Author: Marcus Rossberg

Alienation from Normativity (and Logic?)

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Matthew Chrisman

Robust realists and quasirealist expressivists have both been accused, in different ways, of being committed to an alienated stance towards fundamental oughts, reasons, and values. Either normative facts obtain completely independently of our cares and concerns, in which case, why do we care about them as much as we do? Or their reality is something more like a projection from or construction out of our ways of normative thinking, in which case why should we care about them as much as we do? Sometimes this looks like philosophical bedrock in metaethics. But in this paper I want to explore the possibility that inferentialism offers a way past the impasse. In the first instance, this is by suggesting that normative terms can be viewed analogously to logical terms in getting their meaning neither from what they refer to nor from what attitudes they primarily serve to convey. But I also want to propose a way of thinking of normative/logical facts and normative/logical thinking as reciprocally related to each other in a way that rejects both the realist’s commitment to the explanatory independence of normative/logical facts from normative/logical thinking and the expressivist’s commitment to starting our explanation of normative/logical facts with an account of normative/logical thinking.

What natural language can tell us about natural logic’s contents

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What natural language meanings can tell us about natural logic’s contents

Tyler Knowlton

In contrast to invented logics, which often call for theoretical vocabulary that seems cognitively difficult to master, natural logic is concerned with the foundational building blocks of logical thought that humans naturally/automatically/innately have access to. In this talk, I’ll suggest a new way of getting at the question of what’s in natural logic: investigating the fine detail of the semantic instructions specified by linguistic expressions. Two psycholinguistic case studies in particular suggest potentially counter-intuitive conclusions about the contents of natural logic. First, I’ll use experimental evidence to argue that sentences like “most frogs are green” are understood by speakers in terms of cardinality and subtraction (“the number of green frogs is greater than the total number of frogs minus the number of green ones”) and not in terms of predicate negation (“the green frogs outnumber the non-green frogs”). Combined with typological evidence that natural language seems to eschew predicate negation elsewhere, this finding supports a recent proposal that natural logic lacks a notion predicate negation/set complementation. Second, I’ll use experimental evidence to argue that sentences like “every frog is green” are understood by speakers in terms of applying a predicate to a restricted domain (“the frogs are such that ‘green’ applies universally”) and not in terms of relating two independent sets (“the frogs are a subset of the green things”). Moreover, hypothetical quantifiers whose meanings would require specification in terms of genuine set-theoretic relations seem to be unlearnable by adults and children. These findings suggest that natural logic has the resources to support second-order quantification, but not second-order relations.

The Metalinguistic Construal of Mathematical Propositions

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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.

A relevant framework for barriers to entailment

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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.

Vague Identity: A Uniform Approach

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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

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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

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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.

Towards a Structuralist Metasemantics for Number Words

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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.