Julian C. Cole

In the last two decades, an increasing number of philosophers have endorsed humanistic philosophies of mathematics, i.e., accounts that take humans to be responsible for the subject matter of (contemporary assertoric) mathematical theories. While such accounts have been largely successful in explaining many of the widely accepted features of mathematical theories (e.g., their objectivity, special epistemic status, and applicability, as well as the abstract nature of their subject matters), none have provided a satisfactory explanation of the temporal and modal immutability of such theories, i.e., they are eternally or timelessly and necessarily or modelessly true and have subject matters that exist eternally or timelessly and necessarily or modelessly. Indeed, some humanistic philosophers of mathematics reject their immutability, claiming instead that they enjoy only conditional immutability. In this paper, I show that a humanistic philosophy of mathematics based on Brian Epstein’s Anchoring-Grounding Framework for Social Facts has the resources to explain all widely accepted features of mathematical theories, including, most importantly, their immutability.

Balthasar Grabmayr

The study of computability is traditionally taken to produce absolute truths. According to this standard view, results in computability theory do not depend on contingent cultural practices, such as denoting numbers by Arabic numerals. However, the standard view faces what I call Montague’s Problem. This problem purports to show that computability is inherently relative to notation. Montague’s Problem has led philosophers to argue that classic computability-theoretic results are relative to contingent matters, as opposed to the absolute nature of other mathematical truths. The aim of this talk is twofold. First, I will show that Montague’s Problem generalises to the case of arithmetical definability. Hence, a similar attack can be raised against the standard view that important (un-)definability results express absolute mathematical truths. Second, I will defend the standard view from Montague’s Problem. In doing so, I will introduce absolute notions of computability and definability for important classes of objects.

Sara Ayhan

An account of bilateral proof-theoretic semantics, famously proposed in [3], usually makes two assumptions. First, that the meaning of logical connectives is determined not only by some positive notion, such as assertion or proof, but also by a dual, equally primitive notion, such as denial or refutation. Second, that negation expresses this dual notion in the object language – thereby rejecting Frege’s contrary claim that denial of A is simply assertion of ¬A. This is usually taken to yield a kind of toggle negation – in the classical setting of [3] and in constructive logics capturing the ‘strong negation’ of systems like N4 [1].

I will argue that on such a negation-as-refutation-based account – first explicitly proposed in [2] – one cannot obtain that kind of toggle negation without making highly non-constructive assumptions. So the traditional bilateralist route to N4 and related logics seems blocked. The difference between bilateral representations of N4 and classical logic amounts merely to adding so-called ‘co-ordination principles’: structural rules governing the interaction between proof and refutation rules. The classicist has no problem, of course, admitting non-constructive assumptions into their reasoning. However, I will show that adding co-ordination principles without the explicit aim of recovering classical logic is not straightforwardly justifiable. The cleaner bilateralist option would be to abandon both this kind of negation (others may be available) and co-ordination principles.

References
[1] Ahmad Almukdad and David Nelson, Constructible falsity and inexact predicates, The Journal of Symbolic Logic, 49 (1984), no. 1, pp. 231–233.
[2] Edgar López-Escobar, Refutability and elementary number theory, Indagationes Mathematicae, 75 (1972), no. 4, pp. 362–374.
[3] Ian Rumfitt, ‘Yes’ and ‘No’, Mind, 109 (2000), no. 436, pp. 781–823.

Dean McHugh

This talk brings together two ideas. First, that statements under a modal are interpreted as conditional antecedents. ‘Possibly A’ states that if A were true, there would be some case where the relevant ideals are met. Dually, ‘necessarily A’ states that if A were false, there would be no case where the relevant ideals are met. Second, conditional antecedents are interpreted via sets of alternatives, with some items—such as disjunction and ‘any’—introducing multiple alternatives. Combining them returns, in a uniform and automatic way, a solution to three challenges facing the standard theory of modality: free choice inferences, independence inferences, and counterexamples to substitution of logical equivalents.

Susanne Bobzien

In my talk I offer a mildly revisionist logic of identity that makes room for the possibility of vague objects at the level of logic without undermining the use of the identity relation in mathematics. The paper challenges Evans’s infamous and still widely accepted 1978 argument that indeterminate identity is incoherent and that any coherent notion of identity must be governed by the modal system S5, with a determinateness operator in lieu of the necessity operator. Building on a suggestion by Ken Akiba (2014) and my own view on vagueness (Bobzien 2025), the talk replaces Evans’s S5 with something a little more flexible.

Rashed Ahmad

In assessing different logics, we appeal to the theoretical virtues the logics may enjoy. These theoretical virtues include (but are not necessarily limited to) expressive power, generality, topic-neutrality, simplicity, elegance, and adequacy to the data. In terms of expressive power and generality, Jonathan Erenfryd and I argued in Classical Logic of Paradox (manuscript) that CLP can accommodate naive theories of truth, validity, and paradoxicality without the threat of revenge and metainferential paradoxes, nor of overinternalization of semantic concepts. Additionally, in Issues of Overinternalization: ω-inconsistency (manuscript), I show that first-order CLP can accommodate a theory of (standard) arithmetical truth, and that theory is ω-consistent. These results stand witness to CLP’s expressive power and generality. However, for a logic to be general, it must also be able to accommodate other theories, such as queer feminist and liberation theories. In this paper, we nominate CLP as a good candidate for serving as the basis of a theory of liberation. Before doing so, however, we argue that, given the analysis of Val Plumwood’s argument that classical logic creates a natural hotspot for dualisms that promote oppression, there is a tight relation between theories of liberation and theories of vagueness. So, for a logic to even qualify as a basis for a theory of liberation, it must first be able to be a basis for a theory of vagueness. Thus, we first show how CLP handles vagueness, then present and defend Val Plumwood’s argument against classical logic, and show that classical logic is neither topic-neutral, simple, nor elegant. Finally, we show how a theory of liberation built on CLP fares compared to other proposals in the literature.

John MacKay

One motivation for counterpart theory comes from a family of puzzles about identity and modal variation known as puzzles of tolerance. In these puzzles, an object could have been slightly different but could not have been radically different. It is commonly assumed, both by counterpart theorists and their critics, that the counterpart-theoretic solution to the puzzles of tolerance involves the intransitivity of counterpart relations. In this paper, I argue that this is a mistake, and instead I defend a solution to the puzzles of tolerance that involves transitive counterpart relations with context-sensitive names and variables.

Teresa Kouri Kissel

Ferrari and Orlandelli (2019) propose that an admissibility condition on a proof-theoretic logical pluralism be that the logics in question must be harmonious. For them, this means that they must have connectives which are (a certain brand of) unique and conservative. This allows them to develop an innovative pluralism where the admissible logics are both useful and balanced, which shows variance on two levels: the level of validity and the level of connective meanings.

Here, I will show that we can extend the system one step further, and induce a three-level logical pluralism, which better fits the criteria of usefulness and balance. The first and second levels remain as suggested by Ferrari and Orlandelli (2019), but we can allow for multiple notions of uniqueness in the definition of harmony, or multiple notions of harmony. Either of these options generates a pluralism at the level of our admissbility conditions. This generates a pluralism at three levels: validity, connective meanings, and admissibility conditions. But it still preserves the spirit of Ferrari and Orlandelli (2019): balance and usefulness remain the admissibility constraints across the board.

Filippi Ferrari

In this talk, I address the question of whether logic is intrinsically or extrinsically normative. I begin with some taxonomical remarks and introduce distinctions relevant to assessing the normativity of logic. I then examine a prominent extrinsicist strategy that grounds the normativity of logic in truth and raise a few objections to it. In the final part, I sketch an intrinsicist account of the normative role that logic plays in reasoning.

WooJin Chung, Seoul National

Suppose that Adam is a risk-taker—in general a highly successful one—and Bill prefers to play it safe. You believe that the organization is highly likely to thrive under Adam’s leadership, though there is still a small chance his investment could fail. On the other hand, Bill’s risk-averse strategy will allow the organization to profit to some degree, but it won’t be nearly as successful as in Adam’s best-case—and most likely—scenario. Yet the organization will fare better than Adam’s worst-case—and fairly unlikely—scenario. The following example sounds true in this context:

(1) If Adam takes the lead, the organization will be more successful than (it will be) if Bill takes the lead.

In this talk, I point out that the truth conditions of (1) do not align with the prediction of the standard quantificational view of conditionals (Kratzer 2012), coupled with extant theories on the interaction between comparatives and quantifiers (Schwarzschild & Wilkinson 2002, Heim 2006, Schwarzschild 2008, Beck 2010, among many others). I suggest that a working solution can be developed by dropping the assumption that conditionals are quantifiers over possible worlds.