Reverse mathematics is primarily interested in what set existence axioms are necessary and sufficient in a proof of a theorem. Much work has been done in classifying graph coloring theorems, studying k-regular graphs, k-chromatic graphs, and forests. This article takes inspiration from an old paper by Bean and studies graph coloring theorems restricted to planar graphs. Schmerl showed that these coloring theorems are all equivalent to ${{\mathtt {WKL}}}$ over ${\mathtt {RCA}_{0}}$. In this article, we utilise Weihrauch reducibility to provide a deeper analysis on the uniformity of these implications. Whilst the proofs provided by Schmerl are not obviously non-uniform, we show that in many instances, non-uniformity is indeed necessary.

This paper studies the definability of natural language generalized quantifiers. The semantics of generalized quantifiers are provided by a collection of subsets of the underlying domain. However, the generalized quantifiers appearing in natural language are definable either by first-order quantification or by cardinality notions. This paper provides an explanation for this observed phenomenon. The explanation is that the famous constraints of domain independence and conservativity, when extended to Henkin models, suffice to ensure low-level definability, namely $\Delta ^1_1$-definability or at least $\Sigma ^1_1$-definability; and in most cases this definability can be made to be bounded. This is basically a consequence of Feferman’s Preservation Theorem [15], which Marker [46] has provided a short model-theoretic proof of. Further, we verify that the paradigmatic cardinality quantifiers are indeed $\Delta ^1_1$-definable for a reasonable choice of background theory. Finally, in many other cases, we show that this definability can be lowered to first-order definability.

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 [10], we provided a method for eliminating cuts in non-wellfounded proofs with a local-progress condition, these being the simplest kind of non-wellfounded proofs. The method consisted of splitting the proof into nicely behaved fragments. This article extends our method to proofs based on simple trace conditions. The main idea is to split the system with the trace condition into infinitely many local-progress calculi that together are equivalent to the original trace-based system. This provides a cut-elimination method using only basic tools of structural proof theory and corecursion, which is needed due to working in a non-wellfounded setting. We will employ our method to obtain syntactic cut elimination for K+, a system of modal logic with the master modality.

We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of such a geometry determines the geometry up to definitional equivalence; moreover, if we are given two such geometries $\mathcal {G}$ and $\mathcal {G}'$, then the concepts (explicitly definable relations) of $\mathcal {G}$ are concepts of $\mathcal {G}'$ exactly if the automorphisms of $\mathcal {G}'$ are automorphisms of $\mathcal {G}$. We show this by first proving that a relation is a concept of $\mathcal {G}$ exactly if it is closed under the automorphisms of $\mathcal {G}$ and is definable over the field; moreover, it is enough to consider automorphisms that are affine transformations.

We show how this result can be applied to quickly determine relationships and differences between various geometries and spacetimes, including ordered affine, Euclidean, Galilean, Newtonian, Late Classical, Relativistic and Minkowski spacetimes (we first define these spacetimes and geometries using a Tarskian first-order language centred on the ternary relation ${\mathsf {Bw}}$ of betweenness). We conclude with a selection of open problems related to the existence of certain intermediate geometries.

Kripke’s fixed-point theory of truth relies essentially on the monotonicity of valuation schemes in order to guarantee the existence of fixed points for the truth predicate. Under non-monotonic valuation schemes, Kripke’s jump may fail to generate a monotonic sequence, and fixed points need not exist. Inspired by Leitgeb’s dependence-based reconstruction of fixed points in classical two-valued semantics, this paper investigates whether analogous constructions are possible in three-valued semantics, including under non-monotonic valuation schemes. To this end, we generalize Leitgeb’s semantic dependence relation to three-valued semantics using partial predicates. On this basis, we define a dependence jump operation and a conditional dependence jump, and show how these operations induce monotonic sequences for the truth predicate under arbitrary three-valued valuation schemes. We compare the least fixed points generated by these jumps with those obtained from Kripke’s original jump across a range of standard valuation schemes, including Kleene, supervaluational, Łukasiewicz, and Gödel schemes. Finally, we show that the resulting notion of semantic dependence yields a uniform formal characterization of self-referentiality in three-valued semantics, thereby extending well-known two-valued results to Kripkean truth theories.

We study the reverse mathematics of characterization theorems of regular countable second countable (CSC) spaces. We prove that arithmetic comprehension is equivalent over $\mathbf {RCA}_0$ to every $T_3$ CSC space being metrizable, and we characterize the $T_3$ spaces which are metrizable over $\mathbf {RCA}_0$. We show that Lynn’s theorem for CSC spaces can be carried out in $\mathbf {ACA}_0$, namely that every zero-dimensional separable space is homeomorphic to the order topology of a linear order. We also show that arithmetic comprehension is equivalent to every $T_2$ compact CSC space being well-orderable. From general topology, we know that the locally compact $T_2$ CSC spaces are the well-orderable CSC spaces, and that the $T_3$ scattered CSC spaces are the completely metrizable CSC spaces. We show that these characterizations and a few others are equivalent to arithmetic transfinite recursion over $\mathbf {RCA}_0$. We also find a few statements that are equivalent to $\Pi ^1_1$ comprehension. In particular we show that every $T_3$ CSC space has a Cantor–Bendixson rank and that every $T_3$ CSC space is the disjoint union of a scattered space and dense in itself space are equivalent to $\Pi ^1_1$ comprehension over $\mathbf {RCA}_0$.

Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this article we study classical bounding results for order dimension within the framework of reverse mathematics. We focus on principles asserting that the dimension of a poset can be bounded in terms of the dimension of subposets obtained by removing chains or points, denoted by $\mathsf {DBi_{n}}$, $\mathsf {DBc_{n}}$, and $\mathsf {DB_p}$. We prove that, over $\mathsf {RCA}_0$, both $\mathsf {DBi_{n}}$ and $\mathsf {DBc_{n}}$ are equivalent to $\mathsf {WKL}_0$. To analyze $\mathsf {DB_p}$, we introduce a natural strengthening $\mathsf {DB^+_p}$ and show that both $\mathsf {DB_p}$ and $\mathsf {DB^+_p}$ are provable from $\mathsf {WKL}_0$ and from $\mathsf {I}\boldsymbol \Sigma ^{0}_{2}$, while $\mathsf {B} \boldsymbol \Sigma ^{0}_{2}$ does not suffice to prove $\mathsf {DB^+_p}$. The latter result is obtained by showing that the statement “$\mathsf {DB^+_p}$ is computably true” is equivalent to $\mathsf {I}\boldsymbol \Sigma ^{0}_{2}$.

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.