In this article, we study a degenerate version of Ramsey’s theorem for pairs and two colors (${\mathsf {RT}}^2_2$), in which the homogeneous sets for color 1 are of bounded size. By ${\mathsf {RT}}^2_2$, it follows that every such coloring admits an infinite homogeneous set for color 0. This statement, called ${\mathsf {BRT}}^2_2$, is known to be computably true, that is, every computable instance admits a computable solution, but the known proofs use $\Sigma ^0_2$-induction ($\mathsf {I}\Sigma ^0_2$). We prove that ${\mathsf {BRT}}^2_2$ follows from the Erdős–Moser theorem but not from the ascending descending sequence principle, and that its computably true version is equivalent to $\mathsf {I}\Sigma ^0_2$ over ${\mathsf {RCA}}_0$.

In this article, I investigate the modal logic of exact equivalence (i.e., sameness of exact verifiers). In particular, by building on Kim’s exact truthmaker semantics for modal logic, I provide an answer to the following question: which sentences of the language of propositional modal logic are exactly equivalent by virtue of their logical form?

An apparent issue for the Revision Theory of definitions has long been that its most plausible versions engender $\omega $-inconsistencies. In this paper I develop a new $\omega $-consistent revision theory and use it to argue that revision theorists can and should embrace $\omega $-consistency. I show how my theory, called $\mathbf {S}^{\#N}$, withstands the theoretical pressures towards $\omega $-inconsistency and moreover compares favorably to the best $\omega $-inconsistent theories vis-à-vis several important desiderata. I tentatively conclude that $\mathbf {S}^{\#N}$ is the best known revision theory.

We prove a Thomason-style duality for the category of instantial neighbourhood frames and instantial neighbourhood morphisms, use it to define and investigate ultrafilter extensions, and show that it restricts to Thomason duality for Kripke frames.

In the product $L_1\times L_2$ of two Kripke complete consistent logics, local tabularity of $L_1$ and $L_2$ is necessary for local tabularity of $L_1\times L_2$. However, it is not sufficient: the product of two locally tabular logics may not be locally tabular. We provide extra semantic and axiomatic conditions that give criteria of local tabularity of the product of two locally tabular logics, and apply them to identify new families of locally tabular products. We show that the product of two locally tabular logics may lack the product finite model property. We give an axiomatic criterion of local tabularity for all extensions of . Finally, we describe a new prelocally tabular extension of .

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.

Gödel algebras are the Heyting algebras satisfying the axiom $(x \to y) \vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe free Gödel algebras and coproducts of Gödel algebras. In particular, we realize the Esakia space dual to a Gödel algebra free over a distributive lattice as the, suitably topologized and ordered, collection of all nonempty closed chains of the Priestley dual of the lattice. This provides a tangible dual description of free Gödel algebras without any restriction on the number of free generators, which generalizes known results for the finitely generated case. A similar approach allows us to characterize the Esakia spaces dual to coproducts of arbitrary families of Gödel algebras. We also establish analogous dual descriptions of free algebras and coproducts in every variety of Gödel algebras. As consequences of these results, we obtain a formula to compute the depth of coproducts of Gödel algebras and show that all free Gödel algebras are bi-Heyting algebras.

In Outline of a Theory of Truth, Kripke introduces many of the central concepts of the logical study of truth and paradox. He informally defines some of these—such as groundedness and paradoxicality—using modal locutions. We introduce a modal language for regimenting these informal definitions. Though groundedness and paradoxicality are expressible in the modal language, we prove that intrinsicality—which Kripke emphasizes but does not define modally—is not. This follows from a characterization of the modally definable sets and relations and an attendant axiomatization of the modal semantics.

Many logical properties are known to be undecidable for normal modal logics, with few exceptions such as the consistency and the coincidence with $\mathsf {K}$. This article shows that the property of being a union-splitting in $\mathop {\mathsf {NExt}}{\mathsf {K}}$, the lattice of normal modal logics, is decidable, thus answering Problem 2 in [F. Wolter and M. Zakharyaschev, 2007]. This is done by providing a semantic characterization of union-splittings in terms of finite modal algebras. Moreover, by clarifying the connection to union-splittings, we show that in $\mathop {\mathsf {NExt}}{\mathsf {K}}$, having a decidable axiomatization problem and being a (un)decidable formula are also decidable. The latter answers Problem 17.3 in [A. Chagrov and M. Zakharyaschev, 1997] for $\mathop {\mathsf {NExt}}{\mathsf {K}}$.1

This paper investigates the negation-free fragment of the bi-connexive logic 2C, called 2C$_-$, from the perspective of bilateralist proof-theoretic semantics (PTS). It is argued that eliminating primitive negation has two important conceptual consequences. First, it requires a reconceptualization of contradictory logics: in a bilateralist framework, contradiction need not be understood in terms of negation inconsistency, but rather as the coexistence of proofs and refutations for certain formulas within a non-trivial system. Second, it challenges the standard definition of connexive logics, which typically rely on negation-based schemata. Instead, a rule-based conception of connexivity, grounded in bilateralist PTS, is proposed. This reconception avoids dependence on the validation of specific formula schemata and thereby also dependence on negation. The paper also addresses the issue of proof–refutation duality in the absence of strong negation, which can be formalized and recovered at a meta-level by extending the system with a two-sorted typed $\lambda $-calculus.

In previous work [4] we introduced and examined the class of betweenness algebras. In the current article we study a larger class of algebras with binary operators of possibility and sufficiency, the weak mixed algebras. Furthermore, we develop a system of logic with two binary modalities, sound and complete with respect to the class of frames closely related to the aforementioned algebras, and we prove an embedding theorem which solves an open problem from [4].

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.

Kurt Gödel proved that it is not possible to characterize intuitionistic propositional logic (${IPL}$) by means of finite and deterministic truth-tables. After extending the same result with respect to non-deterministic matrices (Nmatrices), we provide a semantical characterization of ${IPL}$ by means of a $3$-valued Nmatrix with a restricted set of valuations. This structure allows to define an algorithm to delete unsound rows from the non-deterministic truth-tables generated for each formula, which constitutes a new and very simple decision procedure for ${IPL}$. This method can be seen as truth-tables in a broader sense, and a way to overcome Gödel’s limiting result.

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf {CK}$ introduced by Brian Chellas in [3]; however, the main result of the article is that a classically equivalent reformulation of these notions (i.e., of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf {N4CK}$, introduced in [11] into $\mathsf {QN4}$, the paraconsistent variant of Nelson’s first-order logic of strong negation. Thus $\mathsf {N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf {CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf {QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which makes it possible to improve an older result [10, Proposition 7] by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf {FSK}^d$ into $\mathsf {QN4}$.

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.

This paper develops a logic of essence (HLE) in the framework of higher-order logic. The theory aims to provide a general framework for theorizing about the essences of objects, properties, propositions, and logical operations like conjunction, negation, quantification, etc. The first part of the paper presents the formal language and axiom system of HLE. After that, some theorems of the system are proved and it is shown how the logic of metaphysical necessity can be developed within the framework of HLE. The second part of the paper develops a possible worlds semantics for HLE, gives a proof of soundness, and provides examples of models that demonstrate the consistency of some simple essentialist theories.

Recent years have witnessed extensive logical studies of bundled operators. A known difficulty of such studies is how to axiomatize bundled operators. In this paper, we propose a uniform approach to axiomatizing such operators, which we call ‘the method based on almost-definability schemas’. The approach is useful in finding axioms and inference rules, and in defining a suitable canonical relation, thus in the completeness proof of logical systems. This is, hopefully, good news for modal logicians who are interested in axiomatizing bundled operators. To explicate this approach, we choose four bundled operators—the operator N of purely physical necessity, the operator $N'$ called ‘All$_1$ and Only$_2$’, the operator $N"$, and the operator $N"'$, in the literature, where $N\varphi := \Box _1\varphi \land \neg \Box _2\varphi $, $N'\varphi := \Box _1\varphi \land \Box _2\neg \varphi $, $N"\varphi := \Box _1\varphi \vee \neg \Box _2\varphi $, $N"'\varphi := \Box _1\varphi \vee \Box _2\neg \varphi $. This approach can uniformly deal with axiomatizing these bundled operators. Among other contributions, we also answer several open questions, and obtain alternative axiomatizations which are deductively equivalent to the existing ones in the literature.

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.

Normal modal logics extending the logic $\mathsf {K4.3}$ of linear transitive frames are known to lack the Craig interpolation property (CIP), except some logics of bounded depth such as $\mathsf {S5}$. We turn this ‘negative’ fact into a research question and pursue a non-uniform approach to Craig interpolation by investigating the following interpolant existence problem: decide whether there exists a Craig interpolant between two given formulas in any fixed logic above $\mathsf {K4.3}$. Using a bisimulation-based characterisation of interpolant existence for descriptive frames, we show that this problem is decidable and coNP-complete for all finitely axiomatisable normal modal logics containing $\mathsf {K4.3}$. It is thus not harder than entailment in these logics, which is in sharp contrast to other recent non-uniform interpolation results. We also extend our approach to Priorean temporal logics (with both past and future modalities) over the standard time flows—the integers, rationals, reals, and finite strict linear orders—none of which is blessed with the CIP.

Motivated by ideas from the model theory of metric structures, we introduce a metric set theory, $\mathsf {MSE}$, which takes bounded quantification as primitive and consists of a natural metric extensionality axiom (the distance between two sets is the Hausdorff distance between their extensions) and an approximate, non-deterministic form of full comprehension (for any real-valued formula $\varphi (x,y)$, tuple of parameters a, and $r < s$, there is a set containing the class and contained in the class $\{x:\varphi (x,a) < s\}$). We show that $\mathsf {MSE}$ is sufficient to develop classical mathematics after the addition of an appropriate axiom of infinity. We then construct canonical representatives of well-order types and prove that ultrametric models of $\mathsf {MSE}$ always contain externally ill-founded ordinals, conjecturing that this is true of all models. To establish several independence results and, in particular, consistency, we construct a variety of models, including pseudo-finite models and models containing arbitrarily large standard ordinals. Finally, we discuss how to formalize $\mathsf {MSE}$ in either continuous logic or Łukasiewicz logic.