(Cross-posted at NewAPPS)
This is the fourth installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, here is Part II, and here is Part III. In this post I offer a précis of the dialogical account of deduction which I have been developing over the last years, which will then allow me to return to the issue of reductio arguments equipped with a new perspective in the next installments. I have presented the basics of this conception in previous posts, but some details of the account have changed, and so it seems like a good idea to spell it out again.
In this post, I present a brief account of the general dialogical conception of deduction that I endorse. Its relevance for the present purposes is to show that a dialogical conception of reductio ad absurdum arguments is not in any way ad-hoc; indeed, the claim is that this conception applies to deductive arguments in general, and thus a fortiori to reductio arguments. (But I will argue later on that the dialogical component is even more pronounced in reductio arguments than in other deductive arguments.)
Let us start with what can be described as functionalist questions pertaining to deductive arguments and deductive proofs. What is the point of deductive proofs? What are they good for? Why do mathematicians bother producing mathematical proofs at all? While these questions are typically ignored by mathematicians, they have been raised and addressed by so-called ‘maverick’ philosophers of mathematics, such as Hersh (1993) and Rav (1999). One promising vantage point to address these questions is the historical development of deductive proof in ancient Greek mathematics, and on this topic the most authoritative study remains (Netz 1999). Netz emphasizes the importance of orality and dialogue for the emergence of classical, ‘Euclidean’ mathematics in ancient Greece:
Greek mathematics reflects the importance of persuasion. It reflects the role of orality, in the use of formulae, in the structure of proofs… But this orality is regimented into a written form, where vocabulary is limited, presentations follow a relatively rigid pattern… It is at once oral and written… (Netz 1999, 297/8)
Netz’s interpretation relies on earlier work by Lloyd (1996, chap. 10), who argues that the social, cultural and political context in ancient Greece, and in particular the role of the practice of debating, was fundamental for the emergence of the technique of deductive proofs. So from this perspective, it seems that one of the main functions of deductive proofs (then as well as now) is to produce persuasion, in particular what one could call explanatory persuasion: to show not only that something is the case, but also why it is the case. On this conception, a deductive proof corresponds to a dialogue between the person wishing to establish the conclusion (given the presumed truth of the premises), and an interlocutor who will not be easily convinced and will bring up objections, counterexamples, and requests for further clarification. A good proof is one that convinces a fair but ‘tough’ opponent. Now, if this is right, then mathematical proof is an inherently dialogical, multi-agent notion, given that it is essentially a piece of discourse aimed at a putative audience (Ernest 1994).
To be sure, there are different ways in which the claim that deductive proofs are essentially dialogical can be understood. For example, the fact that a proof is only recognized as such by the mathematical community once it has been sufficiently scrutinized by trustworthy experts can also be viewed as a dialogical component, perhaps in a loose sense (the ‘dialogue’ between the mathematician who formulates a proof and the mathematical community). But in what follows I present a more precise rational reconstruction of the (quite specialized) dialogues that would correspond to deductive proofs.
On this conception, proofs are semi-adversarial dialogues of a special kind involving two participants: Prover and Skeptic. Prima facie, the (fictitious) participants have opposite goals, and this is why the adversarial component remains prominent: Prover wants to establish the truth of the conclusion, and Skeptic wants to block the establishment of the conclusion (though not ‘at all costs’). The dialogue starts with Prover asking Skeptic to grant certain premises. Prover then puts forward further statements, which purportedly follow from what has been granted. (Prover may also ask Skeptic to grant additional auxiliary premises along the way.) Ultimately, most of the work is done by Prover, but Skeptic has an important role to play. Skeptic’s moves are: granting premises so as to get the game going; offer a counterexample when an inferential move by Prover is not really necessarily truth-preserving; ask for clarifications when a particular inferential step by Prover is not sufficiently compelling and perspicuous. These three moves correspond neatly to what are arguably the three main features of a deductive proof, so let us comment on each of them in turn.
Accepting premises/assuming hypotheses. One of the key components of deductive reasoning is the dissociation of validity from truth. To reason deductively, one must be prepared to put aside one’s own beliefs regarding the premises to see what follows from them. As we’ve seen above when discussing Maria’s and Fabio’s experiences with reductio arguments, this is a rather demanding cognitive task, one which arguably requires at least some amount of training to be mastered. In a dialogical setting, however, in particular the somewhat contrived form of dialogical interaction that is debating, granting premises ‘for the sake of the argument’ is an integral part of the practice. To assume a hypothesis is thus something like a provisional endorsement of a claim, something like ‘pretending to believe it’ for the sake of the argument (as suggested by Maria) in order to engage with the opponent. Importantly, in a dialogical setting a participant can also draw the consequences of her opponent’s commitments, without having committed to the claim herself.
Adversariality: necessary truth-preservation. In a deductive argument, only necessarily truth-preserving inferential steps are permitted. A dialogical setting allows for the formulation of a compelling rationale for the requirement of necessary truth-preservation: necessarily truth-preserving steps are indefeasible, i.e. there is no counterexample that Skeptic could offer to defeat them. And if each individual inferential step in a proof is indefeasible in this sense, then the proof as a whole is a winning strategy for Prover: no matter what Skeptic does (within the rules of the game), no matter what external information he brings in, Prover will prevail. The idea that a deductive proof corresponds to a winning strategy is of course one of the cornerstones of game-theoretical, dialogical conceptions of logic and proof such as Hintikka’s game-theoretical semantics and Lorenzen’s dialogical logic (see (Hodges 2013) for an overview). On the picture presented here, necessary truth-preservation comes out as the distinctively adversarial component of these dialogues, which determines on who ‘wins’ and who ‘loses’: if every step in the argument is necessarily truth-preserving, then Prover wins; otherwise, and if Skeptic successfully provides a counterexample, then Skeptic wins.
Cooperation: perspicuity as didactic feature. However, the absence of counterexamples to specific inferential steps is not the whole story: each individual step of a proof must also be compelling and persuasive on its own. Indeed, a desideratum for Prover is to break down the argumentation into small inferential steps; a proof where each step is necessarily truth preserving but not sufficiently convincing is not a good proof. Now, purely adversarial considerations cannot explain this feature: big ‘leaps’ are strategically advantageous for Prover. Indeed, in the limit case, Prover could for example get Skeptic to grant the basic axioms of number theory, and then go on to directly state Fermat’s Last Theorem as the conclusion: this ‘inferential step’ would be immune to counterexamples (something we can be sure of since Wiles’ proof of FLT), but obviously such a ‘proof’ would be an utter failure in that it would not achieve the persuasive, explanatory function of a proof. This is why one of the moves available to Skeptic is to request for further clarification whenever Prover moves too quickly, so to speak.
From this point of view, a deductive proof (or a deductive argument more generally) is characterized by a complex interplay between adversariality and cooperation: the participants have opposite goals and ‘compete’ with one another at a lower level, but they are also engaged in a common project to establish the truth or falsity of a given conclusion (given the truth of the premises) in a way that is not only persuasive but also (hopefully) elucidatory. If both participants perform to the best of their abilities, then the common goal of producing novel, verified mathematical knowledge will be optimally achieved.
At this point, the reader may be thinking: this is all very well, but obviously deductive proofs are not really dialogues! They are typically presented in writing rather than produced orally (though of course they can also be presented orally, for example in the context of teaching), and if at all, there is only one ‘voice’ we hear, that of Prover. So at best, they must be viewed as monologues. My answer to this objection is that Skeptic has been ‘silenced’, but he is still alive and well insofar as the deductive method has internalized the role of Skeptic by making it constitutive of the deductive method as such. Recall that the job of Skeptic is to look for counterexamples and to make sure the argumentation is perspicuous. This in turn corresponds to the requirement that each inferential step in a proof must be necessarily truth preserving (and so immune to counterexamples), and that a proof must have the right level of granularity, i.e. it must be sufficiently detailed for the intended audience, in order to achieve its explanatory purpose.
Indeed, the internalization of Skeptic may be identified in the very historical process detailed in (Netz 1999) (and alluded to in the passage quoted above) from proofs orally presented, corresponding to actual dialogues (e.g. Socrates’ proof of how to double the size of a square in the Meno) to written proofs, such as the ones found in Euclid’s Elements. From this perspective, a mathematical proof both is and is not a dialogue: it is a dialogue in that it retains dialogical components (assuming hypotheses, producing indefeasible but convincing arguments), but it is no longer a dialogue properly speaking insofar as one of the participants has been internalized by the method itself, and thus silenced. As Netz puts it, “it is at once oral and written”.
 When it comes to functionalist questions, it makes sense to inquire into what the first practitioners of a given practice thought they were doing, and why they were doing it, when the practice first came about. But this is not to exclude the possibility of shifts of function along the way.
 For Hersh (1993), proof is also about convincing and explaining, but on his account these two aspects come apart. According to him, convincing is aimed at one’s mathematical peers, while explaining is relevant in particular in the context of teaching.
 The ‘semi’ qualification pertains to the equally strong cooperative component in a deductive proof, to be discussed in more detail shortly.
 Compare to what happens in a court of law in adversarial justice systems: defense and prosecution are defending different viewpoints, and thus in some sense competing with one another, but the ultimate common goal is to achieve justice. The presupposition is that justice will be best served if all parties perform to the best of their abilities.