A sheaf representation and duality for finitely presented Heyting algebras

Abstract

A. M.Pitts in [Pi] proved that is a bi-Heyting category satisfying the Lawvere condition. We show that the embedding Φ: → Sh(P0, J0) into the topos of sheaves, (P0 is the category of finite rooted posets and open maps, J0 the canonical topology on P0) given by H ↦ HA(H, (−)) : P0 → Set preserves the structure mentioned above, finite coproducts, and subobject classifier; it is also conservative. This whole structure on can be derived from that of Sh(P0, J0) via the embedding Φ. We also show that the equivalence relations in are not effective in general. On the way to these results we establish a new kind of duality between and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi].

Uniform Interpolation and Propositional Quantifiers in Modal Logics

“…In this case, a semantic argument using bisimulations on Kripke models was given later by Ghilardi and Zawadowski in [10] and independently by Visser in [19]. Pitts' argument uses a simulation of propositional quantifiers in the framework of a sequent proof system.…”

On modal μ-calculus with explicit interpolants

“…In particular, it would be good to know whether the low levels are closed. This closure property is already known for the 0-level of the fixpoint hierarchy, that is, it is already known that modal logic is closed under the existential bisimulation quantifier [4,12]. In this paper we generalize this property to the levels 1 and 2 of the hierarchy.…”

“…Bisimulation quantifiers were first considered in [4] and [12] as a tool for proving uniform interpolation for modal logic, and in [2] to show uniform interpolation for the modal μ-calculus. Given a formula φ, the language of φ is the set of all propositional constants appearing in the formula.…”

Extendible Formulas in Two Variables in Intuitionistic Logic

“…In fact, its positive answer is a direct consequence of Pitts' Uniform Interpolation Theorem. We formulate the latter result in its more general form; see e.g., [19,11].…”