The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined even if it is not sufficiently formal and precise to belong to mathematics proper (in a narrow sense)—and even if (as many have asserted) for that reason the Church–Turing thesis is unprovable. It is proved logically that the class of effective procedures is not decidable, i.e., that no effective procedure is possible for ascertaining whether a given procedure is effective. This (...) undecidability result is proved directly from the notion itself of an effective procedure, without reliance on any (partly) mathematical lemma, conjecture, or thesis invoking recursiveness or Turing-computability. In fact, there is no reliance on anything very mathematical. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of an effective procedure. Though the result that effectiveness is undecidable is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure invariably terminates with the correct verdict. (shrink)

A paradox about sets of properties is presented. The paradox, which invokes an impredicatively defined property, is formalized in a free third-order logic with lambda-abstraction, through a classically proof-theoretically valid deduction of a contradiction from a single premise to the effect that every property has a unit set. Something like a model is offered to establish that the premise is, although classically inconsistent, nevertheless consistent, so that the paradox discredits the logic employed. A resolution through the ramified theory of types (...) is considered. Finally, a general scheme that generates a family of analogous paradoxes and a generally applicable resolution are proposed. (shrink)

This is a non-technical version of "The Decision Problem for Effective Procedures." The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined, even if it does not have a purely mathematical definition—and even if (as many have asserted) for that reason, the Church–Turing thesis (that the effectively calculable functions on natural numbers are exactly the general recursive functions), cannot be proved. However, it is logically provable from the notion of an (...) effective procedure--without reliance on any (partially) mathematical thesis or conjecture concerning effective procedures, such as the Church–Turing thesis--that the class of effective procedures is undecidable, i.e., that no effective procedure is possible for ascertaining whether a given procedure is effective. The proof does not appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of such a procedure. Though this undecidability result itself is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure is, in fact, a decision procedure, i.e., effective—for example, that it invariably terminates with the correct verdict. (shrink)

This article offers an interpretation of a controversial aspect of Frege’s The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for ‘0’, ‘successor’, and ‘number’, and also against a proposed definition for ‘direction’ as applied to lines in geometry. Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain kind. (...) These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define ‘number’ and ‘direction’. That is, the proposed definitions, even when taken together with the extra-definitional facts, fail to fix unique semantic extensions for ‘number’ and ‘direction’, and thereby fail to fix unique truth-values for sentences like ‘Caesar is a number’ and ‘England is a direction’. A minor modification of the criticized definitions well-defines ‘0’, ‘successor’ and ‘number’, thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense. (shrink)

Frege held that the bare demonstrative ‘that’ is incomplete, and that it is the word together with a gesture that serves as the designating expression, and likewise that it is the word ‘yesterday’ together with the time of utterance that designates the relevant day. David Kaplan’s original theory of indexicals holds that Frege’s supplementation thesis is correct about demonstratives but incorrect about ‘yesterday’. Kaplan’s account of demonstratives deviates from Frege’s in treating supplemented demonstratives as directly referential, hence rigid. It is (...) argued here that the gesture or other demonstration that accompanies an utterance of ‘that’ is not part of the designating expression but instead part of the utterance context. (shrink)

Theoria, Volume 88, Issue 2, Page 434-437, April 2022.

University of California, Santa Barbara.

Paradoxes of nested modality, like Chisholm’s paradox, rely on S4 or something stronger as the propositional logic of metaphysical modality. Sarah-Jane Leslie’s objection to the resolution of Chisholm’s paradox by means of rejection of S4 modal logic is investigated. A modal notion of essence congenial to Leslie’s objection is clarified. An argument is presented in support of Leslie’s crucial but unsupported assertion that, on pain of inconsistency, an object’s essence is the same in every possible world. A fallacy in the (...) argument is exposed. Alternative interpretations of Leslie’s objection are provided and are found to involve equivocation between different notions of “essence.” A material artifact’s modal essence, as distinct from its quiddity essence, could have been different than it is. (shrink)

A Russellian notion of what it is for a proposition to be “directly about” something in particular is defined. Various strong and weak, and mediate and immediate, Russellian notions of general aboutness are then defined in terms of Russellian direct aboutness. In particular, a proposition is about something iff the proposition is either directly, or strongly indirectly, about that thing. A competing Russellian account, due to Kaplan, is criticized through a distinction between knowledge by description and denoting by description. The (...) epistemological significance of Russellian aboutness is assessed. A Russellian substitute for de re propositional attitude is considered. (shrink)

Nathan Salmon; XII*—How to Measure the Standard Metre, Proceedings of the Aristotelian Society, Volume 88, Issue 1, 1 June 1988, Pages 193–218.

In this candid autobiographical essay, Nathan Salmon recounts and assesses the impact of various philosophers and events on his philosophical development.

On Kripke’s intended definition, a term designates an object x rigidly if the term designates x with respect to every possible world in which x exists and does not designate anything else with respect to worlds in which x does not exist. Kripke evidently holds in Naming and Necessity, hereafter N&N (pp. 117–144, passim, and especially at 134, 139–140), that certain general terms – including natural-kind terms like ‘‘water’’ and ‘‘tiger’’, phenomenon terms like ‘‘heat’’ and ‘‘hot’’, and color terms like (...) ‘‘blue’’ – are rigid designators solely as a matter of philosophical semantics (independently of empirical, extra-linguistic facts). As a consequence, Kripke argues, identity statements involving these general terms are like identity statements involving proper names (e.g., ‘‘Clark Kent=Superman’’) in that, solely as a matter of philosophical semantics, they express necessary truths if they are true at all. But whereas it is reasonably clear what it is for a (first-order) singular term to designate, Kripke does not explicitly say what it is for a general term to designate. General terms are standardly treated in modern logic as predicates, usually monadic predicates. There are very forceful reasons – due independently to Church and Godel, and ultimately to Frege – for taking predicates to designate their semantic extensions. But insofar as the extension of the general term ‘‘tiger’’ is the class of actual tigers (or its characteristic function), it is clear that the term does not rigidly designate its extension, since the class of tigers in one possible world may differ from the class of tigers in another. What, then, is it for ‘‘tiger’’ to be rigid? (shrink)

The dogma that the propositional logic of metaphysical modality is S5 is rebutted. The author exposes fallacies in standard arguments supporting S5, arguing that propositional metaphysical modal logic is weaker even than both S4 and B, and is instead the minimal and weak metaphysical-modal logic T.

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. (shrink)

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. (shrink)

A sequel to the author’s book /Frege’s Puzzle/ (1986).

A thesis (“weak BCP”) nearly universally held among philosophers of probability connects the concepts of objective chance and metaphysical modality: Any prospect (outcome) that has a positive chance of obtaining is metaphysically possible—(nearly) equivalently, any metaphysically impossible prospect has zero chance. Particular counterexamples are provided utilizing the monotonicity of chance, one of them related to the four world paradox. Explanations are offered for the persistent feeling that there cannot be chancy metaphysical necessities or chancy metaphysical impossibilities. Chance is objective but (...) contrary to popular opinion it is also largely epistemic. Chancy necessities are analogous to necessary a posteriori truths. Philosophy and Phenomenological Research, Volume 99, Issue 3, Page 644-662, November 2019. (shrink)

We defend hylomorphism against Maegan Fairchild’s purported proof of its inconsistency. We provide a deduction of a contradiction from SH+, which is the combination of “simple hylomorphism” and an innocuous premise. We show that the deduction, reminiscent of Russell’s Paradox, is proof-theoretically valid in classical higher-order logic and invokes an impredicatively defined property. We provide a proof that SH+ is nevertheless consistent in a free higher-order logic. It is shown that the unrestricted comprehension principle of property abstraction on which the (...) purported proof of inconsistency relies is analogous to naïve unrestricted set-theoretic comprehension. We conclude that logic imposes a restriction on property comprehension, a restriction that is satisfied by the ramified theory of types. By extension, our observations constitute defenses of theories that are structurally similar to SH+, such as the theory of singular propositions, against similar purported disproofs. (shrink)

A thorough critique (extracted from the author’s 1979 doctoral dissertation) of Kripke’s purported derivation, in footnote 56 of his philosophical masterpiece /Naming and Necessity/, of nontrivial modal essentialism from the theory of rigid designation.

My title is meant to suggest a continuation of the sort of philosophical investigation into the nature of language and modality undertaken in Rudolf Carnap’s Meaning and Necessity and Saul Kripke’s Naming and Necessity. My topic belongs in a class with meaning and naming. It is demonstratives—that is, expressions like ‘that darn cat’ or the pronoun ‘he’ used deictically. A few philosophers deserve particular credit for advancing our understanding of demonstratives and other indexical words. Though Naming and Necessity is concerned (...) with proper names, not demonstratives, it opened wide a window that had remained mostly shut in Meaning and Necessity but that, thanks largely to Kripke, shall forevermore remain unbarred. Understanding of demonstrative semantics grew by a quantum leap in David Kaplan’s remarkable work, especially in his masterpiece “Demonstratives” together with its companion “Afterthoughts.” In contrast to the direct-reference propensities of these two contemporary figures, Gottlob Frege, with his uncompromisingly thoroughgoing intensionalism, shed important light on the workings of demonstratives in “Der Gedanke”—more specifically, in a few brief but insightful remarks from a single paragraph concerning tense and temporal indexicality. (shrink)

Standard compositionality is the doctrine that the semantic content of a compound expression is a function of the semantic contents of the contentful component expressions. In 1954 Hilary Putnam proposed that standard compositionality be replaced by a stricter version according to which even sentences that are synonymously isomorphic (in the sense of Alonzo Church) are not strictly synonymous unless they have the same logical form. On Putnam’s proposal, the semantic content of a compound expression is a function of: (i) the (...) contentful component expressions; and (ii) the expression’s logical form. Kit Fine recently expanded and modified Putnam’s idea into a sweeping theory in philosophy of language and philosophy of mind. The present paper is a detailed critique of Fine’s “semantic relationism.” Fine’s notion of coordination is explained in terms of the familiar pragmatic phenomenon of recognition. A serious error in Fine’s formal disproof of standard Millianism is exposed. It is demonstrated furthermore that Church’s original criticism of Putnam’s proposal can be extended to Fine’s semantic relationism. Finally, it is also demonstrated that the positive position Fine proffers to supplant standard Millianism is in fact exactly equivalent to standard Millianism, so that Fine’s overall position not only does not displace standard Millianism but is in fact inconsistent. (shrink)

Charles Carlini interviews Nathan Salmón about the philosophical work of his mentor and friend, the late Saul Kripke, one of the foremost philosophers of the 20th Century.

An expanded version of the author’s candid, autobiographical “My Philosophical Education”.