Abstract: There is a very natural way to interpret the propositional connectives and quantifiers, relative to the algebra of three semantic values, {0, i, 1} where 0 and 1 are understood as the traditional values of falsity and truth, and the third value is some intermediate value. The evaluation clauses do not, by themselves, determine the logic, because for that, you need to determine how models are used to provide a counterexample to a sequent. If a counterexample is given by a model that assigns every premise the value 1 and assigns every conclusion a value other than 1, the resulting logic is Kleene’s strong three-valued logic, K3. If a counterexample is a model assigning every premise the value 1 or i and every conclusion the value 0, the resulting logic is Priest’s logic of paradox, LP. If a counterexample is a model assigning every premise the value 1 and every conclusion the value 0, you get the logic ST of Strict-Tolerant validity. ST is distinctive, in that it is, in some sense, classical logic—every classically valid sequent in this language is ST-valid—but since it has strictly non-classical models, there are ST theories which are not classical theories.
What does this mean for the logic of identity in a three-valued context?
In this talk, I explain how the classical logic of identity, when interpreted in ST models, gives us a well-behaved class of three-valued models for identity—much larger than the traditional models for identity, used by afficianados of LP or K3—which can be used to model the distinctive non-classical behaviour of identity statements, with a greater degree of freedom than we might have thought.
