General frame

Abstract: This paper presents a new technique for handling modal logics with transitive frames, i.e. extensions of the modal system K4. In effect, the technique is based on the following fundamental result, to be obtained below in §3. Given a formula φ, we can effectively construct finite frames 1, …, n which completely characterize the set of all transitive general frames refuting φ. More exactly, an arbitrary general frame refutes φ iff contains a (not necessarily generated) subframe such that (1) i, for some i ϵ {1, …, n}, is a p-morphic image of (after Fine [1985] we say is subreducible to i), (2) is cofinal in , and (3) every point in that is not in does not get into “closed domains” which are uniquely determined in i, by φ. This purely technical result has, as it turns out, rather unexpected and profound consequences. For instance, it follows at once that if φ determines no closed domains in the frames 1, …, n associated with it, then the normal extension of K4 generated by φ has the finite model property and so is decidable. Moreover, every normal logic axiomatizable by any (even infinite) set of such formulas φ also has the finite model property. This observation would not possibly merit any special attention, were it not for the fact that the class of such logics contains almost all the standard systems within the field of K4 (at least all those mentioned by Segerberg [1971] or Bull and Segerberg [1984]), all logics containing S4.3, all subframe logics of Fine [1985], and a continuum of other logics as well.

Abstract: The previous chapters have concentrated on general frame theory. We have only seen a few concrete frames, and most of them were constructed via manipulations on an orthonormal basis for an arbitrary separable Hilbert space. An advantage of this approach is that we obtain universal constructions, valid in all Hilbert spaces.

Abstract: Publisher Summary It is noted that comparative logic requires, that the propositions should in any case be true or false and this clearly forced the totality of the order in the group-theoretical framework. Going beyond the use of totally ordered abelian groups, so to encompass the lattice-ordered and, more generally, the partially ordered groups has proved very successful. Following this, the setting up of a general frame becomes possible, in which, beside the old comparative logic, other interesting logics also find a quite satisfactory assessment. This chapter focuses at the presentation of this frame however only for the propositional case. The chapter describes basic syntactical notions and facts, basic semantical notions and facts, some other stronger logics, and comparative logic.

Abstract: In this chapter the author explains modal logics of reactive frames. He shows the correspondence between logic, reactive frame property, and the shattered frame unfolded property. He uses classical tools to study completeness in a new interpretation of modal logic, successfully studying some of its subsystems.