# From Giles's game for reasoning in physics to analytic proof systems for fuzzy logics

- 14:00 23rd November 2012 ( week 7, Michaelmas Term 2012 )Tony Hoare Room

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 three fundamental formal 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 demonstrate 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. Moreover, the underlying principle

for transforming semantic games into analytic proof systems can be

generalized to other logics as well.