Vague Identity: A Uniform Approach

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.

Modal QUARC and Barcan

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.

Relevant Logics as Topical Logics

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.

Meaning in Mathematics: a folkloric account

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.

Is the consistency operator canonical?

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.

Reductions between problems in reverse math and computability

Denis Hirschfeldt

Many mathematical principles can be stated in the form “for all X such that C(X) holds, there is a Y such that D(X,Y) holds”, where X and Y range over second-order objects, and C and D are arithmetic conditions. We can think of such a principle as a problem, where an instance of the problem is an X such that C(X) holds, and a solution to this instance is a Y such that D(X,Y) holds. I will discuss notions of reducibility between such problems coming from the closely-related perspectives of reverse mathematics and computability theory.

The interaction between demonstratives and relative clauses – a view from Mandarin

I-Ta Chris Hsieh, National Tsing Hua University, Taiwan

A demonstrative such as that and those may have various uses, be it deictic, anaphoric, or purely descriptive. One of the approaches to demonstratives, namely the Hidden Argument Theory (henceforth, HAT) approach (e.g., King 2001, Blumberg 2020, Nowak 2021, Ahn 2022; a.o.,), is intended to provide a unified account of these uses. In the HAT approach, a demonstrative, unlike a definite article (e.g., the), carries two restrictions; in one recent variant of this approach, namely Ahn (2022, Ling & Phil), the various uses of a demonstrative description are due to the different options to contribute the second restriction, including a deictic demonstration, an index, and a relative clause that adjoins to the demonstrative phrase. In this talk, I examine the interaction between demonstratives and relative clauses and show that current analyses along with the HAT approach could yield some undesirable predictions. Some amendments will be suggested to accurately capture the interpretation of a demonstrative description with a relative clause and at the same time avoid these predictions.

Causal dependence in actuality inferences

Prerna Nadathur

A range of complement-taking predicates give rise to surprising actuality inferences, in which a modally-embedded complement event is understood non-modally, as taking place in the evaluation world.  I argue that actuality inferences can be explained—and unified across predicate classes—on an approach in which the modality of participating predicates is analyzed in causal terms.  This talk focuses on an illuminating case study: enough and too predicates.

Hacquard (2005) observes that, like the ability modals at the heart of the puzzle (Bhatt 1999), enough/too predicates have aspect-sensitive actuality inferences.  Under imperfective marking, French enough predicates like (1a) are compatible with the non-actualization of their complements; their perfective counterparts (as in 1b) show the complement entailment pattern of implicative verbs like French réussir (‘succeed’, ‘manage’; 2).

(1) a. Juno était assez rapide pour gagner la course, mais elle n’a pas gagné.
Juno was-IMPF fast enough to win the race, but she did not win.
b. Juno a été assez rapide pour gagner la course, #mais elle n’a pas gagné.
Juno was-PFV fast enough to win the race, #but she did not win.
(2) Juno { réussissait / a réussi } à gagner la course, #mais elle n’a pas gagné.
Juno { managed-IMPF / managed-PFV } to win the race, #but she did not win.

Despite the contrast in (1)-(2), I argue that enough/too inferences are—semantically speaking—instances of implicativity.  I build on a causal account of implicative lexical semantics (Nadathur 2016, 2019) to show that enough/too actuality inferences arise just in case the compositional interaction between grammatical aspect, modal flavour, and the enough/too matrix adjective reproduces the semantic structure of an implicative: that is, where the matrix adjective denotes an actionable property which is causally involved in realizing the enough/too complement, and the perfective aspect induces an eventive interpretation of the matrix assertion.   Insofar as the implicative analysis explains the aspect-sensitivity of enough/too inferences, I suggest that it naturally extends to ability modals’ actuality inferences, when coupled with a causal approach to ability.

Towards a Structuralist Metasemantics for Number Words

Eric Snyder

According to non-eliminative structuralism, the referents of numerical singular terms, such as the numeral ‘two’ or ‘the number two’, are numbers, construed as positions within the natural number structure. However, a potential problem comes in the form of sentences like ‘{∅, {∅}} is the number two among the von Neumann ordinals’. If this is an identity statement, then its truth would seemingly require identifying the second position of the natural number structure with a particular set, thus giving rise to a version of Benacerraf’s famous Identification Problem. In response, Stewart Shapiro (1997) draws an analogy to expressions like ‘the Vice President’, which are ambiguous between denoting an office-holder (e.g. Kamala Harris) or an office (the office of the Vice Presidency). Similarly, Shapiro suggests that in ordinary arithmetic contexts, such as ‘Two is less than three’, we view positions as analogous to office-holders, while in other contexts, we view them instead as analogous to offices occupied by entities playing the role of numbers, e.g. {∅, {∅}}. However, this suggestion faces two serious challenges. First, what exactly is the nature of this purported ambiguity, and what empirical evidence, if any, is there for it? Second, even if we grant the ambiguity, we appear to get a revenge version of the Identification Problem anyway: just permute the positions within the natural number structure. The purpose of this talk is to defend Shapiro’s ambiguity thesis, by supplying the empirical support required, and explaining how, when appropriately understood, the semantics assumed does not give rise to a revenge form of the Identification Problem.