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.

We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. We also discuss some related questions.

For any $2 \le n < \omega $, we introduce a forcing poset using generalized promises which adds a normal n-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindelöf, which solves an open problem due to Marun. For any $2 < n < \omega $, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal n-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.

We investigate the notion of ideal (equivalently: filter) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell _p$, $c_0$, and James’ space). Those examples lead to distinguishing and characterizing ideals (equivalently: filters) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and ideals (equivalently: filters) on the set of natural numbers.

Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type V-dominated by an independent triple is isolated over the triple. If T has NOTOP, then every model N is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah’s assertion from Chapter XII of [9] that NOTOP implies PMOP (without using NDOP).

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.

We study universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example, we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

Dedicated to the memory of Alexander Prestel (1941–2024)

For cardinals $\mathfrak {a}$ and $\mathfrak {b}$, we write $\mathfrak {a}=^\ast \mathfrak {b}$ if there are sets A and B of cardinalities $\mathfrak {a}$ and $\mathfrak {b}$, respectively, such that there are partial surjections from A onto B and from B onto A. $=^\ast $-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf {ZF}+\mathsf {DC}_\kappa $, where $\kappa $ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165–207 (1984)]. Nevertheless, we show that surjective cardinals form a “surjective cardinal algebra”, whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot \mathfrak {a}=^\ast m\cdot \mathfrak {b}$ implies $\mathfrak {a}=^\ast \mathfrak {b}$ for all cardinals $\mathfrak {a},\mathfrak {b}$ and all nonzero natural numbers m.

We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak {i}<\mathfrak {s}_{1/2}$), Con($\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con($\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of $\mathfrak {s}_{1/2}^{\infty } < $ non$(\mathcal {E})$ and cov$(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where $\mathcal {E}$ is the $\sigma $-ideal generated by closed sets of measure zero.

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 prove that P-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph _{\omega } $-many random reals to a model of CH.These results improve and correct previous theorems that can be found in the literature.

Explanations, and in particular explanations which provide the reasons why their conclusion is true, are a central object in a range of fields. On the one hand, there is a long and illustrious philosophical tradition, which starts from Aristotle, and passes through scholars such as Leibniz, Bolzano and Frege, that give pride of place to this type of explanation, and is rich with brilliant and profound intuitions. Recently, Poggiolesi [25] has formalized ideas coming from this tradition using logical tools of proof theory. On the other hand, recent work has focused on Boolean circuits that compile some common machine learning classifiers and have the same input-output behavior. In this framework, Darwiche and Hirth [7] have proposed a theory for unveiling the reasons behind the decisions made by Boolean classifiers, and they have studied their theoretical implications. In this paper, we uncover the deep links behind these two trends, demonstrating that the proof-theoretic tools introduced by Poggiolesi provide reasons for decisions, in the sense of Darwiche and Hirth [7]. We discuss the conceptual as well as the technical significance of this result.

We show, assuming PD, that every complete finitely axiomatized second-order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second-order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g., supercompact) cardinals does not imply the categoricity of all finitely axiomatizable complete second-order theories. More exactly, we show that a non-categorical complete finitely axiomatized second-order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second-order theories with a model of a certain singular cardinality $\kappa $ of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming $V=L$, that every complete finitely axiomatized second-order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second-order theory with a countable model which is non-categorical.

Let $\mathsf {KP}$ denote Kripke–Platek Set Theory and let $\mathsf {M}$ be the weak set theory obtained from $\mathsf {ZF}$ by removing the collection scheme, restricting separation to $\Delta _0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf {TCo}$). A result due to Kaufmann [9] shows that every countable model, $\mathcal {M}$, of $\mathsf {KP}+\Pi _n\textsf {-Collection}$ has a proper $\Sigma _{n+1}$-elementary end extension. We show that for all $n \geq 1$, there exists an $L_\alpha $ (where $L_\alpha $ is the $\alpha ^{\textrm {th}}$ approximation of the constructible universe L) that satisfies $\textsf {Separation}$, $\textsf {Powerset}$ and $\Pi _n\textsf {-Collection}$, but that has no $\Sigma _{n+1}$-elementary end extension satisfying either $\Pi _n\textsf {-Collection}$ or $\Pi _{n+3}\textsf {-Foundation}$. Thus showing that there are limits to the amount of the theory of $\mathcal {M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann’s theorem. Using admissible covers and the Barwise Compactness theorem, we show that if $\mathcal {M}$ is a countable model $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ and T is a recursive theory that holds in $\mathcal {M}$, then there exists a proper $\Sigma _n$-elementary end extension of $\mathcal {M}$ that satisfies T. We use this result to show that the theory $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Separation}$.

We expand the study of generic stability in three different directions. Generic stability is best understood as a property of types in $NIP$ theories in classical logic. In this article, we make attempts to generalize our understanding to Keisler measures instead of types, arbitrary theories instead of $NIP$ theories, and continuous logic instead of classical logic. For this purpose, we study randomization of first-order structures/theories and modes of convergence of types/measures.

We prove that every $\Sigma ^0_2$ Gale-Stewart game can be won via a winning strategy $\tau $ which is $\Delta _1$-definable over $L_{\delta }$, the $\delta $th stage of Gödel’s constructible universe, where $\delta = \delta _{\sigma ^1_1}$, strengthening a theorem of Solovay from the 1970s. Moreover, the bound is sharp in the sense that there is a $\Sigma ^0_2$ game with no strategy $\tau $ which is witnessed to be winning by an element of $L_{\delta }$.

We prove a rigidity result for maps between Čech–Stone remainders under fairly mild forcing axioms.

Let $\mathcal {M}=(M,<,+, \dots )$ be a weakly o-minimal non-valuational structure expanding an ordered group. We show that the full first-order theory $\operatorname {\mathrm {Th}}(\mathcal {M})$ has definable Skolem functions if and only if isolated types in $S_{n}^{\mathcal M}(A)$ are dense for each $ A\subseteq M $ and $ n\in \mathbb {N} $. Using this, we prove that no strictly weakly o-minimal non-valuational expansion of an ordered group has definable Skolem functions, thereby answering Conjecture 1.7 of Eleftheriou et al. (On definable Skolem functions in weakly o-minimal non-valuational structures. J. Symb. Logic, vol. 82 (2017), no. 4).

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.