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 A ⊢ P and P ⊢ B is valid, while A
⊢ B 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.
