Supervaluationism
Encyclopedia
In logic
, supervaluationism is a semantics for dealing with irreferential singular term
s 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 ∨ ~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—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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, supervaluationism is a semantics for dealing with irreferential singular term
Singular term
There is no really adequate definition of singular term. Here are some definitions proposed by different writers:# A term that tells us which individual is being talked about....
s and vagueness
Vagueness
The term vagueness denotes a property of concepts . A concept is vague:* if the concept's extension is unclear;* if there are objects which one cannot say with certainty whether belong to a group of objects which are identified with this concept or which exhibit characteristics that have this...
. 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 ∨ ~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
Atomic sentence
In logic, an atomic sentence is a type of declarative sentence which is either true or false and which cannot be broken down into other simpler sentences...
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—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.