Abstract
Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. Of course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution.
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Notes
APP’s Note: Theorem 3 of (Suppes 2014) does not hold without suitably modifying the axioms and structural assumptions stated therein. In an effort to charitably reflect Suppes’ intentions, Theorem 3 of (Suppes 2014) has been recast in the present paper with such modifications. The proof of the corrected result proceeds very much in the spirit of the argumentation offered for Theorem 3 of (Suppes 2014).
APP’s Note: The axiomatization advanced in the present paper differs from the axiomatization appearing in the submitted manuscript. The original axiomatization, based on that presented in §6.1 of (Suppes 2014), is incomplete in formulation and execution. In an effort to capture Suppes’s original intentions, an alternative axiomatization that draws upon Suppes’s earlier work (Suppes and Alechina 1994) has been formulated in the present paper. The remarks and technical discussion following Theorem 2 and concluding §4.2 are my own.
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This paper was submitted for the present special issue shortly before Pat Suppes passed away. The manuscript required substantial corrections and revisions, not only for routine copy-editing, but also for mathematical accuracy and completeness. With Pat’s permission, this work was carried out by one of the guest editors, Arthur Paul Pedersen, whose remarks in the following manuscript are denoted by ‘APP’.—Gregory Wheeler, Editor-in-Chief.
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Suppes, P. Qualitative Axioms of Uncertainty as a Foundation for Probability and Decision-Making. Minds & Machines 26, 185–202 (2016). https://doi.org/10.1007/s11023-015-9385-7
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DOI: https://doi.org/10.1007/s11023-015-9385-7