# Supervaluationism

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{{short description|Semantics for dealing with irreferential singular terms and vagueness}}
In [philosophical logic](/source/philosophical_logic), '''supervaluationism''' is a [semantics](/source/Formal_semantics_(logic)) for dealing with irreferential [singular term](/source/singular_term)s and [vagueness](/source/vagueness).<ref>[Shapiro, Stewart](/source/Stewart_Shapiro), "Vagueness and Conversation" in {{cite book |last=Beall |first=Edited |title=Liars and Heaps |publisher=Clarendon |location=Oxford, England |year=2003 |isbn=0-19-926481-3}}</ref> It allows one to apply the [tautologies](/source/Tautology_(logic)) of [propositional logic](/source/propositional_logic) in cases where [truth values](/source/truth_value) are undefined.

According to supervaluationism, a proposition can have a definite truth value even when its components do not. The proposition "[Pegasus](/source/Pegasus) likes [licorice](/source/licorice)", for example, is often interpreted as having no truth-value given the assumption that the name "Pegasus" [fails to refer](/source/failure_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 <math>p \vee \neg p</math> ("''<math>p</math> or not-<math>p</math>''"), so, according to supervaluationism, it should be true regardless of whether or not its [disjuncts](/source/Logical_disjunction) have a truth value; that is, it should be true in all interpretations. If, in general, something is true in all [precisifications](/source/precisification), supervaluationism describes it as "supertrue", while something false in all precisifications is described as "superfalse".<ref>{{cite web |title=Supervaluation: Definition from Answers.com |work=Oxford Dictionary of Philosophy |publisher=Oxford University Press |date=2005 |url=http://www.answers.com/topic/supervaluation |accessdate=2012-03-04}}</ref>

'''Supervaluations'''<!--boldface per WP:R#PLA--> were first formalized by [Bas van Fraassen](/source/Bas_van_Fraassen).<ref>[https://plato.stanford.edu/entries/logic-free/ Free Logic (Stanford Encyclopedia of Philosophy)]</ref>

== Example abstraction ==
Let ''v'' be a [classical](/source/Classical_logic) valuation defined on every [atomic sentence](/source/atomic_sentence) of the language ''L'' and let At(''x'') be the number of distinct atomic sentences in a formula ''x''. There are then at most 2<sup>At(''x'')</sup> 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](/source/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.

== See also ==
*[Kripke semantics](/source/Kripke_semantics)
*[Sorites paradox](/source/Sorites_paradox)
*[Subvaluationism](/source/Subvaluationism)

== References ==
{{reflist}}

== External links ==
* [Stanford Encyclopedia of Philosophy](/source/Stanford_Encyclopedia_of_Philosophy)
** [http://plato.stanford.edu/entries/vagueness/#5 Supervaluationism as a response to vagueness]
** [http://plato.stanford.edu/entries/sorites-paradox/#3.3 Supervaluationism as a response to the Sorites Paradox]

{{Formal semantics}}

Category:Semantics
Category:Theories of deduction

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