A dialogical conception of indirect proofs

In his commentary on Euclid, the 5th century Greek philosopher Proclus defines indirect proofs, or ‘reductions to impossibility’, in the following way (I owe this passage to W. Hodges, from this paper):
Every reduction to impossibility takes the contradictory of what it intends to prove and from this as a hypothesis proceeds until it encounters something admitted to be absurd and, by thus destroying its hypothesis, confirms the proposition it set out to establish. 
Schematically, a proof by reduction is often represented as follows:


It is well know that indirect proofs pose interesting philosophical issues. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? Why assert it in the first place? What kind of speech act is that? It has been pointed out that the initial statement is not an assertion, but rather an assumption, a supposition. But while we may, and in fact do, suppose things that we know are not true in everyday life (say, in the kind of counterfactual reasoning involved in planning), to suppose something precisely with the goal of demonstrating its falsity is a somewhat awkward move, both cognitively and pragmatically.

It seems to me (but this is ultimately an empirical hypothesis to be further investigated) that there are only two situations where this argumentative strategy is regularly used: mathematical and legal contexts (keep legal contexts in mind; they will come back). In other words, my claim is that when people are arguing at the pub or such like, they do not use reductio arguments. (This is something that psychologist David Over and I have a bet on: he thinks people do, I think they don’t. Maybe one day we’ll run a research project together to investigate ‘reductio in the wild’, so to speak.) UPDATE: In comments at NewAPPS, Branden Fitelson mentions this excellent paper on why it is so difficult (it is!) for students to understand the concept of an indirect proof. 

Even in the relevant circles of specialists, quite a few people have issues with indirect proofs, most famously intuitionists who reject double-negation elimination – the crucial step which goes from the rejection of ~A to the assertion of A. It is also often said that Frege’s account of inference as going from true statements to true statements leaves no room for indirect proofs (but here is a recent paper by Ivan Welty countering this claim). So even within mathematics and logic, indirect proofs are somewhat controversial.

If we accept that indirect proofs are a bit of an oddity even within mathematics, it makes sense to ask how on earth this argumentative strategy might have emerged and established itself as one of the most common ways to prove mathematical theorems. Now, as some readers may recall, my current research project focuses on ‘the roots of deduction’, adopting the hypothesis that we need to go back to deduction’s dialogical origins to make sense of the whole thing (as discussed here, for example). And here again, it seems that the dialogical, multi-agent perspective offers fresh insight into the nature of indirect proofs.

Assume a dialectical context in which two participants are disputing on a certain topic, and let us call them 1 and 2 and B to keep it neutral. Then imagine that 1 wants to convince 2 of proposition A; how can she go about? Well, she can propose ~A and see if 2 takes the bait. It is important that ~A be put forward in the form of a question (which is indeed how such disputations often began in ancient Greece, as attested for example by Aristotle’s Topics), so that by accepting ~A, 2 commits to its truth but not 1; 1 has merely put it forward as a question and thus has herself not endorsed ~A. 1 can now proceed to show that something absurd follows from the acceptance of ~A, because this is not her position; it is 2’s position. By showing that something absurd follows from ~A, 1 in fact shows that it was a bad idea for 2 to accept ~A in the first place. There is still the contentious last step which goes from ‘accepting ~A is a bad idea’ to ‘accepting A is a good idea’. But 1 has not done anything pragmatically incoherent because she herself never committed to ~A.

In legal contexts, reductio arguments are used in much the same way. The prosecution may claim A (the defendant was at the crime scene), and the defense may then show that, given additional background information, A leads to absurdity (say, to the possibility of traveling between Paris and London in less than 30 min). (Welty’s paper has a similar legal example.) So what you show as entailing absurdity in a reductio argument is in fact the position of your opponent, not your own position (not even your own assumption). The adversarial, multi-agent component is crucial to understand what it means to prove something indirectly; it makes the postulation of the strange speech-act of supposing precisely that which you want to prove to be false superfluous. In a purely mono-agent context, in contrast, she who formulates an indirect proof has to play awkwardly conflicting roles simultaneously. (Naturally, it is perfectly possible to formulate an indirect proof on your own, but this is a consequence of what I describe as the ‘internalization of opponent’ by the method itself.)

I think that this multi-agent, dialogical account of indirect proofs is conceptually appealing on its own, but within the Roots of Deduction project, we (Matthew Duncombe, Leon Geerdink and myself) are also investigating the historical plausibility of the hypothesis. For now, it is interesting to notice that, in the Prior Analytics, Aristotle makes extensive use of indirect proofs, as is well known, but also that he often uses dialectical vocabulary to explain the concept of an indirect proof. (In fact, he uses dialectical vocabulary throughout the text.) UPDATE: here is a subsequent post I wrote on indirect proof in the Prior Analytics.

(A cool coincidence is that just yesterday Mic Detlefsen invited me to present at his PhilMath Intersem colloquium in Paris in June, precisely on the topic of the history of indirect proofs. So there will be much work to be done on the topic for me, but for now this is my starting point.)


  1. "...when people are arguing at the pub or such like, they do not use reductio arguments."

    Maybe "What if everybody did that?"

    There seems to be a little bit on reductios in classical Chinese logic.

  2. "The adversarial, multi-agent component is crucial to understand what it means to prove something indirectly; it makes the postulation of the strange speech-act of supposing precisely that which you want to prove to be false superfluous."

    As a corollary, in the footnote aesthetic (known as criticism) agency is the expression of individual points of view, so that a refutation is essentially an addition to one's opponent's perspective-train-of-thought, so there is a question of whether critical rhetoric is the same thing as rhetoric prima facie. For example, simply "is" such a rhetorical point of view actually even a secondary argument if one holds a position on one's own in some more formal context? Should we assume that primary arguments are metaphysics, and banish them to the flames, should we adopt linguistic formalism to blur the boundary between criticism and inner dialectic, or should we address rhetoric as a form of agency as you propose?

    Certainly it is still possible to see some rhetorical modes as being personal or personal-extensive rather than dialectical or social-technological, but this does not resolve the problem of agency as you pose it, with the critical boundary between criticism and entity arguments.

    What if rhetoric reduces to "agents", then it seems to reduce to causality. While this and other of my earlier related statements seem to express an obsession that has already been encountered in philosophy, it seems like a potent issue as you suggest that agency could be central to rhetorical forms.

    The question then becomes, what is 'entic agency'? And then, what is entic agency in rhetoric? Perhaps something in these concepts can be linked to Socrates' concept of aporia, to clarify that some concepts 'are real',,,and other concepts merely 'wish they had an argument',,,

    Perhaps it is time for a concept like 'wishism' or 'wishschism'?

    Wishschism, def: The prospective difference between rhetoric and aporia.

    But why keep adding new concepts? There must be something in philosophy that does not require literature---perhaps graphics, or rarification. I adopt both graphics and rarification in my forthcoming book, The Dimensional Philosopher's Toolkit, due to be released in 2013 (Febr. or March supposedly).

  3. What is so awkward about assuming the opposite to see how it works? Wouldn't this only show some humility in the face of your problems to see if the opposite was a feasible strategy?

    Also, I don't this this is uncommon practice at all. An example from physics is that Einstein was said to have argued against Bohr's ghost about quantum mechanics: Even on his deathbed he reasoned (Bohr was long dead at this point), "If Bohr argued this, I'd counter that. If he said this, I'd say that," etc.

    Perhaps this is what you mean of the internalization of the opponent, but, again, I don't think there is anything unusual about it. People train for competition all the time by considering their opponent's moves. It's doing your due diligence when in any sort of competition, or just shadowboxing for training.

    Moreover, the Guinness Book of World Records was started by the Guiness Brewing Company for the sole reason of settling barroom arguments about what is possible and had been done. That is, someone claims they caught the biggeset fish ever... Their friends call BS on that claim... They check the Guinness Book of World Records and see if it shows the claim to be absurd. Their friend is shown to be a liar. QED.

    I hope you didn't bet too much.

    1. My point exactly. In a multi-agent, adversarial situation, it may often happen that you discredit the position of your opponent by showing that it entails an absurdity. A true reductio argument in a bar context would be for someone to claim A, then assume not-A (while nobody else has made commitments regarding either A or not-A), derive an absurdity from your own claim, and conclude A, all on your own, so to speak. That I don't see happening very often in pubs.

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  5. Bah. Left a bit out. The point about the Guinness Book of Records is that people constantly are making ridiculous claims at bars, and everyone assumes the opposite. So there needed to be a way to test who is right and someone is going to be shown a liar.

  6. Baldly stated, the assertion that reductio ad absurdum type arguments only arise in Rule bound disciplines seems difficult to swallow. After all, the Sophists, i.e. the rhetoricians who coached public speakers, used arguments of the form 'X says Y and X is a worthy man. But does X really believe Y? If Y then Z, which is absurd. Thus X is a worthless liar who should be impeached.'
    However, X has a common sense rebuttal- 'Look when I said Y, I was using imperative language. An imperative proposition does not have the sort of logical entailment properties you suggest. If a mother says to her daughter 'if I see you talking to boys I will break your legs' it does not mean she will break her daughter's legs on the honeymoon night because her new husband belongs to the male gender and some conversation tends to break the ice on such occasions.'
    The argument from an internalized opponent too needs some fine-turning. I like ties with cartoon animals on them. My internal opponent warns me not to wear such a tie when going for a job interview and I make the cost-benefit calculation not to do so. No doubt, if I were more eminent in my field, the pay-off matrix would have been the opposite and I'd have worn the garish tie and perhaps my bedroom slippers as well! Clearly the internal opponent has to do with pay-off matrices. As an adolescent, it may be very strong on stuff like what's cool to listen to (or pretend to listen to) and very weak on stuff to do with studies (Teen-agers are happier reading Ayn Rand than Aristotle). Later, the pay-off matrix changes, at least for people in intellectually demanding professions, and we witness the surprising phenomenon of big brain/big ego people becoming convinced of their own errors by purely rational argument. However, this is because they feel the error may hurt them down the road- i.e. they are still 'in it to win it', so this is a purely tactical concession on their part.
    With these two caveats, I think the point you are making, if properly hedged around, makes a lot of intuitive sense and is going to be useful. Still, a dialogic approach will leave untouched and untraversed vast portions of the hidden continent of intentionality unless there is indeed some proper eidetic theory of the object such that there is a severe constraint on Stalnaker-Lewis type closest possible sphere of worlds.
    In Law, there is a notion of harmonious construction and Dworkin's Judge Hercules who can always re-interpret things so that the fabric of the Law suffers no tear or wrinkle- but surely it is a fable or 'Noble Lie'? The fact is different people have such different notions of what Justice aims at, or what states it approves, that you have an extremely inflationist Ontological dysphoria built into the received dialogic.
    In Maths and Physics and so on, similarly, though the volume of consensus-results may increase exponentially, the Ontologically dysphoric 'Life-worlds' of leading agents in these fields probably diverges yet faster.


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