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Article

Local Goldblatt-Thomason Theorem

Logic Journal of the IGPL. 2015. Vol. 23. No. 6. P. 861-880.

The celebrated theorem proved by Goldblatt and Thomason in 1974 gives necessary and sufficient conditions for an elementary class of Kripke frames to be modally definable. Here we obtain a local analogue of this result, which deals with modal definability of classes of pointed frames. Furthermore, we generalize it to the case of n-frames, which are frames with n distinguished worlds. For talking about n-frames, we generalize modal formulas to modal expressions. While a modal formula is evaluated at a single world of a model, a modal expression with n individual variables is evaluated at an n-tuple of worlds, just as a first-order formula with n free variables. We introduce operations on n-frames that preserve validity of modal expressions, and show that closure under these operations is a necessary and sufficient condition for an elementary class of n-frames to be modally definable. We also discuss the relationship between modal expressions and hybrid logic and leave open questions.