A new publication by Philippe Schlenker ["Super Liars" in Review of Symbolic Logic 3(3)] presents an excellent new treatment of that old chestnut, the Liar paradox (exemplified in my post title). Schlenker's insight is to develop a technical logical semantics which, rather than trying to solve the paradox, tries to account for how human languages cope with the paradox while devising a sane system of truth values for human languages.
Schlenker begins with a now standard assumption that an account of Liars and other paradoxes in natural language requires a third truth value. He then shows in an easily readable fashion how this step leads directly to the need for allowing ordinal-many truth values, with a sort of infinite hierarchy of so-called "super liars" arising as well. The resulting treatment is as expressively complete as can be expected, providing the ability for a language to express that any given "super liar" sentence is something other than true.
The semantics of paradox must eventually be integrated into a functioning semantic theory for natural language; this paper makes a good start.
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