At least since the work of Tarski, the Liar paradox has stood in the way of an acceptable account of the notion of truth. It has been less noticed that once one admits a truth predicate into a formal language, along with intuitively valid inferences involving the truth predicate, standard classical logic becomes inconsistent. So, any acceptable account of truth must both explicate how sentences get the truth values they have and amend classical logic to avoid the inconsistency. A natural account of a trivalent semantics arises from treating the problem of assigning truth values to sentences as ... More

At least since the work of Tarski, the Liar paradox has stood in the way of an acceptable account of the notion of truth. It has been less noticed that once one admits a truth predicate into a formal language, along with intuitively valid inferences involving the truth predicate, standard classical logic becomes inconsistent. So, any acceptable account of truth must both explicate how sentences get the truth values they have and amend classical logic to avoid the inconsistency. A natural account of a trivalent semantics arises from treating the problem of assigning truth values to sentences as akin to a boundary‐value problem in physics. The resulting theory solves the Liar paradox while avoiding the usual ‘revenge’ problems. It also suggests a natural modification of classical logic that blocks the paradoxical reasoning. This semantic theory is wedded to an account of the normative standards governing assertion and denial of sentence and a metaphysical analysis truth and factuality. The result is an account in which sentences like the Liar sentence are neither true nor false, and correspond to no facts.