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2015 | OriginalPaper | Chapter

8. Two Principles in Many-Valued Logic

Authors : Stefano Aguzzoli, Vincenzo Marra

Published in: Petr Hájek on Mathematical Fuzzy Logic

Publisher: Springer International Publishing

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Abstract

Classically, two propositions are logically equivalent precisely when they are true under the same logical valuations. Also, two logical valuations are distinct if, and only if, there is a formula that is true according to one valuation, and false according to the other. By a real-valued logic we mean a many-valued logic in the sense of Petr Hájek that is complete with respect to a subalgebra of truth values of a BL-algebra given by a continuous triangular norm on [0, 1]. Abstracting the two foregoing properties from classical logic leads us to two principles that a real-valued logic may or may not satisfy. We prove that the two principles are sufficient to characterise Łukasiewicz and Gödel logic, to within extensions. We also prove that, under the additional assumption that the set of truth values be closed in the Euclidean topology of [0, 1], the two principles also afford a characterisation of Product logic.

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previous chapter The Differential Semantics of Łukasiewicz Syntactic Consequence
next chapter How Do -Groups and Po-Groups Appear in Algebraic and Quantum Structures?
Footnotes
1
It should be emphasised that there is some leeway in formulating the separating conditions \(\mu (\alpha )>0\) and \(\nu (\alpha )=0\) here: see Corollary 8.1 below for equivalent variants.
 
2
Usage of the symbol \(\oplus \) to denote ordinal sums seems fairly standard. It is also standard to use \(\oplus \) to denote Łukasiewicz’s strong disjunction, see Cignoli et al. (2000). This we will do in Sect. 8.4, where context should prevent confusion.
 
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Metadata
Title
Two Principles in Many-Valued Logic
Authors
Stefano Aguzzoli
Vincenzo Marra
Copyright Year
2015
Book
Petr Hájek on Mathematical Fuzzy Logic

Print ISBN: 978-3-319-06232-7

Electronic ISBN: 978-3-319-06233-4

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-06233-4_8

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