By

**Catarina Dutilh Novaes**
(Cross-posted at NewAPPS)

This is the final post in my series on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, here is Part IV, and here is Part V. I now return to the issues raised in the earlier posts equipped with the dialogical account of deduction, and of reductio ad absurdum in particular.

===========================================

A
general dialogical schema for reductio ad absurdum, following Proclus’
description but
inspired by the Socratic elenchus, might look like this:

- Interlocutor 1 commits to A (either prompted by a question from interlocutor 2, or spontaneously), which corresponds to assuming the initial hypothesis.
- Interlocutor 2 leads the initial hypothesis to absurdity, typically by relying on additional discursive commitments of 1 (which may be elicited by 2 through questions).
- Interlocutor 2 concludes ~A.

The
main difference between the monological and the dialogical versions of a
reductio is thus that in the latter there is a kind of

*division of labor*that is absent from the former (as noted above). The agent making the initial assumption is not the same agent who will lead it to absurdity, and then conclude its contradictory. And so, the perceived pragmatic awkwardness of making an assumption precisely with the goal of ‘destroying’ it seems to vanish. Moreover, the adversarial component provides a compelling rationale for the general idea of ‘destroying’ the initial hypothesis; indeed, while the adversarial component is present in all deductive arguments (in particular given the requirement of necessary truth preservation, as argued above), it is even more pronounced in the case of reductio arguments, that is the procedure whereby someone’s discursive commitments are shown to be collectively incoherent since they lead to absurdity. There remains the question of why interlocutor 1 would want to engage in the dialogue at all, but presumably she simply wishes to voice a discursive commitment to A. From there on, the wheel begins to spin, mostly through 2’s actions.
Thus,
the issue of the pragmatic awkwardness of the first speech act is fully
(dis)solved once one adopts the multi-agent, dialogical perspective which
ensures different rationales/motivations for the different steps in the
argument, which are carried out by different agents. In the mono-agent case, as
a result of the process of internalization of one of the participants described
here, one and the same agent has to play conflicting roles, which for
some reasoners (Maria and Fabio, for example) seems to create a situation of
cognitive dissonance.

What
about the three other issues discussed here and here? As already mentioned, the
dialogical perspective does not seem to offer any new resources to tackle the
issue of how to represent the impossible; we are still saddled with this
problem just as we were in the monological case. But the dialogical perspective
does have something to offer with respect to the other two issues, namely the
culprit problem and the ‘act of faith’ problem, even if it does not lead to
fully-fledged (dis)solutions as in the case of the first speech act problem.

Indeed,
the key point is the idea that dialectical refutations are the genealogical ancestors
of reductio ad absurdum arguments. Both the culprit problem and the ‘act of
faith’ problem pertain to the last step in a reductio argument, namely the step
from absurdity to the final conclusion (which is the contradictory of the
initial assumption), and thus to the idea that a positive outcome can be
reached – the establishment of the truth (or falsity) of a given claim. As
we’ve seen, most scholars believe (

*contra*Vlastos) that this last step is absent in dialectical refutations, which can only have the negative outcome of establishing the collective incoherence of a group of beliefs/commitments. Indeed, a reductio argument is much like a dialectical refutation, minus the last step in a reductio. And thus, we could say that, with the addition of the last step, there is a shift of function from the original practice (refutations) to the new one (reductio arguments), but the shift presupposes resources that the original practice lacks. In other words, we could say that reductio arguments overstretch the resources contained in the original matrix, and this gives rise to philosophical quibbles.
This
does not mean that reductio arguments, and in particular the last step, are
never justified. The point is rather that the last step strongly relies on a
number of assumptions, and if these are not in place then the argument does not
go through. Regarding the culprit problem, what is required is that all
auxiliary assumptions/premises used in the argument have a higher degree of
certainty for us than the initial assumption that is singled out to be
rejected. Regarding the act of faith problem, if we can be sure that the
enumeration of cases is truly exhaustive, and that we will not end up in a
situation of aporia where all options lead to absurdity, then we can safely
conclude

*not-p*after showing that*p*leads to absurdity. The dialogical perspective (in particular the comparison with dialectical refutations) allows for the identification of these key assumptions, and this in itself represents a solution of sorts to the issues pertaining to the final step in a reductio argument.
FINAL CONCLUSIONS

This
series started with a discussion of a number of issues arising in connection
with reductio arguments. A reductio ad absurdum may well be a fine weapon, as described
by Hardy, but it is one that brings along a fair amount of perplexity. We’ve
seen that the math education literature seems to suggest that students tend to
find reductio proofs somewhat mystifying, as many of them seem to view such
proofs as unpersuasive. Initially, a brief discussion of two experimental
protocols set the stage for the formulation of four of philosophical issues
arising in connection with reductio arguments: how to represent the impossible;the first speech-act problem; the culprit problem; the ‘act of faith’ problem.

I then presented a brief account of the general dialogical conception of
deductive arguments, which is largely inspired by the historical development of
logic and mathematics in ancient Greece. Equipped with this conception, I then
returned specifically to reductio arguments, firstly by offering a
brief discussion of dialectical refutations, and secondly by reassessing the
issues discussed earlier on now from a dialogical perspective. The conclusion
was that, of the four issues discussed, the dialogical conception can fully
solve one of them, the so-called first speech-act problem; it can further shed
some light on two of them, namely the culprit problem and the ‘act of faith’
problem, giving us further clues as to what is problematic about them and helping
us isolate the assumptions that need to be in place for the last step in a
reductio to be compelling. As for the remaining problem, how to represent the
impossible, at this point it is not clear to me how a dialogical perspective
could contribute towards the formulation of a theory of the impossible and of
representing impossibility, both conceptually and linguistically. But perhaps
further reflection will prove me wrong in this respect at some point in the
future.

Be
that as it may, it seems fair to conclude that the dialogical perspective has
provided a noteworthy contribution towards a better understanding of reductio
ad absurdum arguments. While I’ve adopted a somewhat critical stance at times,
the present analysis is not intended as revisionary of current practices, i.e.
as the plea for a ban on the use of such arguments. Rather, the point is to
outline the assumptions underlying this argumentative strategy by highlighting
its dialogical aspects, and thus hopefully to produce a better understanding of
its nature.

Perhaps
a potential contribution of the present analysis is to the issue of how to

*teach*the technique of reductio arguments, in mathematics as well as elsewhere, in more effective ways. The traditional mode of presentation of reductio proofs, where the theorem to be proved is stated at the very beginning, followed by ‘suppose not’, seems to cause the kind of cognitive dissonance described by Maria and Fabio. Instead, if the gist of a reductio proof is presented in dialogical terms, i.e. the goal being to disprove a commitment undertaken by one’s opponent, then students may well acquire a better grasp of how to produce such arguments and how to interpret them. Ultimately, this is a hypothesis to be tested empirically, for example by teaching the reductio technique to one group of students in dialogical terms and to another group of students in traditional, monological terms, and then comparing the results. But should it prove to be didactically effective, the approach may well make a real difference in how students learn the technique of reductio arguments.
However,
while cognitive and pedagogical elements occupy an important position in the
present investigation, it remains ultimately deeply philosophical in nature.
The goal was to produce a better philosophical understanding of the nature of
reductio ad absurdum arguments, and the main claim is that this is achieved by
adopting a dialogical perspective on deductive arguments in general, and on
reductio ad absurdum in particular.

THE END!

One question is: What is the benefit of the reductio argument over their genealogical ancestors?

ReplyDeleteI take it that dialectical refutation is generally an argument between two people. Even if the argument is meant to be general, it is still only employed against an actual opponent.

By formalizing such arguments using logic, we are able to argue more generally. In terms of the dialogical approach, we are arguing against the Prover or Skeptic, a proxy for a (n idealized) real person. This gives much more power to the argument, since it now applies generally across the population of logicians.

But this benefit comes with logical formalization, and is not specific to the reductio. I have to think that there is some other special reason for the reductio's widespread use and characteristics.

Here is one suggestion:

Since we are arguing against an idealized opponent in the dialogical approach, we have to assume that all the steps in their argument are unassailable. Hence the only thing that could have poisoned their reasoning is the assumption for the reductio, and this is why we are able to reject that one premise.

I think reductios can also be used in dialogical situations of one person arguing against the other, as long as the assumptions I outlined are in place (e.g. that the enumeration of cases is exhaustive). As for your suggestion of a proof applying across the population of logicians, check out Malink's paper in Phronesis this year, which makes a very similar point wrt the differences between the Topics and the Prior Analytics.

DeleteSorry for the length, this is continued in another comment but reductios have always interested me.

ReplyDeleteRegarding the difficulty of representing the impossible:

I've always thought that that difficulty was one of the great virtues of (geometrical) reductios. In studying Euclid and Appolonius, it struck me that unlike regular propositions, some reductios didn't seem to be able to be reduced to the definitions/common notions/postulates.

For example, Euclid's Elements Book III, Proposition 16: The line drawn at right angles to the extremity of the diameter of a circle will fall outside said circle. If not, let it fall within while remaining at right angles to the extremity of the diameter. An absurd situation arises no matter how you draw the line, resulting in a triangle greater than two right angles (!) and therefore, not inside; and not on the circumference either which results in a lesser triangle (putatively) larger than the larger triangle which contains the lesser. Therefore the tangent falls outside the circle.

This example demonstrates the culprit issue you discussed: had our beliefs regarding parallel lines been different, this wouldn't have been absurd at all (leading us to Lobachevskian geometry). Appolonius in his Conics, Book 1, Proposition 17 demonstrates the same: that the tangent must fall outside the parabola.There the parallel postulate's implications are far less discernible.

But in either case, the impossibility of doing the thing asked: of passing a tangent inside the shape, always felt like good evidence for doing that "leap of faith." Every attempt to represent the impossible always resulted in failure, and it could always be shown that all possible types of situation were covered. Hand someone a cone, and a stick (better if its telescopic), and it becomes very easy to corroborate that a line with both extremities within the cone, if extended must at some point pass beyond at least one boundary of any conic section (reductio, Conics, I.18). Very quickly, they'll determine the exercise can't be completed, and I don't think they'd share the same worry about not knowing what they don't know. In as little as a few seconds they will have exhaustively tried the issue and believe it exhaustively tried.

This of course raises an important question regarding geometry's independence from experience, but classical, geometrical reductios always felt like they were utilizing space in a different way than your typical proposition, using the impossibility of doing the task as a way of demonstrating the impossibility of anything that might come out of it: "It's impossible! Go ahead and try it! Therefore, etc. QED." The very paper seemed to resist in a reductio, refusing to allow you to draw the thing you wanted to.

it's clear to me now that (insofar as geometry is concerned) I never successfully enter into that hypothetical realm which may or may not be absurd, remaining in the same world I had attempted to leave after every attempt. I have to disagree with the phenomenological account you offered, insofar as it seemed to suggest we had been invited into an impossible realm and asked to destroy it from within, even according to its own rules, as Maria intuits. But indeed, the thing presents itself to me as if I had been barred access, or more accurately, rerouted back to the place I began. If every time I attempted to draw a line at right angles to the extremity of the diameter of a circle, my purpose was either deflected and it was not the thing I desired to draw, or I succeeded and the tangent was outside the circle, why should I not feel justified that tangents may only exist outside the shape they are tangent to? Moreover, I had not once come into contact with that realm where the tangent falls inside the circle, my will to do so being deflected each time.

DeleteAnd while its obvious that my powerlessness to bring about a state of affairs does not, ipso facto, declare that state of affairs as impossible, (not being able to actualize some potential), the above seems different: to bring about a state of affairs where something becomes possible, that wasn't possible before. (to actualize a potential which previously had no such potency). The latter is pretty rock-solid definition of what's impossible. The particular details of what happens when I fail may be enough to distinguish between my lack of capacity, and the object's lack of receptivity. There may or may not be relevant differences between conceiving the impossible, and attempting to do the impossible.

For similar reasons I've at times felt that Lobachevskian geometry should, strictly speaking, be considered false, unless demonstrated on a saddle-shaped surface, because in every case past the first few theorems, I was asked to demonstrate an impossibility (like parallel lines that met and weren't in perspective), I actually ended up demonstrating its absurdities up on the blackboard, and then was congratulated on proving the thing that was to be demonstrated. I've read somewhere that Kant was the last person who seriously adhered to the idea that diagrams had specific epistemological content that the same proof in mere logic didn't.

P.S.

It seems that you're considering the dual-agent aspect of the dialogical perspective literally, perhaps because of your interest in the pedagogy of reductios. But my suspicion is that the adversarial component does not necessarily lead to two agents. Goethe and, I think, Galileo are on record as seeing Natural Philosophy/science (and its mathematization) as an interrogation of Nature, putting her under duress to reveal her secrets. I think Darwin or some early geneticist is on record as well regarding the difficulty of breeding some animals in captivity as Nature's way of resisting such interrogations into her methods. It may be all anthropomorphized hogwash, but I do think it agrees with much of what a phenomenological approach to intellectual discovery would offer.