Dialogues and Monologues in Logic

Abstract

The dialogical framework is an approach to meaning that provides a pragmatist alternative to both the model-theoretical and the proof-theoretical semantics. However, since dialogic had and still has a bias towards antirealism, it has been quite often seen as a version of the proof-theoretical approach. The main claim of this chapter is that the proof-theoretical approach as displayed by a tableaux system of sequent calculus is, from the dialogical point of view, a monological approach and cannot provide a purely dialogical theory of meaning. Indeed, in general, validity is monological, in the sense that a winning strategy is defined independently of the moves of the Opponent. In the dialogical framework, validity should be based bottom up on a dialogical semantics.

The dialogical approach to logic is not but a semantic rule-based framework where different logics could be developed, combined or compared. But are there any constraints? Can we introduce rules ad libitum to define whatever logical constant? The answer is no, for logical constants must be governed by player-independent dialogical rules. The approach of the present chapter has been influenced by Marcelo Dascal’s reflections on meaning, pragmatics and dialogues. In fact, on my view the dialogical approach to logic offers a framework for developing logic as closest as possible to his own theory of meaning and soft rationality.

Appendix

1.1 Examples

In the following examples, the outer columns indicate the numerical label of the move and the inner columns state the number of a move targeted by an attack. Expressions are not listed following the order of the moves, but writing the defence on the same line as the corresponding attack, thus showing when a round is closed. Recall, from the particle rules, that the sign “—” signalizes that there is no defence against the attack on a negation.

For the sake of a simpler notation, we will not record in the dialogue the rank choices but assume the uniform rank: O: n = 1 P: m = 2.

In the following dialogue played with classical structural rules P’ move 4 answers O’s challenge in move 1, since P, according to the classical rule, is allowed to defend (once more) himself from the challenge in move 1. P states his defence in move 4 though, actually O did not repeat his challenge – we signalize this fact by inscribing the not repeated challenge between square brackets.

O			P
				p∨¬p	0
1	?∨	0		¬p	2
3	p	2		—
[1]	[?∨]	[0]		p	4

1. Classical rules. P wins

In the dialogue displayed below about the same thesis as before, O wins according to the intuitionistic structural rules because, after the challenger’s last attack in move 3, the intuitionist structural rule forbids P to defend himself (once more) from the challenge in move 1.

O			P
				p∨¬p	0
1	?∨	0		¬p	2
3	p	2		—

1. Intuitionist rules. O wins