The aim of this essay is to show that one can give a proof-theoretical semantics to vague predicates (most of the observational
predicates). The starting point is the critic of the truth-conditional analysis of vagueness in terms of degrees of truth,
and the solution proposed is based of the anti-realist principle
meaning is use. Therefore the problem of vague is replaced by the problem of graduality.
Through the example of the predicate to be small are presented the principles of the anti-realist approach. The semantical representation of the predicate is a formula of
the Intuitionistic Type Theory of Martin-Löf (TT). This formula is introduced in a context (in the sense of TT) which formalizes
the conditions of use. Among the contextual parameters is required a contextual “objective” property B which must have bounds. Hence in a given context a paraphrase of the assertion a is small is a is small enough in order to satisfy B.
From this point of view, the logical puzzles called sorits illustrate the relevance of the proof-theoretical approach. They
appear as pedagogical devices which stress the difference between the semantical content of a proposition and its logical
behaviour. The semantical content defined in proof-theoretical terms supports graduality, whereas the logical behaviour forbids
it.