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

[Full paper in
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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.