Intensional completeness in an extension of gödel/dummett logic

Studia Logica 73 (1):51 - 80 (2003)
Abstract
We enrich intuitionistic logic with a lax modal operator and define a corresponding intensional enrichment of Kripke models M = (W, , V) by a function T giving an effort measure T(w, u) {} for each -related pair (w, u). We show that embodies the abstraction involved in passing from true up to bounded effort to true outright. We then introduce a refined notion of intensional validity M |= p : and present a corresponding intensional calculus iLC-h which gives a natural extension by lax modality of the well-known G: odel/Dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : | L |= p : } characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of and completely recovered by a suitable semantic interpretation of proofs.
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