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) ∈ \({\mathbb{N}} \cup\) {∞} for each ⊑-related pair (w, u). We show that ○ embodies the abstraction involved in passing from “ϕ true up to bounded effort” to “ϕ true outright”. We then introduce a refined notion of intensional validity M |= p : ϕ 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 : ϕ | L |= p : ϕ} 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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Fairtlough, M., Mendler, M. Intensional Completeness in an Extension of Gödel/Dummett Logic. Studia Logica 73, 51–80 (2003). https://doi.org/10.1023/A:1022937306253
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DOI: https://doi.org/10.1023/A:1022937306253