Uncertain Inferences and Uncertain Conclusions

Uncertainty may be taken to characterize in­ ferences, their conclusions, their premises or all three. Under some treatments of uncer­ tainty, the inference itself is never character­ ized by uncertainty. We explore both the sig­ nificance of uncertainty in the premises and in the conclusion of an argument that in­ volves uncertainty. We argue that for un­ certainty to characterize the conclusion of an inference is natural, but that there is an in­ terplay between uncertainty in the premises and uncertainty in the procedure of argu­ ment itself. We show that it is possible in principle to incorporate all uncertainty in the premises, rendering uncertainty arguments deductively valid. But we then argue (1) that this does not reflect human argument, (2) that it is computationally costly, and (3) that the gain in simplicity obtained by allow­ ing uncertainty in inference can sometimes outweigh the loss of flexibility it entails. 1 BEING UNCERTAIN AND BEING ABOUT UNCERTAINTY There are many ways of expressing our uncertainty. In what follows, I shall try to ensure that nothing I say depends crucially on whether we express un­ certainty by means of belief functions !Shafer, 1976; Smets and Kennes, 1994], or probability functions, or class es of probability functions [Levi, 1967; Levi, 1980], or probability envelopes [Walley, 1991], or by fuzzi­ ness [Zadeh, 1975], or by possibility measures [Dubois and Prade, 1985] , or any other way. However we construe uncertainty, though, we shall want to make a distinction between a sentence expressing that un­ certainty, and the sentence whose uncertainty may be being expressed. Example 1: "The probability of heads on a toss of a coin is 0.5 ± L" Barring the possibly question-begging occurrences of the indefinite article "a," this is simply a factual assertion about the behavior of coins !Reichen­ bach, 1949]. It embodies uncertainty, because one of the facts about coins is that the results of their flips are uncertain, and in fact uncertain in the way described. For most of us there is little or nothing uncertain about the statement itself. Example !2: "Albert's degree of belief in the occur­ rence of heads on the next toss of this coin is 0.5." Again, this is a factual assertion about Albert; pre­ sumably about his willingness to buy and sell chances based on the outcome of the toss in question [Savage, 1954]. There is nothing special about Albert ; we may in other contexts use the possessive pronouns "your," "our," and "my." A statement like this, in the con­ text of reasoning or decision making, is a statement embodying uncertainty perhaps in a different sense from that embodied in the first example. Contrast those statements with these: Example S: "It is almost certain that the probabil­ ity of heads on the toss of a coin is about a half." Several things may be observed. First, there is the obvious possibility that the interpretation of "almost certain" may be taken from a different representation than is given to probability; there is no reason why this cannot be interpreted subjectively while probabil­ ity is interpreted in terms of frequencies. Second, the role in argument played by the quoted sentence in this example is quite different from the role played by the sentence in the first example. Example 4: "It is very unlikely that Albert's degree of belief in the occurrence of heads on the next toss of this coin is 0.5" Similar observations may be made in this case. "Very unlikely" does not refer to Albert's degree of belief; it may be a value in some objective epistemic representation of uncertainty, or reflect a parameter in a statistical test, or represent the degree of belief of an