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A logic of approximate reasoning. (English) Zbl 0991.03027
The authors consider the problem how to describe inference rules which will, starting from probable premises, produce probable conclusions. As a possible solution, a probabilistic logic is proposed. An axiomatization of the logic, which is proven to be complete, contains the following axiom: \(\bigwedge_{\varepsilon\in Q^+}\bigvee_{\delta\in Q^+, \delta>\varepsilon/2} \bigwedge_{A,B\in\Phi} ((P_{\geq 1-\delta}(\alpha)\wedge P_{\geq 1-\delta}(\alpha\to\beta)) \to P_{\geq 1-\varepsilon}(\beta))\), where \(\Phi\) is a set of formulas belonging to a countable admissible set. It is proven that if \(A_1,\ldots,A_n\vdash B\), then for every \(\varepsilon>0\) there is a \(\delta>\varepsilon/2^m\), \(m\in\omega\) such that \(P_{\geq 1-\delta}A_1,\ldots,P_{\geq 1-\delta}A_n \vdash P_{\geq 1-\varepsilon}B\).

03B48 Probability and inductive logic
03C70 Logic on admissible sets
03C80 Logic with extra quantifiers and operators
68T27 Logic in artificial intelligence
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