Tyler Burge, Haim Gaifman, and the team of Jon Barwise and John Etchemendy have all proposed interesting new solutions to the liar paradox in recent years. The three solutions bear a striking family resemblance. I will explore in this chapter the important similarities and differences among the three approaches. I will also show how Gaifman' s theory can be extended in order to provide the sort of formal pragmatics that the other two solutions require. Finally, I will prove that only two levels of truth (‘true0’ and ‘true1’) are needed in the interpretation of natural language.

The so-called strengthened liar paradox provides an important motivation for all three theories. It is simply an objection to any theory about the paradoxes that tries to distinguish between falsehood and other kinds of failure to be true (having a truth-value gap, or failing to express a proposition). Consider the following liar sentence (A):

(A) Sentence (A) is not true.

The value-gap analyst describes such paradoxical sentences as being neither true nor false (perhaps because they “fail to express a proposition”). Sentence (B) is therefore a consequence of the value-gap position:

(B) Sentence (A) is not true (because gappy).

Sentence (B) is just sentence (A), so the value-gap theorist is forced to describe an integral part of her own theory as being neither true nor false, failing to express a proposition. Gaifman has described the liar paradox as a semantic “black hole,” sucking proposed solutions into the void.