Equivalence and quantier rules for logic with imperfect information
Xavier Caicedo, Francien Dechesne, Theo M.V. Janssen
Abstract:
In this paper, we present a normal form theorem for a version of
Independence Friendly logic, a logic with imperfect
information. Lifting classical results to such logics turns out not to
be straightforward, because independence conditions make the formulas
sensitive for signalling phenomena. In particular, nested
quantfiication over the same variable is shown to cause problems. For
instance, renaming of bound variables may change the interpretations
of a formula, there is only a restricted quantier extraction theorem,
and slashed connectives cannot be so easily removed. Thus we correct
some claims from Hintikka (1996), Caicedo & Krynicki (1999) and Hodges
(1997a). We refine definitions, in particular the notion of
equivalence, and sharpen preconditions, allowing us to restore
(restricted versions of) those claims, including the prenex normal
form theorem of Caicedo & Krynicki (1999). Further important results
are several quantifier rules for IF-logic and a surprising improved
version of the Skolem form theorem for classical logic.