We prove that the satisfaction relation $\mathcal {N}\models \varphi [\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\mathcal {N}$ and the formulas $\varphi $ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\left \langle {\mathbb N},{+},{\cdot },0,1, <\right \rangle $, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have the same $\left \langle {H}_{\omega _2},{\in }\right \rangle $ or the same rank-initial segment $\left \langle {V}_\delta ,{\in }\right \rangle $, yet disagree on which assertions are true in these structures.

On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure ${\mathbb N}=\{\,{0,1,2,\ldots }\,\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

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.

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.

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.

The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality—i.e., the number of urelements—interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal $\kappa $, there is no set of urelements of size $\kappa $. Building on recent work by Hamkins [10], we show that, under certain additional axioms, Basic Law V holds if and only if the urelements form a set. We construct models of urelement set theory in which the Reflection Principle holds while Hume’s Principle fails for sets. Additionally, assuming the consistency of an inaccessible cardinal, we produce a model of Kelley–Morse class theory with urelements that has a global well-ordering but lacks a definable map satisfying Hume’s Principle for classes.

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.

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.

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.

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 $\mathbf{KR}$.

In a recent proof mining application, the proof-theoretical analysis of Dykstra’s cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector’s bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining.

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 $.

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.

Some informal arguments are valid, others are invalid. A core application of logic is to tell us which is which by capturing these validity facts. Philosophers and logicians have explored how well a host of logics carry out this role, familiar examples being propositional, first-order and second-order logic. Since natural language and standard logics are countable, a natural question arises: is there a countable logic guaranteed to capture the validity patterns of any language fragment? That is, is there a countable omega-universal logic? Our article philosophically motivates this question, makes it precise, and then answers it. It is a self-contained, concise sequel to ‘Capturing Consequence’ by A.C. Paseau (RSL vol. 12, 2019).

The generic multiverse was introduced in [74] and [81] to explicate the portion of mathematics which is immune to our independence techniques. It consists, roughly speaking, of all universes of sets obtainable from a given universe by forcing extension. Usuba recently showed that the generic multiverse contains a unique definable universe, assuming strong large cardinal hypotheses. On the basis of this theorem, a non-pluralist about set theory could dismiss the generic multiverse as irrelevant to what set theory is really about, namely that unique definable universe. Whatever one’s attitude towards the generic multiverse, we argue that certain impure proofs ensure its ongoing relevance to the foundations of set theory. The proofs use forcing-fragile theories and absoluteness to prove ${\mathrm {ZFC}}$ theorems about simple “concrete” objects.