Higher-order free logic and the Prior-Kaplan paradox

Bacon, Andrew; Hawthorne, John; Uzquiano, Gabriel

doi:10.1080/00455091.2016.1201387

“…In this section we shall examine one particular 'intensional paradox', attributed to Prior. That it enjoys the kind of status just described is evident from [2], at p. 497. In §2 of that study, with the phrase 'Priorean paradoxes' appearing in the heading, one reads the following: Let's begin by discussing a puzzling result of Prior (1961).…”

Section: On ∀P(qp → ¬P) [= #]mentioning

confidence: 82%

Which ‘Intensional Paradoxes’ are Paradoxes?

Tennant

2024

J Philos Logic

0

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We begin with a brief explanation of our proof-theoretic criterion of paradoxicality—its motivation, its methods, and its results so far. It is a proof-theoretic account of paradoxicality that can be given in addition to, or alongside, the more familiar semantic account of Kripke. It is a question for further research whether the two accounts agree in general on what is to count as a paradox. It is also a question for further research whether and, if so, how the so-called Ekman problem bears on the investigations here of the intensional paradoxes. Possible exceptions to the proof-theoretic criterion are Prior’s Theorem and Russell’s Paradox of Propositions—the two best-known ‘intensional’ paradoxes. We have not yet addressed them. We do so here. The results are encouraging. §1 studies Prior’s Theorem. In the literature on the paradoxes of intensionality, it does not enjoy rigorous formal proof of a Gentzenian kind—the kind that lends itself to proof-theoretic analysis of recondite features that might escape the attention of logicians using non-Gentzenian systems of logic. We make good that lack, both to render the criterion applicable to the formal proof, and to see whether the criterion gets it right. Prior’s Theorem is a theorem in an unfree, classical, quantified propositional logic. But if one were to insist that the logic employed be free, then Prior’s Theorem would not be a theorem at all. Its proof would have an undischarged assumption—the ‘existential presupposition’ that the proposition $$\forall p(Qp\!\rightarrow \!\lnot p)$$ ∀ p ( Q p → ¬ p ) exists. Call this proposition $$\vartheta $$ ϑ . §2 focuses on $$\vartheta $$ ϑ . We analyse a Priorean reductio of $$\vartheta $$ ϑ along with the possibilitate $$\Diamond \forall q(Qq\!\leftrightarrow \!(\vartheta \!\leftrightarrow \! q))$$ ◊ ∀ q ( Q q ↔ ( ϑ ↔ q ) ) . The attempted reductio of this premise-pair, which is constructive, cannot be brought into normal form. The criterion says we have not straightforward inconsistency, but rather genuine paradoxicality. §3 turns to problems engendered by the proposition $$\exists p(Qp\wedge \lnot p)$$ ∃ p ( Q p ∧ ¬ p ) (call it $$\eta $$ η ) for the similar possibilitate $$\Diamond \forall q(Qq\!\leftrightarrow \!(\eta \!\leftrightarrow \! q))$$ ◊ ∀ q ( Q q ↔ ( η ↔ q ) ) . The attempted disproof of this premise-pair—again, a constructive one—cannot succeed. It cannot be brought into normal form. The criterion says the premise-pair is a genuine paradox. In §4 we show how Russell’s Paradox of Propositions, like the Priorean intensional paradoxes, is to be classified as a genuine paradox by the proof-theoretic criterion of paradoxicality.

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“…Andrew Bacon, John Hawthorne, and Gabriel Uzquiano (Bacon, Hawthorne, and Uzquiano 2016) have recently considered a number of ways of rejecting or restricting Universal Instantiation (UI) and argued that they are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior (Prior 1961). I present a novel approach to the paradoxes by describing models that validate a restricted form of UI and avoid their concerns.…”

Section: Introductionmentioning

confidence: 99%

Paradoxes and Restricted Quantification

Tucker

¹

2018

Thought: A Journal of Philosophy

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Andrew Bacon, John Hawthorne, and Gabriel Uzquiano (Bacon, Hawthorne, and Uzquiano 2016) have recently argued that free logics—logics that reject or restrict Universal Instantiation—are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior (Prior 1961). These logics encompass ramified and contextualist approaches to paradoxes, and broadly speaking, there are two kinds of criticism they face. First, they fail to address every version of the Priorean paradoxes. Second, the theoretical considerations behind the logics make absolutely general statements about all propositions, properties of propositions, etc., but because this sort of intensional quantification is always restricted in the logics, they cannot even express those considerations. I present a novel sort of free logic, which rejects the standard Universal Instantiation but validates a restricted form of the rule, and which addresses both kinds of criticism by allowing some propositions to be unrestricted in their quantification.

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“…For discussions of various free higher‐order logics, including positive free higher‐order logic, see Bacon et al. 2016; Besson 2009; Skiba 2020.…”

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confidence: 99%

Higher‐order metaphysics

Skiba

¹

2021

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Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher‐order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico‐philosophical toolbox, while neither being reducible to nor fully explicable in terms of first‐order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, and identity, as well as on the metaphysics of modality.

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Higher-order free logic and the Prior-Kaplan paradox

Cited by 36 publications

References 32 publications

Which ‘Intensional Paradoxes’ are Paradoxes?

Which ‘Intensional Paradoxes’ are Paradoxes?

Paradoxes and Restricted Quantification

Higher‐order metaphysics

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