Paradoxes and Restricted Quantification

Abstract

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.

A Theory of Structured Propositions

“…We have defined the truth value of (∀ σ F )a in terms of the truth value of F [a/c] for arbitrary c. But the term c might itself 64 At least, arbitrarily subject to the constraint that if a 1 ...an are respectively equivalent to a ′ 1 ...a ′ n then |Ra 1 ...an| = |Ra ′ 1 ...a ′ n |. 65 If you want to revisit these assumptions, see for instance Tucker andThomason (2011), Tucker (2018), Kaplan (1995), Deutsch (2014), Bacon et al (2016), Bacon and Russell (2019).…”

Modes of Truth

“…7. Ramification is spelled out as a response to Prior's paradox in Tucker andThomason (2011), andKaplan (1995) and Kripke (2011) express sympathy to this line of response in the context of some related paradoxes. See Tucker (2018) for a non-standard approach to the ban on impredicativity, that does not involve explicit ramification.…”

Against Disquotation

“…Universal instantiation fails, despite there being such a thing as Hesperus/Phosphorus. Opacity-inducedINSTANTIATION-failures are invoked byBacon [2021] as a response to the intensional paradoxes discussed in section 7 and byGoodman [2017] as a Fregean response to Russell's paradox of structured propositions.14 Note that Russell's view was more complicated, involving a hierarchy of such principles, each corresponding to a different quantifier Kaplan [1995],Tucker and Thomason [2011],Kripke [2011], andTucker [2018]. express sympathy for some version of this approach; see alsoBacon et al [2016].15 In particular, the following three schemas are inconsistent in classical first-order logic enriched with a termforming functor 'the proposition that … ':…”