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Additional info for 1998 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '98

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Let W be a denumerable set of labels x , y , . . , and R a binary relation over W . We factor our presentations into two parts, a fixed base system and a varying relational theory, corresponding to the two sorts of syntactic entities we consider: labelled formulas x : A , which pair a label x and a modal formula A (expressing that A holds at world x), and 92 1998 ASSOCIATION FOR SYMBOLIC LOGIC EUROPEAN SUMMER MEETING relational formulas x R y (expressing that ( x ,y ) is in the accessibility relation R ) .

Every argument that has the same logical form is also valid. Of course, if an argument is formally valid, it is also valid. ) The first definition given above uses modal concepts such as possible states of affairs. The second definition uses, in addition, the concept of logical form and with it the distinction between logical and non-logical words.

AtxRz That is, T is K U { r e j ) , K4 is K U {trans), and S4 is T U {trans) or K4 U { r e j ) . To develop complexity results, we perform a fine-grained proof-theoretical analysis of the structural rules of these systems. The key problem to tackle is that although the subformula property (which follows from the absence of cut) bounds the number of different formulae that can appear in a proof, it does not bound the number of times a formula can appear in a sequent. We provide this latter bound by proving that we can bound, and in some cases eliminate altogether, applications of the contraction rules.

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1998 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '98

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