# Supervaluationism

Supervaluationism, in logic, is a semantics for dealing with irreferential singular terms and vagueness.[1]

In the statement "Pegasus likes licorice", for example, the name "Pegasus" fails to refer[clarification needed] and there is nothing in the Pegasus myth that would justify any assignment of values to it, suggesting that the sentence is a vacuous truth. According to supervaluationism, however, such borderline statements lack any truth value.[2] The statement "Pegasus likes licorice or Pegasus doesn't like licorice", however, is an instance of the valid schema [p ∨ ~p] ("p or not-p"), so, according to supervaluationism, it should be true regardless of whether or not its disjuncts have a truth value; that is, it should be true in all interpretations. If, in general, something is true in all precisifications, supervaluationism describes it as "supertrue", while something false in all precisifications is described as "superfalse".[3]

## Example abstraction

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. There are then 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 supertrue (i.e. V(x)=True) if and only if v(x)=True for every v. Likewise for superfalse.

V(x) is undefined when there are exactly 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". There are then exactly two classical valuations v and v* on Lp, namely v(Lp)=True and v*(Lp)=False. So Lp is neither supertrue nor superfalse.