The abstract of this paper begins “the object of this work is the presentation of a mathematically sound basis for the application of the mathematical theory of probability to propositional logic as used in artificial intelligence.” The paper succeeds admirably and makes an important contribution toward rectifying the enduring dilemma that various uncertainty calculi found in artificial intelligence systems are not, by and large, mathematical objects. Extending the work of Nilsson [1], a sample space is defined that allows the derivation of universal stochastic inequalities for propositional logic with (true) probabilities. The construction gives a sufficient basis for a thorough treatment of Dempster’s rule on the combination of evidence as applied to logic [2] and yields some very strong limit theorems. One practical consequence of these results is that in cases where a moderate number of items of information yields a yes/no decision, the straightforward Dempster rule for one proposition can be used even in questions involving more than one proposition. Surprisingly, Dempster’s original formulation appears to be better suited to probabilistic logic than Shafer’s reformulation.

Other methods for reasoning with uncertainty, as given in Freedman [3], are also examined, along with Bayesian updating and fuzzy numbers. An appendix contains a measure theoretic analysis of fuzzy sets using the paper’s framework. An interesting and provocative question is embedded in the paper: can effective reasoning systems be built upon a fixed rule of combination? This question asks whether a dynamic uncertainty method, one which responds to the status of an inference, will be needed. As the author points out, a mathematical version of Halpern and Rabin’s non-numerical approach [4] would be such a method. If this is true, computationally efficient artificial intelligence systems that incorporate reasoning with uncertainty are farther off than we had imagined.

The presentation is mathematically demanding but clear. I recommend this paper, but strongly suggest that the reader first review Nilsson and Dempster’s work.