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Syllogism and quantification

Published online by Cambridge University Press:  12 March 2014

Timothy Smiley
Timothy Smiley*
Affiliation:
Cambridge University
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Extract

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.

In the familiar first-order predicate logic generality is expressed by means of variables and quantifiers, and each interpretation of the system is based upon the choice of some class over which the variables may range, the only restriction placed on this ‘domain of individuals’ being that it should not be empty.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 1 , March 1962 , pp. 58 - 72
Copyright
Copyright © Association for Symbolic Logic 1962

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References

1 Schmidt, Arnold, Uber deduktive Theorien mit mehreren Sorten von Grunddingen, Mathematische Annalen, vol. 115 (1938), pp. 485–605CrossRefGoogle Scholar, and Die Zulässigkeit der Behandlung mehrsortiger Theorien mittels der üblichen einsortigen Prädikatenlogik, Ibid., vol. 123 (1951), pp. 187–200; Wang, Hao, Logic of many-sorted theories, Journal of Symbolic Logic, vol. 17 (1952), pp. 105–116CrossRefGoogle Scholar; Alonzo Church, Introduction to Mathematical Logic, Exercise 55.24. I should add that all these authors impose the restriction that each argument-place of a predicate may only be filled by variables of one particular sort. No such restriction must be made if the system is to be put to the use envisaged here.

2 Cf. Church, p. 172. Following Church I shall not take the existential quantifier as primitive, but will make use of the definition (Εa)φ; =d1 ∼(a)∼φ.

3 E.g. Church's **440 and **453. Cf. Wang, theorems 2.5 and 2.7.

4 For the meaning of ‘├’ here see Church, p. 197.

5 Cf. Church, **453.

6 Strictly in these definitions a ought to be specified to be some particular variable of the A-sort, say the first in an alphabetical ordering, though of course the wff. produced by different choices of variable are synonymous.

7 Jan Łukasiewicz, Aristotle's Syllogistic, §§25–6. Łukasiewicz' lower-case variables correspond to my capitals, except that his variables are replaceable by particular terms whereas I have found it convenient to use ‘syntactical’ variables which stand for predicates. Either system could be re-written to follow either usage. In one respect Łukasiewicz' system is not ‘traditional’, in that he uses the full propositional calculus as an auxiliary. But the reader should be warned that in many other respects the book is firmly in the tradition of Prantl and Maier. Its most persistent failing is the author's ignorance of the idea of a principle of inference (expressed by the sign ‘├’) as opposed to inference (‘∴’) and implication (‘⊃’). It is this which among other things vitiates his criticisms of Aristotle's proofs by reductio ad impossibile and his discussion of syllogistic necessity, and which leads him to confuse his own ‘rejection’ (a brilliant way of formulating elegant decision procedures, based on the fact that an effectively enumerable set with an effectively enumerable complement is effectively decidable) with Aristotle's straightforward a fortiori proofs of non-deducibility. For convenience' sake I have accepted in the text the formulation of the traditional theorems as implications.

8 Shepherdson, J. C., On the interpretation of Aristotelian syllogistic, Journal of Symbolic Logic, vol. 21 (1956), pp. 137–147CrossRefGoogle Scholar, with references to an earlier proof by Słupecki (not available to me).

9 Cf. Church, **466.

10 It might be thought anachronistic to invest Aristotle's method of proceeding with the status of a conscious solution of the decision problem. Łukasiewicz indeed says positively (p. 75) that Aristotle was unaware of the existence of the decision problem, but it is possible that he is prevented from doing Aristotle justice here by his belief that the whole method of providing a concrete counter-example by interpretation is a “flaw of exposition” which brings into logic things “not germane to it”. To me it seems that in a work whose professed purpose was to “state by what means, when, and how every syllogism is produced” (An. Pr. 25b 26) the machinelike alternation of deduction and interpretation stands out as something too important to be under-estimated, at whatever risk of anachronism.

11 Cf. Church, *331.

12 An. Pr. 25a 15–17.

13 This last step recurs in the corresponding analysis of Aristotle's other proof by exposition (An. Pr. 28a 22–6), where the argument — “If both P and R belong to all S, should one of the Ss, e.g. N, be taken, both P and R will belong to this, and thus P will belong to some R” — would be analysed into the steps (1) from (s)P(s) & (s)R(s) to P(n) & R(n), and (2) from P(n) & R(n) to (Er)P(r).

14 Wedberg, A., The Aristotelian theory of classes, Ajatus, vol. 15 (1948), pp. 299–314Google Scholar, and Shepherdson, op. cit., Theorem 6. Since in our system each predicate A″ is equivalent to the original A, I suppose that we need not have posited a whole infinity of derived sorts but could have stopped short at complementary pairs of sorts, defining A″ to be A itself. For an axiomatisation of the traditional theory incorporating this idea, see Thomas, Ivo, CS(n): an Extension of CS, Dominican Studies, vol. 2 (1949), pp. 145–160.Google Scholar

16 Cf. Popper, K. R., The trivialisation of mathematical logic, Proceedings of the Xth International Congress of Philosophy (Amsterdam, 1949), pp. 722–727.Google Scholar

16 In this connexion I find it interesting that the philosopher Locke, having observed that “the common names of substances, as well as other general terms, stand for sorts”, should contrast the formation of complex ideas of substances, when the mind “never puts together any that do not really, or are not supposed to, co-exist”, with that of complex or mixed modes, which are “not only made by the mind but made very arbitrarily, made without patterns, or references to any real existence”. Essay concerning Human Understanding, III.6.1, III.6.29, III.5.3.

17 Cf. Wang, op. cit., theorem 3.2.

8 An. Pr. 24b 28–30. Cf. Collected Papers of Charles Sanders Peirce, §3.396 (a reference suggested, among other helpful criticisms, by Prof. A. N. Prior).

19 Transactions of the Cambridge Philosophical Society, vol. 9 (1856), at p. 91; cited and discussed in Prior, Formal Logic, Pt. II, Ch. II, §4.