**Catarina Dutilh Novaes**

(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.

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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,[1] 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.[2] 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.[3] 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’).[4] 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.[5] 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.[6]

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”.
[1] 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.

[2] 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.

[3] This terminology comes from the computer science
literature on proofs. The earliest occurrence that I am aware of is in
(Sørensen & Urzyczyn 2006), who speak of

*prover-skeptic games*.
[4] The ‘semi’ qualification pertains to the equally
strong cooperative component in a deductive proof, to be discussed in more
detail shortly.

[5] Further discussion on this can be found in (Harris2000; Dutilh Novaes 2013).

[6] 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.

Very astute post Catarina, thank you!

ReplyDeleteOf course Game Semantics puts both dialogical players back 'on the field' (e.g. in existential quantification the Prover gets to choose the x and show that it satisfies P, in universal quantification the Skeptic gets to choose and the Prover has to show that it satisfies...etc).

Matthieu Marion's paper "Why Play Logical Games?" gives a good philosophical overview of the state of the art here.

Thanks, Cathy! And yes, I'm a big fan of this paper by Mathieu as well.

DeleteHere is a difficulty, the clarification of which, I think, will help us both.

ReplyDeleteThe dialogical account you outline can be understood in two ways: in one way, as an account of the

natureof mathematical proof; in another, as an account of itsorigins. I think your exposition vacillates between these two viewpoints.As an account of origins, the account (though I suspect it is correct) would need to be proven in a more empirical way than you’ve attempted here. Furthermore, if granted, it isn’t clear how the result achieved is relevant, to an account of the contemporary practice. This latter problem seems to be a difficulty with genealogical accounts generally.

As an account of the

natureof mathematical proof, the account has a strong and a weak reading. The strong reading is to say that proof is literally a dialogue. The weak reading is one on which the concept of a dialogue provides a useful heuristic for thinking about proofs. The strong reading is false, and you appear to reject it in your remarks about the internalization of the skeptic in section 4. But the weak reading seems too weak: at most, it seems to say that agent and patient remain formally distinct even when materially identical – where Aristotle’s example of this was that of the physician operating upon himself, yours would be the mathematician playing the role of both the convincer and the convinced.The issue you outline is of course the perennial issue for genealogical projects that also seek to be explanatory (as you say yourself). I am convinced that the origins of something will fundamentally influence its nature, whatever that is (as I've argued in my paper on genealogy, which if memory does not fail me you are familiar with as well).

DeleteSpecifically wrt proofs and dialogues, my slogan is 'a proof is and is not a dialogue'. This means that, properly speaking, proofs are no longer dialogues with two active participants (though I still want to emphasize that proof is always a discourse aimed at a putative audience), but since they retain some dialogical components, there is a sense in which it still make sense to view them as dialogues in a weak sense (but which I think would go beyond mere 'heuristic'.)

I read this article and its tell me how write dialogue for summery and how to explain his moral thanks for share it delphi programming help .

ReplyDelete