Normal forms in infinite-valued logic: The case of one variable

Abstract

Let [0,1] be the real unit interval. A Schauder hat is a Λ-shaped function h:[0,1] → [0,1] whose four pieces are given by linear polynomials with integral coefficients. Rose and Rosser gave an effective method to represent every Schauder hat by a sentence in the infinite-valued calculus of Lukasiewicz. We give an effective method to reduce every sentence ψ with one variable, to an equivalent sentence φ which is a disjunction of Schauder hat sentences. Since the equivalence between ψ and φ holds in all n-valued calculi, our normal form reduction may be used for a uniform (i.e., n-free) treatment of deduction in these calculi. For the case under consideration, our methods already yield a self-contained and constructive proof of McNaughton's theorem stating that in the infinite-valued calculus every piecewise linear function with integral coefficients is representable by some sentence.