Academic Research

Aristotle’s Logic (Paolo Crivelli)

  1. I. Propositions
  2. Deleted User — 01/27/2024 6:45 PMPropositions in syllogisms: Aristotle deals with logic in six works: Categories, De Interpretatione, Prior Analytics, Posterior Analytics, Topics, and Sophistical Refutations. The Prior Analytics have the lion’s share. Aristotle’s logic, as it is presented in the Prior Analytics, is mainly about syllogisms. A syllogism is an inference and consists of propositions: one is its conclusion, the others are its premises. In Prior Analytics 1.1 Aristotle offers the following definition of proposition:
    • T1: A proposition is a sentence affirming or denying something about something. (APr I 1 24a16–17)3 (p. 114)
    In De Interpretatione 4 he gives the following definition of sentence:
    • T2 A sentence is a significant utterance one of the parts of which is significant in separation—as a saying, not as an affirmation. (DI 4 16b26–8).
    Since T1 commits Aristotle to the view that all propositions are sentences and T2 to the view that all sentences are significant utterances, the two passages jointly commit him to the view that all propositions are significant utterances, i.e., linguistic expressions endowed with signification. T1 also makes it clear that propositions are predicative declarative sentences and distinguishes between affirmative and negative propositions: an affirmative proposition affirms something about something, a negative one denies something about something. When ‘proposition’ is used to refer exclusively to the premises of a syllogism and is contrasted with ‘conclusion’, this reference is fixed by the context; but there is no implication that the syllogism’s conclusion should not be a proposition nor is there any reason to think that ‘proposition’ is used with a special meaning.ImageImage
  3. Kinds of propositions:
    • Aristotle offers a three-tiered classification of propositions.9 On the first tier, he distinguishes between assertoric, apodictic, and problematic propositions; on the second, between affirmative and negative propositions; on the third, between universal, particular, and indeterminate propositions.10 The distinction drawn on a later tier cuts across each of the kinds in the preceding one.
    • The distinction between assertoric, apodictic, and problematic propositions concerns modality: an assertoric proposition states that something holds, or fails to hold, of something; an apodictic proposition states that something necessarily holds, or fails to hold, of something; and a problematic proposition states that (p. 115) something possibly holds, or fails to hold, of something. I shall focus on assertoric propositions. Following Aristotle’s lead, I shall often use ‘proposition’ to mean ‘assertoric proposition’. A universal affirmative proposition states that something holds of all of something (e.g., ‘Every pleasure is good’); a particular affirmative proposition states that something holds of some of something (e.g., ‘Some pleasure is good’); an indeterminate affirmative proposition states that something holds of something, without specifying whether of all or some of it (e.g., ‘Pleasure is good’); a universal negative proposition states that something holds of none of something (e.g., ‘No pleasure is good’); a particular negative proposition states that something does not hold of some of something (e.g., ‘Some pleasure is not good’); an indeterminate negative proposition states that something does not hold of something, without specifying whether of none or not of some of it (e.g., ‘Pleasure is not good’).11 I shall ignore indeterminate propositions because they are rarely mentioned in Aristotle’s syllogistic. In traditional logic, which developed from Aristotle’s syllogistic and dominated until the end of the 19th century, the status of a proposition as affirmative or negative is called its quality; its status as universal or particular is called its quantity.
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  5. Terms:
    • Terms play a fundamental role in the syllogistic. Here is Aristotle’s definition of term: T3 I call ‘term’ that into which a proposition is dissolved, namely what is predicated and that of which it is predicated, ‘to be’ or ‘not to be’ being added (APr I 1 24b16–18).
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  6. Aristotle’s ‘reversed’ formulation of propositions:
    • Aristotle often adopts a ‘reversed’ formulation of propositions, with the predicate-expression before the subject-expression. Thus, Aristotle prefers ‘ “White” is predicated of all of “horse” ‘ to ‘Every horse is white’, ‘ “White” is predicated of none of “horse” ‘ to ‘No horse is white’, etc. He often uses ‘. . . holds of . . .’ or ‘. . . follows . . .’ or ‘. . . is said of . . .’ in place of (p. 116) ‘. . . is predicated of . . .’.16 In these ‘reversed’ formulations the verbs ‘to be predicated of’ and ‘to be said of’ do not indicate the speech-act of predicating, which can be carried out falsely as well as truly. Rather, in these ‘reversed’ formulations ‘to be predicated of’ and ‘to be said of’ are equivalent to ‘to be true of’.17 They therefore indicate a relation which entails the truth (and excludes the falsehood) of certain speech-acts.
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  7. Given that Aristotle’s ‘reversed’ formulations refer to terms, and given that terms are subject- and predicate-expressions of propositions, it follows that Aristotle’s ‘reversed’ formulations refer to subject- and predicate-expressions of propositions (recall that the subject- and predicate-expressions of propositions (p. 117) are significant utterances). This result is in fact presupposed by my rendering of Aristotle’s ‘reversed’ formulations (e.g., the sentence ‘ “White” is predicated of all of “horse” ‘ refers to the significant utterances ‘white’ and ‘horse’).21 Aristotle does not explain why he adopts these ‘reversed’ formulations. His main reason is probably that he is interested in formulations that display the truth-conditions of different but reciprocally equivalent propositions with the same subject- and predicate-expressions, truth-conditions which mention the terms that are the shared subject- and predicate-expressions of those propositions. For instance, it does not matter whether one uses the proposition ‘Every horse is white’, or ‘All horses are white’, or ‘Each horse is white’, or ‘Any horse is white’. All these propositions have the same truth-conditions and have the terms ‘white’ and ‘horse’ as their predicate- and subject-expression. Their common truthconditions may be specified by saying that each one of them is true just if ‘white’ (the term that is their shared predicate-expression) is predicated of all of ‘horse’ (the term that is their shared subject-expression)22 (in a later subsection I shall explain what it is for a term to be predicated of all of a term). In this sense, all the propositions mentioned are propositions to the effect that ‘white’ is predicated of all of ‘horse’.
  8. The letters adopted by traditional logic:
    • In the later logical tradition, some abbreviations were introduced to allow a compact presentation. The letters ‘a’, ‘e’, ‘i’, and ‘o’ were used to indicate universal affirmative, universal negative, particular affirmative, and particular negative propositions (‘a’ and ‘i’ are the first two vowels in ‘affirmo’, ‘e’ and ‘o’ are the first two in ‘nego’). Contrariety and contradiction. Aristotle distinguishes two relations of opposition which can obtain between propositions with the same subject- and predicate-expressions: contrariety and contradiction.
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  9. In the later logical tradition, the relations of opposition between propositions with the same subject- and predicate-expressions were illustrated by means of a drawn square where the upper corners are taken by universal propositions and the left-hand corners by affirmative propositions. The diagonals would represent contradiction, the upper side contrariety.27 ‘To be predicated of all’ and ‘to be predicated of none’. At the end of Prior Analytics I 1 Aristotle discusses the relations of being-predicated-of-all-of and being-predicated-ofnone-of, which play fundamental roles in his syllogistic:
    • T6 – We use ‘to be predicated of all’ whenever none of the subject can be taken of which the other will not be said.28 Likewise with ‘to be predicated of none’. (APr I 1 24b28–30)
  10. Here are two anodyne paraphrases:
    • [1] For every term P, for every term S, PaS just if it is not the case that for some z, both S is said of z and it is not the case that P is said of z.
    • [2] For every term P, for every term S, PeS just if it is not the case that for some z, both S is said of z and P is said of z.
  11. Characterisations of being-predicated-of-some-of and not-being-predicated-of-some-of: Aristotle never formulates characterisations of being-predicated-of-some-of and not-being-predicated-of-some-of; but some results he claims to be able to establish require them. On the model of [1] and [2], I propose:
    • [3] For every term P, for every term S, PiS just if for some z, both S is said of z and P is said of z.
    • [4] For every term P, for every term S, PoS just if for some z, both S is said of z and it is not the case that P is said of z.
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  12. Contradiction and contrariety:Principles [1]–[4] entail that contradictory propositions are neither true together nor false together. This follows from:
    • [5] For every term P, for every term S, PaS just if it is not the case that PoS. For every term P, for every term S, PeS just if it is not the case that PiS. The proof is an immediate consequence of the fact that the right-hand member of the biconditional embedded in [1] (or, respectively, [2]) is the negation of the right-hand member of the biconditional embedded in [4] (or, respectively, [3]). Let the following thesis be available:
    [6] Every term is said of something. With thesis [6] in place, it can be proved that contrary propositions are not true together. In other words, the following can be proved:
    • [7] For every term P, for every term S, it is not the case that both PaS and PeS.
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  14. The textual evidence relevant to the two interpretations:Some passages tell in favor of the heterodox interpretation. First, in the course of his discussion of modal syllogisms with one apodictic and one assertoric premise, Aristotle says:
    • Since A necessarily holds, or does not hold, of all of B, and C is one of the Bs, it is evident that either of these [sc. being A or not being A] will belong necessarily to C (APr I 9 30a21–3).
    Secondly, in Prior Analytics I 41 (49b14–32) Aristotle distinguishes ‘A holds of all of that of which B holds’ from ‘A holds of all of that of all of which B holds’ and examines their reciprocal entailment. One would expect Aristotle to address such an issue if the heterodox interpretation is right; it is not clear why he should address it if the orthodox interpretation is right. Secondly, in Posterior Analytics 1.4 Aristotle says:
    • I call ‘of every’ what is not about one thing but not about another, nor at one time but not another. For instance, if animal of every man, then if it is true to call this one a man, it is true to call him an animal too [ei alêthes tond’eipein anthrôpon, alêthes kai zôon], and if now the one then the other, and similarly if in every line there is a point (APo I 4 73a28–32).
  15. Singular propositions:
    • Singular propositions, which include propositions whose subjectexpression is a proper name (e.g., ‘Socrates is a man’), do not play a prominent role in the Prior Analytics. But they are mentioned.53 Their status is unclear. One passage (APr II 27 70a24–30) suggests that they should be treated as universal propositions; another (APr I 33 47b21–29) that they should not.54 Perhaps they fall outside the remit of syllogistic because ‘arguments and inquiries are mostly concerned with’ species between individuals and highest genera (APr I 27 43a42–3). If so, the rare references to singular propositions should be regarded as slips of the pen.
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  16. II. Syllogisms
  17. Syllogism defined: In Prior Analytics I 1, Aristotle offers the following definition of syllogism:
    • A syllogism is a discourse in which, certain things having been posited, something different from the things laid down results of necessity due to these things being. By ‘due to these things being’ I mean ‘to result because of these things’, and by ‘to result because of these things’ I mean ‘needing no further term from outside for the necessity to come about’ (APr I 1 24b18–22).5
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  18. The three figures:
    • Aristotle concentrates on syllogisms of a particular kind. These ‘canonical syllogisms’, as I shall call them, involve exactly three propositions: two premises and one conclusion. These propositions involve at most three terms. Not all syllogisms that consist of three propositions involving at most three terms are canonical syllogisms. By focusing on canonical syllogisms, Aristotle does not tacitly restrict the meaning of ‘syllogism’. In fact, in Prior Analytics I 23 he argues that all syllogisms can be brought back to canonical syllogisms.
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  19. Kinds of canonical syllogisms:
    • Aristotle identifies 14 kinds of canonical syllogisms. They are described by the following tables (which include their traditional names).
      • One premiss is a proposition to the effect thatFor some term A, for some term B, for some term CThe conclusion is a proposition to the effect thatPrior Analytics I 4
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    • For some term M, for some term N, for some term X
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    • For some term P, for some term R, for some term S
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  22. Perfect and imperfect syllogisms:
  23. Aristotle distinguishes ‘perfect’ from ‘imperfect’ syllogisms:
    • I call ‘perfect’ a syllogism which needs nothing else apart from the assumptions for the necessity [sc. of something following from these assumptions] to be apparent, whereas I call ‘imperfect’ that which needs one or more things which are necessary because of the underlying terms, but have not been assumed by means of premises (APr I 1 24b22–6).
    It is important that a syllogism’s validity be evident. A syllogism whose validity is not evident is of no use. For we employ inferences to persuade ourselves or others that a certain conclusion follows necessarily from certain premises; but if a syllogism’s validity is not evident, then the syllogism will not persuade anyone. Hence, if a syllogism is imperfect, i.e., fails to be evidently valid, we need ways of transforming it so that its validity becomes evident. Aristotle has a theory of how to achieve this. He uses the verb ‘to perfect’ to describe such a transformation.77 The perfecting of an imperfect syllogism involves interlarding it with intermediate steps which render its validity evident. In many (though not all) cases, the inserted steps evidently follow from what precedes and have the desired conclusion evidently following from them, so that the procedure may be viewed as a breaking down of the originally imperfect syllogism into shorter evidently valid inferences. The result obtained by adding intermediate steps is the same syllogism as the original one (because it has the same premises and conclusion and a syllogism is identified by its premises and conclusion). The difference is a matter of presentation: once perfected, the syllogism is so presented that its validity is evident.
  24. Which canonical syllogisms are perfect?
  25. All first-figure syllogisms (i.e., syllogisms in Barbara, Celarent, Darii, or Ferio, which are represented in Table 6.1) are perfect; all secondfigure syllogisms (i.e., syllogisms in Cesare, Camestres, Festino, or Baroco, which are represented in Table 6.2) and third-figure syllogisms (i.e., syllogisms in Darapti, Felapton, Disamis, Datisi, Bocardo, or Ferison, represented in (p. 130) Table 6.3) are imperfect.79 Since a syllogism is perfect just if it is evidently valid, and imperfect just if it is not perfect, two consequences follow:
    • [8] Every first-figure syllogism is evidently valid.
    • [9] No second- or third-figure syllogism is evidently valid
  26. Conversion: Conversion can be of three types:
    • For all terms S and P, if from certain premises a proposition to the effect that SeP is inferred, then from those premises any proposition to the effect that PeS may be inferred.
    • For all terms S and P, if from certain premises a proposition to the effect that SiP is inferred, then from those premises any proposition to the effect that PiS may be inferred.
    • (p. 132) For all terms S and P, if from certain premises a proposition to the effect that SaP is inferred, then from those premises any proposition to the effect that PiS may be inferred.
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  27. Perfect syllogisms:
  28. Another procedure allowed by Aristotle in perfecting imperfect syllogisms is the application of perfect syllogisms, namely first-figure syllogisms.
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  30. Reduction to the impossible: All syllogisms of twelve of Aristotle’s fourteen kinds can be perfected by the procedures contemplated so far. These procedures are however ineffective for syllogisms of two kinds, namely Baroco (Table 6.2) and Bocardo (Table 6.3). For these something else is needed: If from certain premises a certain conclusion is inferred, then any contradictory of any of those premises may be inferred from the result of replacing that premise with any contradictory or contrary of that conclusion. Syllogisms perfected by PI ‘are brought to a conclusion through the impossible’ (otherwise they ‘are brought to a conclusion ostensively’).95 ‘PI’ abbreviates the Latin ‘per impossibile’. Consider how syllogisms in Baroco are perfected by reduction to the impossible: Again, if M holds of all of N, but not of some of X, it is necessary for N not to hold of some of X; for if it holds of all, and M is also predicated of all of N, then it is necessary for M to hold of all of X; but it was assumed not to hold of some. (APr I 5 27a36–27b1)
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  32. The clause ‘due to these things being’:
  33. In T16 Aristotle remarks that a successful application of the method of rejection by counter-examples shows that ‘nothing (p. 140) necessary results due to these things being [ouden gar anagkaion sumbainei tô(i) tauta einai]’ (26a4–5). This remark recalls T9’s definition of syllogism, according to which in a syllogism ‘something . . . results of necessity due to these things being [ti . . . ex anangkês sumbainei tô(i) tauta einai]’ (24b19–20).106 Note that in the Analytics the formula ‘due to these things being’ occurs nowhere outside T9 and T16. Note also that near the end of T16 Aristotle replaces the formula ‘due to these things being’ with the formula ‘through these things’: for he says that ‘since nothing is necessary through these things [dia toutôn], there will not be a syllogism’ (26a7–8). In the Topics Aristotle offers the following definition of syllogism: ‘A syllogism is a discourse in which, certain things having been posited, something different from the things laid down results of necessity through the things laid down [dia tôn keimenôn]’ (Top. I 1 100a25–7). This definition matches that of the Prior Analytics except that the formula ‘through the things laid down’ replaces the formula ‘due to these things being’. This suggests that the two formulae are equivalent (at least in the context of syllogistic theory). So when, near the end of T16, he replaces the formula ‘due to these things being’ with the formula ‘through these things’, Aristotle is probably helping himself to a stylistic variant.

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