Although we believe the results reported below to have direct philosophical import, we shall for the most part confine our remarks to the realm of mathematics. The reader is referred to [4] for a philosophically oriented discussion, comprehensible to mathematicians, of tense logic.The “minimal” tense logicT0is the system having connectives ∼, →,F(“at some future time”), andP(“at some past time”); the following axioms:(whereGandHabbreviate ∼F∼ and ∼P∼ respectively); and the following rules:(8) fromαandα → β, inferβ,(9) fromα, infer any substitution instance ofα,(10) fromα, (...) inferGα,(11) fromα, inferHα.A tense logic is a systemTwhose language is that ofT0and whose axioms and rules include (1)–(11). The axioms and rules ofTother than (1)–(11) are calledproperaxioms and rules.We shall investigate three systems of semantics for tense logics, i.e. three notions ofstructureand three relations ⊧ which stand between structures and formulas. One reads⊧αas “αis valid in.” A structureis amodelof a tense logicTif every formula provable inTis valid in. A semantics isadequateforTif the set of models ofTin the semantics is characteristic forT, i.e. if wheneverT ∀ αthen there is a modelofTin the semantics such that∀ α. Two structuresand, possibly from different semantics, are calledequivalent(∼) if exactly the same formulas are valid inas in. (shrink)

Given a normal (multi-)modal logic a characterization is given of the finitely presentable algebras A whose logics L A split the lattice of normal extensions of . This is a substantial generalization of Rautenberg [10] and [11] in which is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok [2] that for all cycle-free and finite A L A splits the lattice of normal extensions of K. Although we (...) firmly believe it to be true, we have not been able to prove that if a logic splits the lattice of extensions of then is the logic of an algebra finitely presentable over ; in this respect our result remains partial. (shrink)