In All But Finitely Many Possible Worlds: Model-Theoretic Investigations on ‘Overwhelming Majority’ Default Conditionals

Abstract

Defeasible conditionals are statements of the form `ifAthen normallyB'. One plausible interpretation introduced in nonmonotonic reasoning dictates that ($$A\Rightarrow B$$AźB) is true iff B is true in `most' A-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of `overwhelming majority', defined as `truth in a cofinite subset of$$\omega $$ź', the first infinite ordinal. One approach employs the modal logic of the frame $$(\omega , <)$$(ź,<), used in the temporal logic of discrete linear time. We introduce and investigate conditionals, defined modally over $$(\omega , <)$$(ź,<); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of `majority' as sets cofinal (in $$\omega $$ź) rather than cofinite (subsets of $$\omega $$ź) is examined. For these modal approaches over $$(\omega , <)$$(ź,<), a decision procedure readily emerges, as the modal logic $${\mathbf {K4DLZ}}$$K4DLZ of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity; this allows also for a quick proof of $$\mathsf {NP}$$NP-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of $$\omega $$ź-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of $$\omega $$ź. This approach gives rise to weak conditional logics, as expected. The relative strength of the conditionals introduced is compared to (the conditional logic `equivalent' of) KLM logics and other conditional logics in the literature.