Nicolas Clerbout

First-Order Dialogical Games and Tableaux

We present a new proof of soundness/completeness of tableaux with respect to dialogical games in Classical First-Order Logic. As far as we know it is the first thorough result for dialogical games where finiteness of plays is guaranteed by means of what we call repetition ranks.

Context-Sensitivity in Jain Philosophy: A Dialogical Study of Siddharṣigaṇi’s Commentary on the Handbook of Logic

In classical India, Jain philosophers developed a theory of viewpoints (naya-vāda) according to which any statement is always performed within and dependent upon a given epistemic perspective or viewpoint. The Jainas furnished this epistemology with an (epistemic) theory of disputation that takes into account the viewpoint in which the main thesis has been stated. The main aim of our paper is to delve into the Jain notion of viewpoint-contextualisation and to develop the elements of a suitable logical system that should offer a reconstruction of the Jainas’ epistemic theory of disputation. A crucial step of our project is to approach the Jain theory of disputation with the help of a theory of meaning for logical constants based on argumentative practices called dialogical logic. Since in the dialogical framework the meaning of the logical constants is given by the norms or rules for their use in a debate, it provides a meaning theory closer to the Jain context-sensitive disputation theory than the main-stream formal model-theoretic semantics.

Constructive Type Theory and the Dialogical Approach to Meaning

. November 2013 Volume 8: Games, Game Theory and Game Semantics pages 1-72. DOI: . Also online in: www.thebalticyearbook.org

Linking Game-Theoretical Approaches with Constructive Type Theory Dialogical Strategies, CTT demonstrations and the Axiom of Choice

This title links two of the most dominant research streams in philosophy of logic, namely game theory and proof theory. As the work’s subtitle expresses, the authors will build this link by means of the dialogical approach to logic. One important aspect of the present study is that the authors restrict themselves to the logically valid fragment of Constructive Type Theory (CTT). The reason is that, once that fragment is achieved the result can be extended to cover the whole CTT system. The first chapters in the brief offer overviews on the two frameworks discussed in the book with an emphasis on the dialogical framework. The third chapter demonstrates the left-to-right direction of the equivalence result. This is followed by a chapter that demonstrates the use of the algorithm in showing how to transform a specific winning strategy into a CCT-demonstration of the axiom of choice. The fifth chapter develops the algorithm from CTT-demonstrations to dialogical strategies. This brief concludes by introducing elements of discussion which are to be developed in subsequent work.

Immanent Reasoning or Equality in Action: A Plaidoyer for the Play Level

abstraction absurdum application arbitrary object arbitrary reference assertion assumption Bool Boolean axiom of choice canonical Cartesian case-dependent category categorical choice computation computational rule concession conjunction constructive content context copy-cat core core of the strategy correct naming course of values critical Curry–Howard isomorphism decision definitional definitional equality demonstration dependent dialectical dialogical dialogical roots of equality dialogue disjoint union disjunction dispense double negation ecthesis elimination emergence of equality empty set IMMANENT REASONING OR EQUALITY IN ACTION 263 equality extensive form of a dialogical game extensive form of a strategy function function type game game-tree Geltung global meaning harmony hypothesis identity instruction judgement knowledge local meaning material m-dependent resolution meaning metalanguage metalevel metalogic natural number natural deduction negation nominal object object language ontological Opponent pensée aveugle play local reason posit posit-substitution predicate predication predicator premiss presupposition projection prop Proponent proposition propositional equality quantifier range-course resolution resolution of functions resolution of instructions

Concluding Remarks: A Plaidoyer for the Play Level

To some extent, the criticisms the dialogical approach to logic has been subject to have provided an opportunity for clarifying its basic tenets. Moreover, our responses to the objections have highlighted crucial distinctions constituting the originality and flexibility of this logical framework. We will therefore in this concluding chapter consider some recent objections raised against the dialogical framework in order to pinpoint some of its fundamental features, whose importance may not have appeared clearly enough through the main body of the book; namely, dialogue-definiteness, player-independence, and the dialogical conception of propositions. Showing how and why these features have been developed, and specifying their point and the level they operate on, will enable us to vindicate the play level and thus disarm the objections that have been raised against the dialogical framework for having neglected this crucial level.

Advanced Dialogues: Play Level

This chapter will provide a more technical approach to the standard (non-CTT) dialogical framework at the play level. The next chapter ( 5) will do the same at the strategy level. It will then be possible to introduce local reasons in the dialogues and thus start making it explicit that dialogues are games of giving and asking for reasons; in this sense, the elements contributing to the meaning as use will appear in the object language. The link to equality in action will then be spelled out in the following two chapters ( 6 and 7), based on what will be presented in the next two chapters ( 4 and 5).

Advanced Dialogues: Strategy Level

The strategy standpoint is but a generalisation of the procedure which is implemented at the play level; it is a systematic exposition of all the relevant variants of a game—the relevancy of the variants being determined from the viewpoint of one of the two players. For a more intuitive approach of strategies and a step-by-step introduction of strategies as branching tables, see Sect. 3.5, p. 59. Such trees with branching tables are a good didactic approach to strategies, for the rules in building the tree are the same as those for building the plays: we simply use an algorithm yielding all the relevant plays for a player, keeping the table presentation we use for plays. The link from plays to strategies is thus clearly apparent. This method however is rather cumbersome and becomes unmanageable as soon as we deal with games involving more than two choices, the generated trees taking too much space. We will here present strategies from another perspective, that of extensive forms of dialogical games (more precisely from their core; see below, Sect. 5.3) rather than the table presentation; the extensive form presentation has this advantage over the table presentation that strategies can be linked more straightforwardly to demonstrations, which will be useful in Chap. 9. This link is crucial to the logical framework of dialogues, for the dialogical notion of validity is secured through the notion of a winning strategy for the Proponent. Many metalogical results in the dialogical framework are obtained by leaving the level of rules and plays to move to the level of strategies; winning strategies for a player are one of these metalogical results.

Introduction: Some Brief Historical and Philosophical Remarks

The present volume develops a new way of linking Constructive Type Theory (CTT) with dialogical logic by following these three complementary paths, as mentioned in the preface: A. The path observing that Sundholm’s (1997) notion of epistemic assumption is closely linked to the Copy-cat and Socratic rules and that it provides the dialogical conception of definitional equality; B. the path joining (in principle) Martin-Lof in his (2017a, 2017b) suggestions, according to which the new insights provided by the dialogical framework mainly amount to the following three interconnected points: B.1. the introduction of rules of interaction rather than of rules of inference; B.2. the challenge to the semantization of pragmatics and the claim of the deontic nature of logic; B.3. the central role of the notion of execution in the rules of interaction: executions are responses to questions of knowing how. C. The path stressing the importance of the play level and the associated notion of dialogue-definiteness.

Basic Notions for Dialogical Logic

The dialogical approach to logic is not a specific logical system; it is rather a general framework having a rule-based approach to meaning (instead of a truth-functional or a model-theoretical approach) which allows different logics to be developed, combined and compared within it. The main philosophical idea behind this framework is that meaning and rationality are constituted by argumentative interaction between epistemic subjects; it has proved particularly fruitful in history of philosophy and logic. We shall here provide a brief overview of dialogues in a more intuitive approach than what is found in the rest of the book in order to give a feeling of what the dialogical framework can do and what it is aiming at.