By Johan Van Benthem

The current paintings is a rewritten model of van Benthem's dissertation ``Modal Correspondence Theory'' (University of Amsterdam, 1976) and a supplementary record known as ``Modal common sense as Second-Order Logic'' (University of Amsterdam, 1977).

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Additional resources for Modal Logic and Classical Logic

Example text

The modal collapse of (F, V) with respect to the finite set 2 consisting of ~\

set of universally valid modal formulas is recursively enu­ merable as well (cf. chapters 3 and 5). Therefore, by Post’s Theorem, universal validity of modal formulas is a recursive notion.

To repeat, general frames of the form HGF(X) are descriptive. ) The better-known modal completeness theorems are of yet a diffe­ rent form, however. 5 if and only if it holds in all frames whose alternative relation is an equivalence relation. If this is restated in the present terms, using the fact (cf. chapter 3) that the cha­ racteristic axioms Lp -* p, Lp -* LLp and MLp p o f S3 define reflexivity, transitivity and symmetry — in that order —, then it assumes the following form. 5 and for any modal formula

That LF implies well-foundedness of the converse of R is seen by contraposition: Suppose an / as forbidden exists. Define V(p) = W — \f(n) \n e IN}. For no f( n ) 9 48 MODAL LOGIC A N D CLASSICAL LOGIC (F, V) |= L p [f(n )]9 and, for all v s which are n o t/ (»)’s, (F, V) j= p[v ]. Therefore, (F, V) |= L(Lp - p ) [ f( 0)], and (F, V) \¥> L p [f(0)]: LF has been falsified. , (F, V) |= L(Lp p)[tv] and (F, V) L p [w ])t but (i) holds. An / as forbidden may then be constructed as follows. Set /(0) = w.

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