Supervaluationism

In logic, supervaluationism is a semantics for dealing with irreferential singular terms and vagueness. Consider the sentence 'Pegasus likes licorice' in which the name 'Pegasus' fails to refer. What should its truth value be? There is nothing in the myth that would justify any assignment of values to it. On the other hand, consider 'Pegasus likes licorice or Pegasus doesn't like licorice' which is an instance of the valid schema 'p &or; ~p' (i.e. 'p or not-p'). Shouldn't it be true regardless of whether or not its disjuncts have a truth value? According to supervaluationism, it should be.

Precisely, let v be a classical valuation defined on every atomic sentence of the language L, and let At(x) be the number of distinct atomic sentences in x. Then there are at most 2^At(x) classical valuations defined on every sentence x. A supervaluation V is a function from sentences to truth values such that, x is super-true (i.e. V(x)=True) if and only if v(x)=True for every v; likewise for super-false. Otherwise, V(x) is undefined&mdash;i.e. exactly when there are two valuations v and v* such that v(x)=True and v*(x)=False.

For example, let Lp be the formal translation of 'Pegasus likes licorice'. Then there are exactly two classical valuations v and v* on Lp, viz. v(Lp)=True and v*(Lp)=False. So Lp is neither super-true nor super-false.