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A Curious Dialogical Logic and its Composition Problem

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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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Notes

  1. For more on the history of dialogical logic, see [19, 22].

  2. Two of the current authors have discussed some of these matters further in [3]. The subject is also touched on in [29].

  3. Note that it is not the case that every D-dialogue is an E-dialogue; an IL theorem that displays the divergence of the two is ¬¬(((φ → ψ) → γ) → (((ψ → φ) → γ) → γ)).

  4. Note that these two questions are not, strictly speaking, the same. In [40] a ruleset which generates CL but which does not have E is given.

  5. See also the modal logics, mostly dynamic, in [15, fn. 1].

  6. One can think of logics which do not satisfy unrestricted uniform substitution as being propositional theories, where the interpretation of the atomic variables is fixed in advance.

  7. In many logics, it doesn’t make sense to distinguish a schematic/substitutional notion of consequence from a non-substitutional one. However some logics, such as provability logics, do make such a distinction.

  8. Thanks to one of the anonymous referees on a previous version of this paper for pointing this out to us.

  9. This illustrates nicely Krabbe’s point about the importance of structural rules regulating repetitive behavior (“the bugbear of dialogue theory”) of the players [17, pp. 296, 303].

  10. One reason why N is not a relevance logic is because such formulas as ¬(p ∨ ¬p) → q are valid. However, N could be considered a relevance logic in Rückert’s sense of P-relevance logic, in which P must make all possible attacks and defenses. See [34, ch. 5].

  11. A similar thing happens in linear logic [8, Sections 2.1, 6]; for a dialogical characterization of linear logic, see [4].

  12. More precisely, we attach to the end of f(b) the result of cutting off the root of t, because the root of t and the leaf of b represent assertions by P of identical formulas.

  13. Recall that the Gödel-Gentzen negative translation, however, does not preserve N-validity; see the proof of Lemma 11. This translation does not merely add negations, but also changes the shape of the formula.

  14. Benedikt Löwe suggested this elegant solution.

  15. We do not follow Felscher’s developments of tableaux for intuitionistic logic, which are idiosyncractic for reasons related to his precise proof method.

  16. Such an objection was raised by one of the anonymous referees.

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Acknowledgments

This paper is an extended version of [2], presented at the 4th Indian Conference on Logic and its Applications, Delhi, India, January 2011. The authors would like to thank the anonymous referees who provided many useful comments on earlier versions of this paper, as well as Shahid Rahman, whose suggestion prompted the results in Sections 6 and 7.

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Authors and Affiliations

  1. Exzellenzcluster “Asia and Europe”, Karl Jaspers Centre for Advanced Transcultural Studies, Ruprecht-Karls Universität Heidelberg, Heidelberg, Germany

    Sara L. Uckelman

  2. Center for Artificial Intelligence, New University of Lisbon, Lisbon, Portugal

    Jesse Alama

  3. Department of Philosophy, University of Maryland, College Park, MD, USA

    Aleks Knoks

Authors
  1. Sara L. Uckelman
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  2. Jesse Alama
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Correspondence to Sara L. Uckelman.

Additional information

The first two authors were partially funded by the FCT/NWO/DFG project “Dialogical Foundations of Semantics” (DiFoS) in the ESF EuroCoRes programme LogICCC (FCT LogICCC/0001/2007; LogICCC-FP004; DN 231-80-002; CN 2008/08314/GW).

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Uckelman, S.L., Alama, J. & Knoks, A. A Curious Dialogical Logic and its Composition Problem. J Philos Logic 43, 1065–1100 (2014). https://doi.org/10.1007/s10992-013-9307-1

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