Probabilistic consequence relations

TL;DR

This paper explores probabilistic logical consequence relations in propositional logic, analyzing three types—material, preservation, and symmetric—showing how they relate to classical logic and supervaluationism at different probability thresholds.

Contribution

It extends previous work on probabilistic consequence relations from SET-FMLA to SET-SET settings, examining open and closed thresholds and their logical implications.

Findings

Material consequence aligns with classical logic at any threshold.

Preservation consequence approaches classical logic only at threshold 1.

Symmetric consequence converges to classical logic at threshold 1.

Abstract

This paper investigates logical consequence defined in terms of probability distributions, for a classical propositional language using a standard notion of probability. We examine three distinct probabilistic consequence notions, which we call material consequence, preservation consequence, and symmetric consequence. While material consequence is fully classical for any threshold, preservation consequence and symmetric consequence are subclassical, with only symmetric consequence gradually approaching classical logic at the limit threshold equal to 1. Our results extend earlier results obtained by J. Paris in a SET-FMLA setting to the SET-SET setting, and consider open thresholds beside closed ones. In the SET-SET setting, in particular, they reveal that probability 1 preservation does not yield classical logic, but supervaluationism, and conversely positive probability preservation yields subvaluationism.