Nathan Salmón

This chapter distinguishes two approaches to vagueness. The vagueness-in-language approach sees the world as exact, and all vagueness as inherent in our means of representing the world. By contrast, the vagueness-in-the-world approach sees the world itself as fuzzy, in the sense that for some things and some attributes of things, there is allegedly no fact of the matter whether those things possess or lack those attributes. The former approach is criticized on several grounds. The vagueness-in-language approach is either incoherent or collapses into vagueness-in-the-world. A recent objection to direct-reference theory, based upon the criticized approach, is shown to be fundamentally mistaken.

Nathan Salmón has been conferred the title of Distinguished Professor Emeritus at the University of California, Santa Barbara since 2024. In recognition of this significant milestone, Yuling Zhang conducted an interview with Professor Salmón, whose influential work in the theory of direct reference has cemented his reputation as a leading figure in the field of philosophy of language. The interview focuses on Nathan Salmón's contributions to contemporary philosophy, particularly his work related to Saul Kripke and David Kaplan.

Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Gödel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.

An externalist account of exportation (inferring /de re/ belief from /de dicto/) is outlined. Kripke famously argued in /Naming and Necessity/ (1972) for his controversial thesis concerning certain English sentences that they are contingent /a priori/, in the derivative sense that the proposition semantically expressed is metaphysically contingent yet knowable /a priori/. Kripke also demonstrated that unrestricted exportation in attributions of belief is invalid. It is argued that these conclusions are, nearly enough, inconsistent. Kripke’s argument that belief exportation must be restricted points to a fallacy in his argument that his putative examples of the contingent /a priori/ are genuinely /a priori/.

A new argument is offered which proceeds through epistemic possibility (for all S knows, p), cutting a trail from modality to Millianism, the controversial thesis that the semantic content of a proper name is simply its bearer. New definitions are provided for various epistemic modal notions. A surprising theorem about epistemic necessity is proved. A proposition p can be epistemically necessary for a knowing subject S even though p is /a posteriori/ and S does not know p. The identity relation is well-behaved in metaphysically possible worlds but can go rogue in epistemically possible worlds. Whereas it can be epistemically possible that Lewis Carroll is not Charles Lutwidge Dodgson, this is not epistemically possible in the manner that anti-Millianism requires.

It is widely held that metaphysical modality is the broadest non-epistemic, alethic modality, and that /a posteriori/ modal essentialist truths, like that gold has atomic number 79, enjoy the necessity of the broadest alethic modality. One prominent argument for these conclusions--given by Cian Dorr, John Hawthorne, and Juhani Yli-Vakkuri--rests upon an extremely dubious premise: that certain pairs of properties—e.g., being gold and being made of atoms containing 79 protons—are one and the very same property. The two properties are seen to be distinct on independent philosophical grounds. Metaphysical modality is in fact a restricted alethic modality. In particular, mathematical modality is broader than metaphysical modality. Arguably, the broadest alethic modality is logical modality, which is distinct from metaphysical modality. Even if it is metaphysically necessary that gold have atomic number 79, there is no logical/analytical inconsistency in the supposition that it does not.

Alonzo Church provided three criteria for “strict synonymy”, i.e., sameness of semantic content: Alternatives (0), (1), and (2)--in order of increasing course-grainedness of content. On (2) expressions are strictly synonymous iff they are logically equivalent. (1) is a significant improvement over (2). On (1) expressions are synonymous iff they are lambda-convertible. Even on (1), assuming the Millian account of proper names, ‘Tully admires Cicero’ is deemed synonymous with ‘Cicero is self-admiring’. On (0) expressions are strictly synonymous iff they are “synonymously isomorphic” in Church’s sense. A fourth alternative, here called ‘(3)’, has been maintained by several philosophers. On (3) expressions are strictly synonymous iff they are co-intensional. In particular, sentences are strictly synonymous iff they are true with resect to the same possible worlds. This criterion’s notion of content is even more course grained than that of (2). Several objections to (3) are considered.

A new philosophical analysis is provided of the notorious Sleeping Beauty Problem. It is argued that the correct solution is one-third, but not in the way previous philosophers have typically meant this. A modified version of the Problem demonstrates that neither self-locating information nor amnesia is relevant to the core Problem, which is simply to evaluate the conditional chance of heads given an undated Monday-or-Tuesday awakening. Previous commentators have failed to appreciate the significance of the information that Beauty gains upon waking, and which is relevant to the conditional chance of heads: de re acquaintance with the awakening itself and the non-locating knowledge that it is an experimental awakening. David Lewis and company are committed to several unjustifiable and unacceptable probability assessments. Previous commentators have in effect confused the information that Beauty undergoes this particular experimental awakening for the information that she undergoes some experimental awakening or other. Lewis in particular thereby illegitimately tips the scales both in favor of heads and in favor of Monday. The Sleeping Beauty Problem is equivalent to a ball-in-urn word problem in elementary probability theory.

Toward a theory of n-tuples of individuals and concepts as surrogates for Russellian singular propositions and singular concepts. Alonzo Church proposed a powerful and elegant theory of sequences of functions and their arguments as singular-concept surrogates. Church’s account accords with his Alternative (0), the strictest of his three competing criteria for strict synonymy. The currently popular objection to strict criteria like (0) on the basis of the Russell-Myhill paradox is misguided. Russell-Myhill is not a problem specifically for Alternative (0). Rather it is a disproof of unrestricted concept comprehension. Unrestricted comprehension is also inconsistent with facts about sets of properties. It is demonstrated furthermore that the principal rival conception of propositions, as classes of possible worlds, is subject to a fatal philosophical collapse: It follows on that conception, given that each of us is fallible, that everyone believes everything. Although it is far superior to its principal rival conception, Church’s proposed theory is vulnerable under (0) to a version of Russell’s notorious Gray’s /Elegy/ objection. Some amendments to Church’s proposal are proffered that address Russell’s objection, including an amendment first proposed in the present author’s /Frege’s Puzzle/ (1986). Church’s response (personal correspondence) is considered.

An alternative to the classical Stalnaker-Lewis account of subjunctive conditionals is outlined. A distinction is drawn between a basic notion of “wouldness” and a more full-bloodedly modal variant, each with its own logic. Previous philosophers have challenged the alleged vacuity of counterpossibles using logico-mathematically impossible worlds. Here the vacuity thesis as well as other orthodox alleged logical principles are challenged instead through consideration of a logico-mathematically possible world. The impossible-world theorist’s Strangeness of Impossibility Condition is also challenged using the same logico-mathematically possible world. The dogma that the truth-condition of a subjunctive invokes the antecedent worlds sufficiently “closest” to (most like) the actual world is also challenged through consideration of the same logico-mathematically possible world. The dogma is a variation on Lewis’s counterpart theory, and is flawed in the same way.

A correct observation to the effect that a does not exist, where ‘a’ is a singular term, could be true on any of a variety of grounds. Typically, a true, singular negative existential is true on the unproblematic ground that the subject term ‘a’ designates something that does not presently exist. More interesting philosophically is a singular, negative existential statement in which the subject term ‘a’ designates nothing at all. Both of these contrast sharply with a singular, negative existential in which the subject term is a name from fiction. I argue that such singular, negative existential statements are false. My account of fictional characters differs significantly from Kripke’s. It is shown that an objection to my account rests on a serious misunderstanding. Finally, a crucial aspect of the account is emphasized.