Abstract
We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al. (in: M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of text{ SeqGZL } (A. Ciabattoni, and T. Vetterlein, Fuzzy Sets and Systems 161(14):1941–1958, 2010) and in a sequent-of-relations calculus (M. Baaz, and Ch.G. Fermüller, in: N.V. Murray, (ed.), Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively.
Highlights
The use of games in logic has a long and varied tradition, wherein the types of games considered have been as manifold as the logics to which they have been applied
The main idea of the paper is to start with a semantic game and subsequently lift it to a provability game using disjunctive strategies
We introduce a semantic game for the stepwise reduction of arbitrary truth degree comparison claims to atomic ones
Summary
The use of games in logic has a long and varied tradition, wherein the types of games considered have been as manifold as the logics to which they have been applied. The provability game with disjunctive rules can be interpreted as an analytic calculus, which turns out to be quite close to an existing calculus for Godel logic with involutive negation and constants proposed in [10]. We conclude with a brief summary of our results, followed by suggestions for future research in this area
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