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An alternative definition of quantifiers on four-valued Łukasiewicz algebras. (English) Zbl 1414.03014

G. C. Moisil’s four-valued Łukasiewicz algebras, L\(_4\)-algebras, were introduced in [Ann. Sci. Univ. Jassy, Sect. I, Math. 26, 431–466 (1940; Zbl 0025.00409)]. More recently, the notion of a monadic L\(_4\)-algebra, namely an L\(_4\)-algebra equipped with an additional existential quantifier, has been presented [G. Georgescu et al., Mult.-Valued Log. 3, No. 3, 235–257 (1998; Zbl 0920.06004)]. The paper under review introduces an alternative definition of the existential quantifier for (monadic) L\(_4\)-algebras.
The main results are in Sections 2–5.
Section 2 is devoted to introduce the new \(\frac{2}{3}\) existential quantifier and the subsequent notion of a monadic \(\frac{2}{3}\) L\(_4\)-algebra. A number of related results is presented. The class of monadic \(\frac{2}{3}\) L\(_4\)-algebras forms a variety of algebras denoted by \(\mathbb{M}_\frac{2}{3}\mathbb{L}_4\).
In Section 3, there is a comparison between the algebras of \(\mathbb{M}_\frac{2}{3}\mathbb{L}_4\) and the class of monadic four-valued Łukasiewicz-algebras. It is shown that they are polynomially equivalent.
Section 4 analyzes the differences between some existential quantifiers for L\(_4\)-algebras, namely Boolean, lattice, and the new \(\frac{2}{3}\) existential quantifier.
Finally, in Section 5 a completeness theorem for \(\frac{2}{3}\)-monadic Łukasiewicz predicate logic is presented.

MSC:

03G20 Logical aspects of Łukasiewicz and Post algebras
03B50 Many-valued logic
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
06D35 MV-algebras
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