Probability and Conditionals

TLDR

Absolute probabilities are treated as degrees of rational belief, and the primary semantics for the axiom system is linked to this probabilistic interpretation. The paper aims to connect a semantical theory of conditional statements with the theory of conditional probability. The authors interpret probability calculus as semantics for truth‑functional logic, define conditional probabilities via absolute probabilities, extend it to counter‑factuals, introduce conditional propositions equating absolute probability with the conditional probability of the consequent on the antecedent, and recover an axiom system for this connective.

Abstract

The aim of the paper is to draw a connection between a semantical theory of conditional statements and the theory of conditional probability. First, the probability calculus is interpreted as a semantics for truth functional logic. Absolute probabilities are treated as degrees of rational belief. Conditional probabilities are explicitly defined in terms of absolute probabilities in the familiar way. Second, the probability calculus is extended in order to provide an interpretation for counter-factual probabilities—conditional probabilities where the condition has zero probability. Third, conditional propositions are introduced as propositions whose absolute probability is equal to the conditional probability of the consequent on the antecedent. An axiom system for this conditional connective is recovered from the probabilistic definition. Finally, the primary semantics for this axiom system, presented elsewhere, is related to the probabilistic interpretation.