Kripke's Theory of Truth and the Liar Paradox

Jay Newhard

Kripke has developed a formal theory of truth which captures important intuitions about natural language truth predicates by permitting the truth predicate to be partially defined and to seek its own level. Through the construction of fixed points, Kripke's theory also provides mathematical definitions of an ungrounded sentence and a paradoxical sentence. While the mathematical definition of a paradoxical sentence yields an explanation of the Liar Paradox in terms of the theory--namely, that paradoxical sentences cannot be assigned a truth value in any fixed point--there is not an explanation as to why paradoxical sentences cannot be assigned a truth value in any fixed point. Worse, since the Liar sentence is formulatable in the fixed point languages, Kripke's theory apparently fails to capture every way of ascending to a metalanguage. If the "ghost of the Tarski hierarchy" cannot be dispelled, the many important insights into truth and the Liar Paradox advanced by Kripke‚s theory are undermined.

In this paper it is argued that Kripke's theory does allow a solution to the Liar Paradox, given three plausible moves. The first is to commit to propositions as the proper bearers of truth. The second is to recognize syntactically well-formed sentences which do not express propositions. The third is to distinguish two aspects of interpreted predicates, one a formal aspect, the other a conceptual aspect. These aspects have not yet been distinguished in the literature per se, though the notions upon which each aspect is based are not uncommon. These moves allow us to explain the apparently intractable ascent to a metalanguage, with the result that the Liar sentence, though paradoxical on the mathematical definition, is not pathological. The solution offered here suggests a reformulated version of the Liar Paradox; it is argued that the reformulated Liar sentence may be seen as showing the incompleteness of the extension of the truth predicate, but is not paradoxical, except in Kripke's mathematical sense.