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A Way to Interpret Łukasiewicz Logic and Basic Logic

Thomas Vetterlein
Studia Logica: An International Journal for Symbolic Logic
Vol. 90, No. 3, Vagueness (Dec., 2008), pp. 407-423
Published by: Springer
Stable URL: http://www.jstor.org/stable/40269017
Page Count: 17
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A Way to Interpret Łukasiewicz Logic and Basic Logic
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Abstract

Fuzzy logics are in most cases based on an ad-hoc decision about the interpretation of the conjunction. If they are useful or not can typically be found out only by testing them with example data. Why we should use a specific fuzzy logic can in general not be made plausible. Since the difficulties arise from the use of additional, unmotivated structure with which the set of truth values is endowed, the only way to base fuzzy logics on firm ground is the development of alternative semantics to all of whose components we can associate a meaning. In this paper, we present one possible approach to justify ex post Łukasiewicz Logic as well as Basic Logic. The notion of ambiguity is central. Our framework consists of a Boolean or a Heyting algebra, respectively, endowed with an equivalence relation expressing ambiguity. The quotient set bears naturally the structure of an MV- or a BL-algebra, respectively, and thus can be used to interpret propositions of the mentioned logics.

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