Reasoning from inconclusive evidence, or ‘induction’, is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This Element explores the prospects of using probability theory to provide an inductive logic, a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that overall support for a hypothesis is represented by its probability in light of the total evidence, and incremental support, (...) or confirmation, indicated by an increased probability given the evidence than otherwise. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge from inductive scepticism and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

This is an introductory 2001 textbook on probability and induction written by one of the world's foremost philosophers of science. The book has been designed to offer maximal accessibility to the widest range of students and assumes no formal training in elementary symbolic logic. It offers a comprehensive course covering all basic definitions of induction and probability, and considers such topics as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction. The key features of this (...) book are a lively and vigorous prose style; lucid and systematic organization and presentation of ideas; many practical applications; a rich supply of exercises drawing on examples from such fields as psychology, ecology, economics, bioethics, engineering, and political science; numerous brief historical accounts of how fundamental ideas of probability and induction developed; and a full bibliography of further reading. (shrink)

For a helpful presentation of the various views on probability and inductive logic as well as a thorough survey of the present literature on these topics, one could hardly do better than this work. Kyburg presents, in separate chapters, classical, frequency, logical, subjectivist and epistemological theories of probability, referring to major classical and contemporary works where each of these views is defended. He presents the common criticisms of each view as well as some criticisms of his (...) own and brings out what he takes to be of value in each view. These chapters are preceded by a brief but clear presentation of the probability calculus and a chapter on informal interpretations of probability in ordinary language. A descriptive bibliography follows each chapter in which Kyburg comments on the leading works in each area. The same format applies to his discussion of inductive logic in part II. Inductive inference is discussed in ordinary language and in formal contexts. Statistical inference, confirmation theories, acceptance theories, and demonstrative induction are discussed in separate chapters. There are also discussions of the role of simplicity in inductive inference and the problem of justifying induction. Exercises are added to each chapter and the work might serve well as a text in this field. A forty-eight page bibliography at the end of the book is another one of its assets.--R. H. K. (shrink)

This is an introductory textbook on probability and induction written by one of the world's foremost philosophers of science. The book has been designed to offer maximal accessibility to the widest range of students and assumes no formal training in elementary symbolic logic. It offers a comprehensive course covering all basic definitions of induction and probability, and it considers such topics as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction. The key features of the (...) book are a lively and vigorous prose style; lucid and systematic organisation and presentation of the ideas; many practical applications; a rich supply of exercises drawing on examples from such fields as psychology, ecology, economics, bioethics, engineering, and political science; numerous brief historical accounts of how fundamental ideas of probability and induction developed; a full bibliography of further reading. Although designed primarily for courses in philosophy, the book could certainly be read and enjoyed by those in the social sciences or medical sciences seeking a reader-friendly account of the basic ideas of probability and induction. (shrink)

A Logical Introduction to Probability and Induction starts with elementary logic and uses it as basis for a philosophical discussion of probability and induction. Throughout the book results are carefully proved using the inference rules introduced at the beginning. The textbook is suitable for undergraduate courses in philosophy and logic.

This paper seeks to defend the following conclusions: The program advanced by Carnap and other necessarians for probability logic has little to recommend it except for one important point. Credal probability judgments ought to be adapted to changes in evidence or states of full belief in a principled manner in conformity with the inquirer’s confirmational commitments—except when the inquirer has good reason to modify his or her confirmational commitment. Probability logic ought to spell out the (...) constraints on rationally coherent confirmational commitments. In the case where credal judgments are numerically determinate confirmational commitments correspond to Carnap’s credibility functions mathematically represented by so—called confirmation functions. Serious investigation of the conditions under which confirmational commitments should be changed ought to be a prime target for critical reflection. The necessarians were mistaken in thinking that confirmational commitments are immune to legitimate modification altogether. But their personalist or subjectivist critics went too far in suggesting that we might dispense with confirmational commitments. There is room for serious reflection on conditions under which changes in confirmational commitments may be brought under critical control. Undertaking such reflection need not become embroiled in the anti inductivism that has characterized the work of Popper, Carnap and Jeffrey and narrowed the focus of students of logical and methodological issues pertaining to inquiry. (shrink)

A basic system of inductive logic; An axiomatic foundation for the logic of inductive generalization; A survey of inductive systems; On the condition of partial exchangeability; Representation theorems of the de finetti type; De finetti's generalizations of excahngeability; The structure of probabilities defined on first-order languages; A subjectivit's guide to objective chance.

The λ-continuum of inductive methods was derived from an assumption, called λ-condition, which says that the probability of finding an individual having property $x_{j}$ depends only on the number of observed individuals having property $x_{j}$ and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different from $x_{j}$ have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having (...) some property similar to $x_{j}$ (the most complete proof of this result is presented in Carnap, 1980). The problem thus remained open to find some general condition, weaker than the λ-condition, which would allow for the derivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the λ-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived. In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed. (shrink)

A basic system of inductive logic; An axiomatic foundation for the logic of inductive generalization; A survey of inductive systems; On the condition of partial exchangeability; Representation theorems of the de finetti type; De finetti's generalizations of excahngeability; The structure of probabilities defined on first-order languages; A subjectivit's guide to objective chance.

Calibration inductive logics are based on accepting estimates of relative frequencies, which are used to generate imprecise probabilities. In turn, these imprecise probabilities are intended to guide beliefs and decisions — a process called “calibration”. Two prominent examples are Henry E. Kyburg's system of Evidential Probability and Jon Williamson's version of Objective Bayesianism. There are many unexplored questions about these logics. How well do they perform in the short-run? Under what circumstances do they do better or worse? What (...) is their performance relative to traditional Bayesianism? -/- In this article, we develop an agent-based model of a classic binomial decision problem, including players based on variations of Evidential Probability and Objective Bayesianism. We compare the performances of these players, including against a benchmark player who uses standard Bayesian inductive logic. We find that the calibrated players can match the performance of the Bayesian player, but only with particular acceptance thresholds and decision rules. Among other points, our discussion raises some challenges for characterising “cautious” reasoning using imprecise probabilities. Thus, we demonstrate a new way of systematically comparing imprecise probability systems, and we conclude that calibration inductive logics are surprisingly promising for making decisions. (shrink)

In this thesis I defend and pursue that line about the foundations of probability theory which has come to be known as "the logicist view about probability", and, in particular, the shape which it took in Carnap's Inductive Logic. ;Most philosophers who now deal with probability theory claim that Carnap's program of Inductive Logic has failed. The main aim of my thesis is to show that this judgment is based on a fundamental misunderstanding (...) about the nature and the aim of inductive logic. To that end, I follow, in chapter 1, the events in the history of probability, logic and mathematics which led to choosing certain requirements as the requirements which any theory about the foundations of probability must fulfill in order to be acceptable. ;In chapter 2 I explain how Carnap's inductive logic fulfills those requirements and how the method it uses to give foundations to probability theory can also be used to solve another major problem about probability, namely, the problem of statistical inference. ;Chapter 3 shows how the ideas explained in chapter 2 were developed, formally, by Carnap, and how Carnap's program can be continued in a particular case, that of reasoning by analogy. ;Finally, in chapter 4, I attempt to show that subjectivism about probability theory, the rival theory to Carnap's logicism, does not succeed in meeting the most basic requirement for the acceptability of any theory about the foundations of probability. (shrink)

The project of constructing a logic of scientific inference on the basis of mathematical probability theory was first undertaken in a systematic way by the mid-nineteenth-century British logicians Augustus De Morgan, George Boole and William Stanley Jevons. This paper sketches the origins and motivation of that effort, the emergence of the inverse probability (IP) model of theory assessment, and the vicissitudes which that model suffered at the hands of its critics. Particular emphasis is given to the influence (...) which competing interpretations of probability had on the project, and to the role of the 'lottery' or 'ballot box' metaphor in the philosophical imagination of the proponents of the IP model. (shrink)

Then, in 1960, Carnap drew up a plan of articles for Studies in Inductive Logic and Probability — a surrogate for Volume II of the ...

This book aims to discuss probability and David Hume's inductive scepticism. For the sceptical view which he took of inductive inference, Hume only ever gave one argument. That argument is the sole subject-matter of this book. The book is divided into three parts. Part one presents some remarks on probability. Part two identifies Hume's argument for inductive scepticism. Finally, the third part evaluates Hume's argument for inductive scepticism. Hume's argument that induction must be either (...) deductively valid or circular because based on experience neglects the possibility that it is an argument of non-deductive logic (logical probability, in the sense of Keynes). (shrink)

Finland is internationally known as one of the leading centers of twentieth century analytic philosophy. This volume offers for the first time an overall survey of the Finnish analytic school. The rise of this trend is illustrated by original articles of Edward Westermarck, Eino Kaila, Georg Henrik von Wright, and Jaakko Hintikka. Contributions of Finnish philosophers are then systematically discussed in the fields of logic, philosophy of language, philosophy of science, history of philosophy, ethics and social philosophy. Metaphilosophical reflections (...) on the nature of philosophy are highlighted by the Finnish dialogue between analytic philosophy, phenomenology, pragmatism, and critical theory. (shrink)

In his essay on 'Broad on Induction and Probability' (first published in 1959, reprinted in this volume), Professor G. H. von Wright writes: "If Broad's writings on induction have remained less known than some of his other contributions to philosophy . . . , one reason for this is that Broad never has published a book on the subject. It is very much to be hoped that, for the benefit of future students, Broad's chief papers on induction and (...) class='Hi'>probability will be collected in a single volume . . . . " The present volume attempts to perform this service to future students of induction and probability. The suggestion of publishing a volume of this kind in Synthese Library was first made by Professor Donald Davidson, one of the editors of the Library, and was partly prompted by Professor von Wright's statement. In carrying out this suggestion, the editors of Synthese Library have had the generous support of Professor Broad who has among other things supplied a new Addendum to 'The Principles of Problematic Induction' and corrected a number of misprints found in the first printings of this paper. The editors gratefully acknow ledge Professor Broad's help and encouragement. A bibliography of Professor Broad's writings (up to 1959) has been compiled by Dr. C. Lewy and has appeared in P. A. Schilpp, editor, The Philosophy of C. D. Broad (The Library of Living Philosophers), pp. 833-852. (shrink)

In his essay on 'Broad on Induction and Probability' (first published in 1959, reprinted in this volume), Professor G. H. von Wright writes: "If Broad's writings on induction have remained less known than some of his other contributions to philosophy..., one reason for this is that Broad never has published a book on the subject. It is very much to be hoped that, for the benefit of future students, Broad's chief papers on induction and probability will be collected (...) in a single volume.... " The present volume attempts to perform this service to future students of induction and probability. The suggestion of publishing a volume of this kind in Synthese Library was first made by Professor Donald Davidson, one of the editors of the Library, and was partly prompted by Professor von Wright's statement. In carrying out this suggestion, the editors of Synthese Library have had the generous support of Professor Broad who has among other things supplied a new Addendum to 'The Principles of Problematic Induction' and corrected a number of misprints found in the first printings of this paper. The editors gratefully acknow ledge Professor Broad's help and encouragement. A bibliography of Professor Broad's writings (up to 1959) has been compiled by Dr. C. Lewy and has appeared in P. A. Schilpp, editor, The Philosophy of C. D. Broad (The Library of Living Philosophers), pp. 833-852. (shrink)

After describing the three main kinds of probability theories, the author presents a new logical theory of probability. Probability, for him, is a logical relation between bodies of evidence and statements. The theory is an advance over Carnap's logical theory of probability in that it is not based on a formal language with strong completeness properties. The theory is formalized in Quine's protosyntax, which has two disadvantages: first, long formulas couched in terms of a large number (...) of defined symbols disfigure the book and obscure the exposition; and secondly, it is desirable to keep our theories of inductive logic and deductive logic independent. There seems to be no reason to burden one sort of theory with the other. There is no index.--J. R. W. (shrink)

According to a popular interpretation, Carnap’s interpretation of probability had evolved from a logical towards a subjective conception. However Carnap himself insisted that his basic philosophical view of probability was always the same. I address this apparent clash between Carnap's self-identification and the subsequent interpretations of his work. Following its original intentions, I reconstruct inductive logic as an explication. The emerging picture is of a versatile linguistic framework, whose main function is not the discovery of objective (...) logical relations in the object language, but the stipulation of conceptual possibilities. Within this representation, I map out the changes that the project went through. Seen from such an explication-based perspective, inductive logic becomes quite hard to categorize using the standard labels. (shrink)

A. J. Ayer was one of the foremost analytical philosophers of the twentieth century, and was known as a brilliant and engaging speaker. In essays based on his influential Dewey Lectures, Ayer addresses some of the most critical and controversial questions in epistemology and the philosophy of science, examining the nature of inductive reasoning and grappling with the issues that most concerned him as a philosopher. This edition contains revised and expanded versions of the lectures and two additional essays. (...) Ayer begins by considering Hume's formulation of the problem of induction and then explores the inferences on which we base our beliefs in factual matters. In other essays, he defines the three kinds of probability that inform inductive reasoning and examines the various criteria for verifiability and falsifiability. In his extensive introduction, Graham Macdonald discusses the arguments in _Probability and Evidence_, how they relate to Ayer's other works, and their influence in contemporary philosophy. He also provides a brief biographical sketch of Ayer, and includes a bibliography of works about and in response to _Probability and Evidence_. (shrink)

This paper starts by summarizing work that philosophers have done in the fields of inductive logic since 1950s and truth approximation since 1970s. It then proceeds to interpret and critically evaluate the studies on machine learning within artificial intelligence since 1980s. Parallels are drawn between identifiability results within formal learning theory and convergence results within Hintikka’s inductive logic. Another comparison is made between the PAC-learning of concepts and the notion of probable approximate truth.