On the Strict–Tolerant Conception of Truth

Abstract

We discuss four distinct semantic consequence relations which are based on Strong Kleene theories of truth and which generalize the notion of classical consequence to 3-valued logics. Then we set up a uniform signed tableau calculus(the strict–tolerant calculus), which we show to be sound and complete with respect to each of the four semantic consequence relations. The signs employed by our calculus are , , and , which indicate a strict assertion, strict denial, tolerant assertion and tolerant denial respectively. Recently, Ripley applied the strict–tolerant account of assertion and denial (originally developedby Cobreros et al. [2012] to bear on vagueness) to develop a new approach to truth and alethic paradox, which we call the Strict–Tolerant Conceptionof Truth (STCT). The paper aims to contribute to our understanding of STCT in at least three ways. First, by developing the strict–tolerant calculus.Second, by developing a semantic version of the strict–tolerant calculus(assertoric semantics), which informs us about the (strict–tolerant) assertoric possibilities relative to a fixed ground model. Third, by showing that the strict–tolerant calculus and assertoric semantics jointly suggest that STCT's claim that the strict and tolerant can be understood in terms of one another has to be reconsidered.