Tuesday, 14 July 2015

Problems with reductio proofs: assuming the impossible

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is a series of posts with sections of the paper on reductio ad absurdum from a dialogical perspective that I am working on right now. This is Part II, here is Part I. In this post I discuss issues in connection with the first step in a reductio argument, that of assuming the impossible.


We can think of a reductio ad absurdum as having three main components, following Proclus’ description:

(i) Assuming the initial hypothesis.
(ii) Leading the hypothesis to absurdity.
(iii) Concluding the contradictory of the initial hypothesis.

I discuss two problems pertaining to (i) in this post, and two problems pertaining to (iii) in the next post. (ii) is not itself unproblematic, and we have seen for example that Maria worries whether the ‘usual’ rules for reasoning still apply once we’ve entered the impossible world established by (i). Moreover, the problematic status of (i) arises to a great extent from its perceived pragmatic conflict with (ii). But the focus will be on issues arising in connection with (i) and (iii).

A reductio proof starts with the assumption of precisely that which we want to prove is impossible (or false). As we’ve seen, this seems to create a feeling of cognitive dissonance in (some) reasoners: “I do not know what is true and what I pretend [to be] true.” (Maria) This may seem surprising at first sight: don’t we all regularly reason on the basis of false propositions, such as in counterfactual reasoning? (“If I had eaten a proper meal earlier today, I wouldn’t be so damn hungry now!”) However, as a matter of fact, there is considerable empirical evidence suggesting that dissociating one’s beliefs from reasoning is a very complex task, cognitively speaking (to ‘pretend that something is true’, in Maria’s terms). The belief bias literature, for example, has amply demonstrated the effect of belief on reasoning, even when participants are told to focus only on the connections between premises and conclusions. Moreover, empirical studies of reasoning behavior among adults with low to no schooling show their reluctance to reason with premises of which they have no knowledge (Harris 2000; Dutilh Novaes 2013). From this perspective, reasoning on the basis of hypotheses or suppositions may well be something that requires some sort of training (e.g. schooling) to be mastered.

For our purposes, it may be useful to distinguish a number of different cases involving reasoning with false or impossible hypotheses:
  • The hypothesis may be false or impossible, without me in fact knowing that it is false or impossible (indeed, I may be in the process of investigating precisely that).
  • The hypothesis may be false or impossible, and I know it to be false or impossible, but I want to see what would be the case if it were true (counterfactual reasoning).
  • The hypothesis may be false or impossible, I know it to be false or impossible, and I am assuming it precisely with the goal of showing it to be false or impossible (a reductio argument).
In the narrow sense adopted here, only the last case counts as true reductios ad absurdum. My claim is that, while the first and even the second cases describe cognitive activities that human beings regularly engage in, also outside circles of specialists, the third case introduces a much stronger component of pragmatic awkwardness: to assume something precisely with the goal of showing it to be false. In Leron’s terms (cited in the previous post), this corresponds to the act of postulating a false/impossible world only to proceed to ‘destroy’ it (show it to be false/impossible). Maria describes it as telling herself a ‘lie’ and having to believe it.

Perhaps this pragmatic awkwardness arises only due to a reasoner’s failure to appreciate the difference between a categorical assertion and an assumption/hypothesis. Indeed, in the examples typically offered to illustrate the purported pervasiveness of reductio arguments ‘in the wild’, one usually encounters arguments formulated as a conditional or hypothetical sentence (“If he gets here in time for supper, he'll have to fly like Superman”). However, the conventions for the formulation of a reductio proof in mathematics do not display this general structure perspicuously. Usually, such proofs start with ‘Suppose that A’; now, while the ‘Suppose that…’ operator should function in much the same way, judging from Maria’s and Fabio’s reports it seems that reasoners feel they must truly commit to the truth of the initial hypothesis while leading it towards absurdity. In effect, this is precisely what seems to be so cognitively demanding: the conflicting roles of ‘pretending’ to believe the hypothesis and of working towards its very destruction.

The distinction between the initial assumption being merely false or else impossible is also worth discussing in some detail. On most theories of (mental) propositional attitudes, having an attitude towards a false proposition is for the most part unproblematic: false propositions are much as true propositions in terms of their semantics (spelled out in e.g. truth-conditional terms, or use-based accounts, or what have you). Just as I can believe something that is false, I should be able to suppose, i.e. entertain the possibility of, something that is false (even if I know it to be false, which is not compatible with belief).

But matters become considerably more delicate once we are dealing with impossible propositions. While it is not necessarily the case that every reductio argument will start by assuming an impossibility, this is indeed what happens in many paradigmatic cases – certainly if we accept that mathematical truths are necessary truths, and thus mathematical falsities are necessary falsities, i.e. impossibilities. In the Tractatus, Wittgenstein (in)famously claimed that impossibility cannot be expressed, as it cannot be depicted; impossible thoughts cannot be conceived, and impossible propositions cannot be meaningfully formulated. Even though most of us have surely outgrown the Tractarian conception of propositions, we still seem to be stuck with the problem of providing a satisfactory philosophical account of impossible thoughts. As well put by Jago:
Impossible thoughts might appear nothing more than a quirky feature of the way we can meaningfully represent the world around us. It may come as something of a surprise, therefore, to learn that just about every major philosophical theory of content and meaning is unable to account for impossible thoughts. Moreover, a wide variety of philosophical views converge in a kind of pressure group against the existence of impossible thoughts. (Jago 2014, 3)
Jago then goes on to detail how and why different philosophical frameworks fail miserably to account for impossible thoughts and impossible content. What is particularly unsatisfying is that these frameworks (including the Tractarian framework) tend to treat all impossible thoughts/propositions as having the same content: in possible world semantics, for example, all impossible propositions have the same meaning, namely they correspond to the empty set (the domain being the collection of all possible worlds). But the initial assumptions of reductio proofs in mathematics surely do not all have the same content! If they did, each such reductio proof would be essentially identical to all others, and this is obviously an undesirable result.

In other words, a philosophical account of reductio ad absurdum requires a sufficiently fine-grained account of impossible thoughts/propositions if we are to make sense of the initial step, that of putting forward the initial hypothesis. Naturally, there are many other reasons to look for adequate theories of the impossible, and some proposals have been put forward (e.g. (Jago 2014)). Here, I will not venture into offering yet another theory of impossible thoughts; for my purposes, it is enough to note that this is yet another point where reductio ad absurdum is not a straightforward affair from a philosophical perspective.

In sum, there are two problems with the initial speech act in a reductio: the general problem of representing impossible content so as to make sense of the speech act of assuming an impossibility; and the pragmatic and equally serious problem of the awkwardness of putting forward a proposition (even if as an assumption or hypothesis), ‘pretending’ to believe it, precisely in order to show it to be false/impossible. It is this second problem which will receive a natural solution once one adopts a dialogical perspective.

(To be continued...)


  1. One natural idea for making reductio proofs more palatable to those who find this first stage problematic, which you suggest by mentioning counterfactuals, is to use appropriate "subjunctive" language. For example, instead of saying, "Suppose that the square root of 2 is rational," one would say, "Suppose that the square root of 2 were rational." Then one is not telling a lie any more: one is merely considering the consequences of a hypothesis.

    1. Yes, this seems right. As I suggested, one of the problems with classical presentations of reductio proofs is that they adopt categorical language to talk about the hypothesis to be refuted.

  2. You distinguish between three cases which may occur whenever someone deals with a reductio. It occurred to me that the following may occur: One person claims A to be the case, but his friend insists on ¬A. Genuinely interested in discovering the truth together, they then investigate and by means of a reductio, assert A after assuming ¬A. This is in line with Case 2.

    Here on out, should either party, pedagogically speaking, wish to educate anyone else on the matter, wouldn't they fall under Case 3, when they wish to share unless they specifically had it in mind to act out their former ignorance?

    In other words, it struck me that Case 3 may just be a retelling of activities that fell under case 2 where, and as you point out this is important, the one doing the retelling has conviction regarding the outcome. Depending on what one thinks, it may be the case that *every* event that falls under Case 3 is a retelling of case 2 from a new standpoint of knowledge e.g. a teacher or textbook who has resolved the "final gap problem," or a description of such an event.