ABSTRACT
This chapter provides an application of the external numbers to philosophy. It presents a way to model the Sorites paradox using nonstandard analysis, via the axiom scheme of External Induction and proposes the external numbers as models for the kind of vagueness expressed by the Sorites paradox. Ideal languages as a response to the Sorites paradox seem to have unsatisfying features for, according to R. Keefe, they seem to require giving up fundamental rules of inference such as modus ponens and Mathematical Induction. Natural languages distinguish between intension and extension of terms. The intension is the internal content of a term or concept while the extension is the range of applicability of a term by naming the particular objects that the term denotes. The chapter discusses an application of Kleene’s three-valued logic and applications of fuzzy logics because these seem to be the most relevant in what concerns the phenomenon of vagueness.
