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A note on modal formulae and relational properties

Published online by Cambridge University Press:  12 March 2014

J. F. A. K. van Benthem*
Affiliation:
Instituut Voor Grondslagenonderzoek, Universiteit van Amsterdam, Amsterdam, The Netherlands

Extract

Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators □ and ⋄. The semantic structures are frames, i.e., pairs <W, R> with RW2. Let F, V be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of W. Then the relation

may be defined, for arbitrary formulae α, following the Kripke truth-definition. From this relation we may further define

Now, to every modal formula α there corresponds some property Pα of R. A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first-order predicate. For these particular Pα we have

for all F and wW. These formulae Pα are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when Pα may be taken to be some first-order formula. For example, it can be seen that

for all F and wW. It is customary to talk about a related correspondence, namely when for all F we have

Note that this correspondence holds whenever the first one above holds.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1975

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