Connexive Logic, Probabilistic Default Reasoning, and Compound Conditionals
Niki Pfeifer, Giuseppe Sanfilippo · Studia Logica · 2023
Abstract We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the formif not-A,thenA, should not hold, since the conditional’s antecedentnot-Acontradicts its consequentA. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event $${A| \overline{A}}$$ A|A¯ is $${p(A| \overline{A})=0}$$ p(A|A¯)=0 . In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles:Aristotle’s theses,Aristotle’s Second Thesis,Abelard’s First Principle, andBoethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (likenon-symmetry). Both approaches satisfyminimalrequirements for a connexive logic. Finally, we compare both approaches conceptually.