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