(Cross-posted at NewAPPS)
Some months ago I wrote two posts on the concept of indirect proofs: one presenting a dialogical conception of these proofs, and the other analyzing the concept of ‘proofs through the impossible’ in the Prior Analytics. Since then I gave a few talks on this material, receiving useful feedback from audiences in Groningen and Paris. Moreover, this week we hosted the conference ‘Dialectic andAristotle’s Logic’ in Groningen, and after various talks and discussions I have come to formulate some new ideas on the topic of reductio proofs and their dialectical/dialogical underpinnings. So for those of you who enjoyed the previous posts, here are some further thoughts and tentative answers to lingering questions.
Recall that the dialogical conception I presented in previous posts was meant to address the awkwardness of the first speech act in a reductio proof, namely that of supposing precisely that which you intend to refute by showing that it entails an absurdity. From studies in the literature on math education, it is known that this first step can be very confusing to students learning the technique of reductio proofs. On the dialogical conception, however, no such awkwardness arises, as there is a division of roles between the agent who supposes the initial thesis to be refuted, and the agent who in fact derives an absurdity from the thesis.
However, at that point I did not have a satisfactory answer to the contentious last step in a reductio proof, namely to conclude A from the fact that ~A entails an absurdity, or to conclude ~A from the fact that A entails an absurdity. (It is important to distinguish these two cases, as intuitionists are more than happy to accept the latter but vehemently deny the former.) In more pragmatic terms, the issue with this last step is the contentious move from showing that maintaining thesis A is a bad idea to concluding that maintaining thesis ~A is a good idea. For all we know, maintaining thesis ~A is also a bad idea.
Another philosophical problem with reductio proofs (which was mentioned during the conference by Mathieu Marion, and if my memory does not fail me a point that was raised by Mic Detlefsen when I presented the material in Paris) is that there is no way of knowing who exactly is the 'culprit' that led to absurdity. In a reductio proof, typically a given statement is isolated from the start as the one to be refuted (“Suppose for reductio that…”). But as the proof proceeds, along the way typically use is made of other premises, which may themselves be the source of trouble leading to absurdity. Why single out one particular thesis as the one to be refuted? Why conclude ~A rather than ~B in a reductio proof that starts with an assumption of A, but which uses B as a premise along the way?
Finally, a frequent charge against reductio proofs is that they have lower explanatory value than direct proofs: they show that something is the case, but not why something is the case. This is the gist of Aristotle’s own statement of the superiority of ‘ostensive’ (direct) demonstration over demonstration through the impossible in the Posterior Analytics A 26 (thanks to Marko Malink for the reference).
At the conference earlier this week, I came to realize that these issues arise from the fact that reductio proofs are expected to do a job that their genealogical antecedent, namely dialectical refutations, were not expected to perform. The typical structure of a dialectical exchange (as exemplified in Plato’s dialogues but also in Aristotle’s Topics) is for the questioner to show that the theses jointly maintained by the answerer lead to absurdity; in other words, the goal is to show that the answerer’s epistemic and discursive commitments are jointly incoherent. Indeed, this is the quintessential, and perhaps even the only way to ‘win’ such debates: to force your opponent into an incoherent position.
Now, as argued by L. Castagnoli (one of the conference’s participants) in his work on the notion of self-refutation (book, paper), the goal of reaching absurdity in such dialectical exchanges is not to establish the truth-value of a given thesis (either the falsity of one of answerer’s claims or the truth of the opposite claim), but simply to serve as a dialectical ‘silencer’:
Although ancient self-refutation arguments cannot ‘falsify’ our most radical adversaries’ views (and defuse our own most hyperbolical doubts) by proving that what they envisage is ‘logically impossible’, they can silence them, by delimiting the area of constructive philosophical inquiry and debate. (Castagnoli 2007, 69)
This is exactly what happens, for example, in Zeno’s paradoxical arguments: they show that rejecting Parmenides’ thesis that there is no such thing as movement or change leads to absurd conclusions. The argument is not meant to establish the truth of Parmenides’ position, but simply to discredit the position of his opponents. [UPDATE: this is precisely the position that Plato attributes to Zeno in Parmenides 128 a-e -- thanks to Matthew Duncombe for the reference.]
What this all means for the issue of reductio proofs is that, in a sense, and at least historically, indirect argumentation of this kind – refutation – had pride of place in ancient dialectic, rather than being a derivative concept vis-à-vis direct argumentation. However, these refutations were not expected to establish the truth-value (either truth or falsity) of a given statement: they were only meant to show the incoherence of the answerer’s overall position. In other words, the last contentious step in a reductio proof is conceptually extraneous to the original dialectical framework – no wonder it looks suspicious! Similarly, given that a refutation is intended to discredit the answerer’s overall position, the issue of the actual culprit leading to absurdity does not arise: there is no need to locate the exact source of trouble. Finally, such refutations were not expected to perform an explanatory role; instead, if they were intended as dialectical silencers, they were purely adversarial tools, whereas explanation is essentially a cooperative notion.
The core idea in my current research project is that every proof is and is not a dialogue: it is a dialogue because it retains crucial dialogical/dialectical features, but it is not a dialogue properly speaking because one of the participants has been internalized (the built-in opponent hypothesis). In the case of reductio proofs, their dialogical/dialectical origins are even more acutely perceived, but a key transformation was from the concept of purely negative refutations as ‘dialectical silencers’ to the concept of indirect proofs, capable of establishing positive conclusions. Moreover, as purely eristic (adversarial) dialectic then gave rise to more cooperative, didactic contexts where explanation of causes became the key notion (as in Aristotle’s notion of episteme), direct arguments came to be preferred over indirect ones.
In sum, I submit that reductio proofs are rightly seen as philosophically problematic. Even if the dialogical conception alleviates worries concerning the awkwardness of the first speech-act in a reductio proof, the contentious last step from absurdity to the final conclusion is extraneous to the dialectical framework of refutations. It requires (potentially unfairly) singling out a particular assumption as the culprit, and is based on a pragmatically questionable move (bad idea to maintain A/~A => good idea to maintain ~A/A).
However, while perhaps philosophically suspicious, I think it is fair to say that most of us are not prepared to live without this argumentative strategy in our tool boxes; it is just too damn convenient!
(Caveat: this analysis is based only on elements from the history of logic and philosophy; it would be interesting to see if similar considerations apply to the development of reductio proofs in mathematics.)