Gilio, Angelo and Over, David E. and Pfeifer, Niki and Sanfilippo, Giuseppe (2024) On trivalent logics, probabilistic weak deduction theorems, and a general import-export principle. ARTIFICIAL INTELLIGENCE, 337: 104229. ISSN 0004-3702, 1872-7921
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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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Levels of knowledge; Betting schemes; Coherence; Conditional random quantities; Trivalent logics; Compound and iterated conditionals; Inference rules; Aristotelian syllogisms; Probabilistic entailment; Probabilistic weak deduction theorems; General import-export principle |
| Subjects: | 100 Philosophy & psychology > 100 Philosophy |
| Divisions: | Philosophy, Art History, History, and Humanities > Institut für Philosophie > Lehrstuhl für Theoretische Philosophie (Prof. Dr. phil Hans Rott) |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 30 Oct 2025 07:59 |
| Last Modified: | 30 Oct 2025 07:59 |
| URI: | https://pred.uni-regensburg.de/id/eprint/64663 |
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