Work in Progress Seminar (2025/2026)



Titolo: Inconsistent variants of classical determinate truth  


Abstract: In the paper "Classical Determinate Truth I" Fujimoto and Halbach had introduced a theory CD in which the semantic notions of truth and determinateness are axiomatised as primitive predicates. The truth predicate of that theory is fully classically compositional for the sentences of the whole language and it satisfies Tarski biconditionals for all the determinate sentences. The authors had asked whether that theory can be consistently extended with an axiom saying that for any sentence it is determinate whether it is determinate.   In our talk, we present a recent result obtained by Castaldo, Głowacki, and the author, showing that this extension of CD is in fact inconsistent. We discuss how this result forms a general impossibility result for a broad class of theories of determinateness or groundedness.  



Titolo: Mathematical conventionalism and mathematical practice

Abstract: Mathematical conventionalism claims that mathematical truth is determined by linguistic conventions, but it faces the problem of explaining the existence of mathematical entities. Jared Warren (2020) considers existence to be trivial: if a conventionally adopted theory of arithmetic contains existence claims, then numbers exist. Zeynep Soysal (2025), drawing on a descriptivist account about set-theoretic expressions, argues that more is required. In her view, not every theory—such as an inconsistent one—describes an existing entity. To address this, she links existence to consistency. Like Warren’s, her account is naturalistic in that it is based on the actual linguistic conventions of mathematical practice. Using a dataset from interviews with 28 practicing set theorists (over 500 pages of transcripts and a 100-page systematic summary), we studied set-theoretic practice with a focus on consistency beliefs and forms of pluralism. Our findings support the core conventionalist thesis: mathematicians can imagine that slightly different axioms could have been adopted. They also confirm Soysal’s view: set theorists reject inconsistent theories, and the consistency of their associated theory is evidence enough to use phrases like “sets exist.” Two complications emerged, however. First, our findings reveal divergences among speakers that make it difficult to determine what all set theorists accept no matter what. Second, Soysal’s descriptivism explicitly includes informal descriptions. Yet our data show that informal ways of talking about sets are not always consistent. We will present examples of this phenomenon and offer some preliminary conclusions.






Work in Progress Seminar (2024/2025)



Titolo: Counterpossibles in relative computability theory: a closer look (jww Luca San Mauro and Giorgio Lenta)


Abstract: A counterpossible is a counterfactual with an impossible antecedent. Matthias Jenny has argued that relative computability theory provides examples of false counterpossibles. If Jenny was right, it would be highly significant, since it would follow that the standard analysis of counterfactuals, according to which counterpossibles are all vacuously true, is incorrect. In this paper, we argue against the claim that computability theory provides examples of false counterpossibles. We distinguish two ways of reading the alleged false counterpossibles. Under the first reading, they are indeed false, but, as we will argue, they are not genuine counterpossibles. Under the second reading, they are genuine counterpossibles, but they are true. There is a way to interpret the alleged false counterpossibles as a false, and there is a way to interpret them as a counterpossible, but under none of these two interpretations they are false counterpossible.

Titolo: On Relativism about Uncountability


Abstract: This talk will concern the broadly "Skolemite" idea that transfinite cardinalities are in some sense "unavoidably relative". I'll discuss how recent work on countabilism relates to these ideas, and will also raise and address some criticisms of the relativist position that have been set out recently by Giorgio Venturi.

Titolo: Is the concept of set semantically indeterminate? (jww Pablo Dopico)


Abstract: In the 1995 acceptance lecture for the Rolf Schock Prize Michael Dummett introduced the notion of indefinite precisifiable concept as related to his analysis of vagueness. However, he did not provide any clear example. In this paper we argue that the concept of set provides such an example. Moreover, despite its original relation to vagueness, we argue that this kind of semantic indeterminacy actually differs from it and from the more recently debated notion of open texture. The argument is based on an axiomatic understanding of (the determination of) the extension of the concept of set: no matter how one (recursively enumerably) extends ZF, there will always remain collections the theory cannot settle as belonging or not to the extension of the concept of set. A crucial point is to assess how the notion of indefinite precisification behaves when related to programs such as Woodin’s Ultimate-L Conjecture or Hamkins’ Multiverse. We conclude by sketching possible partial answers based on categoricity or the Iterative Conception of Set.

Titolo: On the complexity of infinite argumentation (jww Uri Andrews)


Abstract: Abstract argumentation theory is a core research area in artificial intelligence, offering a robust framework for reasoning in the presence of conflicting information. While the study of finite argumentation frameworks (AFs) has received extensive attention, the exploration of infinite AFs remains largely underdeveloped. In this work, we take a significant step toward filling this gap by systematically investigating the computational complexity of problems associated with infinite AFs. If time permits, we will also explore intriguing connections between infinite AFs and truth theory. 

Titolo: What is the content of the Continuum Hypothesis?


Abstract: The Continuum Hypothesis featured top of Hilbert's list of 23 problems in 1900. Today, we still consider the question, with various programmes pulling in different directions. This conceptual diversity raises a puzzle; in what sense are our thoughts about sets and the Continuum Hypothesis the same? In what sense do we disagree when we talk about it? A standard assumption takes it that one must either accept the full determinacy or radical indeterminacy of set-theoretic claims; that there is either no disagreement or complete disagreement about a unique intended subject matter. In this paper, I argue for a possible middle-ground. Assuming a representationalist view of how content is determined, I argue that whilst the Continuum Hypothesis can mean different things to different agents, it can also be determinate within certain communities.

Titolo: Global supervenience in inquisitive modal logic


Abstract: 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, … ,Q_m globally supervene on predicates P_1, … ,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 talk I discuss this notion of global supervenience from the perspective of modal logic. First, I will show that claims of global supervenience cannot be expressed in standard modal predicate logic. Second, I will show that they can be expressed naturally in an inquisitive extension of modal predicate logic, where they are captured as strict conditionals involving questions; 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. 

Titolo: Trivalence, conditionals, and epistemic modals


Abstract: In this talk, we explore the prospects for trivalent semantics and logics for theories of conditionals and epistemic modals. We extend well-known trivalent frameworks for conditionals to incorporate epistemic modality, analyzing the resulting logics and probability theories. We focus on some key puzzles involving conditionals and modals, such as triviality results and epistemic contradictions, and present their trivalent resolution.


(This talk is partly based on join work with Paul Égré and Jan Sprenger, and partly based on joint work with Paolo Santorio)



Titolo: On mathematical defining


Abstract: This talk is dedicated to the subject of mathematical definitions from a logical point of view, taking into account its conceptual and technical aspects. Defining is one of the two fundamental types of mathematical practices, the other one being constituted by the practice of proofs. In spite of this, mathematical definitions are not as well understood as mathematical proofs, and we believe that this causes a significant gap in the foundations of mathematics.


Michele Giannone (University of Pisa)


Titolo: Beyond FOL. Where to stop?


Abstract: We critically assess Paseau and Griffiths' (2022) attempt to refine the Tarski-Sher thesis by linking invariance-based logicality to L∞∞ through both top-down and bottom-up arguments. Using Sagi’s (2018) set-theoretic weight and Kennedy & Väänänen’s (2021) definability framework, we highlight key distinctions in their approach. We further refine McGee’s theorem to reveal an asymmetry: while bottom-up reasoning supports their conclusion, Väänänen’s results suggest that extending to L∞∞ is not necessary from a top-down view. This discrepancy challenges their alignment and suggests a form of logical pluralism.



Titolo: Quantified bilateral systems for arbitrary terms


Abstract: Arbitrary objects play an important role in both ordinary and mathematical reasoning. The reason for that is their distinct behavior: an arbitrary object presents those properties common to all individual objects in its range. Many philosophers have argued this precise quality to be contradictory, therefore rejecting arbitrary objects altogether. Each of the arguments stem from some clash between their distinct characteristic and classical logic. For that reason, Kit Fine and Marco Santambrogio defend dropping classical logic when dealing with arbitrary objects, in favour of supervaluationism and intuitionistic logic, respectively. However, such a strong move is not necessary. The arguments assume rejecting properties of an arbitrary object to be expressible in terms of asserting something of it - a unilateral setting. (Yago, forthcoming) shows how those arguments may be rebutted in bilateral systems. More specifically, the author argues Incurvati and Schloder's Weak Rejectivist Logic, on a propositional level, reflects the way in which we assert sentences about arbitrary objects, and further dissolve the counterarguments against them, such as Berkeley's problem, or Lesniewski's. Following that work, we shall offer quantified bilateral systems, which may be seen as first-order extensions of the Weak Rejectivist Logic, to work directly with arbitrary objects while retaining classical logic. One of them, and its corresponding semantics, may be seen as the bilateral counterpart to Fine's work on using models of arbitrary objects for the semantics of natural deduction systems such as Quine's and Gentzen's. The other, however, introduces term-forming operators which allows us to, given a certain condition F, directly talk about arbitrary F's. These new terms may be seen as counterparts to epsilon-terms. We shall consider their related semantics, how they relate to Kit Fine's seminal work on arbitrary objects, and some of the logical aspects of the new systems. The paper is a work in progress.



Titolo: On class hierarchies


Abstract: In her seminal article ‘Proper Classes’, Penelope Maddy introduced a theory of classes validating the naïve comprehension rules. The theory is based on a step-by-step construction of the extension and anti-extension of the membership predicate, which mirrors Kripke’s construction of the extension and anti-extension of the truth predicate. Maddy’s theory has been criticized by Øystein Linnebo for its ‘rampant indeterminacy’ and for making identity among classes too fine-grained. In this paper, I present a theory of classes that builds on Maddy’s theory but avoids its rampant indeterminacy and allows for identity among classes to be suitably coarse-grained. For all the systems I discuss, I provide model theories and proof theories (formulated in bilateral natural deduction systems), along with suitable soundness and completeness results.