We develop duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces. We show that these binary relations are in 1–1 correspondence with subframes of Heyting spaces. We introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear iff it is a subframe variety, and that it is dense nuclear iff it is a cofinal subframe variety. We give an alternative proof that every subframe variety (...) of Heyting algebras is generated by its finite members. (shrink)

Inquisitive logic is a research program seeking to expand the purview of logic beyond declarative sentences to include the logic of questions. To this end, inquisitive propositional logic extends classical propositional logic for declarative sentences with principles governing a new binary connective of inquisitive disjunction, which allows the formation of questions. Recently inquisitive logicians have considered what happens if the logic of declarative sentences is assumed to be intuitionistic rather than classical. In short, what should inquisitive logic be on an (...) intuitionistic base? In this paper, we provide an answer to this question from the perspective of nuclear semantics, an approach to classical and intuitionistic semantics pursued in our previous work. In particular, we show how Beth semantics for intuitionistic logic naturally extends to a semantics for inquisitive intuitionistic logic. In addition, we show how an explicit view of inquisitive intuitionistic logic comes via a translation into propositional lax logic, whose completeness we prove with respect to Beth semantics. (shrink)

This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective N satisfying nucleus property, called here substructural nuclear logics, and its subclass, called here substructural nuclear image-based logics, where N further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce operational Kripke-style semantics for those logics and provide two sorts of completeness results for (...) them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based logics. (shrink)

The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues related to the calculi and not on the significance of adjunctions in modal logic. It then seems a worthy enterprise (both (...) for these contemporary topical pursuits and also for historical interest) to trace the concept of adjunction back to the origins of the algebraic semantics of modal logic and to make explicit its ubiquity in this branch of mathematics. (shrink)

1. The theory referred to by the—perhaps intimidating—main title of this book is an extension of Per Martin-Löf's dependent type theory. Much philosophical work pertaining to dependent type theory...

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into (...) K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a poset, forming what we (...) call a Dragalin frame. This allowed Dragalin to introduce a semantics for intuitionistic logic that generalizes Beth and Kripke semantics. He proved that every spatial locale (locale of open sets of a topological space) can be realized as fixpoints of the nucleus generated by a Dragalin frame. In this paper, we strengthen Dragalin’s result and prove that every locale—not only spatial locales—can be realized as fixpoints of the nucleus generated by a Dragalin frame. In fact, we prove the stronger result that for every nucleus on the upsets of a poset, there is a Dragalin frame based on that poset that generates the given nucleus. We then compare Dragalin’s approach to generating nuclei with the relational approach of Fairtlough and Mendler [11], based on what we call FM-frames. Surprisingly, every Dragalin frame can be turned into an equivalent FM-frame, albeit on a different poset. Thus, every locale can be realized as fixpoints of the nucleus generated by an FM-frame. Finally, we consider the relational approach of Goldblatt [13] and characterize the locales that can be realized using Goldblatt frames. (shrink)