Properties of independently axiomatizable bimodal logics

Abstract

In monomodal logic there are a fair number of high-powered results on completeness covering large classes of modal systems; witness for example Fine [74], [85] and Sahlqvist [75]. Monomodal logic is therefore a well-understood subject in contrast to polymodal logic, where even the most elementary questions concerning completeness, decidability, etc. have been left unanswered. Given that in many applications of modal logic one modality is not sufficient, the lack of general results is acutely felt by the “users” of modal logics, contrary to logicians who might entertain the view that a deep understanding of one modality alone provides enough insight to be able to generalize the results to logics with several modalities. Although this view has its justification, the main results we are going to prove are certainly not of this type, for they require a fundamentally new technique. The results obtained are called transfer theorems in Fine and Schurz [91] and are of the following type. Let L ∌ ⊥ be an independently axiomatizable bimodal logic and L⎕ and L∎ its monomodal fragments. Then L has a property P iff L⎕ and L∎ have P. Properties which will be discussed are completeness, the finite model property, compactness, persistence, interpolation and Halldén-completeness. In our discussion we will prove transfer theorems for the simplest case when there are just two modal operators, but it will be clear that the proof works in the general case as well.

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

“…Unlike the previous combination methods for the word problem in the nondisjoint case [10,11], this approach has the known decidability transfer results for validity in the fusion of modal logics [12,15] as consequences. Our combination result is, however, more general than these transfer results since it applies also to non-normal modal logics-thus answering in the affirmative a long-standing open question in modal logics-and to equational theories not induced by modal logics (see, e.g., Example 3.6).…”

“…This is a more sophisticated example of a Gaussian theory, in which the string of existential quantifiers ∃z in (1) can be both not empty and applied to a non-trivial solver. 12 Next, we give an example of a theory that is not Gaussian.…”

“…Let u z , u 0 , u 1 abbreviate respectively u(x, z), u(x, 0), u(x, 1). If we replace every occurrence of t(x, z), t(x, 0), t(x, 1) in (12) by u z , u 0 , u 1 , respectively, and translate the formula (11) into the signature of BR, we obtain a formula that is equivalent in BR to (12). To see that, consider the following chains of equalities modulo the signature translation and the axioms of BA and BR: 25…”

“…and the relativized validity problem (Does the formula ϕ follow from the global assumption ψ?). There are strong combination results showing that in many cases decidability transfers from two modal logics to their fusion [12][13][14][15][16][17][18]. Again, transfer results for the harder decision problem, relativized validity, are easier to show than for the simpler one, validity.…”

Heterogeneous Fibring of Deductive Systems Via Abstract Proof Systems

“…Among the methods for combining logics, fusion of modalities (logics presented by Hilbert calculi at the deductive level and Kripke structures at the semantic level) [24] is the best understood in what concerns preservation of properties as soundness, weak completeness, uniform Craig interpolation (for theoremhood) and decidability via finite model property (see [26,19,13]). Further results on preservation of weak completeness can be seen in [10].…”