Aristotle’s Modal Proofs

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Adriane Rini

Aristotle’s Modal Proofs

A8 22 in Predicate Logic

123

– Prior Analytics

Dr. Adriane RiniMassey UniversityPhilosophy - HPCPrivate Bag 11-222Palmerston NorthNew ZealandA.Rini@massey.ac.nz

ISBN 978-94-007-0049-9 e-ISBN 978-94-007-0050-5DOI 10.1007/978-94-007-0050-5Springer Dordrecht Heidelberg London New York

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ARISTOTLE’S MODAL PROOFS

Prior Analytics A8 22 in Predicate Logic

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I: Modern Methods for Ancient Logic

Chapter 1 The Non-Modal Syllogistic: An. Pr. A1 7 . . . . . 11Chapter 2 The Assertoric Syllogistic in LPC . . . . . . . . . . . . 21Chapter 3 A Realm of Darkness . . . . . . . . . . . . . . . . . . . . . 32Chapter 4 Technicolour Terms . . . . . . . . . . . . . . . . . . . . . . 39Chapter 5 Representing the Modals . . . . . . . . . . . . . . . . . 45

Part II: Necessity in the Syllogistic:

Chapter 6 Syllogizing in Red: Trivializing the Modals . . . . 63Chapter 7 First Figure Mixed Apodeictic Syllogisms . . . . . 72Chapter 8 Modal Conversion in the Apodeictic Syllogistic 79Chapter 9 Against the Canonical Listings . . . . . . . . . . . . . . 95Chapter 10 Apodeictic Possibility . . . . . . . . . . . . . . . . . . . . 106

Part III: Contingency in the Syllogistic:

Chapter 11 Contingency (A13, A14) . . . . . . . . . . . . . . . . . . 119Chapter 12 Realizing Possibilities . . . . . . . . . . . . . . . . . . . . 135Chapter 13 Barbara XQM . . . . . . . . . . . . . . . . . . . . . . . . . . 146Chapter 14 First Figure X+Q (A15) . . . . . . . . . . . . . . . . . . 157Chapter 15 First Figure L+Q, Q+L (A16) . . . . . . . . . . . . . . 169Chapter 16 Contingency in the 2nd Figure (A17 19) 180Chapter 17 Contingency in the 3rd Figure 194Chapter 18 Summary and Conclusion . . . . . . . . . . . . . . . . . 217

Appendix: The LPC Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

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An. Pr. A13 22

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Acknowledgements

This research is supported by the Marsden Fund Council from Government funding,administered by the Royal Society of New Zealand. And I thank especially PeterGilberd and the Marsden team at the Royal Society for their encouragement and supportfor this research over the years. I am also grateful to VLAC (Vlaams Academisch

The chapters below carry through on a project begun in a series of articles (Rini1995, 1996, 1998, 2000, 2003, 2007). There are two people without whose help andinsight this book could not have been written: Robin Smith and Max Cresswell. Smith’s(1989) translation of the Prior Analytics and his extensive commentary on thesyllogistic have helped to make Aristotle’s technical works more easily accessible andhelp to identify and isolate the stages, methods and problems that characterise themodal syllogistic. Cresswell kept reminding me of the importance of asking whatlogical point is being made in a portion of Aristotle’s text. I also thank an anonymousreader from Springer whose thoughtful comments and criticisms prompted substantialimprovements in the manuscript. a Kocurek for Englishtranslations of passages from Ebert and Nortmann’s (2007) German translation andcommentary on Aristotle’s Prior Analytics. Finally, I thank Springer s editorial and

Readers familiar with the literature on Aristotle’s Prior Analytics willappreciate the influence that earlier studies by Paul Thom, Ulrich Nortmann and othershave had on my thinking. While we are not always agreed on our readings I am greatlyin their debt.

Note on the translations:

Unless otherwise stated all translations from the Prior Analytics are from: Aristotle:Prior Analytics, translated with introduction and commentary by Robin Smith, ©1989by Robin Smith. Reprinted by permission of Hackett Publishing Company, Inc. Allrights reserved.

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I am grateful to Helg

Centrum) Centre for Advanced Studies of the Royal Flemish Academy of Belgiumfor Science and the Arts, which provided support while I was on leave during thefinal production of the manuscript.

’production staff for their courteousness and helpfulness at all stages.

Introduction

Aristotle invented logic. He was the first to devise a way to study not just examples ofhuman reasoning but the very patterns and structures of human reasoning. Thisinvention is the subject of the early chapters of Aristotle’s Prior Analytics. Aristotle iskeenly aware that he has created a new and useful tool – he himself sees his system ofsyllogistic as the foundation of all our scientific reasoning about the world, and he takesgreat care to show how the syllogistic can accommodate the kind of necessity andpossibility that together form the basic building blocks of his science. But hisexploration of syllogisms involving necessity and possibility is generally thought tocast a shadow over his otherwise brilliant innovation. Until recently the standard viewof Aristotle’s syllogistic has been that it separates into two parts. First, there is the basicsyllogistic set out in Prior Analytics A1 7, made up of fourteen argument forms. Thiswas the traditional logic taught in universities right up until it was replaced by modernformal logic in the nd t his b asic s ystem o f s yllogistici sw hat t oday

called non-modal (or sometimes the assertoric) syllogistic. There is a clarity andsimplicity to the non-modal syllogistic that is easy to see – and anyone who thinks logicis beautiful will likely find that beauty alive in Aristotle’s non-modal syllogistic too.By and large a modern reader can approach it with an easy familiarity. It looks in themain how we expect a logic to look. We can see what Aristotle is doing, and we can seewhy it is good. The second part is the modal syllogistic, Prior Analytics A8 22, inwhich Aristotle studies syllogisms about necessity and possibility. While the inventionof the simple system of syllogistic logic – i.e., the non-modal syllogistic – is almostalways rated as one of mankind’s greatest accomplishments, the modal syllogistic hasbeen infamously labelled ‘a failure’, ‘incoherent’, ‘a realm of darkness’

This book is written in the conviction that whether or not the modal syllogisticis a realm of darkness is a question of logic. The Prior Analytics is after all a work oflogic. That is why this must be a logic book. It is a logic book in the tradition of workby McCall (1963), Johnson (1989), Thomason (1993, 1997), Patterson (1995),Nortmann (1996), Thom (1996), Schmidt (2000), Malink (2006), and others, who offerformal modellings of Aristotle’s modal syllogistic. Many of those who produce aformal modelling for the modal syllogistic present a set of axioms or principles fromwhich they derive as theorems formal representations of those and only thosesyllogisms that they consider Aristotle accepted, or ought to have accepted. But themodal syllogistic is more than just a list of valid forms. In it Aristotle tries to explainjust why modal conclusions really do follow from modal premises. I have made it myproject to present and evaluate his proof methods. I have tried to show that there is asimple logical structure to this part of the Analytics. The first and most noticeablerespect in which the present work differs from many of these other moderninterpretations is in the fact that it is concerned with a logical representation of the waysin which Aristotle himself tries to prove his modal syllogisms.

A second respect in which the present work differs from others is that it usesstandard predicate logic translations with only de re modality. The reason for preferring

nineteenth c entury, ais often

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in the course of the following chapters is that when our project is interpretingAristotle’s Prior Analytics A8 22, then we need something which gives a preciserepresentation by formulae which have standard and well-understood meanings. Thesimplest way to achieve that, in my view, is to make use of modern modal predicatelogic. Of the recent scholarly interpretations mentioned above, only Nortmann (1996)and Schmidt (2000) study Aristotle using modern modal predicate logic. But theirinterpretations include in addition to de re modals, also de dicto modals, and thissaddles them with more sophisticated formal techniques than the present study requires.Perhaps most scholars who work on the subject will disagree with this predicate logicapproach. Some resist it strongly. They object to the introduction of such powerful toolsas the individual variable and all that it affords. And, to be sure, Aristotle does not havethe individual variable and the associated combination of unrestricted quantifiers andtruth-functional connectives. But because logicians and philosophers know how tointerpret predicate logic this makes it an especially useful tool. Clearly it is a far morepowerful tool than we want to attribute to Aristotle, but that is a different pointaltogether.

Another major need in any interpretation of Aristotle’s syllogistic is closeadherence to the text of the Prior Analytics. I have based my discussion on Smith’s(1989) translation, and I have consulted other translations and referred to the originalGreek whenever this has seemed necessary. In fact the problems in interpreting A8 22 are primarily logical and philosophical, and are only peripherally illuminated bythe nuances of Aristotle’s Greek. I have of course paid close attention to those caseswhere the Greek itself is vital. The question of adherence to the text takes another formalso. There is a well-known need for an account of how Aristotle’s system of logicrelates to the rest of his philosophy. A careful and conscientious scholar looks toidentify and investigate links between individual parts of the Prior Analytics and otherparts of Aristotle’s works. There is not anyone who disputes this – it is one of the bigquestions within Aristotle scholarship. We all want the links between the PriorAnalytics and the rest explained, and eventually that must be done. But before we canstudy the links, we first need an account of what is going on in An.Pr. A8 22 – anaccount that deals with the modal syllogistic itself, that closely respects Aristotle’stextual discussion in those chapters. Whether we can relate the logic to other parts ofAristotle’s philosophy is an important consideration, but it should not be allowed to getin the way of the more basic project of explaining the modal syllogistic as it is set outin An.Pr. A8 22.

One criticism of the ‘logical’ approach may be that it is going to distortAristotle. Undoubtedly there is much truth in that. Predicate logic translations are onedistortion. In this study I allow what might appear to be a second distortion: Idistinguish different kinds of terms which I label ‘red’ and ‘green’. Red terms are termslike ‘horse’, ‘plant’ or ‘man’. They name things in virtue of features those things must

An.Pr.

predicate logic is philosophically important. Certainly, part of what I want to illustrate

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have. Green terms are terms like ‘moving’, which name things in virtue of their non-necessary features; since a horse might be moving, but need not be moving. I put thisdistinction to work in Part II, showing how it helps to make sense of syllogismsinvolving necessity. It is a clumsy distinction, and Part III shows the need forrefinement in order to accommodate all of the claims Aristotle makes in An.Pr. aboutpossibility.

This view of the syllogistic is contentious. Barnes (2007, p. 133), for example,thinks any such approach wrong-headed:

Of course, Aristotle’s syllogistic is essentially tied to the concept ofpredication; for the argument forms which it examines are fixed by acertain logical structure, namely the subject–predicate structure. Butnothing in the syllogistic requires, or even suggests, any classificationof predicates: that a predicate is substantial or qualitative, relational ora matter of habitus – all that is of supreme indifference to thesyllogistic.

Barnes is undoubtedly right for the non-modal syllogistic. The machinery Aristotledevelops in the non-modal syllogistic is what I later call a colour-blind system – it doesnot take account of any difference between kinds of terms. Red and green are not atplay in the non-modal syllogistic. We can certainly take Barnes’s comments that way.But what is important to consider is that when Aristotle investigates the modalsyllogistic, there is a straightforward way of understanding him according to which heis showing that if you put colour into his basic syllogistic machinery then you getcolour out of his basic syllogistic machinery.

The Prior Analytics is a technical work, and ultimately it is Aristotle’s technicaltools and logical language that need to be interpreted and explained. Part I sets thefoundation, explaining the scholarly tradition, and explaining the basic building blocksof Aristotle’s system. Part II focuses on Aristotle’s syllogistic involving necessity –the apodeictic syllogistic. I will show that Aristotle’s apodeictic syllogistic in PriorAnalytics A8 11 requires nothing more than the simple non-modal syllogistic systemof An.Pr. A1 7, together with a general restriction. The restriction depends on aprinciple that I call the Substance Principle.

The Substance Principle (SP)If n is a red term then n is equivalent to necessarily-n.

Any time a red term applies to something at all it also applies by necessity. There aretwo ways we might understand this. We might take SP to describe two different terms,‘man’ and ‘necessary man’ one a non-modal term and one a modal term. Or we–

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1In An.Post. A22, Aristotle suggests that even though a term like ‘moving thing’ might be thegrammatical subject of the proposition, it is not always appropriate to treat it as the real logical subject.Even so, in the Prior Analytics, Aristotle often uses propositions such as (ii) as premises in the logic, butnot where he requires modal conversion.

might take SP to describe two different relations between an individual and a singleterm for we might say that this same term can apply or can apply by necessity. SPis intended to be neutral on this question. All it insists is that if a term is red then if itapplies at all it applies by necessity.

The restriction applies to one of Aristotle’s most basic proof methods in thesyllogistic what he calls conversion. In establishing the validity of a syllogisticschema Aristotle often converts the terms – that is, he switches the order of the subjectterm and the predicate term. Such conversions are straightforward in the non-modalsyllogistic. Aristotle moves easily from ‘some men are moving’ to ‘some moving thingsare men’. But when modals are involved the conversions sometimes need to becarefully restricted. Aristotle does not give much guidance about restrictions in thePrior Analytics, but he seems to need something like the restriction he describes in apassage in Posterior Analytics A22. In that passage Aristotle explains that the only wayto genuinely predicate is to predicate something of a subject which is identified by aterm for something with an essential nature. Take for example a proposition (i):

(i) All men are necessary-animals

Proposition (i) is an example of a genuine predication – the subject term ‘man’ is a redterm. It names a substance in virtue of its essential nature. Consider another example(ii):

(ii) All moving things are necessary-animals

Proposition (ii) is not genuine. Moving thing is not a red term.1 Aristotle seems to havesomething like genuine predication in mind in the Prior Analytics when he describescertain modal conversion principles.

The Genuineness RequirementIn certain modal conversions the ‘input’ proposition must be genuine — itssubject must be a red term.

For example, when we try to convert propositions (i) and (ii) we get different results:(i) ‘All men are necessary-animals’ converts to give us ‘Some animals are necessary-men.’ But (ii) ‘All moving things are necessary-animals’ does not validly convert – i.e.,we do not get ‘Some animals are necessary-movers.’

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In the case of what I call LE-conversion, in which we convert a privativeuniversal proposition, Aristotle requires a negative form of the Substance Principle:

Neg(SP)Where n is a red term then not only is n equivalent to necessarily-n, but alsonot-n is equivalent to necessarily-not-n.

Aristotle does not appear to realize that Neg(SP) does not follow from the SubstancePrinciple.

The function of the Substance Principle together with the GenuinenessRequirement is to guide our choice of syllogistic terms, making sure that we fit the rightAristotelian terms into the right locations when we are constructing and provingsyllogisms about necessity. The semantic constraints, however, are not made explicitin the Prior Analytics. If we suppose that Aristotle really is a good logician, then theneed in the apodeictic syllogistic for some kind of implicit guiding principles becomesobvious. One consequence of this approach is that the semantic restrictions are doingthe real work and the apodeictic syllogistic does not depend upon principles of modallogic. There are historical antecedents for this, particularly in the work of the medievalphilosophers Averröes and Kilwardby, who distinguish modal and non-modalsyllogisms according to the type of terms occurring in them. (See Thom 2003, andKnuuttila 2008.)

Part III deals with syllogisms involving possibility. This part is traditionallyknown as the problematic syllogistic. It is the subject of An.Pr. A13 22. In Part IIIthere are two different methods at work: ampliation and realization.

AmpliationAristotle makes a distinction (32b24 37) between the following kinds ofconstruction involving possibility:

(i) Some B is a possible A(ii) Some possible B is a possible A

The second, where both the A and B term involve possibility, is, in the medievaljargon, an ampliated proposition. By contrast, (i) is unampliated: only the Aterm – i.e., the predicate term – involves possibility. Using ampliation many ofthe syllogisms in the problematic syllogistic turn out to be substitutioninstances of non-modal syllogisms, and are therefore trivial. Here, too, itemerges that principles of modal logic are not involved.

In cases where ampliation is not available, Aristotle introduces a new method. This is

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easiest to explain with an example. An acorn can become an oak or remain a non-oak.So although an oak is a necessary oak (the Substance Principle guarantees this), a non-oak, for instance an acorn, is not a necessary-non-oak, for an acorn could become anoak its potentiality could be realized.

RealizationGiven a premise that something is possible, assume that the possibility isrealized, and then reason non-modally. Any non-modal proposition obtained inthis way may then be concluded to be possible.

There are undoubtedly places where Aristotle seems not quite as sure footed aswe would like him to be. Often, for instance, Aristotle seems to get into difficultybecause of his views about negation. Another difficulty, which arises in the problematicsyllogistic, is how he handles the tension between realization and the negative form ofthe Substance Principle. But even acknowledging such problems, what I have tried totell in the chapters that follow is how to find a clarity and beauty, and even a surprisingsimplicity, in Aristotle’s modal syllogistic.

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Note on the formal notation:I use standard symbols of the Lower Predicate Calculus (LPC) throughout (i.e. ~, w, &,e, /, �, and �), with L for ‘it is necessary that’ and M for ‘it is possible that’ (in thesense in which Mn / ~L~n) and Q for ‘it is contingent that’, where Qn implies bothMn and M~n. The letters L, M and Q, along with X, are also used to indicate the modalstatus of a proposition, and in that case they are not italicised. I follow McCall’s systemfor classifying the modal status of a syllogism or schema, as described on p. 45. Whereauthors use a different notation, as for instance N for necessity and P for possibility, Ihave translated their notation into my own, unless I am explicitly commenting onnotational matters.

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