Giles's game and the proof theory of Łukasiewicz logic (Q1037588)

Starting in 1974, \textit{R. Giles} proposed in a series of papers [Stud. Log. 33(1974), 397--415 (1975; Zbl 0324.02017); Fuzzy Sets Syst. 2, 233--257 (1979; Zbl 0411.03018); Stud. Log. 38, 49--73 (1979; Zbl 0417.03008); Fuzzy Sets Syst. 25, No. 3, 297--323 (1988; Zbl 0652.90001)] a general treatment of reasoning with vague predicates by means of a formal system based upon a dialogue interpretation, dealing with subjective belief. The main idea is to let a sentence represent a belief by expressing it in the form of a bet. In this setting a sentence \(\psi\) is considered to follow from sentences \(\varphi_1,\dots,\varphi_n\) just when anyone who accepts the bets \(\varphi_1,\dots,\varphi_n\) can at the same time bet \(\psi\) without fear of loss. In this paper, it is shown that `disjunctive strategies' for Giles's game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In particular, such strategies mirror derivations in a hypersequent calculus developed in recent work [see \textit{G. Metcalfe, N. Olivetti} and \textit{D. Gabbay}, Proof theory for fuzzy logics. Applied Logic Series 36. Dordrecht: Springer (2009; Zbl 1168.03002)] on the proof theory of Łukasiewicz logic.