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The $$\mathcal{L}_n^m$$-propositional calculus. (English) Zbl 1349.03090
Summary: T. Almada and J. Vaz de Carvalho [Stud. Log. 69, No. 3, 329–338 (2001; Zbl 0995.03047)] stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the $$\mathcal{L}_n^m$$-propositional calculus, denoted by $$\ell_n^m$$, is introduced in terms of the binary connectives $$\to$$ (implication), $$\twoheadrightarrow$$ (standard implication), $$\wedge$$ (conjunction), $$\vee$$ (disjunction) and the unary ones $$f$$ (negation) and $$D_i$$, $$1\leq i\leq n-1$$ (generalized Moisil operators). It is proved that $$\ell_n^m$$ belongs to the class of standard systems of implicative extensional propositional calculi. Besides, it is shown that the definitions of $$L_n^m$$-algebra and $$\ell_n^m$$-algebra are equivalent. Finally, the completeness theorem for $$\ell_n^m$$ is obtained.
##### MSC:
 03G20 Logical aspects of Łukasiewicz and Post algebras 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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