Equivalence and quantifier rules for logic with imperfect information

Abstract

In this paper, we present a prenex 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 to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore (restricted versions of) those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games.

Conjunction is Parallel Computation

“…A careful discussion of negation in game theoretic semantics and a proper implementation of non-upward monotone quantifiers in this framework would go beyond the limits of this paper; for a discussion of properties of game-theoretic negation, see Hintikka (2002Hintikka ( , 2006Caicedo, Dechesne & Janssen (2009). Let me just note that negation translates into role permutation: Verifier takes the role of Falsifier and vice versa.…”

Compositional Natural Language Semantics using Independence Friendly Logic or Dependence Logic

“…The approach is known as 'trump semantics'. His definition is corrected for some details in [4], where also the equivalence is proven with a game interpretation.…”

Compositionality: Its Historic Context

“…This semantics was not acceptable for Hintikka because of the higher order ontology (Sandu & Hintikka 2001). However, a variant of the compositional semantics enabled Caicedo, Dechesne & Janssen (2009) to prove properties of IF-logic, e.g. a normal form theorem.…”