Series: Penn State Logic Seminar

Date: Tuesday, September 30, 2003

Time: 2:30 - 3:45 PM

Place: 324 Sackett Building

Speaker: Dale Jacquette, Penn State, Philosophy

Title: Denying the Liar


The liar paradox as an informal semantic diagonalization has impressed
philosophers and mathematicians as a challenge to the bivalence of
propositional logic.  A sentence that declares its own falsehood as a
semantic self-non-application that appears to be true if and only if
it is false has served as the model for similar kinds of paradoxes in
the form of Grelling's heterology paradox, Russell's paradox in set
theory with unrestricted descriptive comprehension, and even G"odel's
incompleteness theorems.  Efforts to solve or resolve the liar have
produced Tarski's ascending hierarchy of object and infinitely
iterated metalanguages, Kripke's transfinitely ramified theory of
truth value gaps, and Hartry Field's variations on Kripke's proposal,
among other interesting reactions.  I identify and formalize three
requirements for the liar paradox in the bivalence of classical logic,
the Tarskian deflationary or disquotational truth convention, and the
definability of the liar sentence, and argue that while the first liar
horn goes through, beginning with the assumption that the liar
sentence is true and proving that in that case it is false, the second
horn, from the assumption that the liar sentence is false, fails.  I
diagnose the failure of the second liar paradox dilemma horn as
resulting from the fact the the conditional needed to prove the truth
of the liar from the assumption of its falsehood is contradicted by
the three requirements that have generally been supposed to be
sufficient to generate the liar.  If the reasoning is correct, then
there is no liar paradox to be solved.