Inphilosophical logic,supervaluationism is asemantics for dealing withirreferentialsingular terms andvagueness.[1] It allows one to apply thetautologies ofpropositional logic in cases wheretruth values are undefined.
According to supervaluationism, a proposition can have a definite truth value even when its components do not. The proposition "Pegasus likeslicorice", for example, is often interpreted as having no truth-value given the assumption that the name "Pegasus"fails to refer. If indeed reference fails for "Pegasus", then it seems as though there is nothing that can justify an assignment of a truth-value to any apparent assertion in which the term "Pegasus" occurs. The statement "Pegasus likes licorice or Pegasus doesn't like licorice", however, is an instance of the valid schema (" or not-"), so, according to supervaluationism, it should be true regardless of whether or not itsdisjuncts have a truth value; that is, it should be true in all interpretations. If, in general, something is true in allprecisifications, supervaluationism describes it as "supertrue", while something false in all precisifications is described as "superfalse".[2]
Supervaluations were first formalized byBas van Fraassen.[3]
Letv be aclassical valuation defined on everyatomic sentence of the languageL and let At(x) be the number of distinct atomic sentences in a formulax. There are then at most 2At(x) classical valuations defined on every sentencex. A supervaluationV is a function from sentences to truth values such thatx is supertrue (i.e.V(x)=True)if and only ifv(x)=True for everyv. Likewise for superfalse.
V(x) is undefined when there are exactly two valuationsv andv* such thatv(x)=True andv*(x)=False. For example, letLp be the formal translation of "Pegasus likes licorice". There are then exactly two classical valuationsv andv* onLp, namelyv(Lp)=True andv*(Lp)=False. SoLp is neither supertrue nor superfalse.
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