Wednesday, 15 July 2015

Problems with reductio proofs: "jumping to conclusions"

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the third installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, and here is Part II. In this post I discuss issues pertaining specifically to the last step in a reductio argument, namely that of going from reaching absurdity to concluding the contradictory of the initial hypothesis.

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One worry we may have concerning reductio arguments is what could be described as ‘the culprit problem’. This is not a worry clearly formulated in the protocols previously described, but one which has been raised a number of times when I presented this material to different audiences. The basic problem is: we start with the initial assumption, which we intend to prove to be false, but along the way we avail ourselves to auxiliary hypotheses/premises. Now, it is the conjunction of all these premises and hypotheses that lead to absurdity, and it is not immediately clear whether we can single out one of them as the culprit to be rejected. For all we know, others may be to blame, and so there seems to be some arbitrariness involved in singling out one specific ingredient as responsible for things turning sour.

To be sure, in most practical cases this will not be a real concern; typically, the auxiliary premises we avail ourselves to are statements on which we have a high degree of epistemic confidence (for example, because they have been established by proofs that we recognize as correct). But it remains of philosophical significance that absurdity typically arises from the interaction between numerous elements, any of which can, in theory at least, be held to be responsible for the absurdity. A reductio argument, however, relies on the somewhat contentious assumption that we can isolate the culprit.

However, culprit considerations do not seem to be what motivates Fabio’s dramatic description of this last step as “an act of faith that I must do, a sacrifice I make”. Why is this step problematic then? Well, in first instance, what is established by leading the initial hypothesis to absurdity is that it is a bad idea to maintain this hypothesis (assuming that it can be reliably singled out as the culprit, e.g. if the auxiliary premises are beyond doubt). How does one go from it being a bad idea to maintain the hypothesis to it being a good idea to maintain its contradictory?

 It may be helpful at this point to notice that a reductio ad absurdum is a special form of proof by cases. In a typical proof by cases, if one wants to establish that all x have property Y, for example, it may be helpful to divide the class of x things into groups, and then prove for each of the groups that all its members have property Y. More generally, a proof by cases starts with a presumably exhaustive enumeration of cases – say, A, B, C, and D – and then goes on to show that if A, then Z; if B, then Z; if C, then Z; if D, then Z. Now, on the assumption that the enumeration is exhaustive, if follows unconditionally that Z, in all circumstances.

A reductio ad absurdum also starts with the tacit assumption of an exhaustive enumeration of cases: for a given proposition A, either A is the case or its contradictory is the case, but not both (this follows from excluded middle and the principle of non-contradiction). Once it is shown that A leads to absurdity, by elimination one concludes that its contradictory must be the case. But notice that, other than in a typical proof by cases, one of the options has not been investigated at all; one is not required to investigate the other option, that is to assume the contradictory ~A to see what happens (after having assumed A and led it to absurdity). The presupposition is that, since exactly one of the two propositions must obtain, once one of them has been eliminated, the second becomes established even though it has not itself been investigated.

There are at least two worries one may voice concerning the validity of these assumptions. 1. Is the enumeration really exhaustive? (Maria: “there might be many things that I haven’t seen.”) If it is not, it would be premature to conclude that, since A cannot be the case, then ~A must be the case; there may be some other option B which hasn’t been considered. 2. What if both A and ~A lead to absurdity? We are left in the dark as to whether this might happen if we do not run a similar procedure for ~A.

While 2 may seem somewhat far-fetched, there are actually a number of philosophical examples of situations of exactly this kind, which are often described as ‘aporetic’. Some of Plato’s dialogues, for example the Parmenides and the Theaetetus, can be described in these terms: a (presumably) exhaustive enumeration of cases is presented (e.g. in the Theaetetus, different meanings of ‘knowledge’), all of them are investigated, and all of them are found to lead to absurdity. Another such example are Kant’s antinomies (even though Kant believed that they can ultimately be resolved, once the underlying mistakes are identified).

This is not to say that these issues will affect each and every instance of a reductio argument; we may have rock-solid reasons to believe that the enumeration in question is indeed exhaustive; it may well be the case that A leads to absurdities while we have no reason to think that the same will happen with ~A. But clearly these are strong assumptions which we may have reasons to question, in specific cases at least. Concerns of this nature may well be what leads Fabio to describe the last step as ‘an act of faith’.

In sum, we have identified four problems exerting some pressure on the epistemic status of reductio arguments; this was accomplished partially (though not exclusively) based on cognitive considerations (in particular, the testimonies of Maria and Fabio).
  1. Representing the impossible
  2. Pragmatic awkwardness of the first speech act
  3. The culprit problem
  4. The ‘act of faith’ problem

In upcoming posts I will argue that 2 is fully resolved once one adopts a dialogical conception of reductio arguments; I will also argue that both 3 and 4 arise from the fact that reductio arguments are expected to do something that their genealogical predecessors, dialectical refutations, were not designed to accomplish, namely to establish the truth or falsity of specific theses. As for 1, admittedly the dialogical perspective does not seem to have that much to add towards possible solutions, but this is a problem that goes well beyond issues pertaining to reductio arguments alone; indeed, it is a problem we all struggle with.

2 comments:

  1. "a reductio ad absurdum is a special form of proof by cases"

    There seems to be no good reason to defend the above, on purely logical grounds. Indeed, from a semantical viewpoint, reductio ad absurdum and proof by cases have dual justifications: the former is supported by the principle according to which a sentence and its negation cannot both simultaneously be true; the latter by the principle according to which a sentence and its negation cannot both simultaneously be false.

    (The fact that intuitionistic logic rejects both reductio and proof by cases only makes matters more confuse. There are plenty of logics in the literature that reject only one of these.)

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  2. Well, a reductio proof is not a typical proof by cases anyway, because it fails to examine all the cases (it examines A, reaches absurdity, concludes ~A without having examined it, or vice-versa). But I find it helpful to put things in this way precisely to outline that it is somehow strange not to investigate all cases.

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