Zoe McConaughey

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: n nA. n nThe 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; n n n n nB. n nthe 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: n nB.1. n nthe introduction of rules of interaction rather than of rules of inference; n n n n nB.2. n nthe challenge to the semantization of pragmatics and the claim of the deontic nature of logic; n n n n nB.3. n nthe central role of the notion of execution in the rules of interaction: executions are responses to questions of knowing how. n n n n n n n n n n n nC. n nThe 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.

A Brief Introduction to Constructive Type Theory

Martin-Lof’s Constructive Type Theory (CTT) is a formal language developed in order to reason constructively about mathematics. It is thus a formal language conceived primarily as a tool to reason with rather than a formal language conceived primarily as a mathematical system to reason about. Constructive Type Theory is therefore much closer in spirit to Frege’s ideography and to the language of Russell and Whitehead’s Principia Mathematica than to the majority of logical systems (“logics”) studied by contemporary logicians. Since CTT is designed as a language to reason with, much attention is paid to the explanation of basic concepts. This is perhaps the main reason why the style of presentation of CTT differs somewhat from the style of presentation typically found in, for instance, ordinary logic textbooks. For those new to the system it might be useful to approach an introduction, such as the one given below, more as a language course than as a course in mathematics.

The Remarkable Case of the Axiom of Choice

It is rightly said that the principle of set theory known as the Axiom of Choice is “probably the most interesting and in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s Axiom of Parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel, & Levy, 1973).

The Dialogical Roots of Equality: Dialogues for Immanent Reasoning

In this chapter we will spell out all the relevant rules of dialogues for immanent reasoning, that is, the dialogical framework incorporating features of Constructive Type Theory—a dialogical framework making the players’ reasons for asserting a proposition explicit. The rules can be divided, just as in the standard framework, into rules determining local meaning and rules determining global meaning. These include: n n1. n nConcerning local meaning (Sect. 7.1): n n(a) n nformation rules (p. 105); n n n n n(b) n nrules for the synthesis of local reasons (p. 108); and n n n n n(c) n nrules for the analysis of local reasons (p. 109). n n n n n n n n n n n n2. n nConcerning global meaning, we have the following (structural) rules (Sect. 7.2): n n(a) n nrules for the resolution of instructions (p. 112); n n n n n(b) n nrules for the substitution of instructions (p. 113); n n n n n(c) n nequality rules determined by the application of the Socratic rules (p. 113); and n n n n n(d) n nrules for the transmission of equality (p. 115).

Local Reasons and Dialogues for Immanent Reasoning

In this chapter we will provide the logical framework of dialogues for immanent reasoning, the dialogical framework incorporating features of Constructive Type Theory and making explicit the players’ reasons for asserting a proposition. We will therefore be using the material provided in Chaps. 2, 3, 4, and 5 on CTT and on the standard dialogical framework and assume the reader is familiar enough with it. The framework of dialogues for immanent reasoning takes a further step in the task of making explicit the dynamic foundations of reasoning, based on equality in action: reasons adducing statements are introduced in the object-language, and the Proponent’s task in formal dialogues is to force the Opponent to provide herself the reasons the Proponent needs in order to justify his thesis; once the Opponent has produced her reasons, the Proponent uses equality rules (the Socratic rule for instance) to copy these reasons and use them to his own ends. In this respect, CTT proof-objects are adapted to the dialogical framework, yielding two new elements in our framework: local reasons, the backbone of dialogues for immanent reasoning at the play level which will be introduced in this chapter and the next (Chaps. 6 and 7); and strategic reasons, justifications at the strategy level corresponding to proof-objects which will be the object of Chap. 9. Having local reasons in the dialogical framework provides a structure in which the reasons given and asked for actually appear in the object-language, and the Proponent can then (locally) justify his statements by explicitly copying the Opponent’s reasons for his own statements.