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)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem of finite relation (...) algebras. (shrink)

Gives the first published adaptation of the Lindenbaum/Henkin method of maximal consistent sets for establishing the completeness of modal propositional logics with respect to the relational models of Kripke.

It is shown that the many-dimensional modal logic K n , determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any $n > 2$ . On the other hand, K n is determined by a class of frames satisfying a single first-order sentence.

We give a simple proof of the canonicity of Sahlqvist identities, using methods that were introduced in a paper by Jónsson and Tarski in 1951.

Let be the class of frames satisfying the condition.

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem of finite relation (...) algebras. (shrink)

We prove that every n-modal logic between K n and S5 n is undecidable, whenever n ≥ 3. We also show that each of these logics is non- finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov-Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a (...) reduction of the (undecidable) representation problem of finite relation algebras. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite (...) relation algebras. (shrink)

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae (...) are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

A first-order sentence is quasi-modal if its class of models is closed under the modal validity preserving constructions of disjoint unions, inner substructures and bounded epimorphic images. It is shown that all members of the proper class of canonical structures of a modal logic Λ have the same quasi-modal first-order theory Ψ Λ . The models of this theory determine a modal logic Λ e which is the largest sublogic of Λ to be determined by an elementary class. The canonical (...) structures of Λ e also have Ψ Λ as their quasi-modal theory. In addition there is a largest sublogic Λ c of Λ that is determined by its canonical structures, and again the canonical structures of Λ c have Ψ Λ are their quasi-modal theory. Thus Ψ Λ = Ψ Λ c = Ψ Λ e . Finally, we show that all finite structures validating Λ are models of Ψ Λ , and that if Λ is determined by its finite structures, then Ψ Λ is equal to the quasi-modal theory of these structures. (shrink)

A first-order sentence isquasi-modalif its class of models is closed under the modal validity preserving constructions of disjoint unions, inner substructures and bounded epimorphic images.It is shown that all members of the proper class of canonical structures of a modal logicΛhave the same quasi-modal first-order theoryΨΛ. The models of this theory determine a modal logicΛewhich is the largest sublogic ofΛto be determined by an elementary class. The canonical structures ofΛealso haveΨΛas their quasi-modal theory.In addition there is a largest sublogicΛeofΛthat is (...) determined by its canonical structures, and again the canonical structures ofΛehaveΨΛare their quasi-modal theory. Thus. Finally, we show that all finite structures validatingΛare models ofΨΛ, and that ifΛis determined by its finite structures, thenΨΛis equal to the quasi-modal theory of these structures. (shrink)

We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...) Thomason). The constructions use the result of Erd $\H{o}$ s that there are finite graphs with arbitrarily large chromatic number and girth. (shrink)

We show that every extension of the basic hybrid logic with modal Sahlqvist axioms is complete. As a corollary of our approach, we also obtain the Beth property for a large class of hybrid logics. Finally, we show that the new completeness result cannot be combined with the existing general completeness result for pure axioms.

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)