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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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