Inductive Logic is number ten in the 11-volume Handbook of the History of Logic. While there are many examples were a science split from philosophy and became autonomous (such as physics with Newton and biology with Darwin), and while there are, perhaps, topics that are of exclusively philosophical interest, inductive logic — as this handbook attests — is a research field where philosophers and scientists fruitfully and constructively interact. This handbook covers the rich history of (...) scientific turning points in Inductive Logic, including probability theory and decision theory. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration. (shrink)

Inductive Logic is number ten in the 11-volume Handbook of the History of Logic. While there are many examples were a science split from philosophy and became autonomous (such as physics with Newton and biology with Darwin), and while there are, perhaps, topics that are of exclusively philosophical interest, inductive logic — as this handbook attests — is a research field where philosophers and scientists fruitfully and constructively interact. This handbook covers the rich history of (...) scientific turning points in Inductive Logic, including probability theory and decision theory. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration. (shrink)

My dissertation explores the ways in which Rudolf Carnap sought to make philosophy scientific by further developing recent interpretive efforts to explain Carnap’s mature philosophical work as a form of engineering. It does this by looking in detail at his philosophical practice in his most sustained mature project, his work on pure and applied inductive logic. I, first, specify the sort of engineering Carnap is engaged in as involving an engineering design problem and then draw out the complications of (...) design problems from current work in history of engineering and technology studies. I then model Carnap’s practice based on those lessons and uncover ways in which Carnap’s technical work in inductive logic takes some of these lessons on board. This shows ways in which Carnap’s philosophical project subtly changes right through his late work on induction, providing an important corrective to interpretations that ignore the work on inductive logic. Specifically, I show that paying attention to the historical details of Carnap’s attempt to apply his work in inductive logic to decision theory and theoretical statistics in the 1950s and 1960s helps us understand how Carnap develops and rearticulates the philosophical point of the practical/theoretical distinction in his late work, offering thus a new interpretation of Carnap’s technical work within the broader context of philosophy of science and analytical philosophy in general. (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)

This chapter describes the logic of inductive inference as seen through the eyes of the modern theory of personal probability, including a number of its recent refinements and extensions. The structure of the chapter is as follows. After a brief discussion of mathematical probability, to establish notation and terminology, it recounts the gradual evolution of the probabilistic explication of induction from Bayes to the present. The focus is not in this history per se, but in its use (...) to highlight the key assumptions, criticisms, refinements, and achievements of that theory. Along the way, the structure of the modern theory is presented, and its relation to the problem of induction discussed. (shrink)

The problem of induction : East and West -- The later Nyaya solution -- The method of generalization : Vyaptigrahopayah -- Counterfactual reasoning : Tarkah -- Universal based extraordinary perception : Samanyalaksanapratyaksa -- Earlier views of adjuncts : Upadhivadah -- The accepted view of adjuncts : Upadhivadasiddhantah -- Classification of adjuncts : Upadhivibhagah -- Sriharsa's Khandanakhandakhadyam on pervasion -- Selected passages from Prabhacandra's Prameyakamalamartanda on critique of pervasion and inference -- Selections from Dharmakirti's Nyayabindu on non-perception as a probans.

Francis Bacon, to suit the needs of his time, created materialistic scientific inductivism to make logic serve empirical sciences, thus making a contribution to logic. Because of this contribution, he occupies an illustrious place in the history of both philosophy and science, deserving the encomium of Marx and Engels as "the real founding father of British materialism and the whole of modern empirical sciences".

First published in 1840, this two-volume treatise by Cambridge polymath William Whewell remains significant in the philosophy of science. The work was intended as the 'moral' to his three-volume History of the Inductive Sciences, which is also reissued in this series. Building on philosophical foundations laid by Immanuel Kant and Francis Bacon, Whewell opens with the aphorism 'Man is the Interpreter of Nature, Science the right interpretation'. Volume 1 contains the majority of Whewell's section on 'ideas', in which he (...) investigates the philosophy underlying a range of different disciplines, including pure, classificatory and mechanical sciences. Whewell's work upholds throughout his belief that the mind was active and not merely a passive receiver of knowledge from the world. A key text in Victorian epistemological debates, notably challenged by John Stuart Mill and his System of Logic, Whewell's treatise merits continued study and discussion in the present day. (shrink)

First published in 1840, this two-volume treatise by Cambridge polymath William Whewell remains significant in the philosophy of science. The work was intended as the 'moral' to his three-volume History of the Inductive Sciences, which is also reissued in this series. Building on philosophical foundations laid by Immanuel Kant and Francis Bacon, Whewell opens with the aphorism 'Man is the Interpreter of Nature, Science the right interpretation'. Volume 2 contains the final sections of Part 1, addressing namely the philosophy (...) of biology and palaetiology. Part 2, 'Of Knowledge', includes a selective review of opinions on the nature of knowledge and the means of seeking it, beginning with Plato. Whewell's work upholds throughout his belief that the mind was active and not merely a passive receiver of knowledge from the world. A key text in Victorian epistemological debates, notably challenged by John Stuart Mill and his System of Logic, Whewell's treatise merits continued study and discussion in the present day. (shrink)

In this chapter we will deal with “mechanizing” induction, i.e. with ways in which theoretical computer science approaches inductive generalization. In the field of Machine Learning, algorithms for induction are developed. Depending on the form of the available data, the nature of these algorithms may be very different. Some of them combine geometric and statistical ideas, while others use classical reasoning based on logical formalism. However, we are not so much interested in the algorithms themselves, but more on (...) the philosophical and theoretical foundations they share. Thus in the first of two parts, we will examine different approaches and inductive assumptions in two particular learning settings. While many machine learning algorithms work well on a lot of tasks, the interpretation of the learned hypothesis is often difficult. Thus, while e.g. an algorithm surprisingly is able to determine the gender of the author of a given text with about 80 percent accuracy [Argamon and Shimoni, 2003], for a human it takes some extra effort to understand on the basis of which criteria the algorithm is able to do so. With that respect the advantage of approaches using logic are obvious: If the output hypothesis is a formula of predicate logic, it is easy to interpret. However, if decision trees or algorithms from the area of inductive logic programming are based purely on classical logic, they suffer from the fact that most universal statements do not hold for exceptional cases, and classical logic does not offer any convenient way of representing statements which are meant to hold in the “normal case”. Thus, in the second part we will focus on approaches for Nonmonotonic Reasoning that try to handle this problem. Both Machine Learning and Nonmonotonic Reasoning have been anticipated partially by work in philosophy of science and philosophical logic. At the same time, recent developments in theoretical computer science are expected to trig- ger further progress in philosophical theories of inference, confirmation, theory revision, learning, and the semantic and pragmatics of conditionals. We hope this survey will contribute to this kind of progress by building bridges between computational, logical, and philosophical accounts of induction. (shrink)

This work is a departure from traditional investigations of induction. Rather than consider issues related to the justification or construction of a scientific inductive logic, Professor Will exposes, evaluates, and rejects the epistemological framework within which work in the philosophy of induction is usually conducted. He argues that the frustrating difficulties faced in the philosophy of induction are endemic to that theory of knowledge which "resulted from the empiricist criticism and revision of the basic Cartesian view (...) that knowledge is a kind of vision or intuition, that there are certain first intuitions, and that further knowledge consists of consequent intuitions elicited from or founded upon these." According to this view of knowledge, a knowledge claim can be justified in the face of the challenge of philosophical skepticism only if it is either itself indubitable or rests on some foundation that is indubitable. The burden of Will’s work is to show that this foundation or "justification" picture of knowledge is seriously mistaken and must be replaced with a more adequate and more realistic view. Human knowledge, he stresses, is best seen as analogous to a constitution. As human knowledge differs from area to area these constitutions differ, "just as history differs from horticulture and architecture from aviation." However, "collectively they form a huge sprawling entity that represents what may be thought of as the constitution for the institutionalized activity of human knowledge.". (shrink)

Ethics and jurisprudence are liable to the remark in common with logic. Almost every writer having taken a different view of some of the particulars which ...

Mr. Barker examines the efforts of Keynes, Reichenbach, Carnap, Williams, Popper, Kemeny and others in their search for the rationale of experimental inference. On what paradigm of reasoning does empirical knowledge depend? Some philosophers suppose it to be induction by enumeration, others induction by elimination, but Mr. Barker sees hope in a modified version of the hypothetico-deductive method. Our knowledge, he explains, forms a’ system ‘in which the fates of various bits are bound together. Philosophers are misled when (...) they speak as if we could confirm a hypothesis that stands alone. Instead, we ought to count one hypothesis better confirmed than its rival ‘if and only if there is some system including the former and the evidence which is somehow than any system including the latter and the evidence’. Here he brings in Kemeny’s ‘logical measure’ of simplicity, read in terms of the number of ways in which two systems of hypotheses can be true in an membered universe. Mr. Barker does not go into the application of this enormous Apparat, nor does he explain the sense of ‘system’ which is supposed to go with his examples of hypotheses, e.g., ‘There’s a cat behind the sofa’. (shrink)

John D. Norton is responsible for a number of influential views in contemporary philosophy of science. This paper will discuss two of them. The material theory of induction claims that inductive arguments are ultimately justified by their material features, not their formal features. Thus, while a deductive argument can be valid irrespective of the content of the propositions that make up the argument, an inductive argument about, say, apples, will be justified (or not) depending on facts about apples. The (...) argument view of thought experiments claims that thought experiments are arguments, and that they function epistemically however arguments do. These two views have generated a great deal of discussion, although there hasn’t been much written about their combination. I argue that despite some interesting harmonies, there is a serious tension between them. I consider several options for easing this tension, before suggesting a set of changes to the argument view that I take to be consistent with Norton’s fundamental philosophical commitments, and which retain what seems intuitively correct about the argument view. These changes require that we move away from a unitary epistemology of thought experiments and towards a more pluralist position. (shrink)

A Fortiori Logic: Innovations, History and Assessments is a wide-ranging and in-depth study of a fortiori reasoning, comprising a great many new theoretical insights into such argument, a history of its use and discussion from antiquity to the present day, and critical analyses of the main attempts at its elucidation. Its purpose is nothing less than to lay the foundations for a new branch of logic and greatly develop it; and thus to once and for all (...) dispel the many fallacious ideas circulating regarding the nature of a fortiori reasoning. -/- The work is divided into three parts. The first part, Formalities, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged. -/- The second part of the work, Ancient and Medieval History, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah. -/- The third part of the present work, Modern and Contemporary Authors, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is impressive, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed. -/- This book also contains, in a final appendix, some valuable contributions to general logic, including new analyses of symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox. (shrink)

This book brings together papers presented at a symposium at the University of Pennsylvania and later revised. They are concerned with the concept of rational belief and with the rôle that induction plays in theories of rationality. There are three well-known theories: subjectivism provides the norm that ‘we may believe a proposition if and only if it fits in with those we already believe, and that we must believe it if and only if avoiding the belief would make for (...) some imbalance in the set of all our beliefs’; an empiricist logic argues that probability must be our guide and holds that a rational person believes a proposition if and only if it has a sufficiently high degree of confirmation assessed on the basis of observation; the pragmatist conceives belief as a species of action, assessing it as rational when it contributes to the furtherance of ends. (shrink)

The Collected Works of John Stuart Mill took thirty years to complete and is acknowledged as the definitive edition of J.S. Mill and as one of the finest works editions ever completed. Mill's contributions to philosophy, economics, and history, and in the roles of scholar, politician and journalist can hardly be overstated and this edition remains the only reliable version of the full range of Mill's writings. Each volume contains extensive notes, a new introduction and an index. Many of (...) the volumes have been unavailable for some time, but the Works are now again available, both as a complete set and as individual volumes. (shrink)

Jin Yuelin (1895?1984), a Chinese logician and philosopher, is greatly influenced by Hume's and Russell's philosophies. How should we respond to Hume's problem of induction? This is an important clue to understand Jin's whole philosophical career. The first section of this paper gives a brief historical review of Russell and Jin. The second section outlines Hume's skeptical arguments against causality and induction. The third section expounds Russell's justification of induction by discussing his views on Hume's skepticism, causality, (...) principle of induction, and empirical postulates. The fourth section clarifies Jin's justification of induction by discussing his critique of Hume's epistemology and his arguments for the reliability of causality and the eternal truth and apriority of the principle of induction. The final section compares Jin's justification of induction with Russell's and concludes that there are similarities and differences between their projects and that both their attempts fail. This paper takes the similar responses to the problem of induction by Jin and Russell to demonstrate the communication that there has been between Chinese philosophers and the Western ones. (shrink)

The problem of underdetermination is thought to hold important lessons for philosophy of science. Yet, as Kyle Stanford has recently argued, typical treatments of it offer only restatements of familiar philosophical problems. Following suggestions in Duhem and Sklar, Stanford calls for a New Induction from the history of science. It will provide proof, he thinks, of “the kind of underdetermination that the history of science reveals to be a distinctive and genuine threat to even our best scientific (...) theories” (Stanford 2001, p. S12). This paper examines Stanford’s New Induction and argues that it – like the other forms of underdetermination that he criticizes – merely recapitulates familiar philosophical conundra. (shrink)

Charles Sanders Peirce wrote the article “The probability of induction” in 1878. It was the fourth article of the series “Illustrations of the Logic of Science” which comprised a total of six articles. According to Peirce, to get a clear idea of the conception of probability, one has ‘to consider what real and sensible difference there is between one degree of probability and another.’ He endorsed what John Venn had called the ‘materialistic view’ of the subject, namely that (...) probability is the proportion of times in which an occurrence of one kind is accompanied by an occurrence of another kind. On the other hand, Peirce recognized the existence of a different interpretation of probability, which was termed by Venn the ‘conceptualistic view’, namely the degree of belief that ought to be attached to a proposition. Peirce’s intent on writing this article seems to be to inquire about the claims of the conceptualists concerning the problem of induction. After reasoning on some examples, he concluded on the impossibility of assigning probability for induction. We show here that the arguments advanced in his article are not sufficient to support such conclusion. Peirce’s thoughts on the probability of induction surely may have influenced statisticians and research scientists of the 20th century in shaping data analysis. (shrink)

Schurz (2019, ch. 4) argues that probabilistic accounts of induction fail. In particular, he criticises probabilistic accounts of induction that appeal to direct inference principles, including subjective Bayesian approaches (e.g., Howson 2000) and objective Bayesian approaches (see, e.g., Williamson 2017). In this paper, I argue that Schurz’ preferred direct inference principle, namely Reichenbach’s Principle of the Narrowest Reference Class, faces formidable problems in a standard probabilistic setting. Furthermore, the main alternative direct inference principle, Lewis’ Principal Principle, is also (...) hard to reconcile with standard probabilism. So, I argue, standard probabilistic approaches cannot appeal to direct inference to explicate the logic of induction. However, I go on to defend a non-standard objective Bayesian account of induction: I argue that this approach can both accommodate direct inference and provide a viable account of the logic of induction. I then defend this account against Schurz’ criticisms. (shrink)

Philosophy, Science, and History: A Guide and Reader is a compact overview of HOPOS that aims to introduce students to the groundwork of the field. Part I of the Reader begins with classic texts in the history of logical empiricism, including Reichenbach's discovery-justification distinction. With careful reference to Kuhn's analysis of scientific revolutions, the section provides key texts analyzing the relationship of HOPOS to the history of science, including texts by Santayana, Rudwick, and Shapin and Schaffer. Part (...) II provides texts illuminating central debates in the history of science and its philosophy. These include the history of natural philosophy (Descartes, Newton, Leibniz, Kant, Hume, and du Châtelet in a new translation); induction and the logic of discovery (including the Mill-Whewell debate, Duhem, and Hanson); and catastrophism versus uniformitarianism in natural history (Playfair on Hutton and Lyell; de Buffon, Cuvier, and Darwin). The editor's introductions to each section provide a broader perspective informed by contemporary research in each area, including related topics. (shrink)

Twenty-three years ago Robert Ayers noticed several brief and intriguing comments on miracles in the Collected Papers of Charles Sanders Peirce. Working with just those scraps of information from the CP, he stitched together a rough but helpful starting point for understanding this aspect of Peirce's religious and scientific thought. In the last few years several more articles on this subject have been written, each filling in a gap left by the others: Ayers' is a theological view, based solely on (...) the CP; later articles fill out Peirce's mathematics and his logic. This paper attempts to fill in a genealogical gap by showing how his thought on miracles is directly related to his dialogues with Plato, Hume, and Lutoslawski. My resources are largely unpublished manuscripts, many of which are fragmentary. I show the relationship between these manuscripts and two key published essays, "Philosophy and the Conduct of Life", and "On the Logic of Drawing History from Ancient Documents, Especially from Testimonies", and then show how Peirce, in dialogue with Plato, exposes and overcomes the nominalistic and anti- miracle prejudices of historiography in his day. The resulting view of history is fallibilistic, realistic and evolutionary, in which miracles are not violations of laws of nature but are to be expected as evolutionary variations that form part of the ongoing self-revelation of the cosmos. Miracles, like all events in history, must not be viewed prejudicially by adherents or detractors, but must be taken into careful account in the grand induction of history and science. (shrink)

During the periods when logical positivism and ordinary language philosophy were ascendant, Brand Blanshard was defending necessity in his writing and in his teaching. The last five chapters of the second volume of The Nature of Thought, published in 1940, were devoted to necessity, and no less than four chapters of Reason and Analysis, appearing in 1962, were on the same subject. The new realism that has supplanted positivism and language philosophy on the American scene should, it would seem, be (...) sympathetic to Blanshard’s insistence that there are physical necessities. For, according to the new realism, the constituents of gross things are real entities that account for the behavior of these things. One would think that it would be recognized that this is possible only if that behavior were a necessary outcome of having those constituents. For, were it only contingent, the behavior could not genuinely be laid at the feet of those constituents. All expectations have been disappointed here, for the new realism remains uncompromisingly Humean when it comes to necessity. Necessity is still, as in the days of positivism and language philosophy, a matter of concepts or linguistic terms and not a matter of things. But if I am right that the new realism is thus far developed only in a one-sided way, then in the near future we shall see a more coherent realism that has embodied many of the features of the doctrine of necessity elaborated by Blanshard. (shrink)

We compare Jan ?ukasiewicz's and Karl Popper's views on induction. The English translation of the two ?ukasiewicz's papers is included in the Appendix.

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated (...) by Herbrand’s Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

Hume's Problem comprises two main projects: defending Hume's argument about induction against a dozen or so purported answers, and laying out a logic of induction that incorporates Hume's great insight in a formal theory. In this review, I will look at several instances of Howson's defense of Hume; then I will sketch the broad outlines of Howson's own "answer," the details of which are myriad and sometimes technical.

In lieu of an abstract, here is a brief excerpt of the content:@ @ Notes and Dlscussaons A NOTE ON CHARLES PEIRCE'S THEORY OF INDUCTION By "Peirce's theory of induction," I refer to a system or collection of ideas which Peirce formulated about the nature and validity of inductive inference or inductive reasoning. This system or collection of ideas covers Peirce's writings from 1867 to 1905.1 During this period of his long philosophical career from 1857 to 1914, Peirce (...) wrote his most important papers on the logic of science and probability. ~ These papers, though only a minor portion of Peirce's philosophical works, constitute a self-explanatory unit of study, simply because they deal with one single subject: the validity of synthetic inference. In the year 1867, Peirce wrote his paper "On the Natural Classification of Arguments," which provides a natural introduction to his treatment of inference and its validity in syllogistic terms and to his trichotomy of inference into deduction, induction, and hypothesis. Through the year 1877 to 1878, Peirce wrote a series of papers on "Illustrations of the Logic of Science." Beside the well-known "The Fixation of Belief" and "How to Make our Ideas Clear," there are four other important but comparatively little discussed papers which are essentially concerned with the logic of science and probability. They are in the following order: "Doctrine of Chances" (with corrections, 1893 and notes, 1910), "The Probability of Induction" (with corrections, 1893), "Deduction, Induction, and Hypothesis" (with corrections, 1893), and "The Order of Nature." In these papers, Peirce laid out his rationale for the validity of ampliative or synthetic inference and indicated a justification of induction in terms of probability. He also made explicit his conception of probabilty in terms of empirical frequency. But very often his ideas concerning the validity of synthetic inference appear disconnected and do not fit into a consistent, not to say wellorganized, whole. It was not until 1883 that Peirce wrote a comparatively systematic treatise on his general theory concerning the validity of ampliative or synthetic inference. This is his "A Theory of Probable inference," in which we find Peirce's statement of what a probable inference is and his argument that inductive inference is valid on account of its being a probable inference. After the year 1883, Peirce's philosophical writings appear to bear little upon his earlier problems in logic of science and probubility. But here and there we find a variety of topics under which Peirce touched the problem of justifying kinds of inference. One significant feature among these is Peirce's recurrent distinction between kinds of induction, a distinction which he had never made in his earlier writings. The distinction is between a "crude induction" (or "pooh-pooh" argument ), a "quantitative induction," and a "qualitative induction." This distinc1See the "Chronological Listing of Peirce's Papers Directly Bearing upon Induction and Probability" in the appendix. 2The earfiest of his papers which we have is dated 1857,and his philosophical activity continued until his death in 1914. [361] 362 HISTORY OF PHILOSOPHY tion is found in the following three short papers: "Three Kinds of Induction" (from "The Logic of Drawing History from Ancient Documents," 1901), "Kinds of Induction" (1903), and "The Varieties and Validity of Induction," (c. 1905). What Peiree calls a "crude induction" is an inference which concludes a universal generalization from what has been found in common in a group of instances. It is induction by simple enumeration. A quantitative induction is an inference from our knowledge of a statistical proportion of individual things having a certain character in a given sample to a conclusion regarding that in the population, from which the sample is chosen. ~ A qualitative induction, on the other hand, is a method of ascertaining factual truths by postulating hypotheses and confirming them on the basis of relevant evidences. That all these three forms of inference are considered induction by Peirce is due to the fact that they do not usually have the form of deduction, and they usually give rise to generalizations which are not deductively warranted by their premises. They differ, nevertheless, because they have different bases for generalizations and their generalizations have different contents. 4 In the light of Peirce's distinction between these three forms... (shrink)

A study of the development of Mill's thought through successive editions of _A System of Logic. His view of the genesis of most scientific laws, it is argued, progressively shifted from inductivism to hypothetico-deductivism. Mill's analysis of hypotheses and of methods for their assessment is considered in detail. New light is shed on relations between Mill's metascience and that of William Whewell.

In this paper Venn's account of probability inference and induction is examined, tracing their differences as well as how they ‘co-operate’ in inferences from particulars to particulars. We discuss the role of mathematical idealizations in making probability inferences, the celebrated rule of succession and we delve into the nature of the reference class problem arguing that for Venn it is a common problem for both induction and inference in probability. Our approach is both historical and philosophical attempting to (...) sketch Venn's position both in the philosophy of probability and induction of his time and in relation to the twentieth-century frequentism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume and the Lockean Background: Induction and the Uniformity Principle David Owen Introduction What has come to be called Hume's problem of induction is special in many ways. It is arguably his most important and influential argument, especially when seen in its overall context of the more general argument about causaUty. It has come to be one of the great "standard problems" ofphilosophyandyetis,by most accounts, almost unique (...) in having no ancient precursor. Interestingly, the argument is frequently presented in terms that the author never used. Itis difficult for us not to think of the problem in terms of the contrast between deductive vs. inductive reasoning (or logic, or whatever). But this is a contrast nowhere to be found in Hume: he talks of demonstrative vs. probable (or moral) reasoning. Hume uses the word "induction" only twice in the Treatise1 and in neither case is it clear that the word is used toreportapieceofprobablereasoning.2One ofthefewoccurrences of"deduce" or "deducing" occurs in the second Enquiry, where he talks about "following the experimental method, and deducing general maximsfrom acomparisonofparticularinstances,"3hardlyaparadigm of what we would call deductive reasoning.4 Many modern studies of Hume that make a real attempt to understand what Hume meant by demonstration and probability, in the context of his argument about induction, tend towards the ahistorical,6 treating Hume's writings as contemporary texts. But failure to take into account some of the background concerning the nature of demonstrative and probable reason makes it unlikely, I think, that we can understand Hume's argument. Part ofthe problem is the failure to identify Hume's targets in that famous negative argument about induction (vague reference is sometimes made to the "rationalists") and part of the problem is identifying the role the uniformity principle plays in Hume's famous argumentand thelightitsheds on the conception ofprobablereasoning Hume was attacking. In this paper I want to argue that a careful look at Locke solves some ofthese problems: he has a well developed view ofdemonstrative and probable reasoning, his confidence in conformity as a ground of probability is a Ukely target of Hume's attack and his Volume XVIII Number 2 179 DAVID OWEN accountshedsmuchlightonwhatrole the principleofuniformitymight play in Hume's account.6 Of course, at a certain level of philosophical abstraction it is irrelevant whether we have understood what Hume "really meant." Theproblemofinductionhas emergedas a central onein epistemology, philosophical logic, and philosophy of science. It would be an interesting question in philosophical historiography to trace the development ofthe "problem" from Hume's time to our own but that is not my task here. I trust one needs no justification for taking an historical look at a famous problem rather than tackling the problem head-on in one's own terms. But if one is needed, here is mine: sometimes, by looking at a problem from a different, historical perspective, one sees a different problem or afamiliar problem in a new light. One's philosophical horizons are broadened. Furthermore, one may make an interesting historical discovery, or at least draw some interesting historical connections. I take it as uncontroversial that Locke, following Descartes, rejected a formal, logical account of demonstrative reasoning. Instead ofusing a formal, syllogistic account ofhow two propositions can entail a third, Locke concentrated on the relation two ideas bear one to another.7 Thus a demonstration, for Locke, getsits force, notfrom what we would call a deductive relation between one set of propositions (.the premises) and anotherproposition (the conclusion), butrather from the relation one idea bears to another, either directly, or via a chain of intermediate ideas. It is true that Locke frequently talks of inferring one proposition from another, but this is always cashed out in terms of relations ofideas via a chain ofintermediate ideas.8 It is also true that Locke occasionally uses the terms "deduce" and "deduction." But these are never used in our sense that has to do with formal deductive validity. Rather, they are simply used loosely in the informal sense of "argue" or "infer," and "argument" or "inference."9 In the rest of this paper, all uses ofthe word"deduction" andits cognates shouldbe taken as having the modern sense, unless flagged as having "Locke... (shrink)

The Collected Works of John Stuart Mill took thirty years to complete and is acknowledged as the definitive edition of J.S. Mill and as one of the finest works editions ever completed. Mill's contributions to philosophy, economics, and history, and in the roles of scholar, politician and journalist can hardly be overstated and this edition remains the only reliable version of the full range of Mill's writings. Each volume contains extensive notes, a new introduction and an index. Many of (...) the volumes have been unavailable for some time, but the Works are now again available, both as a complete set and as individual volumes. (shrink)

In this essay, my aim is twofold: to clarify how the late Mill conceived of the certainty of inductive generalizations and to offer a systematic clarification of the limited domain of application of the Mill’s Canons of Induction. I shall argue that Mill’s views on the certainty of knowledge changed overtime and that this change was accompanied by a new view on the certainty of the inductive results yielded by the Canons of Induction. The key message of the (...) later editions of The System of Logic as conceived by the late Mill was no longer that by the Canons of Induction we can establish scientific certainty and true causes, but rather that the Canons are useful in establishing causal laws in a provisional way. (shrink)

hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part of Hilberts (...) Programme. According to Hilberts Programme one could provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics. (shrink)

This article evaluates Karl Popper's contribution to analytic philosophy, and outlines some of the contradictions in his work which make it difficult to locate in any particular tradition. In particular, the article investigates Popper's own claims to be a member of the rationalist tradition. Although Popper described himself as a member of this tradition, his definition of it diverged quite radically from that offered by other supporters of rationalism, like, for example, Mach, Carnap, and the logical positivists of the Vienna (...) Circle. The reason for this was that Popper believed the rationalist tradition, if it were to remain coherent and relevant, needed to overcome the dilemma posed by Hume's problem of induction. Popper believed that this problem rendered conventional understandings of rationalism, science, and inductive reasoning incoherent. This article suggests that Popper's principal contribution to modern philosophy was to reconfigure the rationalist tradition in such a way as to circumvent the problem of induction while preserving the rationalist commitment to reason, rational debate, and objective knowledge. Popper's reconfiguration of the epistemological bases of the rationalist tradition challenged dominant understandings of rationalist and analytic philosophy, and may be appropriately understood as part of a wider move among philosophers like Quine and Putnam to challenge conventional understandings of analytic philosophy, and of what philosophy itself could and could not achieve. It also informed a vision of social and political life (and of the social and political sciences) as rooted in principles of freedom, equality, and rational debate, but which cannot be fit within the traditional ideological landscape. (shrink)