# 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.

## Keywords

Defeasible conditionals Default reasoning Conditional logics of normality## References

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