In the modal literature various notions of completeness have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability (...) and preservation under ultrafilter extensions are introduced and studied as well. (shrink)

In this paper thecanonicalmodal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of anultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable Σ⊿-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any complete modal (...) logic defining a Σ⊿-elementary class of frames is canonical.The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in §2. The concept of a canonical modal logic is introduced and motivated in §3, which also contains the above-mentioned theorems. In §4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions is discussed.The modal language to be considered here has an infinite supply of proposition letters, a propositional constant ⊥, the usual Boolean operators ¬, ∨, ∨, →, and ↔ —with ¬ and ∨ regarded as primitives—and the two unary modal operators ◇ and □ — ◇ being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals. (shrink)

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)

The duality theory established by Halmos in [2] for boolean hemimorphism applies of course to the diagonalizable algebra, because ντν is an hemimorphism. For commodity in working on diagonalizable algebras we recall the basic facts and give the characteristic conditions on the dual of ντν.

In the modal literature various notions of "completeness" have been studied for normal modal logics. Four of these are defined here, viz. completeness, first-order completeness, canonicity and possession of the finite model property -- and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and (...) preservation under ultrafilter extensions are introduced and studied as well. (shrink)