As some of you may remember (I mentioned it here before), in July I am going to start a new 5-year research project with the title “The Roots of Deduction”; the ‘modest’ goal is to try to figure out what deduction is all about by integrating philosophy, psychology and history in the investigation. The starting point will be a reconceptualization of the very notion of deduction that respects its multi-agent, dialogical origins (more on this soon). In this context, the workshop I attended in Tübingen last week, on the connections between proofs and dialogues, was spot-on the general theme of my research project, and I thought it might be of interest to some readers to hear more about it (hence this post). However, what follows is not a report of the workshop properly speaking, but rather a brief discussion of some of the points raised insofar as they are connected to my research interests.
I presented a paper with the title “Every proof is (and isn’t) a dialogue: On the dialogical foundations of logic”. Originally, the ‘(and isn’t)’ clause wasn’t part of the title, but as I started working on it, I realized that, while there is a sense in which every proof is a dialogue, there is also a sense in which proofs are no longer dialogues, properly speaking. I drew on the historical development of the deductive method (as documented in Netz’s The Shaping of Deduction) to argue that a deductive argument is a discourse aimed at compelling the audience to accept (the truth of) the conclusion, if they accept (the truth of) the premises. Crucially, proofs would correspond to adversarial dialogues, as the participants have opposite goals: establishing the conclusion vs. blocking the establishment of the conclusion. Think of Socrates as the opponent: the one constantly looking for flaws (in particular inconsistencies) in the argumentation put forward by proponent in order to argue for the thesis. But proofs are in fact no longer actual dialogues, and a passage by Netz illustrates their hybrid status particularly well:
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)
As I see it, the two main transformations leading from actual dialogues to proofs are: the move from oral to written contexts, as described by Netz; and (this is my own conceptual contribution) the fact that opponent is internalized by the deductive method. The main role of opponent is to look for counterexamples, i.e. situations where the premises hold but the conclusion (of each individual inferential step) does not. But the deductive method has internalized the opponent in the sense that it is now built into the framework: every inferential step must be immune to counterexamples. I refer to this conception as the built-in opponent (BIO) conception of proofs.
This brief description of my own take on the matter is intended to provide the background for what I will now say on the other talks at the workshop. We started with a series of talks by highly respected ‘dialogicians’, i.e. people working within the framework of dialogical logic: H. Rückert, S. Rahman, L. Keiff, T. Tulenheimo. A common thread in their talks (especially Rahman’s and Keiff’s) was the attempt to transpose some of the main themes within proof-theoretic semantics (especially harmony and trivialization) to the dialogical setting. What counts as an adequate dialogue game, as defined by specific procedural and particle rules? Just as not any system of inferential rules will yield an interesting proof-system (contra Carnap’s principle of tolerance), not any description of a game will give you a well-functioning game. So the usual challenges for proof-theoretic semantics (in particular trivializing operators such as tonk and its many cousins) can also be posed to the dialogical logic framework. Compared to the proof-theoretical framework, the dialogicians are just starting to think about these challenges, but some nice ideas and results were already presented. In particular, one way to capture what is wrong with tonk-like connectives in a dialogical setting is to notice that they don’t have player-independent rules, so player-independency may be construed as a counterpart to harmony in the dialogical setting.
The next day started with a fascinating talk by M. H. Sørensen. I didn’t know any of his work before, and I was very impressed. Among other things, he presented a game between Proponent and Skeptic, which looked like a very neat formalization of my general idea of a built-in opponent (the Skeptic). After that, we had a series of talks by the Tübingen crowd themselves (P. Schroeder-Heister, T. Piecha, L. Tranchini) where the notion of implications as rules and the distinction between the categorical and the hypothetical figured prominently. These are concepts that Schroeder-Heister has been talking about for a while, and in particular his critique of the predominance of the categorical over the hypothetical in logic is something that I find very compelling (unfortunately, I can’t do justice to it here, but I do urge interested readers to go look it up themselves). My own suggestion was that, as they stand, dialogue games are also essentially categorical rather than hypothetical, but a move from the level of plays to the level of strategies may be a way to adopt a truly hypothetical perspective in a dialogical setting.
A general conclusion drawn in the workshop is that there are roughly two attitudes one can have with respect to games and dialogues in logic: either as codifications/regimentations of certain practices, or simply as mathematical objects not being committed to describing an intelligible, realistic story about games that one might want to engage in. Andreas Blass pointed this out during Q&A after my talk, and it’s certainly an important distinction to keep in mind. One can refer to the first as the philosophical approach, and to the second as the mathematical approach. I am more interested in the first approach, for fairly obvious reasons (including the fact that I don't quite see myself proving non-trivial mathematical results!), but even the second approach seems to support my general view of deduction as a multi-agent, dialogical notion, as I will now argue.
Some of the most interesting technical results presented were equivalence results between dialogue games (viewed as mathematical objects) and other non-dialogical logical formalisms. B. Więckowski presented dialogue games in sequent-style and proved some neat equivalence results; M. H. Sørensen also dwelled on the correspondences between sequent calculus and formalisms for dialogue games. Chris Fermüller went a step further and presented equivalence results for hypersequents. (An interesting thing about his results is that they suggest that it might be possible to formulate the idea of games of imperfect information in his hypersequent framework, as it is about parallel processes.) I think that these technical results lend support to my ‘built-in opponent’ conception of a proof because they seem to suggest that there really is a built-in opponent operating even in logical formalisms that are not obviously dialogical to start with, such as sequent calculi. But for now, this is just a hunch that I would have to think about more carefully.
There were other fascinating talks that I have not been able to include in my discussion here, but this has gone on for too long already for a blog post, so I shall leave it at this. However, I am just getting started thinking about these issues, so I certainly welcome comments/feedback from readers!
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