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

Abstract

Defeasible conditionals are statements of the form ‘if A then normally B’. One plausible interpretation introduced in nonmonotonic reasoning dictates that ($A\Rightarrow B$) 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 $\omega$’, the first infinite ordinal. One approach employs the modal logic of the frame $(\omega , <)$, used in the temporal logic of discrete linear time. We introduce and investigate conditionals, defined modally over $(\omega , <)$; 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 $\omega$) rather than cofinite (subsets of $\omega$) is examined. For these modal approaches over $(\omega , <)$, a decision procedure readily emerges, as the modal logic ${\mathbf {K4DLZ}}$ 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 $\mathsf {NP}$-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of $\omega$-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of $\omega$. 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.