Synopsis of Jacques Brunschwig, ‘La proposition particulière et les preuves de non-concluance chez Aristote’
[‘Aristotle on Particular Propositions and Proofs of Inconclusiveness’]
Born in 1929, Jacques Brunschwig is currently Emeritus Professor of the History of Ancient Philosophy at the University of Paris I, where he taught for much of his career. A student of the Ecole Normale, his first publications, on a wide range of topics in ancient Greek and early modern philosophy, date from 1960. Today he is best known to English readers as the author of Papers in Hellenistic Philosophy (1994) and as an editor of the landmark compendium, Greek Thought: A Guide to Classical Knowledge (2000).
Brunschwig’s contribution to the Cahiers is a detailed analysis of a fairly specific problem in Aristotle’s logic, with particular reference to several obscure passages in the Prior Analytics (notably 26a39-26b14, 27b9-23, and 35a24-35b23). In his Prior Analytics, Aristotle sets out to establish an abstract system of general formal logic, based on a theory of deductive reasoning (syllogismos); his Posterior Analytics then relies on such reasoning in order to help develop a systematic account of scientific knowledge. A ‘syllogism’ or deduction, Aristotle explains, ‘is a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so’ (Prior Analytics, 24b19-20). After considering ‘perfect’ or self-evident deductions, Aristotle considers various cases that are more complex and problematic. Brunschwig is interested here in inconclusive deductions which involve particular as opposed to universal (or indefinite) propositions, i.e. propositions which concern ‘some’ rather than ‘all’ members of a given class.
Brunschwig begins his article, via a reference to Jan Lukasiewicz’s study of Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic (1957), by directly addressing the general question at issue in this tenth volume of the Cahiers: unlike the formalist logic developed by the Stoics, ‘Aristotelian logic is formal without being formalistic.’ It is formal insofar as its deductive laws do not apply directly to concrete terms or variables but only to the logical ‘places’ that might be occupied by such terms, indicated by abstract symbols (letters). Aristotle’s logic is formal, in other words, in the sense that it concerns form as opposed to matter, form in the sense, for instance, of a geometric shape or design (CpA 10.1:3, 5). Formalism, however, ‘requires that the same thought always be expressed by means of a series of words that remain exactly the same, and ordered in exactly the same way’, such that a formalist proof of the validity of a deduction can be verified via analysis of its abstract form alone, ‘without reference to the signification of the terms used in the proof’ (Lukasiewicz, quoted page 4). Rather than develop a formalist symbolic system, Aristotle’s formal logic uses natural language, in a relatively loose, ad hoc way, guided by the priority he gives to the signified rather than the signifier (4). Aristotle’s logic, in short, is not a rigorously formalist method of calculation or calcul (5).
In order to describe ‘the effects of this formality without formalism’, Brunschwig proposes to consider the problems that Aristotle encounters when he tries to specify the meaning of particular propositions, in particular when these are used in invalid or inconclusive deductions or syllogisms. Brunschwig hopes that by studying these problems he can point to a ‘significant shift’ in Aristotle’s approach over the course of the Prior Analytics, as he moves from a position initially compromised by reliance on ‘the equivocal quality of natural language’ to one that is more conscious of such equivocation and better able to control it. ‘At the end of this evolution, the particular proposition abandons some of its usual connotations, which make its logical manipulation difficult, and comes to be defined solely by its place in a system of oppositions, with all of the consequences that this implies’ (5).
Early in the Prior Analytics, Aristotle distinguishes a particular from a universal proposition: ‘A proposition is a sentence affirming or denying something of something; and this is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, e.g. “contraries are subjects of the same science”, or “pleasure is not good”’ (24a16-21). Brunschwig notes that the truth of a particular proposition (e.g. ‘some As are B’), thus defined, might appear to be involved in three incompatible sorts of relations: (a) its truth can seem to contradict the truth of a universal proposition concerning the opposite quality; (b) its truth can seem to be implied by that of a universal proposition concerning the same quality; (c) its truth can seem to imply, and be implied by, the truth of another particular proposition concerning the opposite quality (6). Since we cannot affirm all three of these relations together, we need to choose between what Brunschwig distinguishes as ‘minimal’ and ‘maximal’ interpretations of particular propositions. A minimal interpretation implies that ‘at least one A is B (without excluding the possibility that all As are B)’; a maximal interpretation implies that ‘at least one but no more than one A is B’ (7-8).
Aristotle himself privileges the minimal interpretation, Brunschwig notes, but he doesn’t seem to have taken on board the full ‘implications and requirements’ of this choice. His failure to make a clear ‘initial decision’ on this point is the source of multiple complications and confusions, such that on occasion his use of particular propositions appears equivocal (9-10). (For instance, people who believe that ‘some As are B’ might be told that they are quite right, since in fact all As are B – or alternatively, they might be accused of wrongly implying that since ‘only’ some As are B, some others may not be...). Introducing the second half of his article, Brunschwig notes that these confusions are especially obvious ‘in the domain of proofs of inconclusiveness [preuves de non-concluance]’, i.e. syllogisms that are ‘incapable of authorising a conclusion’ (11).
Aristotle usually demonstrates that the combination of a pair of propositions (a major premise followed by a minor premise) regarding any three terms (e.g. animal, man, horse, or animal, white, man) remains inconclusive via a ‘proof by contrasting instances’, such that none of the four standard relations (conventionally listed as a, e, i, o: every B is A; no B is A; some B is A; some B is not A) apply. (As Robin Smith explains, in a recent overview of Aristotle’s logic, ‘Aristotle proves invalidity by constructing counterexamples. This is very much in the spirit of modern logical theory: all that it takes to show that a certain form is invalid is a single instance of that form with true premises and a false conclusion. However, Aristotle states his results not by saying that certain premise-conclusion combinations are invalid but by saying that certain premise pairs do not “syllogize”: that is, that, given the pair in question, examples can be constructed in which premises of that form are true and a conclusion of any of the four possible forms [a, e, i, o] is false. When possible, he does this by a clever and economical method: he gives two triplets of terms, one of which makes the premises true and a universal affirmative “conclusion” true, and the other of which makes the premises true and a universal negative “conclusion” true. The first is a counterexample for an argument with either an E or an O conclusion, and the second is a counterexample for an argument with either an A or an I conclusion’ [Smith, ‘Aristotle’s Logic’]).
Brunschwig is especially interested, however, in an alternative approach that Aristotle sometimes adopts, which he dubs ‘proof by the undetermined [l’indéterminé]’. Here, ‘the possibility of deducing, on the basis of an already established inconclusion [non-concluance], a new inconclusion, depends on what Aristotle calls the indetermination of the particular proposition, i.e. on the fact that it may be true whether or not its subalternate proposition [sa subalternante, i.e. the universal proposition concerning the same quality] is false or true. In other words, if we assume AoB is true (i.e. A does not apply to some Bs), this doesn’t imply or exclude the truth of AeB (i.e. A may either apply to no Bs and thus a fortiori not apply to some Bs, or else not apply to some Bs while applying to some other Bs)’ (13). Such indetermination only applies to particular propositions interpreted in the minimal sense, i.e. as meaning that ‘at least one A is B (without excluding the possibility that all As are B)’.
Aristotle never developed a systematic account of such ‘proof by indetermination’, but has recourse to it ‘when his usual procedure, proof by contrasting instances, runs into obstacles relative to the question of how to interpret particular propositions’ (14). Brunschwig identifies seven instances in the Prior Analytics where Aristotle uses his proof by indetermination, and then proceeds to analyse the three most significant cases.
In the first case, Aristotle asks us to consider the triplet of terms ‘animal, man, white: next take some of the white things of which man is not predicated – swan and snow: animal is predicated of all of the one, but of none of the other. Consequently there cannot be a deduction’ (26b6-10). But Brunschwig notes, after Günther Patzig, that the substitution of swan or snow for white, in the minor premise, converts it from a particular into a universal proposition; because he remains attached here to the ‘maximal’ connotations of the particular proposition, Aristotle is effectively trying to ‘reconcile unreconcilables’, and only succeeds in ‘obscuring the situation’ (15-16). In the second case Brunschwig considers (27b9-23), Aristotle appears to have ‘a much clearer understanding of the problem’, precisely because he is careful to exclude a maximal interpretation of the particular proposition in question (17). The third and most instructive case (35a24-35b23), Brunschwig says, is unique in Aristotle’s logic. It involves the combination of different modal categories (assertoric and problematic): here proof by indetermination is no longer a substitute for the ‘proof by contrasting instances’ but is rather a clarified application of one and the same proof. Aristotle’s text may be worth quoting at length:
Whenever the major premiss is universal, but assertoric [or simple], not problematic [possible], and the minor is particular and problematic, whether both premisses are negative or affirmative, or one is negative, the other affirmative, in all cases there will be an imperfect deduction [syllogism]. Only some of them will be proved per impossibile, others by the conversion of the problematic premiss, as has been shown above. And a deduction will be possible by means of conversion when the major premiss is universal and assertoric, whether positive or negative, and the minor particular, negative, and problematic, e.g. if A belongs to all B or to no B, and B may possibly not belong to some C. For if BC is converted in respect of possibility, a deduction results. But whenever the particular premiss is assertoric and negative, there cannot be a deduction. As instances of the positive relation we may take the terms white-animal-snow; of the negative, white-animal-pitch. For the demonstration must be made through the indefinite nature of the particular premiss. But if the minor premiss is universal, and the major particular, whether either premiss is negative or affirmative, problematic or assertoric, in no way is a syllogism possible. Nor is a syllogism possible when the premisses are particular or indefinite, whether problematic or assertoric, or the one problematic, the other assertoric.
A particular proposition, as considered here, ‘no longer has any meaning other than that which follows from its status as a simple negation of the universal. The “logical” version of a particular proposition has had to go to some trouble to kill off the “natural” version, but it has got there in the end’ (21).
The last stage of Brunschwig’s argument demonstrates that insofar as Aristotle sometimes has recourse to indetermination of the particular ‘without expressly saying so’, so then his initial reliance on the maximal (and ‘natural’) interpretation of particular propositions has not only been killed off but has been ‘well and truly buried’ (22). As his analysis of inconclusive propositions proceeds and ‘evolves’ over the course of the Prior Analytics, ‘Aristotle has thus progressively and simultaneously liquidated the maximal connotations of the particular [proposition], abolished the distinction between a “proof by contrasting instances” and a “proof by the indeterminate”, and softened the criteria whereby we might recognise that two concrete terms “satisfy” a given relation.’ In the process, Aristotle never lost the ‘good taste’ required to submit to the obligation, imposed by the constraints of the logical problem (or ‘thing itself [la chose même]’), to modify his initial positions and to follow the logic of the problem to the end (25-26).
References to this text in other articles in the Cahiers pour l’Analyse:
- Aristotle. Prior Analytics (Analytica Priora) [c. 350 BCE], trans. A.J. Jenkinson, revised by Jonathan Barnes, in The Complete Works of Aristotle, ed. Jonathan Barnes. Princeton: Princeton University Press, 1984. Vol. 1, pp. 39-113. Jenkinson’s original translation is online at http://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html.
Selected secondary literature:
- Owen, Gwilyn E.L. Logic, Science, and Dialectic. Ithaca, NY: Cornell University Press, 1986.
- Brunschwig, Jacques. Papers in Hellenistic Philosophy. Cambridge: Cambridge University Press, 1994.
- Brunschwig, Jacques. Les Stoïciens et leur logique. Paris: Vrin, 2006.
- Brunschwig, Jacques, G. E. R. Lloyd, and Pierre Pellegrin, eds. Greek Thought: A Guide to Classical Knowledge. Harvard University Press Reference Library. Cambridge, Mass.: Harvard University Press, 2000.
- Brunschwig, Jacques, Monique Canto-Sperber, and Pierre Pellegrin. Le Style de la pensée: Recueil de textes en hommage à Jacques Brunschwig. Paris: Belles lettres, 2002.
- Smith, Robin. ‘Aristotle’s Logic’, The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), Edward N. Zalta(ed.), online at http://plato.stanford.edu/archives/spr2009/entries/aristotle-logic/.