Abstract
We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, and B→C have probabilities a, b, c, r, and s, respectively, then for probability p of A→C, we have f(a,b,c,r,s)≤p≤g(a,b,c,r,s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner.
Highlights
The main part of probabilization of logical inference rules is defining the corresponding best possible bounds for probabilities of propositions
We present some probabilistic forms of the hypothetical syllogism rule covering both Hailperin’s approach, dealing with the case when probability P( A)
The most famous inference rules in propositional logics are modus ponens (B follows from A and A → B) and modus tollens (¬ A follows from ¬ B and A → B)
Summary
The main part of probabilization of logical inference rules is defining the corresponding best possible bounds for probabilities of propositions. Wagner’s paper [1] on probabilistic versions of modus ponens and modus tollens inference rules influenced this work immediately. This paper presents the first step in the process of the probabilization of a complete inference rule system for classical propositional calculus. We present some probabilistic forms of the hypothetical syllogism rule covering both Hailperin’s approach, dealing with the case when probability P( A). The given statement generalizes and contains both results of Hailperin’s modus ponens probabilized and Wagner’s modus tollens probabilized forms (see [1,10]). We obtained the probabilistic versions of Suppes-style modus ponens and modus tollens rules (see [1,10]) as an immediate consequence of our result. Our result will be of importance in the context of statistical, probabilistic, and approximate reasoning
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