Friday, 11 January 2013

Indirect proofs in the Prior Analytics

(Cross-posted at NewAPPS)

A few days ago I wrote a post on a dialogical conceptualization of indirect proofs. Not coincidentally, much of my thinking on this topic at the moment is prompted by the Prior Analytics, as we are currently holding a reading group of the text in Groningen. We are still making our way through the text, but here are some potentially interesting preliminary findings.
I am deeply convinced that the emergence of the technique of indirect proofs marks the very birth of the deductive method, as it is a significant departure from more ‘mundane’ forms of argumentation (as I argued before). So it is perhaps not surprising that the first fully-fledged logical text in history, the Prior Analytics, offers a sophisticated account of indirect proofs.

The first chapters of the Prior Analytics focus on showing which combinations of pairs of premises of the four categorical propositional forms (a: Every A is B; i: Some A is B; e: No A is B; o: Some A is not B) produce conclusions that follow ‘of necessity’. He first argues (chap. 4) that the so-called first-figure syllogisms are perfect (or complete, in Smith’s translation) because their validity is immediately apparent to us: it follows from the meaning of ‘Every’ and ‘No’ (Dici de omni/de nullo). What determines the figure of a syllogism is the position of the middle term with respect to the two other terms. (I am adopting the ‘A is B’ formulation, but Aristotle famously also uses the ‘B belongs to A’ schema: I use M for middle term, S for subject of the conclusion, and P for predicate of the conclusion).

First                Second                  Third
M/P                P/M                       M/P
S/M                S/M                       M/S
------              -------                     -------
S/P                 S/P                         S/P

What he does next is to show that valid syllogisms in the second and third figures can be shown to be valid by means of a process of ‘perfection’ (or ‘completion’), which consists in applying a few rules of inference (the perfect syllogisms themselves, conversion and subalternation) to pairs of premises so as to obtain a conclusion (see a paper of mine with Edgar Andrade for further details). So for example, the pair ‘No P is M, Every S is M’ can be shown to produce the conclusion ‘No S is P’ by an application of conversion to the first premise, which results in ‘No M is P’, and then we have Celarent, which is one of the first-figure perfect syllogisms.

Conversion consists in switching positions for subject and predicate. Naturally, since it is a matter of changing the relative disposition of the terms in the premises (so as to obtain the disposition that characterizes the first-figure syllogisms), conversion is the key device. But only the e and the i propositions convert simpliciter: from ‘No A is B’ we can infer ‘No B is A’ (and vice-versa), and the same for ‘Some A is B’. The a and o propositions do not convert (the a propositions are said to convert accidentally: ‘Every A is B’ converts to ‘Some B is A’). So when we have a pair of a and/or o premises, the proof-theoretical framework of syllogistic does not offer any devices to ‘perfect’ the pair in question, even though some such combinations do produce conclusions, such as Baroco (second figure: ‘Every P is M, some S is not M, thus some S is not P’) and Bocardo (third figure: ‘Some M is not P, every M is S, thus some S is not P’).

This is where indirect proofs come in. To perfect such syllogisms, what Aristotle calls the ‘ostensive’ (direct) approach (in the Striker translation; ‘probative’ in the Smith translation) will not do, simply because the proof-theoretical power of syllogistic is quite limited.

Aristotle contrasts the idea of an ostensive argument with that of an argument from an assumption/hypothesis (chapter A 23). Arguments leading to the impossible, which correspond to our notion of an indirect proof, are for him a kind of argument from an assumption/hypothesis. To illustrate an argument leading to the impossible, Aristotle actually offers a mathematical example, namely the proof of the incommensurability of the diagonal (41a26-28). This is important, as it suggests more than casual contact between mathematicians and philosophers at the time when the deductive method was taking shape almost simultaneously in both disciplines.

Elsewhere in the text, he uses the same approach to perfect the syllogisms that cannot be perfected ‘ostensively’ (directly), i.e. those containing premises and conclusions that cannot be converted. His usual procedure is the following: if you want to show that premises A and B produce conclusion C, you take A and the contradictory of the conclusion, not-C, and show that you can deduce not-B from A and not-C. The general idea can be represented as follows (I owe this schema to Leon Geerdink):
[not-C]            A
:
:
--------------
                 not-B               B
                     ----------------------
          ⊥
            --------------
         C
You can thus deduce C from A and B, but with the auxiliary hypothesis/assumption of the contradictory of C. The subproof from [not-C] and A to not-B is constituted of direct applications of the usual rules of inference of the system (conversion and/or one of the perfect syllogisms), and the fact that [not-C] is treated as a hypothesis is at this point an extra-logical, quasi-pragmatic property of the proof. (He uses the phrase ‘reached through an agreement’ to refer to the status of the hypothesis, which clearly has a dialectical flavor).

There is a beautiful formulation of the different stages of the proof in 41a23-26 (Striker translation):
All those who reach a conclusion through the impossible deduce the falsehood by a syllogism, but prove the initial thesis from a hypothesis, when something impossible results from the assumption of the contradictory.
He thus distinguishes the act of deducing (which corresponds above to going from not-C and A to not-B, which is the falsehood) from the act of proving, which refers to the whole argument leading to the main conclusion C, the ‘initial thesis’. The act of deducing here is an ostensive argument and corresponds to a subproof, whereas the act of proving (showing) corresponds to the whole demonstration. For the act of deducing, the status of the premises (taken as hypotheses or asserted) is irrelevant, but for the act of proving it makes all the difference that not-C is merely taken as a hypothesis at the beginning. This is an important distinction to keep in mind, and one of which Aristotle was already keenly aware.

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