Zeynep Soysal

In this talk I will defend the metalinguistic solution to the problem of mathematical omniscience for the possible-worlds account of propositions. The metalinguistic solution says that mathematical propositions are possible-worlds propositions about the relation between mathematical sentences and what these sentences express. This solution faces two types of problems. First, it is thought to yield a highly counterintuitive account of mathematical propositions. Second, it still ascribes too much mathematical knowledge if we assume the standard possible-worlds account of belief and knowledge on which these are closed under entailment. I will defend the metalinguistic construal of mathematical propositions against these two types of objections by drawing upon a conventionalist metasemantics for mathematics and an algorithmic model of belief, knowledge, and communication.

Nicole Cruz (with Michael Lee)

People draw on event co-occurrences as a foundation for causal and scientific inference, but in which ways can events co-occur? Statistically, one can express a dependency between events A and C as P(C|A) != P(C), but this relation can be further specified in a variety of ways, particularly when A and/or C are themselves conditional events. In the psychology of reasoning, the conditional P(C|A) is often thought to become biconditional when people add the converse, P(A|C), or inverse, P(not-C|not-A), or both, with the effects of these additions largely treated as equivalent. In contrast, from a coherence based logical perspective it makes a difference whether the converse or the inverse is added, and in what way. In particular, the addition can occur by forming the conjunction of two conditionals, or by merely constraining their probabilities to be equal. Here we outline four distinct ways of defining biconditional relationships, illustrating their differences by how they constrain the conclusion probabilities of a set of inference forms. We present a Bayesian latent-mixture model with which the biconditionals can be dissociated from one another, and discuss implications for the interpretation of empirical findings in the field.

Fabrizio Cariani

I identify and develop a solution to the puzzle of anankastic conditionals that is novel in the sense that it has gone largely unnoticed, but also well-worn in that the materials for it have long been available. The solution involves an integration of the classical Kratzerian premise semantics and a default theory of reasons (such as the one presented in Horty, 2012, leveraging several decades of research on default logic). To stress-test the proposal I also investigate how it might be applied to the variety of anankastic-adjacent data discussed by Condoravdi and Lauer (2016). The resulting approach is not just an exceptional fit for the problem of anankastic conditionals, but also an independently promising upgrade on the classical account of modals.

Yale Weiss

In her recent book, Russell (2023) examines various so-called “barriers to entailment”, including Hume’s law, roughly the thesis that an ‘ought’ cannot be derived from an ‘is’. Hume’s law bears an obvious resemblance to the proscription on fallacies of modality in relevance logic, which has traditionally formally been captured by the so-called Ackermann property. In the context of relevant modal logic, this property might be articulated thus: no conditional whose antecedent is box-free and whose consequent is box-prefixed is valid (for the connection, interpret box deontically). While the deontic significance of Ackermann-like properties has been observed before, Russell’s new book suggests a more broad-scoped formal investigation of the relationship between barrier theses of various kinds and corresponding Ackermann-like properties. In this talk, I undertake such an investigation by elaborating a general relevant bimodal logical framework in which several of the barriers Russell examines can be given formal expression. I then consider various Ackermann-like properties corresponding to these barriers and prove that certain systems satisfy them. Finally, I respond to some objections Russell makes against the use of relevance logic to formulate Hume’s law and related barriers.

Xinhe Wu

There are numerous apparent examples of vague identity, i.e., examples where two objects appear to be neither determinately identical nor determinately distinct. Philosophers disagree on whether the source of vagueness in identity is semantic or ontic/metaphysical. In this talk, I explore the use of Boolean-valued models as a many-valued semantic framework for identity. I argue that this semantics works well with both a semantic and ontic conception of vague identity. I also discuss, in the context of Boolean-valued logic, responses to the Evans’ argument under the two conceptions.

Jonas Raab

I develop a modal extension of the Quantified Argument Calculus (QUARC)—a novel logical system introduced by Hanoch Ben-Yami. QUARC is meant to better capture the logic of natural language. The purpose of this paper is to develop a variable domain semantics for modal QUARC (M-QUARC), and to show that even if the usual restrictions are imposed on models with variable domains, M-QUARC-analogues of the Barcan and Converse Barcan formulas still are not validated. I introduce new restrictions—restrictions on the extension of the predicates—and show that with these in place, the Barcan and Converse Barcan formulas are valid. The upshot is that M-QUARC sheds light on the in-/validity of such formulas.

Andrew Tedder

There is a simple way of reading a structure of topics into the matrix models of a given logic, namely by taking the topics of a given matrix model to be represented by subalgebras of the algebra reduct of the matrix, and then considering assignments of subalgebras to formulas. The resulting topic-enriched matrix models bear suggestive similarities to the two-component frame models developed by Berto et. al. in Topics of Thought. In this talk I’ll show how this reading of topics can be applied to the relevant logic R, and its algebraic characterisation in terms of De Morgan monoids, and indicate how we can, using this machinery and the fact that R satisfies the variable sharing property, read R as a topic-sensitive logic. I’ll then suggest how this approach to modeling topics can be applied to a broader range of logics/classes of matrices, and gesture at some avenues of research.

Ainsley May

Current accounts of meaning in mathematics face a dilemma between triviality and over-specificity. On the one hand, intensional accounts of meaning such as possible world semantics give the trivial result that every mathematical theorem has the same meaning since they are all necessarily true. This triviality is unsatisfactory because we clearly hold some mathematical theorems have different meanings from others. On the other hand, hyperintensional accounts like impossible worlds and structured propositions allow us to distinguish between necessary truths. However, they are so fine-grained that it becomes difficult to uniformly identify the salient semantic features.

In response to this dilemma, I propose an account of mathematical meaning called the folkloric account. On the folkloric account the content of a mathematical theorem is the collection of models, within some reference class of models, that make the theorem true. The appeal of this account is partly that it retains central aspects of world-based accounts, such as evaluation within a model. Yet it overcomes their limitations by incorporating more models to represent different mathematical theories and structures without allowing absolutely every such structure. Here, I introduce the folkloric account and use examples to highlight some of its strengths and identify weaknesses to address in future research.

James Walsh

It is a well-known empirical phenomenon that natural axiomatic theories are well-ordered by consistency strength. The restriction to natural theories is necessary; using ad-hoc techniques (such as self-reference and Rosser orderings) one can exhibit non-linearity and ill-foundedness in the consistency strength hierarchy. What explains the contrast between natural theories and axiomatic theories in general?

Our approach to this problem is inspired by work on an analogous problem in recursion theory. The natural Turing degrees (0,0′,…,Kleene’s O,…,0#,…) are well-ordered by Turing reducibility, yet the Turing degrees in general are neither linearly ordered nor well-founded, as ad-hoc techniques (such as the priority method) bear out. Martin’s Conjecture, which is still unresolved, is a proposed explanation for this phenomenon. In particular, Martin’s Conjecture specifies a way in which the Turing jump is canonical.

After discussing Martin’s Conjecture, we will formulate analogous proof-theoretic hypotheses according to which the consistency operator is canonical. We will then discuss results—both positive and negative—within this framework. Some of these results were obtained jointly with Antonio Montalbán.