# 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.

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