It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (Implication via (...) spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category $${\mathcal {S}}$$ S, provided that $${\mathcal {S}}$$ S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication. (shrink)

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein (...) and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting–Brouwer logic [math] that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from extensions of [math] to extensions of [math].

A Heyting algebra is supplemented if each element a has a dual pseudo-complement \, and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the (...) hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally supplemented extension. This provides an algebraic description of the hyper-MacNeille completion of a Heyting algebra, allows development of further properties of the hyper-MacNeille completion, and provides new examples of varieties of Heyting algebras that are closed under hyper-MacNeille completions. In particular, connections between the centrally supplemented extension and Boolean products allow us to show that any finitely generated variety of Heyting algebras is closed under hyper-MacNeille completions. (shrink)

When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret (...) ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that ${X \in {\bf H}_{n}}$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$ , and for a limit ordinal α, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. (shrink)

This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].

For any $n<\omega $ we construct an infinite $(n+1)$ -generated Heyting algebra whose n-generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$. From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$ -generated algebra, but which contains only finite n-generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

In this paper we study the variety \(\textsf{WL}\) of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the \(\{\vee,\wedge,\Rightarrow,\bot,\top \}\) -fragment of the arithmetical base preservativity logic \(\mathsf {iP^{-}}\). The variety \(\textsf{WL}\) properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means of WL-frames. We extended (...) this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic \(\textsf{iP}^{-}\). (shrink)

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for$\forall $) and “sometime in the past” (for$\exists $). It is well known that Prior’s intuitionistic modal logic${\sf MIPC}$axiomatizes the monadic fragment of the intuitionistic predicate logic, and that${\sf MIPC}$is translated fully and faithfully into the monadic fragment${\sf MS4}$of the predicate${\sf S4}$via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension${\sf TS4}$of${\sf S4}$and provide a full and faithful translation (...) of${\sf MIPC}$into${\sf TS4}$. We compare this new translation of${\sf MIPC}$with the Gödel translation by showing that both${\sf TS4}$and${\sf MS4}$can be translated fully and faithfully into a tense extension of${\sf MS4}$, which we denote by${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for${\sf MS4.t}$using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. (shrink)

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

Sahlqvist theory is extended to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic.

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer (...) a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics. (shrink)

The paper studies Heyting algebras within the framework of computable structure theory. We prove that the class _K_ containing all Heyting algebras with distinguished atoms and coatoms is complete in the sense of the work of Hirschfeldt et al. (Ann Pure Appl Logic 115(1-3):71-113, 2002). This shows that the class _K_ is rich from the computability-theoretic point of view: for example, every possible degree spectrum can be realized by a countable structure from _K_. In addition, there is no simple syntactic (...) characterization of computably categorical members of _K_ (i.e., structures from _K_ possessing a unique computable copy, up to computable isomorphisms). (shrink)

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety (...) theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1. (shrink)

\(\nabla \) -algebra is a natural generalization of Heyting algebra, unifying many algebraic structures including bounded lattices, Heyting algebras, temporal Heyting algebras and the algebraic presentation of the dynamic topological systems. In a series of two papers, we will systematically study the algebro-topological properties of different varieties of \(\nabla \) -algebras. In the present paper, we start with investigating the structure of these varieties by characterizing their subdirectly irreducible and simple elements. Then, we prove the closure of these varieties under (...) the Dedekind-MacNeille completion and provide the canonical construction and the Kripke representation for \(\nabla \) -algebras by which we establish the amalgamation property for some varieties of \(\nabla \) -algebras. In the sequel of the present paper, we will complete the study by covering the logics of these varieties and their corresponding Priestley-Esakia and spectral duality theories. (shrink)

In this paper we continue the investigation of monadic Heyting algebras which we started in [2]. Here we present the representation theorem for monadic Heyting algebras and develop the duality theory for them. As a result we obtain an adequate topological semantics for intuitionistic modal logics over MIPC along with a Kripke-type semantics for them. It is also shown the importance and the effectiveness of the duality theory for further investigation of monadic Heyting algebras and logics over MIPC.

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis closely related to that (...) for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising. (shrink)

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give the bi-tense (...) logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

In this paper we continue the study of the variety \ of monadic Gödel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of Gödel logic, which is equivalent to the one-variable monadic fragment of first-order Gödel logic. We show three families of locally finite subvarieties of \ and give their equational bases. We also introduce a topological duality for monadic Gödel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of (...) the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic Gödel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over n generators. (shrink)

The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?

This paper is the concluding part of [1] and [2], and it investigates the inner structure of the lattice (MHA) of all varieties of monadic Heyting algebras. For every n , we introduce and investigate varieties of depth n and cluster n, and present two partitions of (MHA), into varieties of depth n, and into varieties of cluster n. We pay a special attention to the lower part of (MHA) and investigate finite and critical varieties of monadic Heyting algebras in (...) detail. In particular, we prove that there exist exactly thirteen critical varieties in (MHA) and that it is decidable whether a given variety of monadic Heyting algebras is finite or not. The representation of (MHA) is also given. All these provide us with a satisfactory insight into (MHA). Since (MHA) is dual to the lattice NExtMIPC of all normal extensions of the intuitionistic modal logic MIPC, we also obtain a clearer picture of the lattice structure of intuitionistic modal logics over MIPC. (shrink)

The Gödel translation provides an embedding of the intuitionistic logic$\mathsf {IPC}$into the modal logic$\mathsf {Grz}$, which then embeds into the modal logic$\mathsf {GL}$via the splitting translation. Combined with Solovay’s theorem that$\mathsf {GL}$is the modal logic of the provability predicate of Peano Arithmetic$\mathsf {PA}$, both$\mathsf {IPC}$and$\mathsf {Grz}$admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions$\mathsf {MIPC}$,$\mathsf {MGrz}$, and$\mathsf {MGL}$of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version (...) of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds$\mathsf {M^{+}IPC}$into$\mathsf {M^{+}Grz}$and the splitting translation embeds$\mathsf {M^{+}Grz}$into$\mathsf {MGL}$. As proven by Japaridze, Solovay’s result extends to the monadic system$\mathsf {MGL}$, which leads us to a provability interpretation of both$\mathsf {M^{+}IPC}$and$\mathsf {M^{+}Grz}$. (shrink)