Sequence and Consequence

TL;DR

This paper explores the logic of omega-sequence semantics for conditionals, axiomatizing it, analyzing its properties, and generalizing to transfinite sequences, while addressing its motivation and validity.

Contribution

It axiomatizes the logic of omega-sequence semantics, introduces new axioms, and generalizes the semantics to transfinite sequences, clarifying their logical properties.

Findings

The logic of omega-sequence semantics is characterized by adding Flattening and Sequentiality to C2.

Sequentiality is complex and likely invalid, while Flattening is more plausible.

Generalizing to transfinite sequences simplifies the logic by adding only Flattening.

Abstract

In the course of proving a tenability result about the probabilities of conditionals, van Fraassen (1976) introduced a semantics for conditionals based on omega-sequences of worlds, which amounts to a particularly simple special case of ordering semantics for conditionals. On that semantics, 'If p, then q' is true at an omega-sequence just in case q is true at the first tail of the sequence where p is true (if such a tail exists). This approach has become increasingly popular in recent years. However, its logic has never been explored. We axiomatize the logic of omega-sequence semantics, showing that it is the result of adding two new axioms to Stalnaker's logic C2: one, Flattening, which is prima facie attractive, and, and a second, Sequentiality, which is complex and difficult to assess, but, we argue, likely invalid. But we also show that when sequence semantics is generalized from omega-sequences to arbitrary (transfinite) ordinal sequences, the result is a more attractive logic that adds only Flattening to C2. We also explore the logics of a few other interesting restrictions of ordinal sequence semantics. Finally, we address the question of whether sequence semantics is motivated by probabilistic considerations, answering, pace van Fraassen, in the negative.