Classicality Lost: K3 and LP after the Fall

Abstract

It is commonly held that the ascription of truth to a sentence is intersubstitutable with that very sentence. However, the simplest subclassical logics available to proponents of this view, namely K3 and LP, are hopelessly weak for many purposes. In this paper, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality o ered by proponents of LP are far less promising than those available to proponents of K3. This undermines the ability of proponents LP to engage in public reasoning in classical domains. IntroductionGottlob Frege famously held that "nothing is added to [a] thought by . . . ascribing to it the property of truth" ( , ). This idea is commonly expressed with the slogan that truth is transparent: φ and T r ( φ ) -the sentence that says that φ is true-are fully intersubstitutable in extensional contexts. Unfortunately, in classical logic, the law of excluded middle, i.e.φ ∨ ¬φ, and the rule of explosion, i.e. φ, ¬φ ∴ ψ, allow us to derive any sentence from the liar sentence if we have transparency. It's tempting to put the blame on transparency here.However, it isn't entirely obvious what to replace transparency with. That is why a number of authors have instead blamed classical logic. Saul Kripke ( ), Robert Martin and Peter ' φ ' is a term for φ in the object language. Note that the corner quotes here are Gödel quotes, not Quine quotes. To be fully precise, I would need to use both kinds of quotes; however, since corner quotes are commonly used for both, I allow myself the usual use-mention sloppiness here and throughout.See McGee ( ) and Halbach ( ) for surveys of some of the options.