Kleene's “strong” three-valued logic

We began Chapter 1 by noting that sentences concerning borderline cases of vague predicates pose counterexamples to the Principle of Bivalence. For example, the sentence Mary Middleford is tall appears to be neither true nor false. We begin our exploration of logics for vagueness by dropping the Principle of Bivalence and allowing sentences to be either true (T), false (F), or neither true nor false (N – if you like, you may also say that N is neutral). This gives rise to three-valued (trivalent) systems of logic. We use the same language as classical propositional logic. Truth-value assignments can now assign N (as well as T or F) to atomic formulas, and we'll use this value to signal the application of a vague predicate to a borderline case.

How are the truth-functions for the standard propositional connectives defined over the three values? There are several plausible choices, and the set of truth-functions we choose will define a specific system of three-valued logic. In this chapter we present four well-known systems of three-valued logic. Many others have been developed, but these four systems are sufficient to explore the flavor of three-valued logics and how they might be used to tackle problems associated with vagueness.