Title

Logic and Interaction: The Dialogical Turn

Abstract Course

A brief examination on the most recent literature in logic will make it apparent that a host of research in this area is devoted to the study of the interface between games, logic and epistemology. These studies provide the basis of ongoing enquiries in the history and philosophy of logic, going from the Indian, the Greek, the Arabic, the Obligationes of the Middle Ages to the most contemporary developments in the fields of theoretical computer science, computational linguistics, artificial intelligence, social sciences and legal reasoning. In fact, a dynamic turn, as Johan van Benthem puts it, is taking place where the epistemic aspects of inference are linked with game theoretical approaches to meaning. In regard to the birth of this turn, it could be placed around the 1960's when Paul Lorenzen and Kuno Lorenz developed dialogical logic --- inspired by Wittgenstein's language games and mathematical game theory --- and when some time later on Jaakko Hintikka combined game-theoretical semantics with epistemic (modal) logic. If we were to pinpoint a precise date, the very beginnings of the dynamic turn could be situated in 1958, with Lorenzen's talk ``Logik und Agon''. However, the dynamic take on the epistemic conception of logic in both of its brands, the dialogical one and the one based on Hintikka's GTS, disregarded a major advance in precisely the epistemic approach to logic, namely, the development by Per Martin-Löf of Constructive Type Theory (CTT) --- with the sole exception of the pioneering paper of Aarne Ranta [1988]. This frame, that provides a type theoretical development of the Curry-Howard-isomorphism and introduces dependent-types, leads to the formulation of a fully-interpreted language --- a language with content that challenges the standard metalogical approach to meaning of model theoretic semantics in general and of the modal-interpretation of epistemic logic in particular. Furthermore, an inferential and contentual language based on CTT (Sundholm[1986,2001], Ranta[1994]) has now been successfully applied not only to the semantics of natural languages but also to the foundations of logic, computer sciences and constructive mathematics. Philosophically speaking, CTT shares the Kantian view that judgements, rather than propositions, constitute the foundation of knowledge. According to this perspective, the basic ontology is determined by the two fundamental forms of judgement, namely categorical judgements with independent proof-objects and hypothetical judgements with dependent proof-objects (i.e., functions). See chapter 1. Interesting is the fact that, as discussed by Rahman & Clerbout [2013], CTT furnishes the basis for the research on those fields where the dialogical framework --- after some initial analyses --- stopped to work on, namely the foundations of mathematics and the development of a general dialogical theory of meaning. The, up to now, lack of interface between the game theoretical approaches and CTT is particularly striking because of the common philosophical grounds of the dialogical framework and those of constructive logic. Let us very briefly develop the latter point. One possible way to put it is to follow Mathieu Marion's proposal and to make use of Robert Brandom's [1994,2000] pragmatist take on inferentialism, which is led by two main insights of Kantian origin and one that stems from Brandom's reading of Hegel:

• That judgements are the fundamental units of knowledge, and
• That human cognition and action are characterized by certain sorts of normative assessment.
• Communication is mainly conceived as cooperation in a joint social activity rather than on sharing contents.

The crucial point of the epistemic approach is, as mentioned above, that assertion or judgement amounts to a knowledge claim and this is independent of classical or intuitionistic views --- cf. Prawitz[2012, p.47]. So, if the meaning of an expression is deployed from its role in assertions, then an epistemic approach to meaning results. In relation to the second point, according to Brandom, the normative aspect is implemented via W. Sellar's notion of games of giving and asking for reasons, which deploy the intertwining of commitments and entitlements. Indeed, on Brandom's view, it is the chain of commitments and entitlements in a game of giving and asking for reasons that tights up judgement and inference. Sundholm[2013] provides the following formulation of the notion of inference in a communicative context that can be also seen as describing the core of Brandom's pragmatist inferentialism:

When I say ``Therefore'' I give others my authority for asserting the conclusion, given theirs for asserting the premsisses.

This is quite close to the main tenet of the dialogical approach to meaning, but with one important and crucial difference. The pragmatist approach to meaning of the dialogical framework shares with Brandom's pragmatist inferentialism the claim that the meaning of linguistic expressions is related to their role in games of questions and answers, and also endorses Brandom's notion of justification of a judgement as involving the interaction of commitments and entitlements. The important difference is that dialogicians maintain that more fundamental lower-levels should be distinguished --- as discussed in chapter 2. Those lower-level semantic levels include (i) the description of how to formulate a suitable question to a given posit and how to answer it, and (ii) the development of plays, constituted by several combinations of sequences of questions and answers brought forward as responses to the posit of a thesis. From the dialogical perspective, the level of judgements corresponds to the final stage of the chain of interactions just mentioned. More precisely, the justifications of judgements correspond to the level of winning strategies, that select those plays that turn out to be relevant for the drawing of inferences. Furthermore, the game theoretical take on dependent types is rooted on choices dependences, that can be seen as a result of the intertwining of games of questions and answers. Let us point out that the distinctions drawn within the dialogical framework between local meaning, play level and strategy level seem to provide an answer to Brandom's question involving his claim that the ``grasp of concepts'' amounts to the mastery of inferential roles but that this

[...] does not mean that in order to count as grasping a particular concept an individual must be disposed to make or otherwise endorse in practice all the right inferences involving it. To be in the game at all, one must make enough of the right moves --- but how much is enough is quite flexible. Brandom [1994, p.636]

 Indeed, from the dialogical point of view, in order to grasp the meaning of an expression, the individual does not need to know the moves ensuring his victory (he must not have a winning strategy) and does not even need to win at all. What is required is that he knows what are the relevant moves he is entitled and committed to (local meaning) in order to develop a play --- in a similar way to knowing how to play chess does not necessarily mean to actually be in possession of a winning strategy. Knowing how to play allows to know what can count as a winning strategy, when there is one: strategic legitimacy (Geltung) is not to be found at the level of meaning-explanation. Thus, one way to see the motivations that animate the proposal to link CTT and games is to furnish the technical elements that bind the pragmatist approach to the grasp of concepts in Brandom's style with the proof-theoretical CTT take on meaning.

    At this point of the discussion, we hope that the grounds --- or at least a glimpse of them --- for working out systematically the links between game theoretical approaches and CTT should be clear enough. A pending task, that we will not undertake here, is to discuss how the rigorous elaboration of a fully-interpreted language in the terms of CTT fits with Brandom's pragmatic inferentialism. However, as stressed in chapter 4, a full-interpreted language lies at the core of the dialogical take on meaning. More generally and summing up, the development of a dialogical approach to Constructive Type Theory can be motivated by the following considerations:

References

• Clerbout, N. 2013a. Etude sur quelques sémantiques dialogiques: Concepts fondamentaux et éléments de metathéorie. London/ C. Publication.
• ———. 2013b. “First-Order Dialogical Games and Tableaux”. Journal of Philosophical Logic. Online-first Publication – DOI: 10.1007/s10992-013-9289-z.
• Fontaine, M. 2013. Argumentation et engagement ontologique. London/ C. Publications
• Keiff, L. 2009. “Dialogical Logic”. In: E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), URL = . First published 2009.
• Lorenz, K. 1970. Elemente der Sprachkritik: Eine Alternative zum Dogmatismus und Skeptizismus in der Analytischen Philosophie. Frankfurt/Main: Suhrkamp.
• ———. 2001. “Basic objectives of dialogue logic in historical perspective”. In: Rahman and Rückert 2001, 255–263.
• Lorenzen, P. and K. Lorenz. 1978. Dialogische Logik. Darmstadt: Wissenschaftliche Buchgesellschaft.
• Magnier, S. 2013. Approche dialogique de la dynamique épistémique et de la condition juridique. London: College Publications.
• Martin-Löf, P. 1984. Intuitionistic Type Theory: Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980. Naples: Bibliopolis.
• Rahman, S. and Keiff, L. 2004. “On How to Be a Dialogician”. In: D. Vanderveken (ed.), Logic, Thought and Action, Dordrecht: Kluwer, pp. 359–408.
• Redmond, J. 2010. Logique dynamique de la fiction: Pour une approche dialogique. London: College Publications.
• ———, and M. Fontaine. 2011. How to Play Dialogues: An Introduction to Dialogical Logic. London: College Publications.

Contact

shahid.rahman@univ-lille3.fr