Chapter 7: Logics for Reasoning about Uncertainty

Overview

Proof, n. Evidence having a shade more of plausibility than of unlikelihood. The testimony of two credible witnesses as opposed to that of only one.

—Ambrose Bierce, The Devil's Dictionary

The previous chapters considered various issues regarding the representation of uncertainty. This chapter considers formal logics for reasoning about uncertainty. These logics provide tools for carefully representing arguments involving uncertainty, as well as methods for characterizing the underlying notion of uncertainty. Note that I said "logics," not "logic." I consider a number of logics. The choice of logic depends in part on (1) the underlying representation of uncertainty (e.g., is it a probability measure or a ranking function?), (2) the degree to which quantitative reasoning is significant (is it enough to say that U is more likely than V, or is it important to be able to talk about the probability of U?), and (3) the notions being reasoned about (e.g., likelihood or expectation). In this chapter, I consider how each of these questions affects the choice of logic.

As I said in Chapter 1, a formal logic consists of an appropriate syntax (i.e., a language) and semantics, essentially, models that are used for deciding if formulas in the language are true and false. Quite often logics are also characterized axiomatically: a collection of axioms and inference rules is provided from which (hopefully all) valid formulas (i.e., formulas that are true in all semantic models) can be derived.

The various types of frames that were discussed in Chapter 6 (epistemic frames, probability frames, etc.) provide the basis for the semantic models used in this chapter. Thus, not much work needs to be done on the semantic front in this chapter. Instead, I focus is on issues of language and, to a lesser extent, on axiomatizations.

Perhaps the simplest logic considered in the literature, and the one that most students encounter initially, is propositional logic (sometimes called sentential logic). It is intended to capture features of arguments such as the following:

Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence this thing is mimsy.

While propositional logic is useful for reasoning about conjunctions, negations, and implications, it is not so useful when it comes to dealing with notions like knowledge or likelihood. For example, notions like "Alice knows it is mimsy" or "it is more likely to be mimsy than not" cannot be expressed in propositional logic. Such statements are crucial for reasoning about uncertainty. Knowledge is an example of what philosophers have called a propositional attitude. Propositional attitudes can be expressed using modal logic.

Since not all readers will have studied formal logic before, I start in this chapter with a self-contained (but short!) introduction to propositional logic. (Even readers familiar with propositional logic may want to scan the next section, just to get comfortable with the notation I use.) I go on to consider epistemic logic, a modal logic suitable for reasoning about knowledge, and then consider logics for reasoning about more quantitative notions of uncertainty, reasoning about independence, and reasoning about expectation. The following chapters consider yet other logics; for example, Chapter 8 considers logics for reasoning about defaults and counterfactuals, and Chapter 9 considers logics for reasoning about belief revision. All the logics considered in the next three chapters are propositional; they cannot express quantified statements like "there exists someone in this room who is very likely to be a millionaire within five years" nor can they express structure such as "Alice is a graduate student and Bob is not." First-order logic can express such structure; it is considered in Chapter 10.