The LPi and LPi1/2 logics: two complete fuzzy systems joining Lukasiewicz and Product Logics
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Title: | The LPi and LPi1/2 logics: two complete fuzzy systems joining Lukasiewicz and Product Logics | |||
Journal: | Archive for Mathematical Logic | |||
Volume | 40 | |||
Number | 1 | |||
Pages: | 39-67 | |||
Year: | 2001 |
Abstract
In this paper we provide a finite axiomatization (using two finitary ryles only) for the propositional logic (called LPi) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from LPi by the adding of a constant symbol and of a defining axiom for 1/2, called LPi1/2. We show that LPi1/2 contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with Delta, and the Product and Gödel's Logics with Delta and involution. Standard completeness results are proved by means of investigating the algebras corresponding to LPi and LPi1/2. For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebra can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z.