This paper draws attention to key logical relations between two distinct and independently intelligible notions, compatibility and implication. We identify a set of inference rules governing compatibility and show how crucial principles concerning conditionals can be derived from these rules, on the assumption that implication is definable in terms of compatibility. As will emerge — and this is the main point of the paper — the logical relations considered are to a large extent neutral as to the underlying logic, as they hold in a wide range of non-classical systems.
If Socrates is running slowly, he must be running. Likewise, if he’s not running, he must not be running slowly. Haze [2] proposes an extension of first-order logic inspired by this characteristic inferential behaviour of words like ‘slowly’: FOL-SA (first-order logic with scoped adverbs). Haze presents the logic model-theoretically using a hierarchy of models, where the level of a model corresponds to the number of nestings of adverb formulas within adverb formulas, and leaves open the investigation of its proof theory. In this paper we develop a semantic tableaux proof system for FOL-SA, prove its soundness and completeness, and outline some directions for further research.
The object language of Kripke’s 1975 semantic theory of truth, based on the Strong Kleene valuation scheme, cannot contain a predicate that expresses the notion "ungroundedness" that Kripke provides an analysis of. This is unfortunate; it means that, in the object language of Kripke’s theory, there is no obvious way to express Kripke’s diagnostic insight about what causes semantic pathology. This paper shows how to introduce a “groundedness” predicate, G, to a Kripkean theory of truth that can fill this expressive gap. In the fixed-point construction that gives an interpretation for G, G’s anti-extension tracks networks of sentences that, due to predications of truth, result in non-terminating graphs of semantic dependence. In the fixed-point models that provide a class of intended interpretations for G: (i) every sentence with a classical semantic value is in the extension of G, (ii) every sentence in the anti-extension of G receives the value 1/2 , and (iii) the anti-extension of G includes all the sentences that receive 1/2 in the corresponding model of Kripke’s original theory. A language augmented with predicates for truth and groundedness possesses sufficient expressive resources to articulate Kripke’s diagnostic insight as it applies to itself.
This paper investigates Bourbaki’s concepts of set-theoretical structure and isomorphism. First, I introduce Bourbaki’s formalism in a modern guise. Second, I show that there is a one-to-one correspondence (up to logical equivalence) between theories consisting of finitely many axioms expressed in some formal (first-order or higher-order) language and t-species of set-theoretical structure—that is, species of structure S which are transportable in the sense that the extension of the predicate “being a structure of species S” is closed under isomorphisms. Third, I examine the significance of the formal definitions and results presented in this paper for the philosophy of mathematics and the philosophy of the empirical sciences. In particular, I argue that: (1) since definitions of predicates “being a structure of species S” are conservative and eliminable relative to ZFC, such predicates can be used to achieve a reduction of (large parts of) mathematics to ZFC; (2) since t-species of structure are transportable, they can provide a basis for a set-theoretical form of structuralism by enabling us to “forget” those aspects of structures that are not isomorphism-invariant; and (3) the correspondence between t-species of structure and formal axiomatic theories shows that there is no deep divide between the so-called semantic and syntactic approaches in the philosophy of science.
This paper provides a formal characterization of the phenomenon of ‘semantic pollution’ relative to proof systems for modal logic and Kripke semantics. We propose that semantic pollution can be made precise by several properties of syntax occurring in proof systems. First, our base requirement for semantic pollution is given by the property of violating invariance results under Kripke model equivalences. On top of that, the distinction between local and global syntax, and between syntax that is dependent on and independent from the propositional valuation, induce four levels of semantic pollution: weak pollution, global pollution, local pollution and strong pollution. We analyze several main proof systems for (extensions of) modal logic in terms of these levels of pollution: the display calculus, the hybrid calculus and the labeled calculus. The results show that the display calculus is only weakly semantically polluted, while the hybrid calculus has formulas introducing several types of pollution. The only calculus that is strongly semantically polluted is the labeled calculus. These formal results are in line with general intuitions about semantic pollution. Additionally, our formal framework for semantic pollution is suitable for applications to other logics and semantics. Finally, we comment on the relation of our framework to the philosophical debate surrounding semantic pollution.
The thesis underlying this paper is that informal rigour – a term famously coined by Kreisel in [63] – corresponds to a more definite method than has previously been recognized. We will illustrate this by first presenting a scheme for what we will refer to as an informally rigorous argument and then showing in detail how the three central examples considered by Kreisel [64] conform to this. These respectively pertain to the analysis of first-order validity, the status Markov’s Principle in intuitionistic analysis, and the definiteness of the Continuum Hypothesis. Upon comparing these arguments systematically, we will suggest that the latter is the most tendentious in virtue of the status which Kreisel himself came to assign to the notion of second-order validity. We conclude by providing some brief observations about the scope and continued significance of Kreisel’s method.
This is a transcript of a conversation held between Professor W. Hugh Woodin and the participants of the XVIII Workshop in Set Theory at CIRM, Luminy, France on November 5th, 2025. The moderators were Dr. Corey Bacal Switzer and Professor Jouko Väänänen.
This article provides a comprehensive logical study of the metaethical principle referred to as “ought implies can” (OiC, for short). We analyze and formalize ten OiC interpretations from the philosophical literature and develop a class of sound and strongly complete deontic agency logics axiomatizing these ten principles. To do so, we adopt a non-normal modal logic approach to the agency formalism of ‘seeing to it that,’ that is, logic. The resulting logics are employed to demonstrate a formal taxonomy of OiC in , logically determining the (in)dependencies between the various interpretations. Additionally, we extend the class of logics with four other prominent metaethical principles discussed in the literature, analyzing their relation to OiC. These include the principles of “no vacuous commands,” “deontic contingency,” “deontic consistency,” and “no deontic dilemmas.” Last, to address a common objection, we extend the developed class of logics with restricted forms of monotonicity and aggregation that take into account the variety of OiC. Through this, we restore and enhance some of the inferential power lost in adopting a non-normal modal approach.
I present a counter-example to Nash’s derivation of utility theory in ‘The Bargaining Problem’.
This paper develops a unified formal framework for understanding open texture. Building on Kit Fine’s specification space approach, I formalise three distinct accounts: open texture as perpetual openness, as freedom to diverge rationally, and as defeasibility. I show that these formalisations can come apart, while constraints on admissible precisifications create entailment relations between them. I then illustrate the framework’s utility by applying it to Horty’s Reason Model of legal precedent. Finally, the framework highlights a principled difference between open texture and vagueness: whilst supervaluationist approaches to vagueness assume that vague predicates admit complete precisifications that preserve penumbral connections, predicates exhibiting perpetual openness preserve penumbral connections only in limited cases.
Our usual explication of a logic through an axiomatic calculus is structurally foundationalist. Justification gets transferred by applying rules finitely many times from already justified sentences to new sentences in a linear-like fashion, by starting from a base of non-to-be justified sentences (i.e. the axioms). In contrast, this paper develops a coherentist approach to classical propositional logic, by explaining the fundamental notion of deducibility in terms of a primitive notion of logical coherence. This is done by introducing a calculus consisting of rules that capture the properties of classical consistency. The basic logical reasoning process explicated is that of constructing consistent sets of formulas. I show how to define classical deducibility in terms of the primitive relation of logical coherence. As applications, I use the coherence calculus to provide a non-semantic proof of the consistency of an axiomatic calculus for classical propositional logic, I show how the system avoids an impossibility result for semantic resemblance proven by Leitgeb, I show that the strategy of defining deducibility from coherence does not work for intuitionistic logic and I briefly compare my approach to the Simple Theory of Propositions put forward by Stalnaker and Fritz.
We further investigate the metalogical properties of the ordered fragment. First, we provide a simplified proof of the satisfiability invariance under A. Herzig’s translation of the ordered fragment into modal logic KD . Second, based on the notion of bisimulation developed by B. Bednarczyk and R. Jaakkola, we show that each ordered formula is equivalent to a disjunction of ‘ordered types’. Third, we show that the fragment enjoys uniform interpolation, and that uniform interpolants can be effectively constructed from ‘ordered types’. Finally, we establish the Łoś-Tarski Preservation Theorem for the fragment, and therefore conclude that the ordered fragment is nice.
Weber argues in Paradoxes and Inconsistent Mathematics (2021) for a substructural foundation of mathematics. The mathematical theories investigated by Weber are all axiomatized using the paraconsistent logic subDLQ. This paper shows how to modify the logic so as to ensure that the deduction theorem Weber claims to hold in fact does. What is a more significant contribution is the suggestion for how to analyze Weber’s notion of “bad” assumptions. A new form of a Hilbert calculus is presented in which premises are taken as a pair consisting of a premise set—the formulas of which can be drawn upon unrestrictedly—and a premise multiset—the formulas of which can be used at most as many times as they occur in the multiset. The idea is that “bad” assumptions go into the multiset, whereas non-bad assumptions go into the set. It is shown that the consequence relation restricted to only “non-bad” axiom sets is fully structural. Weber’s mathematical theories are non-classical; some are even contra-classical in that they are provably inconsistent. Since Weber regards the axioms of his mathematical theories as non-bad, they are, however, non-classical in a more standard sense than what Weber seems to claim: they are fully structural, Tarskian, and closed under both modus ponens and adjunction.
The no-no paradox and its generalizations (called no-no type) are self-referential statements whose paradoxicality arises from the impossibility of symmetric truth-value assignments imposed by syntactical symmetry. Their syntactic symmetry can be characterized by permutation groups. This paper addresses whether any permutation group can be represented by a no-no type paradox, in the sense that the symmetry of the paradox is precisely characterized by the group. By introducing the notion of invariance preorder for binary sequences, we extend the algebraic techniques from research on the representability of the permutation groups by Boolean functions. We prove that for n≥ 4 , the alternating group A_n and the symmetric group S_n have the same invariance preorder. Consequently, we establish the main result of this paper: for any n≥ 4 , the alternating group A_n cannot be represented by any Boolean system of self-referential statements — and a fortiori, by any no-no type paradox. With the information of the invariance preorder for S_n , we give a systematic construction of both Boolean paradoxes and no-no type paradoxes with maximal symmetry. Our results reveal a profound connection between algebraic symmetry and semantic pathology.
Forcing is a fundamental set-theoretic technique, with which many independence results can be established. A famous example is the independence of the continuum hypothesis in ZFC set theory. Forcing is also well-known to be complex, and therefore difficult to master. Here, we provide a gentle introduction of forcing, by developing forcing for second-order logic. Second-order logic can be interpreted as a rudimentary kind of set theory. Although very limited as a theory of sets, second-order logic is rich enough to capture a form of the continuum hypothesis, as well as the generalized continuum hypothesis, which is slightly easier to state. We develop forcing for second-order logic in the form of possibility semantics, and use this to show that a second-order version of the generalized continuum hypothesis cannot be derived in a standard proof system for second-order logic. Mathematically, the results we obtain in this way are much weaker than standard independence results in set theory. However, as a consequence, we are able to avoid many technical complexities in the presentation of forcing. In this way, we hope that forcing for second-order logic can serve an expository function, of providing an accessible route towards understanding some of the central ideas behind forcing.
The notion of normality has gained significant attention in recent works in epistemology. Goodman and Salow (Philosophical Studies, 175, 183–196. 2018, 2021, Philosophical Review, 132(1), 89–145. 2023) introduced an abstract class of structures, namely normality structures, and a framework based on them that can support a diverse range of normality-based epistemic theories. In this work, we provide a formal study of the normality structures introduced by Goodman and Salow (2021). There are two general motivations for such a study: first, to explore the structures as mathematical entities, and second, to study them as semantic structures appropriate for reasoning about epistemic and doxastic concepts. To pursue these aims, we propose two distinct languages: a modal language and a conditional one. We then develop the logic of normality structures in each language and establish their soundness and completeness with respect to normality structures. Within the second language we propose, not only can we define certain epistemic modalities that satisfy the desired conditions, but we can also potentially provide the logic for a more generalized class of frames, suitable for a more abstract notion of normality that relaxes some of the more controversial conditions imposed by Goodman and Salow. Finally, we briefly demonstrate how these formal tools can contribute to ongoing debates in formal epistemology.
A logic L has the disjunction property just in case _Lφ∨ψ implies _Lφ or _Lψ . This property is important to constructivists and is a well-known feature of intuitionistic logic. In this paper we use model-theoretic techniques to show that the disjunction property holds in Urquhart’s operational relevance logics R_U^+ , T_U^+ , RW_U^+ , TW_U^+ , and E_U^+ . These results suggest that operational relevance logics merit further attention from a constructivist perspective. Along the way, we also provide a novel proof that the disjunction property holds in intuitionistic logic.
This paper develops a sequent calculus framework for representing locally and globally valid metainferences through which contraction-free sequent calculi for the modal logics S5 and the propositional fragment of Carnap’s C are obtained. The sequent calculi allow for strongly terminating and backtracking-free proof search, features which in turn arguably explain why the decision problems for S5 and the propositional fragment of Carnap’s C are reducible to that of propositional classical logic.
Clarke-Doane and Ash (2024) argues that mathematics and philosophy are on a par as a priori disciplines. In particular, each fails to be objective. Should this be so, it is unclear that philosophy can do anything for set theory or that new axioms can ever be rationally justified. Blue (2024) explicates a methodology for rationally justifying new axioms. I will argue against (Clarke-Doane and Ash 2024) by buttressing (Blue 2024), describing how it accounts for the case for Definable Determinacy and how it might extend to a potential case for Baire category principles. Along the way, I will describe a not-yet-refuted scenario for positively answering Todorčević’s question “In order to have the true structure theory of ℘ (ω _1) do we really need to retreat to an inner model of the universe of sets?” (Todorcevic 2024, Question 4.7).
An object is gunky in virtue of how it decomposes. In particular, an object is gunky if and only if all its parts have proper parts. Since Anaxagoras, philosophers have appealed to the existence of gunk to support a range of metaphysical views. These discussions raise questions about the composition of gunk: How is gunk generated? How do we get gunk? Obviously, gunk cannot be composed of atoms. Otherwise, we have admitted objects into our ontology (i.e. atoms) with no proper parts. This has led to the widespread belief that gunk cannot be generated. It must be given. In this paper, we prove this to be false. Though gunk cannot be generated by atoms, we show that it can nevertheless be generated by some fundamental entities. We apply Weyl’s Equidistribution Theorem to produce a model of a universe which is gunky yet generated by a single entity. This dispels other misconceptions about gunk and provides a new perspective on debates about metaphysical fundamentality.
