Varieties of logical matrices
The paper introduces some tools in the study of (superintuitionistic) propositional logics. In the language non-standard connectives \(f_ 1,...,f_ k\) of any arity are allowed, besides the usual ones. A logic is a set of formulas containing all theorems of some fixed ''minimal'' logic \(L(f_ 1,...,f_ k)\) (whose axioms contain some superintuitionistic logic, and whose rules are: replacement, modus ponens, and congruence rules for the \(f_ i's)\) and closed under replacement and m.p. (not necessarily under congruence rules for the \(f_ i's)\). A logical matrix is defined as usual as a pseudo-Boolean algebra with operators and with a unary predicate of membership to a filter. A variety of l.m.'s is the class of all logical matrices satisfying a given set of formulas. The lattices of all logics and of all varieties of l.m.'s are isomorphic under the correspondence ''logic- characteristic (Lindenbaum) logical matrix''. Operators on classes of l.m.'s are defined which allow the description, in Birkhoff's style, of the variety generated by a class of l.m.'s. The study of Post complete logics is reduced to that of atoms of the lattice of varieties of l.m.'s, which are the varieties generated by 0- generated matrices. Thus some generalization of results in superintuitionistic logic are obtained, e.g. that any Post complete logic derives \(p\vee \neg p\), or that any classical logic is an intersection of logics which have just one Post complete extension. With regard to modal logics, the author introduces \(\omega\)-scales as a tool enriching the usual Kripke semantics, motivated by the fact that the logic \(GL'=GL+\square p\to p\) has no Kripke semantics (it is ''Kripke contradictory'' in the sense that for no usual Kripke frame (or scale) A is \(A\vDash GL')\). It is shown that GL' is complete for \(\omega\)-scales. By building logical matrices from certain \(\omega\)-scales, it is shown that there is a continuum of Kripke contradictory logics. By a complicated example it is shown that for each x there is a modal logic whose minimal (in cardinality) scale or \(\omega\)-scale has cardinality \(H_ x\) \((H_ 0=\aleph_ 0\), \(H_{x+1}=2^{H_ x}\), \(H_ y=\lim \{H_ x/\) \(x<y\}\) for y a limit). Other results: there is a proper extension of the logic Grz of Grzegorczyk, without the rule p/\(\square p\), which is a counterpart of intuitionistic logic, under the usual translation; and there are at least six pretabular extensions of S4.
lattice of varieties of logical matrices
scientific article; zbMATH DE number 3999894
scientific article