Dialogue Games for Minimal Logic

Alexandra Pavlova

DOI: http://dx.doi.org/10.12775/LLP.2020.022

Abstract


In this paper, we define a class of dialogue games for Johansson’s minimal logic and prove that it corresponds to the validity of minimal logic. Many authors have stated similar results for intuitionistic and classical logic either with or without actually proving the correspondence. Rahman, Clerbout and Keiff [17] have already specified dialogues for minimal logic; however, they transformed it into Fitch-style natural deduction only. We propose a different specification for minimal logic with the proof of correspondence between the existence of winning strategies for the Proponent in this class of games and the sequent calculus for minimal logic.


Keywords


dialogue logic; sequent calculi; minimal logic

Full Text:

PDF

References


Alama. J., A. Knoks and S.L. Uckelman, “Dialogue games for classical logic”, pages 82–86 in M. Giese and R. Kuznets (eds.), TABLEAUX 2011: Workshops, Tutorials, and Short Papers, 2011.

Alama, J., and S.L. Uckelman, “What is dialogical about dialogical logic?”, pages 207–222 in H.J. Ribeiro (ed.), Inside Arguments: Logic and the Study of Argumentation, Newcastle: Cambridge Scholars Publication, 2012.

Clerbout, N., “First-order dialogical games and tableaux”, Journal of Philosophical Logic 43, 4 (2014): 785–801. DOI: http://dx.doi.org/10.1007/s10992-013-9289-z

Clerbout, N., “Finiteness of plays and dialogical problem of decidability”, IfCoLog Journal of Logics and their Applications 1, 1 (2014): 115–130.

Felscher, W., “Dialogues, strategies, and intuitionistic provability”, Annals of Pure and Applied Logic 28 (1985): 217–254. DOI: http://dx.doi.org/10.1016/0168-0072(85)90016-8

Fermüller, C.G., “Parallel dialogue games and hypersequents for intermediate logics”, pages 48–64 in M.C. Mayer and F. Pirri (eds.), TABLEAUX 2003 Automated Reasoning with Analytic Tableaux and Related Methods, 2003. DOI: http://dx.doi.org/10.1007/978-3-540-45206-5_7

Gentzen, G., and K. Erich, “Untersuchungen über das logische Schließen. I”, Mathematische Zeitschrift 39, 2 (1934): 176–210.

Gentzen, G., and Karl Erich, “Untersuchungen über das logische Schließen. II”, Mathematische Zeitschrift 39, 3 (1935): 405–431.

Hintikka, J., The Principles of Mathematics Revisited, Cambridge: Cambridge University Press, 1996. DOI: http://dx.doi.org/10.1017/CBO9780511624919

Johansson, I., “Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus”, Compositio Mathematica 4 (1937): 119–136.

Kleene, S.C., Introduction to Metamathematics, the Netherlands, 1952.

Krabbe, E.C.W.. “Dialogue logic”, pages 665–704 in D.M. Gabbay and J. Woods (eds.), Handbook of the History of Logic, Vol. 7, New York: Elsevier, 2006. DOI: http://dx.doi.org/10.1016/S1874-5857(06)80035-X

Lorenzen, P., and K. Lorenz, Dialogische Logik, Wissenschaftlische Buchgesellschaft, Darmstadt, 1978.

Pavlova, A.M., “Truth in dialogue logic and game-theoretical semantics (GTS)”, Logical Investigations 21, 2 (2015): 107–133.

Naibo, A., and P. Mattia, “Are uniqueness and deducibility of identical the same?”, Theoria 81, 2 (2015): 143–181. DOI: http://dx.doi.org/10.1111/theo.12051

Rahman, S., and W.A. Carnielli, “The dialogical approach to paraconsistency”, Synthese 125 (2000): 201–232. DOI: http://dx.doi.org/10.1023/A:1005294523930

Rahman, S., N. Clerbout, and L. Keiff, “On dialogues and natural deduction”, pages 301–355 in G. Primeiro and S. Rahman (eds.), Acts of Knowledge: History, Philosophy and Logic, London: College Publications, 2009.

Rahman, S., and T. Tulenheimo, “From games to dialogues and back: Towards a general frame for validity”, in Games: Unifying logic, Language and Philosophy, Springer, 2009. DOI: http://dx.doi.org/10.1007/978-1-4020-9374-6_8

Sørensen, M.H., and P. Urzyczyn, “Sequent calculus, dialogues, and cut elimination”, pages 253–261 in Reflections on Type Theory, λ-Calculus, and the Mind, Universiteit Nijmegen, 2007.

Troelstra, A.S., and H. Schwichtenberg, Basic Proof Theory, 2nd edn, Cambridge University Press, 2000. DOI: http://dx.doi.org/10.1017/CBO9781139168717








ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism