Lessons of the Liar

Abstract

Abstract In proving that the language of arithmetic does not contain its own truth predicate, Tarski demonstrated that the claim that a language both satisfies certain minimal conditions and contains its own truth predicate leads to a contradiction – a result that can seem puzzling in light of the fact that it seems obvious that English does satisfy the relevant conditions, while containing its own truth predicate (though of course this cannot be). Chapter 5 explores the well‐known response to this problem (a version of the Liar paradox), which maintains that English is really an infinite hierarchy of languages defined by a hierarchy of Tarski‐style truth predicates. The construction of the hierarchy is explained, and the ways in which it is used to block different versions of the paradox are illustrated. The discussion then turns to problems with the approach, the most serious being the irresistible urge to violate the hierarchy's restrictions on intelligibility in the very process of setting it up – something we tend to forget because we imagine ourselves taking a position outside the hierarchy from which it can be described. Once we realize that the hierarchy is supposed to apply to the language we are using to describe it, the paradox returns with a vengeance, threatening to destroy the very construction that was introduced to avoid it.