Semantic games and hypersequents: a case study in many valued reasoning

Colloquium Mathematical Philosophy, Chris Fermüller (Vienna) gives a talk at the MCMP Colloquium (2 May, 2013) titled "Semantic games and hypersequents: a case study in many valued reasoning". Abstract: For a quite a while it had been an open problem whether there is an analytic (cut-free) calculus for infinite valued Lukasiewicz logic, one of threefundamental many valued logics that lie at the centre of interest in contemporary mathematical fuzzy logic. The hypersequent calculus HL presented by Metcalfe, Gabbay, and Olivetti in 2004/5 settled the question positively; but HL did not fit well into the family of sequent and hypersequent systems for related nonclassical logics. In particular it remained unclear in what sense HL provides an analysis of logical reasoning in a many valued context. On the other hand, already in the 1970s Robin Giles had shown that a straightforward dialogue game, combined with a specific way to calculate expected losses associated with bets on the results on `dispersive experiments' leads to a characterisation of Lukasiewizc logic. We illustrate how these seemingly unrelated results fit together: the logical rules of HL naturally emerge from a systematic search for winning strategies in Giles's game. This amounts to a rather tangible interpretation of hypersequents that can be extended to other logics as well.