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A Curious Dialogical Logic and its Composition Problem


Uckelman, Sara L; Alama, Jesse; Knoks, Aleks (2014). A Curious Dialogical Logic and its Composition Problem. Journal of Philosophical Logic, 43(6):1065-1100.

Abstract

Dialogue semantics for logic are two-player logic games between a Proponent who puts forward a logical formula φ as valid or true and an Opponent who disputes this. An advantage of the dialogical approach is that it is a uniform framework from which different logics can be obtained through only small variations of the basic rules. We introduce the composition problem for dialogue games as the problem of resolving, for a set S of rules for dialogue games, whether the set of S-dialogically valid formulas is closed under modus ponens. Solving the composition problem is fundamental for the dialogical approach to logic; despite its simplicity, it often requires an indirect solution with the help of significant logical machinery such as cut-elimination. Direct solutions to the composition problem can, however, sometimes be had. As an example, we give a set N of dialogue rules which is well-justified from the dialogical point of view, but whose set N of dialogically valid formulas is both non-trivial and non-standard. We prove that the composition problem for N can be solved directly, and introduce a tableaux system for N.

Abstract

Dialogue semantics for logic are two-player logic games between a Proponent who puts forward a logical formula φ as valid or true and an Opponent who disputes this. An advantage of the dialogical approach is that it is a uniform framework from which different logics can be obtained through only small variations of the basic rules. We introduce the composition problem for dialogue games as the problem of resolving, for a set S of rules for dialogue games, whether the set of S-dialogically valid formulas is closed under modus ponens. Solving the composition problem is fundamental for the dialogical approach to logic; despite its simplicity, it often requires an indirect solution with the help of significant logical machinery such as cut-elimination. Direct solutions to the composition problem can, however, sometimes be had. As an example, we give a set N of dialogue rules which is well-justified from the dialogical point of view, but whose set N of dialogically valid formulas is both non-trivial and non-standard. We prove that the composition problem for N can be solved directly, and introduce a tableaux system for N.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:06 Faculty of Arts > Institute of Philosophy
Dewey Decimal Classification:100 Philosophy
Scopus Subject Areas:Social Sciences & Humanities > Philosophy
Language:English
Date:1 December 2014
Deposited On:12 Sep 2020 07:27
Last Modified:14 Sep 2020 10:56
Publisher:Springer
ISSN:0022-3611
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s10992-013-9307-1

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