# Aristotle's Modal Syllogistic

https://www.jstor.org/stable/j.ctt6wpq7x

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Acknowledgments
(pp. ix-x)
4. Abbreviations of Aristotle’s Works
(pp. xi-xiv)
5. Introduction
(pp. 1-18)

Aristotle was the first to undertake a systematic study of deductive inference. He is therefore considered the founder of logic. At the heart of his mature logical theory lies the assertoric syllogistic, presented in chapters 1.1–2 and 1.4–7 of thePrior Analytics.The assertoric syllogistic deals with inferences that consist of nonmodal propositions such as ‘A belongs to all B’ or ‘A does not belong to some B’. Now, Aristotle is the founder not only of logic but also of modal logic. He developed a system of modal syllogistic, presented inPrior Analytics1.3 and 1.8–22. There...

6. I. The Assertoric Syllogistic
• [I Introduction]
(pp. 19-22)

Aristotle’s syllogistic is concerned with categorical propositions. These have a tripartite syntax, consisting of a subject term, a predicate term, and a copula. For example, the assertoric proposition ‘A belongs to all B’ consists of the subject term B, the predicate term A, and the copula ‘belongs to all’. Assertoric propositions are not modalized; that is, they contain no modal qualifiers. In the first seven chapters of thePrior Analytics, Aristotle develops a deductive system of these propositions, based on conversion rules and perfect first-figure schemata (Chapter 1).

Aristotle does not describe the semantics of his assertoric propositions in any...

• 1 Categorical Propositions
(pp. 23-33)

In the first chapter of thePrior Analytics, Aristotle begins his investigation of deductive inference by clarifying what a proposition (πρότασις) is:

A proposition is a sentence affirming or denying something of something.¹ (APr. 1.1 24a16–17)

Propositions are sentences, that is, linguistic expressions of a certain language.² Every proposition contains two constituents, one of which is affirmed or denied of the other. Aristotle calls these constituents ‘terms’ (ὅροι): I call a term that into which a proposition may be broken up, that is, both what is predicated and what it is predicated of, with the addition of ‘to be’...

• 2 The dictum de omni
(pp. 34-44)

At the end of the first chapter of thePrior Analytics,Aristotle offers an explanation of the semantics of aX-propositions. This explanation, which has come to be known as thedictum de omni,reads as follows:

We say “predicated of all” when none of those of the subject can be taken of which the other will not be said.¹ (APr.1.1 24b28–30)

First it should be noted that there is a question about the Greek text of this passage. Most manuscripts do not have the phrase “of those,” translating the plural article τɷν. But since this article is attested...

• 3 The Orthodox dictum Semantics
(pp. 45-62)

The abstractdictumsemantics appeals to pluralities associated with terms. What are these pluralities? A common answer is that they are sets of individuals, consisting of exactly those individuals that fall under the term in question. On this view, the plurality associated with the term ‘man’, for example, is the set of individual men such as Socrates, Kallias, and Mikkalos. The plurality associated with the term ‘walking’ is the set of all walking individuals. Accordingly, the abstractdictum de omnistates that an aX-proposition is true just in case every individual that falls under the subject term falls under the...

• 4 The Heterodox dictum Semantics
(pp. 63-72)

The heterodoxdictumsemantics is based on the assumption that the plurality associated with a term consists of exactly those items of which the term is aX-predicated. The relation of aX-predication is treated as a primitive preorder, in terms of which eX-, iX-, and oX-predication are defined. The aim of the present chapter is to introduce the heterodoxdictumsemantics and to defend it against some objections that have been raised against it. Moreover, we will see how this semantics determines a natural class of first-order models, which may be called the preorder semantics.

INTRODUCING THE HETERODOXDICTUM SEMANTICS. I...

• 5 The Preorder Semantics
(pp. 73-85)

This chapter provides a closer look at the preorder semantics and some of its logical properties. We will consider how the preorder semantics relates to Aristotle’s claims of validity and invalidity in the assertoric syllogistic and how it compares to the set-theoretic semantics. The discussion will be less exegetical and somewhat more technical than it has been so far.

The preorder semantics treats categorical terms as zero-order individual terms. In the standard models of first-order logic, the semantic value of these terms is a single primitive item, or at least it is considered as such. The semantic value assigned to...

• 6 Ecthesis
(pp. 86-102)

In order to establish the validity of imperfect syllogistic moods, Aristotle employs direct or indirect deductions based on conversion rules and perfect moods. In the assertoric syllogistic, all valid imperfect moods are proved to be valid by such deductions. For some of these moods, however, Aristotle indicates alternative proofs based on the method of what he calls ecthesis. These proofs by ecthesis have often been interpreted in a way that conflicts with the heterodoxdictumsemantics and the preorder semantics. For example, they have been taken to rely on the strong principle of oX-ecthesis mentioned above.

My aim in this...

7. II. The Apodeictic Syllogistic
• [II Introduction]
(pp. 103-106)

Aristotle’s modal syllogistic begins with what is known as the apodeictic syllogistic (Prior Analytics1.3 and 1.8–12). The apodeictic syllogistic is concerned with necessity propositions, that is, with propositions whose copula contains a modal qualifier such as ‘necessarily’. Aristotle focuses on four kinds of necessity propositions:

AaNB A necessarily belongs to all B

AeNB A necessarily belongs to no B

AiNB A necessarily belongs to some B

AoNB A necessarily does not belong to some B

The apodeictic syllogistic consists of three parts. In the first part, Aristotle states the conversion rules for necessity propositions. These rules exactly mirror...

• 7 The Apodeictic dictum de omni
(pp. 107-113)

Aristotle justifies the validity of Barbara NXN inPrior Analytics1.9, in a passage I discussed in connection with thedictum de omni(p. 52). Alexander and others take his justification of Barbara NXN to be based on a version of thedictum de omnithat characterizes the semantics of aN-propositions.¹ As we saw above, Aristotle justified the validity of assertoric Barbara by means of the assertoricdictum de omni,which characterizes the semantics of aX-propositions.² It is therefore plausible that he would justify the validity of Barbara NXN by means of a corresponding apodeicticdictum de omnifor aN-propositions....

• 8 Barbara NXN and the Four Predicables
(pp. 114-133)

Aristotle’s modalized propositions have a tripartite syntax, consisting of a predicate term, a subject term, and a copula. For example, the aN-proposition ‘A necessarily belongs to all B’ consists of the predicate term A, the subject term B, and the aN-copula ‘necessarily belongs to all’ (pp. 23–28). In order to specify the semantics of Aristotle’s modalized propositions, it is vital to give a semantic interpretation of the modal copulae occurring in them. Each of these copulae stands for a relation between terms; for example, the aN-copula stands for the relation of aN-predication. Thus, giving a semantic interpretation of the...

• 9 Categories in the Topics
(pp. 134-151)

In a well-known passage from chapter 4 of theCategories,Aristotle introduces the ten categories as follows:

Of things said without any combination, each signifies either substance or quantity or quality or a relative or where or when or being-in-a-position or having or doing or being-affected. To give a rough idea, examples of substance are man, horse; of quantity: four-foot, five-foot; of quality: white, grammatical; of a relative: double, half, larger; of where: in the Lyceum, in the market-place; of when: yesterday, last-year; of being-in-a-position: is-lying, is-sitting; of having: has-shoes-on, has-armor-on; of doing: cutting, burning; of being-affected: being-cut, being-burned. (Cat....

• 10 Essence Terms and Substance Terms
(pp. 152-167)

As we have seen, Aristotle’s account of categories in theTopicsrelies on the distinction between essence terms like ‘justice’ and nonessence terms like ‘just’. The purpose of the present chapter is to show that his modal syllogistic, too, relies on it. Aristotle makes use of this distinction inPrior Analytics1.34 to reject a purported counterexample to Celarent NXN. I argue that the distinction can be used in a similar way to reject two of Theophrastus’s and Eudemus’s counterexamples to Barbara NXN.

I then discuss in detail the role of essence terms in the modal syllogistic. Every subject of...

• 11 Universal Negative Necessity Propositions
(pp. 168-176)

Having considered aN-predication, we now turn to the relations of eN-, iN-, and oN-predication. Aristotle does not discuss the nature of these relations in any detail. However, his claims of validity and invalidity in the apodeictic syllogistic impose several requirements on them. In particular, they require these relations to validate both the perfect NXN-moods stated inPrior Analytics1.9 and the conversion rules stated in chapter 1.3. It is not obvious whether and how these requirements can be satisfied simultaneously. Some commentators have thought that this is not possible. By contrast, I argue that it is possible. I want to...

• 12 Particular Necessity Propositions
(pp. 177-190)

Finally, let us consider the relations of iN- and oN-predication in Aristotle’s apodeictic syllogistic. I begin with iN-predication, adopting what is known as the disjunctive strategy for defining iN-predication. With regard to oN-predication, a major problem is Aristotle’s claim that Baroco XNN and Bocardo NXN are invalid. As we will see, the invalidity of these two moods is in some tension with Aristotle’s use of the method of ecthesis in the apodeictic syllogistic.

Aristotle does not mention an apodeicticdictum de aliquothat would characterize the relation of iN-predication. Nor is it obvious what such an apodeicticdictum de aliquo...

8. III. The Problematic Syllogistic
• [III Introduction]
(pp. 191-194)

Aristotle’s apodeictic syllogistic is followed by what is known as the problematic syllogistic (Prior Analytics1.3 and 1.13–22). The problematic syllogistic is concerned with possibility propositions, that is, with propositions that contain a modal qualifier such as ‘possibly’. Aristotle distinguishes two kinds of possibility propositions, traditionally referred to as one-sided and two-sided possibility propositions. Being two-sided possible means being neither impossible nor necessary, while being one-sided possible simply means being not impossible. Thus two-sided possibility precludes necessity, whereas one-sided possibility does not. For example, the statement ‘Possibly no man is a horse’ is true if understood as a one-sided...

• 13 Modal Opposition
(pp. 195-210)

I begin by considering evidence that suggests that Aristotle endorses a number of principles of modal opposition. I then show that, although these principles are intrinsically plausible, Aristotle’s claims of invalidity and inconcludence in the problematic syllogistic commit him to denying some of them. This leads to various interpretive problems, for which I suggest solutions.

At the beginning of the problematic syllogistic, inPrior Analytics1.13, Aristotle introduces the distinction between two- and one-sided possibility. He regards the former as the primary notion of possibility, and the latter as a secondary notion resting on an equivocation of the term ‘possible’:...

• 14 Establishing Inconcludence
(pp. 211-222)

Aristotle’s justification of the inconcludence of ea-2-QN and ae-2-NQ may be approached with high expectations; for in order to be convincing, it should explain how eQ-propositions can be compatible with aN-propositions. It should thereby also help us see what led Aristotle to his asymmetric treatment of modal opposition. Now, Aristotle’s justification of the inconcludence of these two premise pairs is more complex than his usual proofs of inconcludence in the modal syllogistic. We will therefore first have a look at his usual method of establishing inconcludence and then consider the more complex case of ea-2-QN and ae-2-NQ.

A premise pair...

• 15 A Deductive System for the Modal Syllogistic
(pp. 223-231)

Aristotle’s syllogistic can be viewed as a deductive system of categorical propositions. Corcoran (1972) and Smiley (1973) have specified suitable deductive systems for the assertoric syllogistic (see pp. 31–33 above). These systems are based on Aristotle’s conversion rules and perfect first-figure moods. They include indirect deductions based on the standard principles of contradictoriness between assertoric propositions. These systems are adequate with respect to Aristotle’s assertoric syllogistic in that every assertoric mood held to be valid by Aristotle is deducible in them, and no assertoric mood held to be invalid by him is deducible in them.

On the other hand,...

• 16 The Validity of XQM-Moods
(pp. 232-247)

InPrior Analytics1.15, Aristotle is concerned with first-figure premise pairs of the form QX and XQ. He holds that premise pairs of the former kind yield a two-sided possibility conclusion, whereas those of the latter kind yield only a one-sided possibility conclusion. Thus, Aristotle asserts the validity of Barbara, Celarent, Darii, and Ferio of the form QXQ and XQM. The four QXQ-moods are regarded as perfect by Aristotle. By contrast, the four XQM-moods are not regarded as perfect, but as being in need of proof. Aristotle establishes their validity by means of indirect proofs that are considerably more complex...

• 17 Two-Sided Possibility Propositions
(pp. 248-260)

The purpose of this chapter is to develop an interpretation of Aristotle’s two-sided possibility propositions by giving a definition of a Q and iQ-predication. I begin by introducing a formal framework for these definitions. This framework, which I call the predicable semantics, is based on three primitive relations: aX-predication, aN-predication, and a strengthened version of aN-predication which is incompatible with oM-predication. The three primitive relations will suffice to formulate definitions of aQ-and iQ-predication. I discuss some consequences of these definitions and explain how they relate to Aristotle’s treatment of Q-propositions and to a distinction he draws inprior Analytics1.13...

• 18 One-Sided Possibility Propositions
(pp. 261-272)

The purpose of this final chapter is to formulate an interpretation in the predicable semantics of M-propositions. I also reconsider N-propositions and discuss how they can be interpreted within the predicable semantics.

DEFINING AM-PREDICATION. If Aristotle accepted the principles of M-N-contradictoriness, the four kinds of M-predication could be simply defined as the contradictories of the four kinds of N-predication. However, as we have seen, Aristotle is committed to denying at least some of these principles (p. 202). Therefore, M-N-contradictoriness cannot serve as a guide in defining the four kinds of M-predication. Instead, the definitions may be guided by the principles...

9. Appendix A: Aristotle’s Claims of Validity, Invalidity, and Inconcludence
(pp. 273-285)
10. Appendix B: The Predicable Semantics of the Modal Syllogistic
(pp. 286-325)
11. Appendix C: Aristotle’s Terms
(pp. 326-336)
12. Bibliography
(pp. 337-348)
13. Index of Names
(pp. 349-352)
14. Index of Passages
(pp. 353-360)
15. Index of Subjects
(pp. 361-367)