# Models for Identity in Three-Valued Logics

## September 5, 2024

Abstract: There is a natural way to interpret the propositional connectives and quantifiers in terms of the three semantic values 0, i, and 1, where 0 and 1 are understood as falsity and truth, and i is understood as some intermediate value. These three-valued valuations do not, by themselves, determine a logic, because for that, you need to settle how models are used to provide a counterexample to a sequent.

If you take a counterexample to the argument from A to B to be a model that assigns A the value 1 and B some value other than 1 (either 0 or i), the resulting logic is Kleene’s strong three-valued logic, K3. If a counterexample is a model assigning A the value 1 or i and B the value 0, the resulting logic is Priest’s logic of paradox, LP. If a counterexample is a model assigning A the value 1 and B the value 0, then the result is the logic ST of Strict–Tolerant validity. The three logics are different generalisations of two-valued Boolean logic to a tri-valuational setting.

The logic ST is distinctive, in that it is, in some sense, a reformulation of classical logic—every classically valid sequent in this language is ST-valid—but since ST allows for strictly non-classical models, there are ST theories which are not classical theories. The Cut rule is not unrestrictedly valid in ST. For example, if the formula P takes the value i in every model, and in the resulting theory, each sequent AP and PB is valid, while AB need not be valid.

There have been a number of different proposals concerning the logic of the identity predicate in this three-valued setting, mostly involving making minimal changes to the classical evaluation conditions, given the underlying ideology of K3 or LP evaluations and their respective treatments of the indeterminate semantic value i. In this talk, I will use the relationship between ST evaluations and classical logic to show how there is a well-behaved class of three-valued models for the identity predicate that is much wider than has been previously proposed.

The key result involves characterising the three-valued models that provide no ST counterexamples to sequents valid in classical first-order predicate logic with identity. In this talk, I provide independent characterisation of such models, showing how the class generalises prior three-valued models for identity, and exploring how these models can be understood from the point of view of the logics ST, K3 and LP.