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)

In the following we will use the well-known correspondence between modal logics and varieties of modal algebras in our investigation of the function which assigns to a modal logic its degree of incompleteness. A modal algebra is an algebra A = where is a Boolean algebra and is a unary operation satisfying 1 = 1 and = x y ; is called a modal operator. A variety of algebras is a class of algebras closed under the operations of forming homomorphic (...) images, subalgebras as and direct products, and if K is a class of algebras then V denotes the smallest variety containing K. The variety of modal algebras is denoted by M, the subvariety of M dened by the equation x x = x by MR and the subvariety of M dened by the equation x n = x n1 , n a natural number, by Mn . Here x 0 = x, x n = , n a natural number. We write for the lattice of subvarieties of a variety K. If K, K 0 are varieties such that K $ K 0 but for no variety K 00 K $ K 00 $ K 0 then we say that K 0 is a cover of K. The smallest normal modal logic { containing the axiom ! and closed under the inference rules of modus ponens, substitution and necessitation { will be denoted by K; the lattice of modal logics which are extensions of K by. (shrink)