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Logic Journal of IGPL Advance Access originally published online on January 23, 2009
Logic Journal of IGPL 2009 17(1):91-129; doi:10.1093/jigpal/jzn030
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Equivalence and quantifier rules for logic with imperfect information

Xavier Caicedo*, Francien Dechesne{dagger} and Theo M.V. Janssen{ddagger}

*Dept. of math., Univ. de los Andes, Bogotá, Colombia, xcaicedo{at}uniandes.edu.co
{dagger}Dept. math. & comp.sc., Techn. Univ, Eindhoven, the Netherlands, f.dechesne{at}tue.nl
{ddagger}ILLC, Universiteit van Amsterdam, Amsterdam, the Netherlands, t.m.v.janssen{at}uva.nl


  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.

Received for publication 19 January 2007.
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