', in two equivalent ways:

The law of double negation can be diagrammed in three steps:

(I) • --p

(2)

-p·

(3) •p

With negation and conjunction, we could define all other possible truthfunctional connectives. But let us introduce the conditional on its own. To claim that q on the condition that p, or 'p::~ q', is to claim that all p-states are q-states. So we can diagram such a conditional as

----·'----•q (Notice how the common diagrams for I categoricals and conjunctive statements, and for A categoricals and conditional statements, reinforce Sommers's claim that the logic of compound statements is part of a logic of

225

CH~4

terms.) I now introduce some other important principles of equivalence, which will apply to our diagrammatic analysis of inferences to come. We saw above that 1--------•p

and

-p·-------1

are equivalent. If we take T and F to be contradictions of one another, then we have, in effect, allowed a pair of points on a line segment to exchange places while reversing their signs of polarity (negatedlunnegated). Indeed, this is just the principle of contraposition:

__f __.q

-

_ _-.q_ _ •-p

which holds for any pair of points on a given line segment. Another principle governing my diagrams holds that any pair of intersecting line segments can each rotate by any degree around the point of intersection so that they exchange relative positions. So, in general,

x: x: -

and specifically,

v·p ~·q

v·q

-

~·p 226

IT ALL ADDS UP

This "principle of rotation" guarantees commutability for conjunction. Finally, a third "principle of composition/decomposition" holds that

v·p

"'·q

-

This principle guarantees laws of conjunctive addition and simplification. It also guarantees conjunctive association, since the following three are equivalent:

•p •q

•r

L---·p r----•(q&r) We will now examine some examples of inferences in order to illustrate how they are analysed by the use of linear diagrams. Again, the diagramming of all premises is followed by an inspection ofthe resulting diagram to see whether or not the conclusion has been diagrammed already. If so, the inference is valid; otherwise. it is invalid. A simple modus ponens argument form is easily diagrammed to show its validity. First, we diagram the unconditional premise, •p•, as

227

CH~4

The conditional premise demands that all p-states be q-states, so that the p-line must be a (possibly proper) part of the q-line. Adding this gives us

Since the T-point extends at virtually any angle to the right, 'q' is on a line intersecting T. So the conclusion, 'q', has already been diagrammed. While the diagram for a modus ponens argument form falls in the T-field, one for a modus to/lens argument form will fall in the F-field. The premise '-q' can be diagrammed in two equivalent ways:

(i)~·-q (ll)q·~ We choose the second way in this case, since it allows us to add the diagram for the conditional premise in an easy and obvious way:

(Using (i) and contraposition would not have meant much more difficulty.) Consider next an example of affirming the consequent. Diagramming our unconditional, 'q', we have

228

IT ALL ADDS UP

According to the conditional premise every p-state is a q-state, so that the p-line must be a part of the q-line. So, either

p·+~

. ~f

••

But, as we cannot know which, we cannot have already diagrammed the conclusion. A second way to illustrate the invalidity of affirming the con~ sequent would be to diagram the unconditional and then the contrapositive of the conditional-that is,

-----•q

-----·-----•p -q Since the 'q' and '-q' lines must be parallel, the conclusion cannot be diagrammed. Hypothetical syllogisms are diagrammed in an especially simple and obvious way. Let 'p::>q' be our first premise.

p

-----· -----·q Adding our second conditional premise, 'q::> r', gives us p

q

- - - · • - - - • - - - •r

And we see at once that our conclusion, 'p::>r', is already diagrammed. Let us look now at an argument of the form pvq ~

q

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CHAPTER4

First, we paraphrase the disjunction as a conditional ('-p=> q'). Our diagram, then, is a familiar one:

-p ~------------·------------·q

illustrating validity. Let us consider next a (slightly) more complex example: p & (r v q) -pv-r

-q First, we diagram the conjunctive premise as

•(r v q)

Next, the second premise is paraphrased as a conditional, 'p=> -r', and diagrammed:

·------·-r p

•(rv q}

Paraphrasing 'r v q' as '-r=>q' now requires

230

IT ALL ADDS UP

p

-r

r-----·------·--------·q

which shows that the argument is invalid, since the conclusion is not -indeed, cannot be-diagrammed. Finally, one last example: (p&q)=>r -r p

The third premise is diagrammed as

The second premise is diagrammed as

which, added to the first premise, yields

•p

•-r

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CHAPTER4

The first premise can be added to give us

•p

-~ •r which we know is equivalent by contraposition to

., • -r ------·-(p&q)

From this we know, by modus ponens, that

1-------·p r-----•-(p&q)

Thus the p- and q-lines must be parallel. So:

··-+-·· The argument is valid since the conclusion has been diagrammed. Unlike the case for syllogistic, there have actually been a number of attempts to use linear diagrams of one sort or another to display and analyse inferences involving compound statements. A truth tree, for example, could be viewed as a kind of linear diagram. Frege•s well-known

232

IT ALL ADDS UP

but rarely used ideography introduced in the Begriffschrift (cf. Murawski, 1988-89), while not meant to serve as a method for diagramming, could, however, be taken as another example. As we have seen, the system of symbolization for logic invented by Frege was never, in fact, accepted by anyone else. Peano's scheme became the dominant system of notation for the majority of mathematical logicians; Russell's popularity and the influence of Principia Mathematica, which made use ofPeano's notation, were mainly responsible for this. But Frege's attitude toward his critics and his excessive pride in his system also contributed to the general refusal by others to use it. Most of those who considered Frege's symbolization claimed to reject it because of its great complexity and because, they often said, it was too difficult to print. Peano's system, in contrast, was simple and was easily printed. Nonetheless, Frege's system owed much of its apparent complexity to the fact that, unlike alternatives, it was not only a system oflogical notation but, simultaneously, a system of illustration of the content of every statement using horizontal and vertical lines. Suffice it to say, when seeking clarity (and support) it is probably best to keep the tasks of symbolization and diagrammatic analysis separate. Let us review very briefly Frege's way of symbolizing (i.e., diagramming in the sense mentioned above) statements and their compounds. First, he makes use of negation and conditionalization as his only primitives. Each statement is represented as a horizontal line (as in my system), and the sign of negation is a small vertical line orthogonal to and below the statement line. The sign of assertion is a vertical line to which the statement line is perpendicular at its left terminus. It resembles my T-line. Subordinate sentences (antecedents) are subtended to main sentences (consequents). Here is a comparison of some ofFrege's diagrams with my own. Formula

Frege's

Mine

p

-p

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CHAPTER4

p

q

--·--· pvq (-p => q)

~p

-p

q

--·--· or -q

p

--·--· p&q ( -(p => -q))

k_p

K·p •q

p

----. q

(Note that in these diagrams Frege has need only for unnegated sentences. This is because he takes the positive as neutral and negation as an operation on the neutral.) One of the advantages of my system of linear diagrams is that it allows a simple and uniform diagrammatic method for all kinds of inferences (i.e., those involving categoricals, singulars. relationals, or compounds). I conclude by showing how the two kinds of linear diagrams (those used for analysed statements and those used for unanalysed, compound statements) can be integrated. To do this I will offer an inference that, when appropriately analysed, involves a simple categorical, singulars, relationals, and a compound statement. The inference is Every logician is wise. Ed is a logician. Some mathematician admires whatever is wise. If Ed is admired by any mathematician, then some 234

IT ALL ADDS UP

mathematician is a fool. Therefore, some mathematician is a fool. The first three premises are diagrammed together as (i)

For now, let us diagram the last premise, the conditional, as (ii)

p

- - - - - · -----··q Using the rules and conventions established for diagramming analysed statements (especially: if an arrow terminus touches a line it touches every point on that line to the left, and a converse of a relational results from reading its terms in an order other than tail-to-head), we can conclude from (i):

(i.e., 'Ed is admired by some mathematician'). Indeed, since all the premises in (i) are asserted (claimed as true), we can add the T-line to (i), and therefore to (iii) as well. Since (iii) is the antecedent of (ii)-that is, (iii. I)

235

CHAPTER4

by modus ponens from (iii. I) and (ii) we get

(iv)

our conclusion.

236

Notes for Chapter 4

A substantial portion of this section comprises Englebretsen ( 1992). It appears here by permission of the editors of the Notre Dame Journal of Formal Logic and the University of Notre Dame. 2 See especially Abeles ( 1991 ), Armstrong and Howe (1990), De Morgan (1966, esp. "On the Syllogism II"), Edwards (1989), Gardner and Haray (1988), Hubbeling (1965), Kamaugh (1953), Keynes ( 1906), Marquand ( 1883 ), Peirce ( 193 1-3 5, "Existential Graphs" in vol. 4), Shin (1995), Smyth (1971), and, of course, Venn (1971). A thorough survey of diagram systems for logic is found in Gardner (1982). A system of diagrams motivated by purely ontological rather than logical considerations is found in Smith (1992). 3 Discussions of possible Aristotelian diagrams are found in Ross (1965. 30 1-2) and in Kneale and Kneale ( 1962: 71-72). See also Flannery (1987). 4 See Leibniz (1966b: 292-324). 5 Lambert (1965: Ill-50). Critiques of Lambert's system are found in Venn (1971: 504-27), Peirce ("Existential Graphs" in 1931-35, and Keynes (1906: 243-47). 6 Rybak and Rybak ( 1976, 1984a, 1984b) use a version of Kamaugh Maps to analyse inferences involving more than four terms, singular terms, compounds, and relationals. However, their system is far from simple and perspicuous, requiring arguments with many terms to be "split" (1976: 473), large maps to contain discontinuous cells (1976: 472), and arguments with relationals to require, in addition to a series of supplementary rules, a complex "streaming procedure" ( 1984b: 269-71 ). As an example of how my system compares favourably to the Rybaks', consider their diagram (1984b: 272, fig. I) for the argument: 'Some botanists are eccentric women. Some botanists do not like any eccentric person. Therefore, some botanists are not liked by all botanists.' The linear diagram is simpler, faster to construct, and far easier to read: 1

--~--------~-·8

Band v/

7

nonl

Geach (1962: 116ft) has an important discussion of such sentences.

237

CONCLUSION Logic, like whiskey, loses its beneficial effect when taken in too large quantities. Lord Dunsany

Logic is logic. That's all I say. Oliver Wendell Holmes

Georgie, put down the book and turn out the light. It's time to sleep. Author's mother

In introducing this essay, I briefly summarized Geach's attack on Aristotelian logic. As a committed Fregean, Geach is quite correct in his insistence that his preferred theory has been challenged by the persistence of a logic that was based on a theory of logical syntax radically different from Frege's. In calling the old logic a "two term" theory, Geach rightly sees that at the heart of that logic is a syntactic theory that isolates in any given statement a pair of terms that are interchangeable, logically homogeneous (what he calls "Aristotle's thesis of interchangeability"). The fact is that the radical syntactic difference between term logic and modem predicate logic is due primarily to the difference between a logic that sees any statement as a pair of grammatically distinct but logically homogeneous terms bound together by a logical copula (unsplit in the case of the ternary theory; split for the quaternary theory) and a logic that sees any statement that is not composed of one or more other statements as a pair of terms that are not only grammatically but logically heterogeneous (viz., noun-namesubject and verb-predicate), fit to unite with one another in the absence of any kind of logical connector-that is, the binary theory. These two very different theories of logical syntax underlie two very different systems of formal logic, each with its own distinct symbolic algorithm. It is tempting to see these differences as reflective of a radical difference between two philosophies of logic and language. But the fact that, for the most part, it is possible to translate back and forth between formulae in one system and formulae in the other, the fact that a very large

239

CONCLUSION

number of logical tasks can be carried out in either logic, and the fact that a number of logicians have felt and even seen significant affinities between the two, counsel against such a radical divide. Still. the differences are important. Standard predicate logic may very well be the best tool available for exploring or establishing the foundations of mathematics. A logic of terms makes no claims here, yet what can be claimed on behalf of a term logic such as the one outlined in this essay is that for those areas where both it and modem predicate logic claim to exhibit the forms of logical expression and inference-rational uses of natural language-it is simpler, more natural, and more powerful. The plus-minus calculus, in particular, achieves an enormous advantage of simplicity over the standard alternative by trading on its adoption of simple, familiar principles of arithmetic and algebra. For some time now. neo-Fregean philosophers and logicians have tried to convince us that. in spite of the wide gulf between natural syntax and the syntax embodied in the first-order predicate calculus with identity, the underlying logic of natural language happens to be that standard logic. This view is urged on the basis of no good arguments, in the face of appearances and common intuition, and the results of the best linguistic understanding to the contrary. The system of logic begun by Frege and perfected by a host of able mathematicians and logicians throughout the twentieth century is a thing of beauty and power. Few. as yet, wish to see it either dismantled or discarded. But what is reasonable is the wish for a broader recognition of its limitations and, more importantly, a larger measure of attention to the still insufficiently understood or appreciated strengths of term logic. While this essay has tried to place the results of Sommers's work toward a fully articulated logic of terms within a broad historical perspective, I have no wish to see a return to the "bad old days" of traditional syllogistic as taught in Geach's "Colleges of Unreason." Our fear, however. is that many see today's versions of term logic as little more than slightly polished versions of the old logic. Yet Sommers has built a term logic that, while borrowing from and building on traditional logic, goes far beyond it in every important way. This essay has been an attempt to show this. The process of bringing a logic of terms, a revitalized syllogistic, to perfection is far from complete. Indeed, there is some indication that interest in such studies is just now beginning to increase as we leave both the nineteenth and twentieth centuries behind. First of all, there is Quine's Predicate Functor Algebra. Though Quine sees it only as a device for casting light on the role of bound variables in the predicate calculus, others (e.g., Noah) see it as a genuine version of term logic. capable of standing independently of the standard system. In recent years there has also been a new interest in "generalized quantifier theory," which, like the new term logic, seeks to reveal the formal properties common to all quantifier-like expressions ("determiners"). The work by Barwise and Cooper and by van Benthem is especially important here. As well, a number of researchers have recently attempted to revitalize and strengthen syllogistic by extending 240

CONCLUSION

it to statements involving nonclassical quantifiers. They have been struck, just as have others before them, by such valid arguments as Most Canadians like the cold. All those who like the cold are well-insulated. So most Canadians are well-insulated. where the validity depends (in part) on a nonclassical quantifier ('most'). Indeed, for some of these logicians the classical quantifiers need only be supplemented by 'most', 'few', and 'many' (e.g., Peterson, 1979; Thompson, 1982). An alternative approach is to introduce an infinite number of quantifiers. One version of this approach takes quantifiers to range from zero (none) to one (all), expressed as either percentages (e.g., Thompson, 1986, 1993) or fractions (e.g., Peterson, 1985, 1991, 1995; Carnes and Peterson, 1991 ). According to this approach, 'SO%', 'half, 'at least six', and so on, are quantifiers. A second version of this approach admits an infinite number of quantifiers because each quantifier is formed relative to a natural number (Murphree, 1991 ). While work on nonclassical quantifiers is not uninteresting, it does countenance quantifiers that cannot be taken as signs of opposition, an essential feature of any formative, including quantifiers, in our own system. Finally, Parry and Hacker's recent Aristotelian Logic ( 1991 ), while flawed in many ways, shows that there is interest in teaching the old logic from a non-neoscholastic perspective. Such developments cannot help but lead eventually to interest in attempts to build an even better version of term logic. In the sixteenth and seventeenth centuries, formal logic in general was degraded and ignored, kept alive only by a few champions of reason such as Leibniz. The late twentieth century has witnessed, for a wide variety of reasons, renewed attempts to displace formal logic in favour of something else. Those of us who look to Aristotle and Abelard, Leibniz, Doole and De Morgan, Frege and Russell, Quine and Sommers as our philosophical forebears cannot sit idly by. The forging of tools of formal reckoning is a task ennobled both by its history and by its goal of a more accurate account of reason.

241

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267

INDEX OF NAMES A

Abel. N.H., 56 Abelard, 4, 16, 18-20, 30, 85, 184. 241 Abeles, F., xxi. 237 Ackrill, J., 6 Aldrich, H., 41 Alexander of Aphrodisias, 15 Allwood, J., I Anderson. L.G., I Angelelli, 1., 49-50 Aristotle,xvii-xix, 3-5,9-16. 18-22,24,27,28, 30, 34, 47, 49, 50, 54,60-62,75, 82, 85, 91, 103, 106, 107, 109. 119-22. 124. 132. 135, 139, 143. 145, 146, 153, 172, 173. 179. 184. 188.241 Armstrong, R. L.. 237 Arnauld. A.. 50 Ashworth. E.• 50 8

Barnes, J.. 49 Barwise, J., 240 Bazan, C.,49 Bird.0.. 49 Black. M.. 45. 61. 63. 70. 75, 180. 181. 186 Bochenski. I.M .• 6 Boethius, 15 Boh.1.,23 Bolyai, N .• 55 Boole. G.• 41-44.47,48,56. 57. 63. 74.92. 109. 143. 148,241 Bouwsma. O.K.. xx Bradley, F. H., 64-66, 75. 77. 93. 94 Braine, M.D., 148 Byrne. R.J., 148

c Carnes, R. D.• 241 Carroll, Lewis (C.L. Dodgson). 53, 69. 142. 188 Castaneda, H.-N., 50. 51 Chastain, C., 148 Chomsky, N., 144 Clark, J.T., 93

269

INDEX OF NAMES Coats, T .• xxi Cooper, R., 240 Copernicus, N., 55 Copi, 1., 50, 77 Corcoran. J., xxi, 49 Couturat, L., 37, 38 Crooks, J., xx Curry, H., 95 Czezowski, T., 50

D Dahi,O., I Dascal, M., 50, 145 Davidson, 0., 90, 186 De Libera, A., 49 De Morgan, A., ix, 2, 16, 38,41-48,51.53, 56, 57, 63, 64, 92, 109, 150, 151, 154, 160,237,241 Descartes, R., xvii, 24. 25, 27, 33. 55 Dipert, R.• 148 Dummett, M .• 59. 60. 93, 106. 124 Durr. K., 49 E Edwards, A. W. F .• 237 Englebretsen, G., 49-51.93-95, 147, 148.237 Evans, J., 148 F Feys, R., 95 Flannery, K., 23 7 Frede, M., 49 Frege, G .• xviii, xix, 12. 41. 47. 53. 54.56-67.69-71,73-78,80.82-87.89.93.94. 101. 104. 107-9. 119-21. 127. 128, 143. 144. 146, 148. 149. 172. 175. 184, 186.233.234,240.241 Friedman, W.H., 95, 147 Furth, M., 93 G Galileo. G., 26 Gardner, M., xxi, 195,237 Gaukroger, S., 50 Gauss, K., 55 Geach, P., xvii, 9-11, 32, 33. 49, 61, 63, 69, 75. 78, 79.82-87. 89,93-95, 123, 124, 127, 134, 179.237.239 Grimm, R.H., 93 H Haack, S., xxi Hadgopoulos, D.J., 49 Hale, R., 93

270

INDE'iQEJ!AMES Halsal, A., xxi Hamilton, W.R., 41, 42, 87, 177 Heim, I.R., 148 Heintz. J., 93 Henle, M., 148 Henry, D.P., 41,49 Higgenbotham, J.• 9S Hintikka, J., 90, 9S Hobbes, T., xix, 27, 30.33. 47. 14S, ISO Hood, T.,xxi Horn, L., 49, 93 Howe, L. W., 237 Hubbeling, H.G., 237 Hume, D., 187 Hungerland, I.C., SO Hunter, G., xxi, 49, 93 I lshiguro, H., S I. 73 J

Jardine, L., 50 Jevons, S., 41, 92, 99 Johnson, W.E., 67, 68, 93, 148, 162 Johnson-Laird, P., 148 K

Kahn, C.H .• 6. 18 Kant. I., 9, 16. 100. 128, 134. 186, 187 Kaplan, D., 94 Kamaugh, M., 237 Katz. B.• 9S Keen, G.B., 23 Kelly, C., xxi Kepler, J., SS Keynes, J.N., 41.237 Klima. G., xxi Kneale. M .• 6, 49, 93, 237 Kneale. W., 6, 49, 93.237 Kripke, S., 131, 17S. 177. 178 Kuhn. T., 99 L

Lafleur, C., 49 Lakatos, 1., 99, I00 Lakoff, G., 123 Lambert, J.H., 188.237 Lancelot, C., 27, SO Laplace, P., SS, 99 Lear, J.,49

271

INDEX OF NAMES Leibniz, G.W., xix, 2, 9, 16, 23, 24, 27,30-42,48,50, 51. 55-57, 63, 64, 69, 72,73, 78, 91. 92, 94, 103, 107, 109, 115, 119. 120, 124, 127. 128, 135, 139, 144-146, 150. 151, 155. 160. 174. 184, 188.237,241 Lewis, H.A., 87 Linsky, L., 93,94 Locke, J., 9, 27, 41, 50 Lockwood, M., I 48 Lukasiewicz, J., 14, 49 M

Makinson, D., 95 Marquand, A., 237 Martinich, A. P., 95 Mates, B., I McRae, R., 50 Mendelsohn, R.• 93 Merrill, D.O., 46. 47, 51 Moody, E. A.• 49 Miii,J.S., 10,17,41,64,143 Murawski, R., 233 Murphree, W., xxi, 241 N

Nemirow, L., 93 Newton, 1., 26, 55 Nicole, P., 27,50 Noah, A.• xxi, 95, 96, 240 Nuchelman, G., 49

0 Ockham, 16,23,30,50,184 Osherson. D., 148 p

Padley, G. A., 27 Palmer, A., 94 Parkinson, G.H.R., 50 Parry, W.T., 241 Patzig, G., 6, 49. 51 Peirce, C. S.• 41, 47, 48, 63, 129, 237 Peterson, P.• xxi, 95, 241 Plato, 3, 4, 9, 19, 30, 31, 54. 58, 60, 62. 71, 84, 105, 121, 173, 183 Purdy, W., xxi Putnam, H., 175 Q Quine, W.V., I, 78-80, 87-92.93, 104, 106, 108. 125, 126, 132, 138, 172, 178, 184, 186, 240, 241

272

INDE'i.Qfl:!.AMES R Ramsey, F.P., 64, 67-69, 75-77, 79, 80, 83, 93, 94 Ramus, Peter, 24, 25, 33 Rearden, M.• 95 Rhodes, M., xxi Richards, T. 95 Riemann, G.F.B., SS Ross, W.O., 49, 237 Russell, B., I, 14, 33, 35, 36, 41, 49, SS, 66-76, 78, 79. 87, 94, 99, 104, 108, 109, 121, 130, 135, 138, 175, 184, 185,241 Rybak, Janet. 237 Rybak,John,237

s Sainsbury, M., 70, 71,94 Salmon, N., 175 Sanchez, V., SO Sanford, D., 93 Sayward, C., xx Scanlan, M., 49 Schroeder, E., 41 Scott. G., xxi Seale, D., xxi Shearman, A. T., 51.93 Shearson, W.,xx Slater, B.H., xxi Sluga, H., 94 Smyth, M.B., 237 Sommers, F., xix, xx, 2, 35, 36, 38, 49, SO, 63, 90, 91, 92, 93, 95, 99, 103-12, 11416, 118-24, 126-39. 141, 142. 144-46. 147, 148, 149, 151, 160, 162, 165, 172, 174. 177. 240,241 Sosa, E., 147 Spade, P.V., 49 Sprigge, T., 94 Staal, J.F., 103 Stalnaker. R., 95 Stout, D.• xxi Strawson, P.F., 69, 78-82, 84, 85, 89. 90.93-95, 123, 130. 132, 136. 138, 179, 183 Suppes, P., 122

T Tarski, A., I, 135, 137 Theophrastus, IS Theron, S., xxi Thorn, P.,49 Thompson, B., xxi, 241 Toms, E., 95

273