Paradoxes of Logical Equivalence and Identity

Assuming the unrestricted application of classical logic, the paradoxes of truth, sets and properties make trouble for näıve intersubstitutivity principles, such as the principle that allows one to substitute, in non-intentional contexts, the claim that φ for the claim that ‘φ’ is true, or the claim that x is F for the claim that x has the property of being F . One way to respond to these paradoxes is to reject the logical assumptions the paradoxes rest on, allowing one to instead accept näıve intersubstitutivity principles governing sets, truth and properties. In this paper I show that even in these weakened logics intersubstitutivity principles can wreak havoc. In these discussions intersubstitutivity principles are normally formulated using relatively weak metalinguistic rules; the following paradoxes arise when a stronger intersubstitutivity axiom, formulated entirely in the object language, is assumed. In section 1 I outline a few different applications of these paradoxes: truth theorists, for example, might want to endorse a principle stating that logically equivalent sentences are substitutable salve veritate. Property and set theorists might want to endorse a version of Leibniz’s law. In section 2 I present two paradoxes that show that in either case they cannot endorse the principle in question, given background assumptions. The second of these paradoxes uses very little in the way of logical machinery, and thus applies to most logics developed to deal with the semantic and set theoretic paradoxes (see for example Bacon [1], Beall [3], Brady [5], Priest [14].) In the second half of section 2 I note that both paradoxes, when interpreted in terms of the notion of logical equivalence, are similar in spirit to recent versions of Curry’s paradox that employ the notion of a valid argument (see, for example, [19] and [4].) I then show that the present paradoxes can be formulated so as not to depend on any distinctive structural rules and, as a result, they are problematic for recent approaches to the validity Curry paradox that relax the rule of structural contraction (Zardini [20], Priest [15], Murzi & Shapiro [12].)