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.

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, $C^*$-algebras and the measure algebra of an infinite compact group.

Many-valued logics, in general, and real-valued logics, in particular, usually focus on a notion of consequence based on preservation of full truth, typically represented by the value $1$ in the semantics given in the real unit interval $[0,1]$. In a recent paper [Foundations of Reasoning with Uncertainty via Real-valued Logics, Proceedings of the National Academy of Sciences 121(21): e2309905121, 2024], Ronald Fagin, Ryan Riegel, and Alexander Gray have introduced a new paradigm that allows to deal with inferences in propositional real-valued logics based on a rich class of sentences, multi-dimensional sentences, that talk about combinations of any possible truth values of real-valued formulas. They have proved a strong completeness result that allows one to derive exactly what information can be inferred about the combinations of truth values of a collection of formulas given information about the combinations of truth values of a finite number of other collections of formulas. In this paper, we extend that work to the first-order (as well as modal) logic of multi-dimensional sentences. We give a parameterized axiomatic system that covers any reasonable logic and prove a corresponding completeness theorem, first assuming that the structures are defined over a fixed domain, and later for the logics of varying domains. As a by-product, we also obtain a zero-one law for finitely-valued versions of these logics. Since several first-order real-valued logics are known not to have recursive axiomatizations but only infinitary ones, our system is by force akin to infinitary systems.

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.

In this paper, we show that $\mathsf {ZFC}+\mathsf {WA}_{n+1}$ implies the consistency of $\mathsf {ZFC}+\mathsf {WA}_n$ for $n\ge 0$. We also prove that $\mathsf {ZFC}+\mathsf {WA}_n$ is finitely axiomatizable, and $\mathsf {ZFC}+\mathsf {WA}$ is not finitely axiomatizable unless it is inconsistent.

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.

We study the computational problem of rigorously describing the asymptotic behavior of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and we prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constraints and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non-computable.

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).

Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant subsets of the Cantor cube in the Sacks model.

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.

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.

It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set X of finitely many Cohen reals, by showing that, in the forcing extension, the transcendence degree of the reals over a combination of the reals in the extension given by each proper subset of X is also maximal. This answers a question of Kanovei and Schindler [2].

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in \mathbb {N}$ and any countable model of $\mathrm {B}\Sigma _{n+2}$, we construct a proper $\Sigma _{n+2}$-elementary end extension satisfying $\mathrm {B}\Sigma _{n+1}$, which answers a question by Clote positively. We also give a characterization of the countable models of $\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of $\mathrm {B}\Sigma _{n+2}$. Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.

A Constructive Logic of Evidence and Truth ($\mathsf {LET_C}$) is introduced. This logic is both paraconsistent and paracomplete, providing connectives for consistency and determinedness that enable the independent recovery of explosiveness and the law of excluded middle for specific propositions. Dual connectives for inconsistency and undeterminedness are also defined in $\mathsf {LET_C}$. Evidence is explicitly formalised by integrating lambda calculus terms into $\mathsf {LET_C}$, resulting in the type system $\mathsf {LET_C^{\lambda }}$. In this system, lambda calculus terms represent procedures for constructing evidence for compound formulas based on the evidence of their constituent parts. A realisability interpretation is provided for $\mathsf {LET_C}$, establishing a strong connection between deductions in this system and recursive functions.

There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include $\{\mathord {\downarrow }, \mathord {@}\}$ with or without nominals (Areces, Blackburn, Marx), $\mathord {@}$ with or without nominals (ten Cate), and $\mathord {\downarrow }$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation theorem for all standard fragments of hybrid logic, in particular for the case with $\mathord {\downarrow }$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense.

We investigate Steel’s conjecture in ‘The Core Model IterabilityProblem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$-iterable, $1$-small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$, if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$. We will show that the conjecture fails, assuming that thereis an iterable premouse M which models KP and which has a-Woodin cardinal. On the other hand, we show thatassuming there is no transitive model of KP with a Woodin cardinal theconjecture holds. In the course of this we will also show that ifM is a premouse which models KP with a largest, regular,uncountable cardinal $\delta $, and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$, and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$. Additionally, we study the preservation of admissibilityunder iteration maps. At last, we will prove a fact about the closure of the setof ordinals at which a weasel has the S-hull property. Thisanswers another question implicit in remarks in [10].

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we extend this result to $\kappa $-prime models, for $\kappa $ an uncountable cardinal or $\aleph _\varepsilon $.

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.

We argue that some of Brouwer’s assumptions, rejected by Bishop, should be considered and studied as possible axioms. We show that Brouwer’s Continuity Principle enables one to prove an intuitionistic Borel Hierarchy Theorem. We also explain that Brouwer’s Fan Theorem is useful for a development of the theory of measure and integral different from the one worked out by Bishop. We show that Brouwer’s bar theorem not only proves the Fan Theorem but also a stronger statement that we call the Almost-fan Theorem. The Almost-fan Theorem implies intuitionistic versions of Ramsey’s Theorem and the Bolzano-Weierstrass Theorem.