Abstract
Section 1 of this paper provides an introduction to this new “algebra of conditionals”, addresses various plausibility tests for such an algebra, provides a Venn diagram disproving a supposed counter-example, and answers various other objections raised in the literature about the efficacy of this algebraic extension of logic and conditional probability. Section 2 greatly simplifies the calculation of the implications of a set of conditional propositions or conditional events. These results depend on defining a deductive relation for conditionals (actually two have been found) with the property that the conjunction of two conditionals implies each of its components. That seemingly innocuous property assures that the deductively closed set implied by a finite set of n conditionals with respect to the deductive relation is implied by the single conditional formed by conjoining all n of them. The results are illustrated by solving several examples of deduction with several uncertain conditionals.
Highlights
The general topic of conditional events and conditional probability continues to be an active area of research, [1, 2, 6]
The main purpose of this paper is to greatly simplify my previous expositions of deduction with partially true conditionals1. This simplification is due to the discovery of two new deductive relations on conditional events that greatly simplify the problem of computing the implications of a set of two or more possibly uncertain conditionals
It will be shown that including this additional conditioning of the premise conditional by the condition d of the conclusion conditional (c|d) in the definition of deduction for conditionals, makes all the difference when calculating the implications of a set of conditionals
Summary
The general topic of conditional events and conditional probability continues to be an active area of research, [1, 2, 6]. The main purpose of this paper is to greatly simplify my previous expositions of deduction with partially true conditionals1 This simplification is due to the discovery of two new (at least to me) deductive relations on conditional events (and conditional propositions) that greatly simplify the problem of computing the implications of a set of two or more possibly uncertain conditionals. This problem arose when two Boolean conditionals (a given b) & (c given d) were “quasi-conjoined” and expressed as a single Boolean conditional (ab or cd) given (b or d), which (depending on the implication relation adopted) might not imply one or both of its components! This paper will eliminate that issue and answer some objections to the idea of treating conditionals whose components are Boolean propositions or probabilistic events as trivalent ordered pairs with operations that extend those of Boolean algebra
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