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

Abstract

Defeasible conditionals of the form ‘if A then normally B’ are usually interpreted with the aid of a ‘normality’ ordering between possible states of affairs: $A\Rightarrow B$ is true if it happens that in the most ‘normal’ (least exceptional) A-worlds, B is also true. Another plausible interpretation of ‘normality’ introduced in nonmonotonic reasoning dictates that $A\Rightarrow B$ is true iff B is true in ‘most’ A-worlds. A formal account of ‘most’ in this majority-based approach to default reasoning has been given through the usage of (weak) filters and (weak) ultrafilters, capturing at least, a basic core of a size-oriented approach to defeasible reasoning. 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 all these modal approaches over $(\omega , <)$, a decision procedure readily emerges, as the modal logic $\mathbf {KD4LZ}$ of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity. A second approach employs the conditional version of Scott-Montague semantics, in the form of $\omega$, endowed with neighborhoods populated by its cofinite subsets. Again, different conditionals are introduced and examined. Although it is not feasible to obtain a completeness theorem, since it is not easy to capture ‘cofiniteness-in-$\omega$’ syntactically, this research reveals the possible structure of ‘overwhelming majority’ conditionals, whose relative strength is compared to (the conditional logic ‘equivalent’ of) KLM logics and other conditional logics in the literature.