A tempting solution to problems of semantic vagueness and to the Liar Paradox is an appeal to truth-value gaps. It is tempting to say, for example, that, where Harry is a borderline case of bald, the sentence
(1) Harry is bald
is neither true nor false: it is in the ‘gap’ between these two values, and perhaps deserves a third truth-value. Similarly with the Liar Paradox. Consider the following Liar sentence:
(2) (2) is false.
That is, sentence (2) says of itself that it is false. If we accept the Tarskian schema
where ‘S’ is a name of a sentence ‘p,’ we are led into paradox. Both the assumption that (2) is true, and the assumption that (2) is false lead us, via (T), to
(3) (2) is true if and only if (2) is false.
Given this result, a natural reaction is to place (2) in a ‘gap’ between true and false.