Table Of ContentAristotle’s Modal Proofs
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Texts and Studies in the History of Philosophy
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Adriane Rini
Aristotle’s Modal Proofs
Prior Analytics
A8 22 in Predicate Logic
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123
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ISBN978-94-007-0049-9 e-ISBN978-94-007-0050-5
DOI10.1007/978-94-007-0050-5
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ARISTOTLE’S MODAL PROOFS
Prior Analytics A8–22 in Predicate Logic
Acknowledgements ............................................... vi
Introduction ...................................................... 1
Part I: Modern Methods for Ancient Logic
Chapter 1 The Non-Modal Syllogistic: An. Pr. A1–7 ..... 11
Chapter 2 The Assertoric Syllogistic in LPC ............ 21
Chapter 3 A Realm of Darkness ..................... 32
Chapter 4 Technicolour Terms ...................... 39
Chapter 5 Representing the Modals .................. 45
Part II: Necessity in the Syllogistic: An. Pr. A8–12
Chapter 6 Syllogizing in Red: Trivializing the Modals .... 63
Chapter 7 First Figure Mixed Apodeictic Syllogisms ..... 72
Chapter 8 Modal Conversion in the Apodeictic Syllogistic . 79
Chapter 9 Against the Canonical Listings .............. 95
Chapter 10 Apodeictic Possibility .................... 106
Part III: Contingency in the Syllogistic: An. Pr. A13–22
Chapter 11 Contingency (A13, A14) .................. 119
Chapter 12 Realizing Possibilities .................... 135
Chapter 13 Barbara XQM .......................... 146
Chapter 14 First Figure X+Q (A15) .................. 157
Chapter 15 First Figure L+Q, Q+L (A16) .............. 169
Chapter 16 Contingency in the 2nd Figure (A17–19) ...... 180
Chapter 17 Contingency in the 3rd Figure (A20– 22 ) ...... 194
Chapter 18 Summary and Conclusion ................. 217
Appendix: The LPC Framework .................................... 232
Bibliography ................................................... 238
Index ......................................................... 241
v
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 Peter
Gilberd and the Marsden team at the Royal Society for their encouragement and support
for this research over the years. I am also grateful to VLAC (Vlaams Academisch
Centrum) Centre for Advanced Studies of the Royal Flemish Academy of Belgium
for Science and the Arts, which provided support while I was on leave during the
final production of the manuscript.
Thec hapters below carry through ona project begun in a serieso fa rticles( Rini
1995, 1996, 1998, 2000, 2003, 2007). There are two people without whose help and
insight 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 the
syllogistic have helped to make Aristotle’s technical works more easily accessible and
help to identify and isolate the stages, methods and problems that characterise the
modal syllogistic. Cresswell kept reminding me of the importance of asking what
logical point is being made in a portion of Aristotle’s text. I also thank an anonymous
reader from Springer whose thoughtful comments and criticisms prompted substantial
improvements in the manuscript. I am grateful to Helga Kocurek for English
translations of passages from Ebert and Nortmann’s (2007) German translation and
commentary on Aristotle’s Prior Analytics. Finally, I thank Springer’s editorial and
production staff for their courteousness and helpfulness at all stages.
Readers familiar with the literature on Aristotle’s Prior Analytics will
appreciate the influence that earlier studies by Paul Thom, Ulrich Nortmann and others
have had on my thinking. While we are not always agreed on our readings I am greatly
in 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, ©1989
by Robin Smith. Reprinted by permission of Hackett Publishing Company, Inc. All
rights reserved.
vi
Introduction
Aristotle invented logic. He was the first to devise a way to study not just examples of
human reasoning but the very patterns and structures of human reasoning. This
invention is the subject of the early chapters of Aristotle’s Prior Analytics. Aristotle is
keenly aware that he has created a new and useful tool – he himself sees his system of
syllogistic as the foundation of all our scientific reasoning about the world, and he takes
great care to show how the syllogistic can accommodate the kind of necessity and
possibility that together form the basic building blocks of his science. But his
exploration of syllogisms involving necessity and possibility is generally thought to
cast a shadow over his otherwise brilliant innovation. Until recently the standard view
of Aristotle’s syllogistic has been that it separates into two parts. First, there is the basic
syllogistic set out in Prior Analytics A1–7, made up of fourteen argument forms. This
was the traditional logic taught in universities right up until it was replaced by modern
formal logic in then ineteenthc entury, andt hisb asics ystemo fs yllogisticis w hatto day
is often called non-modal (or sometimes the assertoric) syllogistic.T herei sa c laritya nd
simplicity to the non-modal syllogistic that is easy to see – and anyone who thinks logic
is 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 the
main how we expect a logic to look. We can see what Aristotle is doing, and we can see
why it is good. The second part is the modal syllogistic, Prior Analytics A8–22, in
which Aristotle studies syllogisms about necessity and possibility. While the invention
of the simple system of syllogistic logic – i.e., the non-modal syllogistic – is almost
always rated as one of mankind’s greatest accomplishments, the modal syllogistic has
been infamously labelled ‘a failure’, ‘incoherent’, ‘a realm of darkness’.
This book is written in the conviction that whether or not the modal syllogistic
is a realm of darkness is a question of logic. The Prior Analytics is after all a work of
logic. That is why this must be a logic book. It is a logic book in the tradition of work
by McCall (1963), Johnson (1989), Thomason (1993, 1997), Patterson (1995),
Nortmann (1996), Thom (1996), Schmidt (2000), Malink (2006), and others, who offer
formal modellings of Aristotle’s modal syllogistic. Many of those who produce a
formal modelling for the modal syllogistic present a set of axioms or principles from
which they derive as theorems formal representations of those and only those
syllogisms that they consider Aristotle accepted, or ought to have accepted. But the
modal syllogistic is more than just a list of valid forms. In it Aristotle tries to explain
just why modal conclusions really do follow from modal premises. I have made it my
project to present and evaluate his proof methods. I have tried to show that there is a
simple logical structure to this part of the Analytics. The first and most noticeable
respect in which the present work differs from many of these other modern
interpretations is in the fact that it is concerned with a logical representation of the ways
in which Aristotle himself tries to prove his modal syllogisms.
A second respect in which the present work differs from others is that it uses
standard predicate logic translations with only de re modality. The reason for preferring
11
INTRODUCTION
predicate logic is philosophically important. Certainly, part of what I want to illustrate
in the course of the following chapters is that when our project is interpreting
Aristotle’s Prior Analytics A8–22, then we need something which gives a precise
representation by formulae which have standard and well-understood meanings. The
simplest way to achieve that, in my view, is to make use of modern modal predicate
logic. Of the recent scholarly interpretations mentioned above, only Nortmann (1996)
and Schmidt (2000) study Aristotle using modern modal predicate logic. But their
interpretations include in addition to de re modals, also de dicto modals, and this
saddles them with more sophisticated formal techniques than the present study requires.
Perhaps most scholars who work on the subject will disagree with this predicate logic
approach. Some resist it strongly. They object to the introduction of such powerful tools
as the individual variable and all that it affords. And, to be sure, Aristotle does not have
the individual variable and the associated combination of unrestricted quantifiers and
truth-functional connectives. But because logicians and philosophers know how to
interpret predicate logic this makes it an especially useful tool. Clearly it is a far more
powerful tool than we want to attribute to Aristotle, but that is a different point
altogether.
Another major need in any interpretation of Aristotle’s syllogistic is close
adherence 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 original
Greek whenever this has seemed necessary. In fact the problems in interpreting An.Pr.
A8–22 are primarily logical and philosophical, and are only peripherally illuminated by
the nuances of Aristotle’s Greek. I have of course paid close attention to those cases
where the Greek itself is vital. The question of adherence to the text takes another form
also. There is a well-known need for an account of how Aristotle’s system of logic
relates to the rest of his philosophy. A careful and conscientious scholar looks to
identify and investigate links between individual parts of the Prior Analytics and other
parts of Aristotle’s works. There is not anyone who disputes this – it is one of the big
questions within Aristotle scholarship. We all want the links between the Prior
Analytics and the rest explained, and eventually that must be done. But before we can
study the links, we first need an account of what is going on in An.Pr. A8–22 – an
account that deals with the modal syllogistic itself, that closely respects Aristotle’s
textual discussion in those chapters. Whether we can relate the logic to other parts of
Aristotle’s philosophy is an important consideration, but it should not be allowed to get
in the way of the more basic project of explaining the modal syllogistic as it is set out
in An.Pr. A8–22.
One criticism of the ‘logical’ approach may be that it is going to distort
Aristotle. Undoubtedly there is much truth in that. Predicate logic translations are one
distortion. In this study I allow what might appear to be a second distortion: I
distinguish different kinds of terms which I label ‘red’ and ‘green’. Red terms are terms
like ‘horse’, ‘plant’ or ‘man’. They name things in virtue of features those things must
2
