In this paper we recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. Then, we show the equivalence between bets on conditionals and conditional bets, by reviewing de Finetti's trivalent analysis of conditionals. But our approach goes beyond de Finetti's early trivalent logical analysis and is based on 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 our approach to compound conditionals. We prove a Probabilistic Deduction Theorem for conditional events. After that, we study some probabilistic deduction theorems, by presenting several examples. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic 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 the Import-Export principle. We also introduce a General Import-Export principle and we illustrate it by examining some p-valid inference rules of System P. Finally, we briefly discuss some related work relevant to AI.

Deduction Theorem: The Problematic Nature of Common Practice in Game Theory Holger I. MEINHARDT † August 2, 2019 We consider the Deduction Theorem that is used in the literature of game theory to run a purported proof by contradiction. In the context of game theory, it is stated that if we have a proof of φ null ϕ, then we also have a proof of φ ϕ. Hence, the proof of φ ϕ is deduced from a previous known statement. However, we argue that one has to manage to prove that the clauses φ and ϕ exist, i.e., they are known true statements in order to establish that φ null ϕ is provable, and that therefore φ ϕ is provable as well. Thus, we are only allowed to reason with known true statements, i.e., we are not allowed to assume that φ or ϕ exist. Doing so, leads immediately to a wrong conclusion. Apart from this, we stress to other facts why the Deduction Theorem is not applicable to run a proof by contradiction. Finally, we present an example from industrial cooperation where the Deduction Theorem is not correctly applied with the consequence that the obtained result contradicts the well-known aggregation issue. MS Classifications 2010: 03B05, 91A12, 91B24 Keywords: Propositional Logic, Deduction Theorem, Herbrand Theorem, Proof by Contradiction, TU Games, Cooperative Oligopoly Games, Partition Function Approach, γ -Belief, Nash Equilibrium, Aggregation across Firms. 1 Introduction We review a common practice in the literature of game theory of applying the Deduction Theorem (Herbrand Theorem, 1930) on a purported proof by contradiction.