A Primer of Probability Logic
Adams (Ernest)
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BOOK ABSTRACT:

Back Cover Blurb

  1. If the premises of a valid inference are not perfectly certain, how probable is its conclusion guaranteed to be? This important question arises when elementary probability is applied to deductive logic. In the case of ‘simple’ inferences, this question has a straightforward, if not an obvious answer. However, in the case of inferences involving conditional propositions, profound implications arise, not only for probability but for truth itself. These questions, along with probabilistic decision theory, are explored in this primer, leading to a consideration of the practical values of truth and probability.
  2. This subject has profound implications even for elementary logic, and is the subject of currently active research in formal logic, artificial intelligence, linguistics, psychology, decision theory, and various parts of philosophy. This primer attempts to provide a unified exposition of results and methods, complemented by numerous exercises, in this area that, is accessible to readers with only an elementary acquaintance with logic. The appendices discuss more advanced aspects of the subject, and the copious bibliographic citations provide a foundation for even further exploration.
  3. Ernest W. Adams is Professor Emeritus of Philosophy at the University of California. Berkeley.
  4. 'In my view, the material covered in this book (and this book alone among books designed for students) should be part of every logic course. Ernest Adams's important theory of conditionals comes from a concern with reasoning from uncertain premises. Here he presents the results in friendly textbook form, with indispensable exercises, and philosophical commentary, clearly indicating where this is controversial.’
    Dorothy Edgington, Oxford University1

Foreward
  1. An adequate logic of science cannot rest solely on the concept of deductive validity. It must also deal with error and uncertainty — matters best addressed in terms of the theory of probability. This was already taken as a first principle in the 19th century by LaPlace, Venn, de Morgan, Peirce and Poincare. And the logic of probability has been seen as central to practical reasoning since its inception. (And perhaps before its inception on a generous reading of Aristotle.) But contemporary logic texts tend to treat probability as an addendum or afterthought rather than as an integral part of the subject.
  2. This book is different; probability is at its center. The very structure of the language investigated is pragmatically based on the probability concepts involved. The conditional is construed as the bearer of conditional probability, along the lines of Adams' eye-opening research on the logic of conditionals. Arguments containing such conditionals are evaluated from a pragmatic probabilistic perspective. The connections to epistemology and practical reason are evident throughout.
  3. As a Primer, this text is written so that it is accessible to a student new to the subject matter. In another sense it can serve as a primer for professional philosophers who are not already familiar with Adams’ original contributions to probability logic. Here these are given an elementary systematic exposition. Readers will be introduced to a new and fascinating perspective on the function of language.
    Brian Skyrms

1.5 Summary and Limitations of This Work
  1. Our primary concern in this work will be with applications of probability to deductive logic, and especially with the fact that the premises of reasoning in everyday life are seldom certainties and what this implies about the certainty of the conclusions that are deduced from them. But this leads to technical questions like asking, for instance, how probable the premises of an inference have to be if you want to assure that its conclusion is at least 99% probable. This is analogous to asking how accurate the measurements of the height and width of a rectangle have to be if you want to estimate its area with 99% accuracy. The minimal knowledge of formal probability needed to answer this question in the probability case will be given in chapters 2 and 42, which will differ from the material in ordinary courses in mathematical probability mainly in not dealing with the kinds of ‘combinatorial' problems that typically arise in games of chance and in the theory of sampling3.
  2. The probabilistic background given in chapters 2 and 4 is applied in chapters 3 and 5 to answering the question of how probable the conclusion of a deductively valid inference must be, given the probabilities of its premises4. The chapters differ in that while chapter 3 looks at things ’statically’, and leaves out of account the fact that probabilities can change as a result of gaining information or ‘adding new premises’, chapter 5 introduces the ‘dynamical dimension’ that is associated with the nonmonotonicity that was commented on in earlier sections. This brings in the first of the controversial principles that will be involved in our discussion, because the dynamical theory is based on Bayes’ Principle, which is a general law describing how probabilities change when information is acquired. This principle is ‘critiqued’ in section 4.7*, and it is the first of three fundamental but controversial principles that will guide the theoretical developments with which this text is concerned.
  3. Chapters 6 and 7 focus on the logic of conditional propositions, and this discussion is based on the second major controversial assumption of this text: namely that the probabilities of conditional propositions are conditional probabilities. Not only do these probabilities not conform to the laws presupposed in chapters 3 and 5, but they lead to a ‘logic’ of conditionals that differs radically from the logic that is based on the assumption that the truth value of an ‘if-then’ proposition is the same as that of a material conditional. This leads to an argument that is set forth in chapter 8, which is a variant due to Alan Hájek of David Lewis’ fundamental ‘Triviality Argument’, that shows that, given certain plausible ‘technical assumptions’, if the ‘right’ measure of a conditional’s probability is a conditional probability then its probability cannot equal the probability of its being true, no matter how its truth might be defined. Chapter 9 gives a brief introduction to the theory of practical reason, showing how conclusions about propositions and their probabilities influence actions, and how the success of the actions in attaining the goals of the persons who perform them depends on the ‘rightness’ of the probabilities acted upon. This chapter ends with the most ‘philosophical’ section of this work, which concerns the ‘pragmatics of probability’. This has to do with how probabilities ought to be measured, assuming that persons are guided in their decisions by them and ‘right probabilities’ provide good guidance. This is a question of ‘the meaning of probability’, and, as would be expected, it is the most controversial question of all. Our equally controversial assumption is that good or ‘real’ probabilities are frequencies with which human beings can correctly predict things, with the corollary that they are best guided in the long run if their estimates of probabilities correspond to these frequencies.
  4. Finally, something should be said about the limitations of this text. It is meant to be a primer, that is, as an introduction to a subject that even now, in what appears to be its infancy, has many ramifications of both a mathematical and a philosophical character. But the fact that the subject has both a philosophical and a mathematical side means that, unlike other introductory works on probability and its applications, our primer will introduce formal mathematical problem-solving techniques only to the extent that they contribute to the understanding of matters that are not fundamentally formal5. In fact, significant mathematical developments in probability logic are passed over because their bearing on applications is less direct than the materials that will be covered here. Some of these, e.g., the extension of the theory to infinitesimal probabilities, are sketched briefly in appendices, some are brought up in exercises, and some are alluded to only in footnotes. It is hoped that by this means, students will be made aware of some of the important ramifications of the subject, current research in it, and they will have some indications to recent literature on it.



In-Page Footnotes ("Adams (Ernest) - A Primer of Probability Logic")

Footnote 1: Footnote 2:
  • Students who have already had courses in probability theory need only skim the basic probability ideas introduced in these chapters, though they will need to pay attention to their applications to deductive logic.
Footnote 3:
  • More generally, courses in mathematical probability tend to focus on applications to certain kinds of reasoning that are inductive in a broad sense, and to make assumptions about probabilities that are themselves not valid a priori, the not making of which actually simplifies our theory.
Footnote 4:
  • In spite of the obviousness of this question, it has not to the writer’s knowledge been raised before in texts either on probability or deductive logic.
Footnote 5:
  • In fact, in the author’s opinion, emphasis on formal techniques tends more to obscure than to clarify the very phenomena that theory is meant to help us understand, which in this case have to do with human reason.

BOOK COMMENT:

CSLI Publications, Stanford, California, 1998. CSLI Lecture Notes Number 68



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