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.
Dedicated to Petr Hájek
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Notes
- 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.
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Acknowledgments
We are grateful to two anonymous referees for several remarks on an earlier version of this chapter that led to improvements in exposition, and to shorter proofs of some of the results given here.
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Aguzzoli, S., Marra, V. (2015). Two Principles in Many-Valued Logic. In: Montagna, F. (eds) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-06233-4_8
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DOI: https://doi.org/10.1007/978-3-319-06233-4_8
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