Schlenker, Philippe. 2009. "Super Liars". Manuscript (43 pages), Institut Jean-Nicod and NYU (revised and expanded, May 2009)

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Abstract: Kripke's theory of truth offered a trivalent semantics for a language which, like English, contains a truth predicate and means of self-reference; but it did so by severely restricting the expressive power of the logic.  In Kripke's analysis, the Liar (e.g. This very sentence is not true) receives the indeterminate truth value, but this fact cannot be expressed in the language; by contrast, it is straightforward to say in English that the Liar is something other than true. Kripke’s theory also fails to handle the Strengthened Liar, which can be expressed in English as: This very sentence is something other than true. We develop a theory which seeks to overcome these difficulties, and is based on a detailed analysis of some of the linguistic means by which the Strengthened Liar can be expressed in English.  In particular, we propose to take literally the quantificational form of the negative expression something other than true. Like other quantifiers, it may have different implicit domain restrictions, which give rise to a variety of negations of different strengths (e.g. something other than true among the values {0, 1}, or among {0, 1, 2}, etc). This analysis naturally leads to a logic with as many truth values as there are ordinals - a conclusion reached independently by Cook 2008a. We develop the theory within a generalization of the Strong Kleene Logic, augmented with negations that each have a non-monotonic semantics. We show that fixed points can be constructed for our logic, and that it enjoys a limited form of ‘expressive completeness’. Finally, we discuss the relation between our theory and various alternatives, including one in which the word true (rather than negation) is semantically ambiguous, and gives rise to a hierarchy of truth predicates of increasing strength.