Deciding Koopman's qualitative probability
Section snippets
Introduction: conditional qualitative probability
In their 2019 paper [14] in this journal, the authors investigate a theory of qualitative probability, generalizing the traditional approach where formulas are equipped with a binary relation ≼ stating that α is no more probable than β.
In [14] one also finds a discussion of the role of qualitative probability in Artificial Intelligence. This will not be paraphrased here. Rather, we will only quote the first lines of the abstract of Nilsson's paper [36] published in this journal in 1986:
Koopman's conditional probability, [31–33]
The fundamental viewpoint of the present work is that the primal intuition of probability expresses itself in a (partial) ordering of eventualities: A certain individual at a certain moment considers the propositions , the meanings of which he apprehends, and which he considers to be determinate (i.e., either true or false) without (in general) knowing whether they are true or false (for the sharpening of these ideas, see §2). Then the phrase
“a on the presumption that h is true is
The bare minimum on boolean algebras
Boolean algebras and their ideals were extensively used by Koopman to frame his qualitative comparative probability. To help the reader, this section is devoted to a succinct exposition of finite boolean algebras, which is all we need in this paper. The relationship between the basic vocabularies of boolean logic and algebra is illustrated in Table 1. Free boolean algebras are just boolean formulas up to logical equivalence. Koopman's terse elementary algebraic setup dispenses with all
The syntax of Koopman's comparative probability
Koopman's universe of discourse is a “boolean ring determined by all the experimental propositions in a given discussion”, [32, p.766]. Since algorithmic issues are not considered in his papers [31], [32], [33], there is no need for Koopman to present his universe of discourse as the Lindenbaum-Tarski algebra determined by pre-conditions .
On the contrary, for our decidability results to make sense, all elements of boolean algebras (≅ boolean rings) considered in this paper must be coded
Semantics: consistency and inference in probability
The consistency problem in Koopman's probability has the following formal definition:
Boolean formulas coding the pre-conditions , with . A system of inequalities, with each of the form It is assumed that all and are boolean formulas in the variables for some fixed .
: With the filter of generated by , does the Lindenbaum-Tarski algebra
De Finetti's semantics, Boole, and PSAT
Theorem 5.3 shows that Koopman's qualitative probability theory allows the formalization of quantitative probabilistic inferences concerning conditionals and independence. In this section we will discuss the equivalent quantitative operational semantics for Koopman's comparative probability theory provided by de Finetti.
Examples
In this section some examples are given of consistency and inference problems in Koopman's probability. Building on the syntactic/semantic setup of Sections 3 and 4, detailed solutions are provided paralleling the steps in the proof of Theorem 5.1, Theorem 5.3. De Finetti's consistency notion (6.1-6.2) will have a key role in Example 7.3.
To better appreciate the relative simplicity of Koopman's probability, the reader is encouraged to work out these exercises in the framework of other more
Concluding remarks
A comprehensive account of comparative conditional probability would be beyond the scope of this paper. The interested reader is referred to [15], [18], [26] for detailed information. In this final section we will touch upon a few selected topics directly related to Koopman's qualitative probability theory and their relations with de Finetti's work on probability.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The author thanks the three referees and the associate editor for their insightful comments and valuable suggestions for improvement.
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