Interpreting connexive principles in coherence-based probability logic

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.

• The paper introduces two approaches in coherence-based probability logic to interpret connexive principles using probabilistic constraints and three-valued conditionals.
• It validates classical theses from Aristotle, Abelard, and Boethius while revealing limitations in certain connexive claims.
• The study leverages de Finetti's coherence to handle zero-probability events, offering practical insights and avenues for future research.

Analyzing Connexive Principles in Coherence-Based Probability Logic

The paper "Interpreting Connexive Principles in Coherence-Based Probability Logic" by Niki Pfeifer and Giuseppe Sanfilippo presents a detailed exploration of connexive principles through the lens of coherence-based probability logic. This research analyzes the validity of selected connexive principles, which arise from the intuition that conditionals with contradictory antecedents and consequents should not hold, such as "if not-$A$, then $A$". The principles examined include well-known theses attributed to philosophers such as Aristotle, Abelard, and Boethius.

The authors propose two distinct approaches to interpret connexivity within coherence-based probability logic. The first approach employs probabilistic constraints on conditional events, conceptualizing conditionals as defaults or negated defaults. Notably, it interprets a conditional $A \rightarrow C$ as a default $A\normally C$, corresponding to the probabilistic constraint $p(C|A)=1$, while $\lsim (A \rightarrow C)$ translates to $p(C|A) \neq 1$. Within this framework, the paper validates several connexive principles (such as Aristotle's Theses and Boethius' original theses) but finds others, including Aristotle’s Second Thesis and certain variations of Boethius' theses, to be non-valid.

The second approach takes a different path, treating basic conditionals as three-valued conditional events, and interprets logical operations among them as conditional random quantities. This method unifies the treatment of iterated and non-iterated connexive principles under a single validity definition. Here, logical operations are explored by mapping them to compounds of conditionals and iterated conditionals. Despite this alternative perspective, it arrives at conclusions similar to the first approach for some principles while differing in others, particularly validating some reversed Boethius' theses that were not validated under the first approach.

Both approaches share a rigorous application of coherence-based probability logic, which offers a general framework superior to classical logic, particularly in handling conditioning events with zero probability. Moreover, this perspective circumvents certain classical probabilistic complications, such as Lewis' triviality results, by relying on the principle of coherence formulated by Bruno de Finetti.

The exploration demonstrates the versatility of coherence-based probability logic in addressing non-classical logics and highlights its rich applicability for connexive logic's needs. Future directions proposed by the authors include extending these methodologies to other principles in alternative logical systems. This could potentially establish new connections and enrich the theoretical landscape of probability logic, thus enhancing its applicability in reasoning about conditionals with complex dependencies.

Overall, the paper provides insightful numerical analyses and explores the theoretical underpinnings of connexive logic principles, contributing to a nuanced understanding of coherence-based probability logic's role in validating such principles. While it acknowledges the limitations within each approach's scope, it simultaneously invites further investigation into refining these approaches or developing new frameworks to unify connexive principles under a coherent probabilistic interpretation.