We continue in this article the abstract algebraic treatment of quantum sentential logics Wil. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic - AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultrapower, Frege relation, Leibniz congruence, Suszko congruence, Leibniz operator) are applied to quantum logics. We also proved several equivalences between state property systems (Jauch-Piron-Aerts line of investigations) and AAL treatment of quantum logics (corollary 18 and 19). We show that there (...) exist the uniquely defined correspondence between state property system and consequence relation defined on quantum logics. We also signalize that a metalogical property - Lindenbaum property does not hold for the set of quantum logics. (shrink)

The first order modal logic of closure spaces belongs to the class of equationally definable standard modal logics . One can say, it satisfies no version of the deduction lemma. Nevertheless the Robinson and Beth theorems can be proved by means of an interpretation of modal theories in classical ones. LCS is described in [1], [2], [3], [4]. The logics obtained by adjoining axioms of quasi-equality or of equality to LCS are denoted by LCSQE and LCSE.

Closure spaces are generalizations of topological spaces, in which the Intersection of two open sets need not be open. The considered logic is related to closure spaces just as the standard logic S4 to topological ones. After describing basic properties of the logic we consider problems of representation of Lindenbaum algebras with some uncountable sets of infinite joins and meets, a notion of equality and a meaning of quantifiers. Results are extended onto the standard logic S4 and they are valid (...) also for some other standard modal logics. (shrink)