Aristotle, Logic, and QUARC

Abstract

The goal of this paper is to present a new reconstruction of Aristotle's assertoric logic as he develops it in Prior Analytics, A1-7. This reconstruction will be much closer to Aristotle's original text than other such reconstructions brought forward up to now. To accomplish this, we will not use classical logic, but a novel system developed by Ben-Yami [2014. ‘The quantified argument calculus’, The Review of Symbolic Logic, 7, 120–46] called ‘QUARC’. This system is apt for a more adequate reconstruction since it does not need first-order variables (‘x’, ‘y’, …) on which the usual quantifiers act—a feature also not to be found in Aristotle. Further, in the classical reconstruction, there is also need for binary connectives (‘∧’, ‘→’) that don't have a counterpart in Aristotle. QUARC, again, does not need them either to represent the Aristotelian sentence types. However, the full QUARC is also not called for so that I develop a subsystem thereof (‘QUARC’) which closely resembles Aristotle's way of developing his logic. I show that we can prove all of Aristotle's claims within this systems and, lastly, how it relates to classical logic.

Acknowledgments

I would like to thank Hanoch Ben-Yami, Hannes Leitgeb, and Norbert Gratzl for many helpful comments and support, Ran Lanzet for sharing his manuscript, and an anonymous referee of this journal for the many comments. This paper has been written while I was a student at the Munich Center for Mathematical Philosophy.

ORCID

Jonas Raab http://orcid.org/0000-0003-4292-9566

Notes

1 Cf. also Read ms, Section 1.

2 This might not be a stipulation at all, but it does not matter in the following. Since Aristotle only considers such sentences, we can interpret it as being stipulated.

3 . The Greek text is taken from Aristotelis 1964.

4 All translations of Book A of the Prior Analytics are Striker's as printed in Aristotle 2009. In the following, I will suppress the ‘APr’.

5 Cf. Aristotle 2009, p. 77, and Crivelli 2012, p. 115. See, however, A7, 29a27ff.: ‘It is also clear that an indeterminate premiss put in the place of a positive particular premiss will produce the same syllogism in all the figures []’.

6 .

7 [] .

8 As pointed out by Wedin (1990, pp. 134f., 141), Aristotle's canonical way of referring to an o-statement differs in his writings; in his De Interpretatione 17b16–25, he introduces it as explicit negation of the corresponding a-statement, whereas in his Prior Analytics he lists the following three (instead of two) as particular statements: ‘belonging to some, or not to some, or not to all []’ (A1, 24a19) and then uses it like in ‘A does not belong to some of the Bs []’ (A2, 24b22f.). Since the topic of this paper is Aristotle's Prior Analytics, I will stick to the latter.

9 (A1, 24b18-23)

10 .

11 ‘Hence a syllogistic premiss in general will be an affirmation or denial of something about something in the way mentioned, and it will be demonstrative if it is true and accepted on the basis of the initial assumptions []’.

12 ‘We will have to discuss demonstration later. Syllogism must be discussed before demonstration because syllogism is more universal than demonstration, for a demonstration is indeed a kind of syllogism, but not every syllogism is a demonstration []’.

13 ‘[W]hat it is for this to be or not to be in that as in a whole []’.

14 “to be predicated of all' or ‘of none’ []'.

15 ‘For one thing to be in another as in a whole is the same as for the other to be predicated of all of the first []’.

16 [] .

17 Cf. also Striker's commentary in Aristotle 2009, pp. 83f. The set-inclusion semantics is the orthodox one, but there have been proposed several others; for one alternative, see Malink 2013, pp. 63ff.; for others, see Andrade and Becerra 2008. Further, see Andrade-Lotero and Dutilh Novaes 2012 for a discussion of what significance the availability of different semantics has (thanks to an anonymous referee for this reference). Andrade-Lotero and Dutilh Novaes argue that ‘there is as of yet no uncontroversial candidate for the semantic side of a technical analysis of the notion of syllogistic validity, precisely because there are no clear guidelines or criteria of what it means for a semantics to be adequate’ (p. 416); the underlying assumption of this paper is that one such criterion is the minimal fit to the original text: even though there are many semantics available, the ‘correct’ one should not rely on techniques that are not to be found in Aristotle (a criterion not discussed in Andrade-Lotero and Dutilh Novaes 2012; it is also not a purely semantical criterion). This, of course, does not necessarily lead to a unique best fit, but rules out some of the semantics considered in Andrade-Lotero and Dutilh Novaes 2012.

18 This does not mean that we can reduce the sentence types to one another—unless we have a negation in our language.

19 See Corcoran 1972, p. 696, Martin 1997, p. 6 (note, too, the new footnote added to the reprint cited here), Smiley 1973, p. 144, and Smith 2018, §5.2.

20 See, for example, Malink 2013, pp. 81f., and Wedin 1978, 1990.

21 Whether or not he actually proves all of them without circularity does not matter for us. It seems that Aristotle proves first (e-e-conv) and uses either something like (i-i-conv) (which is not supported by the text and leads to a circularity) or, as Striker argues (in Aristotle 2009, pp. 86ff.), ecthesis. However, the latter interpretation has its own problems; Striker argues that the term Aristotle uses in the ecthesis is an individual term since otherwise we'd just run into a different circle than with the (i-i-conv) option (pp. 87f.). But I don't find it plausible at all that Aristotle would use something like existential specialization and not include it in his logic or even just presuppose it. For a different approach, see Malink 2013, pp. 39f. See also the discussion below Theorem 3.11.

22 Cf. also Malink 2013, pp. 86–101. I endorse most of what he says there, but, since I reject his semantic reconstructions (they employ, for example, existential instantiation), I don't follow him with respect to his claim that Aristotle is not committed to the o-ecthesis as presented here, but to a weaker one.

23 On the discussion how Aristotle arrives at three (and not four) figures, see Crivelli 2012, pp. 125f.

24 .

25 In chapter 1 of the Prior Analytics, Aristotle also introduced the term ‘perfect’: ‘Now I call a syllogism perfect if it requires nothing beyond the things posited for the necessity to be evident; I call a syllogism imperfect if it requires one or more things that are indeed necessary because of the terms laid down, but that have not been taken among the premisses.[]’ (24b23-26). However, since this does not play any further role, I excluded it from the presentation.

26 Even though the name seems to imply a calculus, I will use ‘QUARC’ and ‘QUARC’ as names for the logics and not just the calculi.

27 Note that we could drop the parentheses and just write, for example, ‘’ instead of ‘’, but I keep them to indicate that the argument position is written in front of the predicate symbol since the QUARC way of thinking about these formal expressions is unfamiliar to most readers. In particular, it makes conspicuous negative predication: ‘’.

28 ‘In the middle figure a deduction can be made both of opposites and of contraries. Let A stand for good, let B and C stand for science. If then one assumes that every science is good, and no science is good, A belongs to every B and to no C, so that B belongs to no C; no science, then, is a science []’. The translation is A. J. Jenkinson's as printed in Barnes (1995).

29 As Ben-Yami (2004, pp. 59ff., 2012, pp. 49ff.) argues, there is no need for a domain. Lanzet (2017) similarly develops QUARC without one. However, I am not convinced that (i) there is no domain nonetheless and (ii) that one needs no domain (cf. also Westerståhl 2012). Regarding (i), Lanzet uses interpretations that map predicates to extensions; but without a domain, where exactly is the extension? where does the interpretation map to? And, regarding (ii), even if this approach is successful in getting rid of a domain, it seems that the meaning of a predicate becomes context-sensitive (cf. Ben-Yami 2012, p. 50) because the predicates get assigned different extensions. Even though two co-extensional predicates do not have to have the same meaning, they surely don't have the same meaning if they aren't. This is why I chose to include a domain. Note also Definition 3.28 where we need it to assign the proper extensions to complex terms.

30 If we drop the non-emptiness requirement, we have to replace (2) with:

31 If we assume (2*) rather than 3.4, this condition becomes instead:

32 Again, using (2*) instead of 3.4, we get:

33 Similarly, this holds if we adopt (a^*₊) and (o^*₊) instead of (a₊) and (o₊).

34 We can also derive (o^*₊) from (a^*₊) and (a_-):

“⇒”	Let . Then, by (a-), , so, by (a*+), or .
“⇒”	Let or . Then, by (a*+), , so, by (a-), .

35 This is also true if we drop the non-emptiness requirement, as long as we interpret the negations as predicate-negations and not as term-building operations; see also Lemma 3.29 in Section 3.5.

36 Dropping the non-emptiness requirement, this does not longer hold; see the footnote to Theorem 3.9.

37 In the case of (a^*₊), inclusion if A is non-empty.

38 In the case of (o^*₊), non-inclusion if A is non-empty.

39 All of this still holds if we drop the non-emptiness requirement.

40 If we drop the non-emptiness requirement, we have to change the proof as follows: Let . Then, by (a^*₊), and . Thus, . So, by (i₊), . Thus, by (e_-), .

41 Dropping the non-emptiness requirement, we get instead: Let , that is, . Suppose that . Then, by (a^*₊), and . Thus, , a contradiction. Therefore, .

42 This passage is quoted above in footnote 28.

43 This does not hold if terms can be empty: Let be an -structure such that . Then, (a^*₊) is not satisfied, so . Further, , so that (i₊) is likewise not satisfied. Indeed, by (o^*₊), , and, since , by (i_-), . See also Read 2015, pp. 541f.

44 Even though this is not logically valid without the non-emptiness requirement, we can weaken it as follows: This holds trivially true under (a₊), but also if we use (a^*₊). For, let be an -structure such that . Then, by (i₊), . Therefore, , so that, by (a^*₊), .

45 This is still valid if we drop the non-emptiness requirement: Let be an -structure such that . Then, by (a^*₊), . Suppose that . Then, , contradiction. Therefore, , that is, by (i₊), .

46 This is clearly still valid if we drop the non-emptiness requirement.

47 This is not valid without the non-emptiness requirement: Let be an -structure such that . Then, by (o^*₊), . However, by (a^*₊), for any term C, iff. and . Thus, only if . But, since , . Therefore,.

To reconcile this, we have to change (ec(o)) to: Note that we don't have to specify that since this is required by (a^*₊); this is also the reason why we need to include ‘’ because .

48 This still holds if we drop the non-emptiness requirement: is only true in if .

49 Cf. Aristotle's remarks regarding at A6, 28b20f.

50 See Section 3.5 for this notation.

51 Nonetheless, we will be more general in Section 3.5.

52 See, for example, Raab 2016, Sections 3.4 and 4.3.

53 Thanks to an anonymous referee for pointing this out.

54 This observation loses some of its force if we replace (a₊) and (o₊) by (a*₊) and (o^*₊) since these rely on conjunction and disjunction. Nonetheless, the point still applies for the specialization/generalization cases. Note also that Malink (2013, p. 89), says explicitly after using ‘rules of classical propositional and quantifier logic’ that ‘[he does] not want to suggest that Aristotle had a clear grasp of all these rules’. However, I also do not want to claim that Aristotle did not have any grasp of it; the point is rather that he did not include them in his formal logic as developed in his Prior Analytics, A1–7.

55 The proof stays essentially the same if we adopt (ec(o)*): Since , .

56 (A2, 25a14–17)

57 (A2, 25a17ff.)

58 Dropping the non-emptiness assumption, we can invoke ((a^*₊)) instead of the definition of -structure.

59 (A2, 25a20ff.)

60 (A2, 25a12f.)

61 If we choose, as Aristotle indeed does, to be non-empty, this is also a countermodel with the non-emptiness requirement dropped.

62 I am following Troelstra and Schwichtenberg 2000.

63 If the terms are not assumed to be non-empty, we replace (A-A) by

64 The following remarks on the rules and their interderivability mostly still apply if we allow for empty terms, that is, drop (A-A). Of course, without (A-A), we cannot drive what is called (Inst) below.

65 See B11, 61a19f., where Aristotle explains to assume ‘the contradictory of the conclusion []’ to derive a contradiction. Cf. also Crivelli 2012, p. 133, and Malink 2013, pp. 31ff.

66 An anonymous referee has pointed out that Martin (1997, p. 7) introduces just one general reductio rule which shows that the multiplication of such rules as above is unnecessary. However, Martin is able to reduce it because he introduces a ‘syntactic negation’ (p. 5) which is exactly not done here since Aristotle does not work with such a concept and without such, a more general form of the reductio rules is not viable (Smith 2018, §4, explicitly tells us that Aristotle ‘does not view negations as sentential compounds’). Opting for the mentioned extended language, viz., a language including negation, we can formulate a general reductio rule.

67 Again, if we allow empty terms, this is no longer the case.

68 (A5, 27a5–9)

69 () (A5, 27a9–14)

70 (A5, 27a32–36)

71 (A5, 27a36–27b1)

72 (A6, 28a17-22)

73 (A6, 28a22-26)

74 (A6, 28a26-29)

75 (A6, 28a29f.)

76 (A6, 28b7-11)

77 (A6, 28b14f.)

78 (A6, 28b11-14)

79 (A6, 28b14f.)

80 (A6, 28b17-20)

81 (A6, 28b20f.)

82 (A6, 28b33ff.)

83 ‘But one can also reduce all syllogisms to the universal ones in the first figure []’.

84 (A7, 29b6ff.).

85 (A7, 29b8-11)

86 (A7, 29b11-15)

87 This also means that we do not obtain a completeness result for QUARC as it stands.

88 Dropping the non-emptiness requirement, we get instead:

89 Dropping the non-emptiness requirement, we only get and .

90 Using (a^*₊) and (o^*₊) instead of (a₊) and (o₊), we can easily construct a countermodel: Let . Then, , that is, by (e₊), . However, since , by (o^*₊), , so, by Lemma 3.8, .

91 Using (o^*₊) instead of (o₊), we can construct a countermodel as before: Let . Then, by (o^*₊), . But so that, by (e₊), and, thus, by (i_-), .

92 Note that none of the following works if we adopt (2^*) instead of (2) in Definition 3.28. However, we could change QUARC in similar fashion and translate it to classical logic by adding the requirement of non-emptiness in the case of a-statements, and translating o-statements to the negation of the corresponding a-statements.

93 See Raab 2016 and, for a similar result using different resources, Lanzet and Ben-Yami 2004.

94 See Raab 2016. For a translation using a three-valued QUARC, see Lanzet 2017.

95 Strictly speaking, the general translation is ‘’ where is the translation-mapping and ‘’ means that we replace all the occurrences of the variable ‘x’ by the anaphora ‘α’ (as soon as the recursive translation bottoms out). However, it can be shown that this is logically equivalent to the given translation.

96 Again, dropping the non-emptiness requirement for QUARC, we have to translate ‘’ to ‘’.