On the complexity of propositional quantification in intuitionistic logic

Abstract

We define a propositionally quantified intuitionistic logic Hπ+ by a natural extension of Kripke's semantics for propositional intuitionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π+, S4π+, S4.2π+, K4π+, Tπ+, Kπ+ and Bπ+, studied by Fine.

Uniform Interpolation and Propositional Quantifiers in Modal Logics

“…The second order modal systems over logics K, T, K4, S4 obtained this way are recursively isomorphic to full second order classical logic. This was proved independently by Fine and Kripke shortly after Fine's paper [8] was published, as Kremer remarked in [11]. Also Kremer's strategy from [12] can be extended to prove the same result, as he claims in [11].…”

“…This was proved independently by Fine and Kripke shortly after Fine's paper [8] was published, as Kremer remarked in [11]. Also Kremer's strategy from [12] can be extended to prove the same result, as he claims in [11]. In particular it means that these systems are undecidable while their propositional counterparts are decidable.…”

“…In the framework of Kripke semantics they are defined as ranging over propositions, i.e., sets of possible worlds. This definition is used in Fine [8], see also Bull [3] and Kremer [11]. The second order modal systems over logics K, T, K4, S4 obtained this way are recursively isomorphic to full second order classical logic.…”

Modeling the Beliefs of other Agents

“…The logic could be decidable, or it could be equivalent to second-order modal logic. (See Fine [8] and Kremer [11].) As far as I know, the case with multiple modalities has not been investigated to any extent, but I have an unpublished proof that, even with the modalities that are decidable in the monomodal case, multimodal logics with propositional quantifiers are equivalent to full second-order logic.…”

Completeness of Second-Order Propositional S4 and H in Topological Semantics

“…Kit Fine and Saul Kripke independently discovered, but did not publish, proofs of this for the modal logics in the 1970s, though in[1], Fine provides an earlier proof sketch that second-order arithmetic can be recursively embedded into these modal logics [5]. provides published proofs for the modal results [7]. gives a proof for H. It should be noted that[1] and[6] independently provide axiomatizations of the principal interpretation, in the Kripke semantics, of S5.…”