Truth and the Liar Paradox

Chapter 5 discusses two issues involving truth, the disquotation schema, and the Boolean many-valued approach. In the Boolean many-valued approach, the (deductive) law of excluded middle is retained but the (semantic) principle of bivalence is abandoned. However, Timothy Williamson has argued against the tenability of this option, contending that excluded middle must imply bivalence. The first half of the chapter refutes Williamson’s argument by showing that truth in a Boolean many-valued semantics, i.e., the highest value of a Boolean algebra, is not disquotational truth that Williamson needs in his argument. The former truth predicate is called the correspondence truth predicate and distinguished from the disquotational truth predicate. The second half of the chapter offers the Boolean 4-valued solution to the Liar Paradox, in which each ungrounded atomic sentence such as the Liar sentence is given one of the two intermediate values between truth and falsity. The disquotation schema breaks down for the ungrounded atomic sentences, and therefore the paradox is avoided. The last section concludes the entire book.