Modal logics with functional alternative relations.
Krister Segerberg · Notre Dame Journal of Formal Logic · 1986
Some families of modal logics form lattices; particularly important examples are the set of extensions of a modal logic and the set of normal extensions of a normal logic.One traditional way of studying such lattices, falling back on previous work in algebra, seeks to establish general properties of very big lattices.Thanks to Kit Fine, Wim Blok, Johan van Benthem, and others, this tradition is very much alive.But there is also an earlier tradition, related to but perhaps possible to distinguish from the one mentioned, where the ambition is to map out in complete detail sufficiently small lattices.The first work in this vein was Scroggs's celebrated [15], followed by Bull's equally celebrated [1].Other investigations in the same tradition are exemplified by [2], [6], [18], [19], [21], [22]; and works such as [5], [7], [11] also bear on it.In view of how enormously complicated the big lattices are, this tradition can never hope to develop very far.Nevertheless, where it is viable there may still be some interest in seeing it pursued.In this paper we will offer one such example, exploring the lattice of extensions of the normal modal logic KD C , where the schema D c .OA D DA is the converse of the wellknown "deontic" schema D. DA D OA.At the outset we may note that the only extensions of KD C other than the Inconsistent Logic (the normal extension of K by J.) which seem to have been described in the literature are KDΪ = KDD C (the smallest normal logic to contain both D and £> c ), the Trivial Logic (the normal extension of Kby the schema DA = A), and the Verum Logic (the normal extension of ΛΓby D±).The relationship between these logics is set out by the chart in Fig. 1.This, then, is the map whose white patches we propose to fill.The research reported in this paper was prompted by Brian Chellas's interest in functional modal logics, as explained in [4], and the author would like to acknowledge fruitful exchanges with him, Max Cress well, and Steve Thomason on this topic, in conversation and in correspondence.