Extensions of the ℵ0-Valued Lukasiewicz Propositional Logic

MV-algebras were introduced by Chang in 1958 preliminary to his providing an algebraic completeness proof for the K0-vaΓued Lukasiewicz propositional logic, LXo. In this paper a method is given for determining, for an arbitrary normal extension of LKo, an MV-algebra characteristic for the extension. The characteristic algebras are finite direct products of two sets of linearly ordered MV-algebras identified by Komori. Conversely, it is shown that, given an algebra which is isomorphic to the direct product of elements from these two sets of linearly ordered MV-algebras, a single axiom can be determined which, when added to the axioms for LXo, yields an axiomatization sound and weakly complete for the given algebra. As a consequence of the lattice ordering of these products of MV-algebras the cardinal and ordinal degrees of completeness of any normal extension of LKo can be determined. / Introduction In [4] Komori proves that any proper extension of LN o has a characteristic matrix isomorphic to a finite direct product of elements of two fundamental types of MV-algebras, thus providing a type of characterization for any axiomatic extension of LK o. We will provide a somewhat stronger result in Theorem 4.4 below as the principal result of this paper. Our result improves on Komori's by giving a method for determining which MV-algebras occur in the finite products. Using the notion of the genus of a formula, given in Rose [11], we show that a formula is satisfied in the Komori characterization if and only if it is of a corresponding genus. By connecting Komori's characterization with Rose's notion of genus it is possible to link characteristic matrices with extensions of LK o. This result amounts to a soundness and weak completeness theorem for the characterizations provided by Komori. 2 Lukasiewicz matrices The ^-valued Lukasiewicz propositional calculi (for n either an integer greater than 1 or Ko) were defined with the aid of logical Received October 26, 1992; revised March 10, 1993