This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat Ab as the base system and refer to its expansions as superabelian logics. The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist $2^{2^\omega }$ distinct logics in this family. Second, we introduce pointed Abelian logic (${\text {pAb}}$), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes Łukasiewicz unbound logic. We provide axiomatizations for its finitary and infinitary versions as extensions of ${\text {pAb}}$ and establish their precise relationship with standard Łukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of these properties for substructural logics and varieties of pointed residuated lattices (their algebraic semantics) is far less well understood. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule—algebraically, the commutativity law—in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly 60 varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly 60 axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly 60 varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

Central to certain versions of logical atomism are claims to the effect that every proposition is a truth-functional combination of elementary propositions. Assuming that propositions form a Boolean algebra, we consider a number of natural formal regimentations of informal claims in this vicinity, and show that they are equivalent. For a number of reasons, such as the need to accommodate quantifiers, logical atomists might consider only complete Boolean algebras, and take into account infinite truth-functional combinations. We show that in such a variant setting, some of the regimentations come apart, and explore how they relate to each other. We also discuss how they relate to the claim that propositions form a double powerset algebra, which has been proposed by a number of authors as a way of capturing the central logical atomist idea.

We extend the framework of abstract algebraic logic to weak logics, namely, logical systems that are not necessarily closed under uniform substitution. We interpret weak logics by algebras expanded with an additional predicate, and we introduce a loose and strict version of algebraizability for weak logics. We study this framework by investigating the connection between the algebraizability of a weak logic and the algebraizability of its schematic fragment, and we then prove a version of Blok and Pigozzi’s Isomorphism Theorem in our setting. We apply this framework to logics in team semantics and show that the classical versions of inquisitive and dependence logic are strictly algebraizable, while their intuitionistic versions are only loosely so.

This paper explores the mathematical connections between the algebraic and relational semantics of Lewis’s logics for counterfactual conditionals. Specifically, we introduce topological variants of Lewis’s well-known possible-worlds semantics—based on spheres, selection functions, and orders—and establish duality results with respect to varieties of Boolean algebras equipped with a counterfactual operator, which serve as the equivalent algebraic semantics of Lewis’s main systems. These results aim to provide a solid mathematical foundation for the study of Lewis’s logics, and offer a new perspective on the most well-known possible worlds-based models. In particular, we write explicit proofs for several results that are often assumed without proof in the literature. Leveraging these duality results, we also derive alternative proofs of strong completeness for Lewis’s variably strict conditional logics with respect to their intended models, and clarify the role of the limit assumption in sphere semantics.

The Lindenbaum lemma saying that completely meet-irreducible closed sets form a basis of any finitary closure system is an easy-to-prove yet crucial result transcending algebraic logic. While the finitarity restriction is crucial for its usual proof, it is not necessary: there are indeed works proving it (or its variant for a larger class of finitely meet-irreducible closed sets) for non-finitary closure systems arising from particular infinitary logics (i.e., substitution-invariant consequence relations). There is also a general result proving it for a wide class of logics with strong p-disjunction and a countable Hilbert-style axiomatization. Identifying the essential properties of strong p-disjunctions we prove a variant of the Lindenbaum lemma for closure systems which are 1) defined over countable sets, 2) countably axiomatized, and 3) frames (in the order-theoretic sense) but not necessarily substitution-invariant.

A three-valued logic is subclassical when it is defined by a single matrix having the classical two-element matrix as a subreduct. In this case, the language of can be expanded with special unary connectives, called external operators. The resulting logic is called the external version of , a notion originally introduced by D. Bochvar in 1938 with respect to his weak Kleene logic. In this paper we study the semantic properties of the external version of a three-valued subclassical logic . We determine sufficient and necessary conditions to turn a model of into a model of . Moreover, we establish some distinctive semantic properties of .

We extend the logical categories framework to first-order modal logic. In our modal categories, modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation theorem into relational structures formalizing a ‘counterpart’ notion. We investigate saturation conditions related to definability questions and we enrich our framework with quotients and disjoint sums, thus leading to the notion of a modal (quasi) pretopos. We finally show a way to build syntactic categories out of first-order modal theories.

An étale structure over a topological space X is a continuous family of structures (in some first-order language) indexed over X. We give an exposition of this fundamental concept from sheaf theory and its relevance to countable model theory and invariant descriptive set theory. We show that many classical aspects of spaces of countable models can be naturally framed and generalized in the context of étale structures, including the Lopez-Escobar theorem on invariant Borel sets, an omitting types theorem, and various characterizations of Scott rank. We also present and prove the countable version of the Joyal–Tierney representation theorem, which states that the isomorphism groupoid of an étale structure determines its theory up to bi-interpretability; and we explain how special cases of this theorem recover several recent results in the literature on groupoids of models and functors between them.

Beth’s theorem equating explicit and implicit definability fails in all logics between Meyer’s basic logic ${\mathbf B}$ and the logic ${\mathbf R}$ of Anderson and Belnap. This result has a simple proof that depends on the fact that these logics do not contain classical negation; it does not extend to logics such as $\mathbf{KR}$ that contain classical negation. Jacob Garber, however, showed that Beth’s theorem fails for $\mathbf{KR}$ by adapting Ralph Freese’s result showing that epimorphisms may not be surjective in the category of modular lattices. We extend Garber’s result to show that the Beth theorem fails in all logics between ${\mathbf B}$ and $\mathbf{KR}$.

Clans are categorical representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel–Ulmer duality. Extending Gabriel–Ulmer duality to account for this additional information, we present a duality theory between clans and locally finitely presentable categories equipped with a weak factorization system of a certain kind.

This paper focuses on the structurally complete extensions of the system $\mathbf {R}$-mingle ($\mathbf {RM}$). The main theorem demonstrates that the set of all hereditarily structurally complete extensions of $\mathbf {RM}$ is countably infinite and forms an almost-chain, with only one branching element. As a corollary, we show that the set of structurally complete extensions of $\mathbf {RM}$ that are not hereditary is also countably infinite and forms a chain. Using algebraic methods, we provide a complete description of both sets. Furthermore, we offer a characterization of passive structural completeness among the extensions of $\mathbf {RM}$: specifically, we prove that a quasivariety of Sugihara algebras is passively structurally complete if and only if it excludes two specific algebras. As a corollary, we give an additional characterization of quasivarieties of Sugihara algebras that are passively structurally complete but not structurally complete. We close the paper with a characterization of actively structurally complete quasivarieties of Sugihara algebras.

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

Glivenko’s theorem says that classical provability of a propositional formula entails intuitionistic provability of the double negation of that formula. This stood right at the beginning of the success story of negative translations, indeed mainly designed for converting classically derivable formulae into intuitionistically derivable ones. We now generalise this approach: simultaneously from double negation to an arbitrary nucleus; from provability in a calculus to an inductively generated abstract consequence relation; and from propositional logic to any set of objects whatsoever. In particular, we give sharp criteria for the generalisation of classical logic to be a conservative extension of the one of intuitionistic logic with double negation.

The logico-algebraic study of Lewis’s hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work starts filling this gap by providing a logico-algebraic analysis of Lewis’s logics. We begin by introducing novel finite axiomatizations for Lewis’s logics on the syntactic side, distinguishing between global and local consequence relations on Lewisian sphere models on the semantical side, in parallel to the case of modal logic. As first main results, we prove the strong completeness of the calculi with respect to the corresponding semantical consequence on spheres, and a deduction theorem. We then demonstrate that the global calculi are strongly algebraizable in terms of a variety of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local ones are generally not algebraizable, although they can be characterized as the degree-preserving logic over the same algebraic models. This yields the strong completeness of all the logics with respect to the algebraic models.

Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$. Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP).

We show that the conceptual distance between any two theories of first-order logic is the same as the generator distance between their Lindenbaum–Tarski algebras of concepts. As a consequence of this, we show that, for any two arbitrary mathematical structures, the generator distance between their meaning algebras (also known as cylindric set algebras) is the same as the conceptual distance between their first-order logic theories. As applications, we give a complete description for the distances between meaning algebras corresponding to structures having at most three elements and show that this small network represents all the possible conceptual distances between complete theories. As a corollary of this, we will see that there are only two non-trivial structures definable on three-element sets up to conceptual equivalence (i.e., up to elementary plus definitional equivalence).

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$-valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics.

Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate logic: the first is sound for Takeuti’s quantum set theory, and the second is sound for a variant of Weaver’s quantum logic.