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.

Two Principles in Many-Valued Logic / S. Aguzzoli, V. Marra (OUTSTANDING CONTRIBUTIONS TO LOGIC). - In: Petr Hájek on Mathematical Fuzzy Logic / [a cura di] F. Montagna. - Dordrecht : Springer, 2015. - ISBN 978-3-319-06233-4. - pp. 159-174 [10.1007/978-3-319-06233-4_8]

Two Principles in Many-Valued Logic

S. Aguzzoli
;
V. Marra
2015

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.
Basic logic ; Łukasiewicz logic ; Gödel logic ; Product logic ; Triangular norms ; Deduction theorem
Settore MAT/01 - Logica Matematica
Settore INF/01 - Informatica
2015
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/242051
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