From Semantic Games to Provability: The Case of Gödel Logic

Abstract

AbstractWe 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 $$\text{ SeqGZL } $$ 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.