On trivalent logics, probabilistic weak deduction theorems, and a general import-export principle

In this paper we first recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. We discuss the equivalence between conditional bets and bets on conditionals, and review de Finetti's trivalent analysis of conditionals. But we go beyond de Finetti's early trivalent logical analysis and his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with the approach to compound conditionals introduced by Gilio and Sanfilippo within the framework of conditional random quantities. As we use the notion of p-entailment, the full deduction theorem does not hold. We prove a Probabilistic Weak Deduction Theorem for conditional events. After that we study some variants of it, with further results, and we present several examples. Moreover, we illustrate how to derive new inference rules related to selected Aristotelian syllogisms. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Weak Deduction Theorem. We use the inference from a disjunction, A or B, to the conditional, if not-A then B, as an example to show the invalidity of this principle. We introduce a General Import-Export principle by examining examples and counterexamples. In particular, when considering the inference rules of System P, we find that a General Import-Export principle is satisfied, even if the assumptions of the Probabilistic Weak Deduction Theorem do not hold. We also deepen further aspects related to p-entailment and p-consistency. Finally, we briefly discuss some related work relevant to AI.