In days of yore, logic was neatly divided into two parts, Deductive Logic and Inductive Logic (Mill 1949/1843). The two parts were often taught as parts of a single course. Inductive logic has faded away, and now the very term has acquired a slightly antiquated patina. It has also acquired a number of quite specific modern meanings.

One very narrow and very specific meaning is that inductive inference is the inference from a set of observations or observation sentences (crow #1 is black; crow #2 is black; …; crow #n is black) to their universal generalization (all crows are black). There is not much logic here, but there is a big problem: to determine when, if ever, such an inference is “justified.”

A somewhat less ambitious construal of “induction” is as the inference from a statistical sample to a statistical generalization, or an approximate statistical generalization: from “51% of the first 10,000 tosses of this coin yielded heads,” to “Roughly half of the tosses of the coin will, in the long run, yield heads.” So construed (as by Baird 1992), the line between the logic of induction and the mathematics of statistics is a bit vague. This has been of little help to inductive logic, since the logic of statistical inference has itself been controversial.

THE HORSE OR THE CART?

John Maynard Keynes (Keynes 1952) proposed that probability should be legislative for rational belief. He also proposed that probabilities should form only a partial order: There were to be incomparable pairs of probabilities where the first is not larger than the second, the second not larger than the first, yet the two probabilities are not equal.

Frank Plumpton Ramsey objected (Ramsey 1931), quite correctly, that any such scheme depended on being able to relate beliefs and probabilities. He disregarded the second proposal, and so took probabilities to be numbers, so that what he took to be necessary was a way of measuring degrees of belief.

Ramsey offered a somewhat naïve operational way of measuring beliefs. He himself took it to be no more than approximate (“I have not worked out the mathematical logic of this in detail, because this would, I think, be rather like working out to seven places of decimals a result only valid to two” (ibid., p. 180). What was important about Ramsey's proposal was that it also suggested why beliefs (assuming their measurability) should satisfy the probability calculus.

Ramsey's approach became the model for later “subjectivistic” approaches. First, we think about ways in which to measure degrees of belief; second, we consider why those degrees should satisfy the probability calculus; and third, we consider how those probabilities should be updated in the light of new evidence.

Introduction

Classical logic—including first order logic, which we studied in Chapter 2—is concerned with deductive inference. If the premises are true, the conclusions drawn using classical logic are always also true. Although this kind of reasoning is not inductive, in the sense that any conclusion we can draw from a set of premises is already “buried” in the premises themselves, it is nonetheless fundamental to many kinds of reasoning tasks. In addition to the study of formal systems such as mathematics, in other domains such as planning and scheduling a problem can in many cases also be constrained to be mainly deductive.

Because of this pervasiveness, many logics for uncertain inference incorporate classical logic at the core. Rather than replacing classical logic, we extend it in various ways to handle reasoning with uncertainty. In this chapter, we will study a number of these formalisms, grouped under the banner nonmonotonic reasoning. Monotonicity, a key property of classical logic, is given up, so that an addition to the premises may invalidate some previous conclusions. This models our experience: the world and our knowledge of it are not static; often we need to retract some previously drawn conclusion on learning new information.

Logic and (Non)monotonicity

One of the main characteristics of classical logic is that it is monotonic, that is, adding more formulas to the set of premises does not invalidate the proofs of the formulas derivable from the original premises alone. In other words, a formula that can be derived from the original premises remains derivable in the expanded premise set.

This book is the outgrowth of an effort to provide a course covering the general topic of uncertain inference. Philosophy students have long lacked a treatment of inductive logic that was acceptable; in fact, many professional philosophers would deny that there was any such thing and would replace it with a study of probability. Yet, there seems to many to be something more traditional than the shifting sands of subjective probabilities that is worth studying. Students of computer science may encounter a wide variety of ways of treating uncertainty and uncertain inference, ranging from nonmonotonic logic to probability to belief functions to fuzzy logic. All of these approaches are discussed in their own terms, but it is rare for their relations and interconnections to be explored. Cognitive science students learn early that the processes by which people make inferences are not quite like the formal logic processes that they study in philosophy, but they often have little exposure to the variety of ideas developed in philosophy and computer science. Much of the uncertain inference of science is statistical inference, but statistics rarely enter directly into the treatment of uncertainty to which any of these three groups of students are exposed.

At what level should such a course be taught? Because a broad and interdisciplinary understanding of uncertainty seemed to be just as lacking among graduate students as among undergraduates, and because without assuming some formal background all that could be accomplished would be rather superficial, the course was developed for upper-level undergraduates and beginning graduate students in these three disciplines. The original goal was to develop a course that would serve all of these groups.

Introduction

In Chapter 3, we discussed the axioms of the probability calculus and derived some of its theorems. We never said, however, what “probability” meant. From a formal or mathematical point of view, there was no need to: we could state and prove facts about the relations among probabilities without knowing what a probability is, just as we can state and prove theorems about points and lines without knowing what they are. (As Bertrand Russell said [Russell, 1901, p. 83] “Mathematics may be defined as the subject where we never know what we are talking about, nor whether what we are saying is true.”)

Nevertheless, because our goal is to make use of the notion of probability in understanding uncertain inference and induction, we must be explicit about its interpretation. There are several reasons for this. In the first place, if we are hoping to follow the injunction to believe what is probable, we have to know what is probable. There is no hope of assigning values to probabilities unless we have some idea of what probability means. What determines those values? Second, we need to know what the import of probability is for us. How is it supposed to bear on our epistemic states or our decisions? Third, what is the domain of the probability function? In the last chapter we took the domain to be a field, but that merely assigns structure to the domain: it doesn't tell us what the domain objects are.

There is no generally accepted interpretation of probability.