By **Catarina Dutilh Novaes**

(Cross-posted at M-Phi)

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.

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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!

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