Coping with uncertainty is a necessary part of ordinary life and is crucial to an understanding of how the mind works. For example, it is a vital element in developing artificial intelligence that will not be undermined by its own rigidities. There have been many approaches to the problem of uncertain inference, ranging from probability to inductive logic to nonmonotonic logic. Thisbook seeks to provide a clear exposition of these approaches within a unified framework. The principal market for the book (...) will be students and professionals in philosophy, computer science, and AI. Among the special features of the book are a chapter on evidential probability, which has not received a basic exposition before; chapters on nonmonotonic reasoning and theory replacement, matters rarely addressed in standard philosophical texts; and chapters on Mill's methods and statistical inference that cover material sorely lacking in the usual treatments of AI and computer science. (shrink)

There are a number of reasons for being interested in uncertainty, and there are also a number of uncertainty formalisms. These formalisms are not unrelated. It is argued that they can all be reflected as special cases of the approach of taking probabilities to be determined by sets of probability functions defined on an algebra of statements. Thus, interval probabilities should be construed as maximum and minimum probabilities within a set of distributions, Glenn Shafer's belief functions should be construed as (...) lower probabilities, etc. Updating probabilities introduces new considerations, and it is shown that the representation of belief as a set of probabilities conflicts in this regard with the updating procedures advocated by Shafer. The attempt to make subjectivistic probability plausible as a doctrine of rational belief by making it more flowery -- i.e., by adding new dimensions -- does not succeed. But, if one is going to represent beliefs by sets of distributions, those sets of distributions might as well be based in statistical knowledge, as they are in epistemological or evidential probability. (shrink)

The dominant argument for the introduction of propensities or chances as an interpretation of probability depends on the difficulty of accounting for single case probabilities. We argue that in almost all cases, the "single case" application of probability can be accounted for otherwise. "Propensities" are needed only in theoretical contexts, and even there applications of probability need only depend on propensities indirectly.

Ralph Waldo Emerson said, “a Foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines.” The alleged evidence has mounted that ordinary folk are prone to inconsistency, and particularly that they are prone to inconsistency when it comes to probabilistic judgments. I write “alleged,” because it is open to question whether the experiments that provide this evidence are well designed—in particular whether Quine’s principle of logistical charity has been followed. I also do so because (...) in some cases of probability judgments the untutored intuitions of ordinary people seem to be at least as good as, and perhaps better than, the intuitions of those who run the experiments. (shrink)

If someone comes to my house, saying, "Here is a bone; I hope Obrecht likes it," I might answer with a deductive argument: "You may rest assured on that score. Obrecht is a dog, and all dogs like bones; therefore Obrecht will like it." We may formalize this argument as follows: Let G be the bone, O be Obrecht, D be the class of dogs, B be the class of bones, and, finally, let L be the class of ordered pairs (...) such that x likes y. The premises are then: x ∈ D ⋅ y ∈ B ⋅ ⊃ ∈ L) G ∈ B O ∈ D. (shrink)

These essays are characterized by meticulous argument in the analytical tradition. The book concerns matters of metaphysics in a broad sense: philosophy of mind and the problems of subjectivity; questions concerning nomological and statistical laws of nature; and, as we would expect, chance and induction.

Ralph Waldo Emerson said, “a Foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines.” The alleged evidence has mounted that ordinary folk are prone to inconsistency, and particularly that they are prone to inconsistency when it comes to probabilistic judgments. I write “alleged,” because it is open to question whether the experiments that provide this evidence are well designed—in particular whether Quine’s principle of logistical charity has been followed. I also do so because (...) in some cases of probability judgments the untutored intuitions of ordinary people seem to be at least as good as, and perhaps better than, the intuitions of those who run the experiments. (shrink)

This collection represents the best recent work on the subject and includes essays by Clark Glymour, James H. Fetzer, and Wesley C. Salmon.

There are two main classes of interpretations of probability. The first are those that rely on a measure of frequency. The other is those that take a logical or subjective view of a unique event, independent of past or future events. The interpretation of probability which is used in the book is then defined as evidential probability, a function based on a set of known statements based on frequency or measure. The properties of probability are then enumerated and explained. Probabilities (...) can also be based on statistical inference. Statistical inferences are statistical properties with high probability based on statistical evidence. (shrink)

The distinction between cause and effect has been viewed as crucial to scientific thinking. David Hume dedicates many pages of his “Enquiry” to the argument of causality, and it appears to be of central vitality to our understanding of the world, despite the fact that he can find nothing to the notion. In Hume's prose, one senses both disappointment and heroic resignation. Some philosophers view causality—sometimes even universal causality—as a needed assumption or basic “presupposition” of science. It is sometimes argued (...) that universal causation, or at least its probability, is required for the induction's justification. (shrink)

The majority of the philosophical attention on induction has been connected with universal conventions: “All crows are black,” “All emeralds are green,” “Every creature with kidneys is a creature with a heart,” and others. It has been observed that if it can be shown how and why such conventions can be given rational justification by our restrained observations of the world, the outcome will be simpler. It is felt that it is but a small step from here to quantitative laws (...) and only a little-longer jump to general theories. After achieving an understanding of general theories, we can go back, in a mopping-up approach, to handle such irrelevant trivia as statistical conventions. (shrink)

Probabilistic connections are simple to reproduce counterfactually or hypothetically, since this involves simply adding the required statements to our evidential corpus without worrying about erasing some statements. In the serious uniform causal connections' case, the problem is complicated by the fact that some erasures will almost always have to be made, and that leads to the problems of intention and vagueness. On the other hand, uniform causal connections, considered both counterfactually and hypothetically, are exactly the connections needed to be taken (...) into account in life. Finally, however, some dispositions can be no more than hard facts. (shrink)

Efforts to get scientific knowledge reap valuable knowledge about the world. It is often rewarding simply to know more, but a greater benefit of knowing more is that, knowing the future, one can make sound decisions. There is an easy and unified decision theory that, if only it applied to everything, would solve all the decision problems humans face. It is known as “Bayesian” decision theory because it requires a set of probabilities determined over the world states.

Induction is the inference from a sample to a population, regardless of the possible existence of exceptions. Induction is used in the practice of science and engineering based on knowledge that can be accepted as evidence. There are two bodies of knowledge: evidential corpus, a set of propositions acceptable as evidence in a certain context; and practical corpus, a set of propositions counting as “practically certain” in that context. There are five kinds of induction described: statistical, universal, nomic, theoretical, and (...) instantial. (shrink)

In order for an “ideal” to make relevance, it is not mandatory, certainly, to be able to reach it. It is enough that it is possible to approach it, and even only to a certain extent. To be able to approach the ideal arbitrarily closely is not needed, even “theoretically.” To make sense of the “improvement” we can get in approaching an ideal, the measure of how close the ideal needs to be must be determined. In the thermometer's case, the (...) measure was determined by the bias in the error distribution at different points on the scale. An essentially internal measure was given, a measure related closely to the pertinent kind of measurement. (shrink)

This chapter discusses the philosophy and logic behind the mathematical ideas of first-order logic, metalanguages, arithmetic, and geometry. It familiarizes the reader to these mathematical ideas, which will be used in succeeding chapters. The chapter also contrasts that mathematical or logical theories can utilize these mathematical ideas, while scientific theories will find them inapplicable. First-order logic involves conclusions based on premises which are given to be true with no room for imprecision or vagueness, both of which are crucial to the (...) productive use of scientific language. Metalanguage is contrasted with object language, in which scientific laws are expressed. Both arithmetic and geometry are used with idealized, abstract objects and have a limited physical application. (shrink)

In general, quantities should be interpreted in the same way as random quantities or random variables are interpreted in statistics: namely, as functions from a domain to a special set of objects. The fact that they reflect to some level the structure of a set of mathematical objects makes the range of these functions extraordinary. Measurement, meanwhile, is not a process of “assigning numbers to objects,” but rather of formulating the values of quantity functions given to objects. More briefly, it (...) is giving magnitudes to objects. (shrink)

One problem that has been plaguing for the long term is the problem of choosing the levels of rational corpora. Since what goes into a corpus is what has a probability higher than the index of that corpus, that index has a bearing on what is in a corpus. We have two levels to deal with, since the focus is both with the evidential corpus and with the practical corpus. What principles can be used to select these levels? A practical (...) way to approach this question is through the analysis of full belief or acceptance. Partial belief is distinguished in terms of a propensity to make, or fail to make, bets. (shrink)

The chapter begins by explaining the concepts of quantity and magnitude. It then presents the method of measurement without using magnitude. This method of direct measurement can be achieved through the observation of the transitive relation among objects. A particular set of equivalence classes is selected to serve as a unit of measurement and is assigned magnitude. The concept of measurement error and approximation is then introduced. In some cases, such as temperature, indirect measurement, or measurement in terms of a (...) different quantity, yields less error and proves more accurate and convenient than direct measurement. In general, indirect measurements of different kinds are appropriate at different regions of the scale. (shrink)

Knowledge is distinguished between knowledge about the relations of ideas, like mathematical truths and logic, and knowledge of matters of fact or empirical knowledge of the world, which is derived from “sense experience.” Observational sentences are based on sense experience and can sometimes be judged to be true or false. Thus, the possibility of errors in observational judgment must be allowed. We can acknowledge the existence of observational error in general, and even be unable to specify any particular observation statement (...) that is certain, and yet be able to accept most observation statements on the basis of the corresponding reports and to reject others. (shrink)

This chapter chronicles the complex relationship between philosophy and science throughout history. It illustrates how they have mutually influenced each other in modern times. Philosophy and science are thought to be polar opposites, but they are not as different as they seem to be. Philosophy is considered part of the humanities and not the sciences. However, it can be argued that schools of science branched off from the domain of philosophy. Scientific studies start as or are inspired by philosophical ideas. (...) Mathematics, the very paradigm of a logical and rational discipline that much of science embodies, is a lot like philosophy in that it involves ideas that have no empirical basis and uses logic. (shrink)

In the previous chapter, it was considered in very broad terms what can happen when the ties are cut between observability and certainty. More needs to be proved, however, before the framework developed can be applied to the types of real theories of interest: quantum mechanics, relativity, and other such highbrow creations. Particularly, close observation at the nature of those statements is needed in the corpus of practical certainties that comprise the analytical observational content of the corpus that we get (...) from the sorts of theories accepted. Consideration of choices among distinct theories is needed, if such there be, having the same analytical observational content. (shrink)

Speculation is often opposed with scientific knowledge. Life existence on other planets of the solar system is through “sheer speculation,” while the chemical nature of the other planets' atmospheres is a matter of scientific knowledge. The present chapter does not dispute the distinction but it aims to examine the importance to science of speculation. The most vital role of speculation is that it provides us with scientific theories.

An answer to the fact that it is very complex to find convincing grounds for considering in universal deterministic uniformity has been to suggest that causality is indeed universal: all events are caused—but many, if not all, causal laws are statistical or probabilistic in character. Thus, a law of causality does not spell out what will be the effect of a given cause in a particular case; it just provides a probability of a given effect when the cause is determined. (...) This seems suitable not only in the rarified air of quantum mechanics, but particularly in the swampy territory of human activity: in epidemiology, psychology, economics, and the like. One way of interpreting statistical relationships as causal is through the law of large numbers. (shrink)