Interpreting connexive principles in coherence-based probability logic

TL;DR

This paper explores how coherence-based probability logic can be used to evaluate connexive principles, emphasizing the probabilistic constraints that uphold the intuition that certain conditionals should not hold when their antecedent contradicts their consequent.

Contribution

It introduces two probabilistic approaches within coherence-based logic to analyze connexive principles, including new notions of validity and connections between conditionals.

Findings

Probabilistic assessment of $A|ar{A}$ must be 0 for coherence.

Connexive principles can be analyzed via probabilistic constraints on conditionals.

Coherence-based logic provides a rich framework for connexive logic analysis.

Abstract

We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form "If $\sim A$, then $A$", should not hold, since the conditional's antecedent $\sim A$ contradicts its consequent $A$. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event $A|\overline{A}$ is $p(A|\overline{A})=0$. Moreover, connexive logics aim to capture the intuition that conditionals should express some "connection" between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle's Theses, Aristotle's Second Thesis, Abelard's First Principle and selected versions of Boethius' Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles.