This paper offers a substantial improvement in the revision-theoretic approach to conditionals in theories of transparent truth. The main modifications are (i) a new limit rule; (ii) a modification of the extension to the continuum-valued case; and (iii) the suggestion of a variation on how universal quantification is handled, leading to more satisfactory laws of restricted quantification.

Two salient notions of sameness of theories are synonymy, aka definitional equivalence, and bi-interpretability. Of these two definitional equivalence is the strictest notion. In which cases can we infer synonymy from bi-interpretability? We study this question for the case of sequential theories. Our result is as follows. Suppose that two sequential theories are bi-interpretable and that the interpretations involved in the bi-interpretation are one-dimensional and identity preserving. Then, the theories are synonymous.

The crucial ingredient of our proof is a version of the Schröder–Bernstein theorem under very weak conditions. We think this last result has some independent interest.

We provide an example to show that this result is optimal. There are two finitely axiomatized sequential theories that are bi-interpretable but not synonymous, where precisely one of the interpretations involved in the bi-interpretation is not identity preserving.

Can we quantify over absolutely every set? Absolutists typically affirm, while relativists typically deny, the possibility of unrestricted quantification (in set theory). In the first part of this article, I develop a novel and intermediate philosophical position in the absolutism versus relativism debate in set theory. In a nutshell, the idea is that problematic sentences related to paradoxes cannot be interpreted with unrestricted quantifier domains, while prima facie absolutist sentences (e.g., “no set is contained in the empty set”) are unproblematic in this respect and can be interpreted over a domain containing all sets. In the second part of the paper, I develop a semantic theory that can implement the intermediate position. The resulting framework allows us to distinguish between inherently absolutist and inherently relativist sentences of the language of set theory.

We investigate a system of modal semantics in which $\Box \phi $ is true if and only if $\phi $ is entailed by a designated set of formulas by a designated logics. We prove some strong completeness results as well as a natural connection to normal modal logics via an application of some lattice-theoretic fixpoint theorems. We raise a difficult problem that arises naturally in this setting about logics which are identical with their own ‘meta-logic’, and draw a surprising connection to recent work by Andrew Bacon and Kit Fine on McKinsey’s substitutional modal semantics.

We study the Lyndon interpolation property (LIP) and the uniform LIP (ULIP) for extensions of $\mathbf {S4}$ and intermediate propositional logics. We prove that among the 18 consistent normal modal logics of finite height extending $\mathbf {S4}$ known to have CIP, 11 logics have LIP and 7 logics do not. We also prove that for intermediate propositional logics, the Craig interpolation property, LIP, and ULIP are equivalent.

The Kruskal–Friedman theorem asserts: in any infinite sequence of finite trees with ordinal labels, some tree can be embedded into a later one, by an embedding that respects a certain gap condition. This strengthening of the original Kruskal theorem has been proved by I. Kříž (Ann. Math. 1989), in confirmation of a conjecture due to H. Friedman, who had established the result for finitely many labels. It provides one of the strongest mathematical examples for the independence phenomenon from Gödel’s theorems. The gap condition is particularly relevant due to its connection with the graph minor theorem of N. Robertson and P. Seymour. In the present article, we consider a uniform version of the Kruskal–Friedman theorem, which extends the result from trees to general recursive data types. Our main theorem shows that this uniform version is equivalent both to $\Pi ^1_1$-transfinite recursion and to a minimal bad sequence principle of Kříž, over the base theory $\mathsf {RCA_0}$ from reverse mathematics. This sheds new light on the role of infinity in finite combinatorics.

An original family of labelled sequent calculi $\mathsf {G3IL}^{\star }$ for classical interpretability logics is presented, modularly designed on the basis of Verbrugge semantics (a.k.a. generalised Veltman semantics) for those logics. We prove that each of our calculi enjoys excellent structural properties, namely, admissibility of weakening, contraction and, more relevantly, cut. A complexity measure of the cut is defined by extending the notion of range previously introduced by Negri w.r.t. a labelled sequent calculus for Gödel–Löb provability logic, and a cut-elimination algorithm is discussed in detail. To our knowledge, this is the most extensive and structurally well-behaving class of analytic proof systems for modal logics of interpretability currently available in the literature.

This paper shows how to set up Fine’s “theory-application” type semantics so as to model the use-unrestricted “Official” consequence relation for a range of relevant logics. The frame condition matching the axiom $(((A \to A) \land (B \to B)) \to C) \to C$—the characteristic axiom of the very first axiomatization of the relevant logic E—is shown forth. It is also shown how to model propositional constants within the semantic framework. Whereas the related Routley–Meyer type frame semantics fails to be strongly complete with regards to certain contractionless logics such as B, the current paper shows that Fine’s weak soundness and completeness result can be extended to a strong one also for logics like B.

The family of finite subsets s of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $\omega $-large sets in Logic. We formulate and prove the generalizations of Friedman’s Free Set and Thin Set theorems and of Rainbow Ramsey’s theorem to colorings of the Schreier barrier. We analyze the strength of these theorems from the point of view of Computability Theory and Reverse Mathematics. Surprisingly, the exactly $\omega $-large counterparts of the Thin Set and Free Set theorems can code $\emptyset ^{(\omega )}$, while the exactly $\omega $-large Rainbow Ramsey theorem does not code the halting set.

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties—requiring that in valid entailments $A\to B$, some atom must appear in both A and B with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf {BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf {B}$) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan’s notion of hyperformalism, and for the very definition of a relevant logic itself.

In previous publications, it was shown that finite non-deterministic matrices are quite powerful in providing semantics for a large class of normal and non-normal modal logics. However, some modal logics, such as those whose axiom systems contained the Löb axiom or the McKinsey formula, were not analyzed via non-deterministic semantics. Furthermore, other modal rules than the rule of necessitation were not yet characterized in the framework.

In this paper, we will overcome this shortcoming and present a novel approach for constructing semantics for normal and non-normal modal logics that is based on restricted non-deterministic matrices. This approach not only offers a uniform semantical framework for modal logics, while keeping the interpretation of the involved modal operators the same, and thus making different systems of modal logic comparable. It might also lead to a new understanding of the concept of modality.

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 $\mathbf{KR}$.

We present a family of minimal modal logics (namely, modal logics based on minimal propositional logic) corresponding each to a different classical modal logic. The minimal modal logics are defined based on their classical counterparts in two distinct ways: (1) via embedding into fusions of classical modal logics through a natural extension of the Gödel–Johansson translation of minimal logic into modal logic S4; (2) via extension to modal logics of the multi- vs. single-succedent correspondence of sequent calculi for classical and minimal logic. We show that, despite being mutually independent, the two methods turn out to be equivalent for a wide class of modal systems. Moreover, we compare the resulting minimal version of K with the constructive modal logic CK studied in the literature, displaying tight relations among the two systems. Based on these relations, we also define a constructive correspondent for each minimal system, thus obtaining a family of constructive modal logics which includes CK as well as other constructive modal logics studied in the literature.

The distinction of the semantic spaces of elements and types is common practice in practically all type systems. A few type systems, including some early ones, have been proposed whose semantic space has functions only, i.e., depending on the context functions may play element roles as well as type roles. All of these systems are either lacking expressive power, in particular, polymorphism, or they violate uniqueness of types. This work presents for the first time a function-based type system in which typing is a relation between functions and which is using an ordering of functions to introduce bounded polymorphism. The ordering is based on an infinite set of top objects, itself strictly linearly ordered, each of which characterizes a certain function space. These top objects are predicative in the sense that a function using some top object cannot be smaller than this object. The interpretation of proposition as types and elements as proofs remains valid and is extended by viewing the ordering between types as logical implication. The proposed system can be shown to satisfy confluence and subject reduction. Furthermore one can show that the ordering is a partial order, every set of expressions has a maximal element, and there is a (unique) minimal, logically strongest, type among all types of an element. The latter result implies an alternative notion of uniqueness of types. Strong normalisation is the deepest property and its proof is based on a well-founded relation defined over a subsystem of expressions without eliminators. Semantic abstraction of the objects involved in typing, i.e., to use functions in element as well as type roles in a relational setting, is the major contribution of function-based type systems. This work shows that dependent products are not necessary for defining type systems with bounded polymorphism, rather it presents a consistent system with bounded polymorphism and minimal types where typing is a relation between partially ordered functions.

In this paper we study logical bilateralism understood as a theory of two primitive derivability relations, namely provability and refutability, in a language devoid of a primitive strong negation and without a falsum constant, $\bot $, and a verum constant, $\top $. There is thus no negation that toggles between provability and refutability, and there are no primitive constants that are used to define an “implies falsity” negation and a “co-implies truth” co-negation. This reduction of expressive power notwithstanding, there remains some interaction between provability and refutability due to the presence of (i) a conditional and the refutability condition of conditionals and (ii) a co-implication and the provability condition of co-implications. Moreover, assuming a hyperconnexive understanding of refuting conditionals and a dual understanding of proving co-implications, neither non-trivial negation inconsistency nor hyperconnexivity is lost for unary negation connectives definable by means of certain surrogates of falsum and verum. Whilst a critical attitude towards $\bot $ and $\top $ can be justified by problematic aspects of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations for these constants, the aim to reduce the availability of a toggling negation and observations on undefinability may also give further reasons to abandon $\bot $ and $\top $.

The notion of global supervenience captures the idea that the overall distribution of certain properties in the world is fixed by the overall distribution of certain other properties. A formal implementation of this idea in constant-domain Kripke models is as follows: predicates $Q_1,\dots ,Q_m$ globally supervene on predicates $P_1,\dots ,P_n$ in world w if two successors of w cannot differ with respect to the extensions of the $Q_i$ without also differing with respect to the extensions of the $P_i$. Equivalently: relative to the successors of w, the extensions of the $Q_i$ are functionally determined by the extensions of the $P_i$. In this paper, we study this notion of global supervenience, achieving three things. First, we prove that claims of global supervenience cannot be expressed in standard modal predicate logic. Second, we prove that they can be expressed naturally in an inquisitive extension of modal predicate logic, where they are captured as strict conditionals involving questions; as we show, this also sheds light on the logical features of global supervenience, which are tightly related to the logical properties of strict conditionals and questions. Third, by making crucial use of the notion of coherence, we prove that the relevant system of inquisitive modal logic is compact and has a recursively enumerable set of validities; these properties are non-trivial, since in this logic a strict conditional expresses a second-order quantification over sets of successors.

This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional discrete formulations are reliant on Hölder’s representation theorem for ordered Archimedean groups. While this is provable in $\mathsf {RCA}_0$, we also consider two forms of the sorites which rest on non-constructive principles: the continuous sorites of Weber & Colyvan [35] and a variant we refer to as the covering sorites. We show in the setting of second-order arithmetic that the former depends on the existence of suprema and thus on arithmetical comprehension ($\mathsf {ACA}_0$) while the latter depends on the Heine–Borel theorem and thus on Weak König’s lemma ($\mathsf {WKL}_0$). We finally illustrate how recursive counterexamples to these principles provide resolutions to the corresponding paradoxes which can be contrasted with supervaluationist, epistemicist, and constructivist approaches.

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.

We say that a Kripke model is a GL-model (Gödel and Löb model) if the accessibility relation $\prec $ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots $, and $t_\omega $ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_\omega $. A non-normal modal logic $\mathbf {D}$, which was studied by Beklemishev [3], is characterized as follows. A formula $\varphi $ is a theorem of $\mathbf {D}$ if and only if $\varphi $ is true at $t_\omega $ in any D-model. $\mathbf {D}$ is an intermediate logic between the provability logics $\mathbf {GL}$ and $\mathbf {S}$. A Hilbert-style proof system for $\mathbf {D}$ is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for $\mathbf {D}$, and show the cut-elimination theorem. We also introduce new Hilbert-style systems for $\mathbf {D}$ by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $\omega $. Finally, we show a general result as follows. Let X and $X^+$ be arbitrary modal logics. If the relationship between semantics of X and semantics of $X^+$ is equal to that of $\mathbf {GL}$ and $\mathbf {D}$, then $X^+$ can be axiomatized based on X in the same way as the new axiomatization of $\mathbf {D}$ based on $\mathbf {GL}$.

The class of all $\ast $-continuous Kleene algebras, whose description includes an infinitary condition on the iteration operator, plays an important role in computer science. The complexity of reasoning in such algebras—ranging from the equational theory to the Horn one, with restricted fragments of the latter in between—was analyzed by Kozen (2002). This paper deals with similar problems for $\ast $-continuous residuated Kleene lattices, also called $\ast $-continuous action lattices, where the product operation is augmented by residuals. We prove that, in the presence of residuals, the fragment of the corresponding Horn theory with $\ast $-free hypotheses has the same complexity as the $\omega ^\omega $ iteration of the halting problem, and hence is properly hyperarithmetical. We also prove that if only commutativity conditions are allowed as hypotheses, then the complexity drops down to $\Pi ^0_1$ (i.e., the complement of the halting problem), which is the same as that for $\ast $-continuous Kleene algebras. In fact, we get stronger upper bound results: the fragments under consideration are translated into suitable fragments of infinitary action logic with exponentiation, and our upper bounds are obtained for the latter ones.