Pure Inductive Logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years, plus the main contributions of the authors and their collaborators over the last decade, to present a comprehensive account of the discipline within a single unified context.

The current state of inductive logic is puzzling. Survey presentations are recurrently offered and a very rich and extensive handbook was entirely dedicated to the topic just a few years ago [23]. Among the contributions to this very volume, however, one finds forceful arguments to the effect that inductive logic is not needed and that the belief in its existence is itself a misguided illusion , while other distinguished observers have eventually come to see at least the label (...) as “slightly antiquated” .What seems not to have lost any of its currency is the problem which inductive logic is meant to address. Inference from limited ascertained information to uncertain hypotheses is ubiquitous in learning, prediction and discovery. The logical insight that such kind of inference is fallible m .. (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.

Inductive Logic is a ‘thematic compilation’ by Avi Sion. It collects in one volume many (though not all) of the essays, that he has written on this subject over a period of some 23 years, which all demonstrate the possibility and conditions of validity of human knowledge, the utility and reliability of human cognitive means when properly used, contrary to the skeptical assumptions that are nowadays fashionable. A new essay, The Logic of Analogy, was added in 2022.

Hempel's paradox of the ravens arises from the inconsistency of three prima facie plausible principles of confirmation. This paper uses Carnapian inductive logic to (a) identify which of the principles is false, (b) give insight into why this principle is false, and (c) identify a true principle that is sufficiently similar to the false one that failure to distinguish the two might explain why the false principle is prima facie plausible. This solution to the paradox is compared with a (...) variety of other responses and is shown to differ from all of them. (shrink)

This chapter contains sections titled: Good Old‐Fashioned Inductive Logic (GOFIL): Carnap's Program Subjectivized Inductive Logic (SIL): De Finetti Regnant New‐Fangled Inductive Logic (NFIL).

Among the various meanings in which the word ‘probability’ is used in everyday language, in the discussion of scientists, and in the theories of probability, there are especially two which must be clearly distinguished. We shall use for them the terms ‘probability1’ and ‘probability2'. Probability1 is a logical concept, a certain logical relation between two sentences ; it is the same as the concept of degree of confirmation. I shall write briefly “c” for “degree of confirmation,” and “c” for “the (...) degree of confirmation of the hypothesis h on the evidence e“; the evidence is usually a report on the results of our observations. On the other hand, probability2 is an empirical concept; it is the relative frequency in the long run of one property with respect to another. The controversy between the so-called logical conception of probability, as represented e.g. by Keynes, and Jeffreys, and others, and the frequency conception, maintained e.g. by v. Mises and Reichenbach, seems to me futile. These two theories deal with two different probability concepts which are both of great importance for science. Therefore, the theories are not incompatible, but rather supplement each other. (shrink)

This book constitutes the refereed proceedings of the 10th International Conference on Inductive Logic Programming, ILP 2000, held in London, UK in July 2000 as past of CL 2000. The 15 revised full papers presented together with an invited paper were carefully reviewed and selected from 37 submissions. The papers address all current issues in inductive logic programming and inductive learning, from foundational aspects to applications in various fields like data mining, knowledge discovery, and ILP system design.

We consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability.

We propose an Analogy Principle in the context of Unary Inductive Logic and characterize the probability functions which satisfy it. In particular in the case of a language with just two predicates the probability functions satisfying this principle correspond to solutions of Skyrmsʼ ‘Wheel of Fortune’.

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)

Sections 1 through 3 present all of the main ideas behind the probabilistic logic of evidential support. For most readers these three sections will suffice to provide an adequate understanding of the subject. Those readers who want to know more about how the logic applies when the implications of hypotheses about evidence claims (called likelihoods) are vague or imprecise may, after reading sections 1-3, skip to section 6. Sections 4 and 5 are for the more advanced reader who (...) wants a detailed understanding of some telling results about how this logic may bring about convergence to the truth. (shrink)

Logic is a field studied mainly by researchers and students of philosophy, mathematics and computing. Inductive logic seeks to determine the extent to which the premises of an argument entail its conclusion, aiming to provide a theory of how one should reason in the face of uncertainty. It has applications to decision making and artificial intelligence, as well as how scientists should reason when not in possession of the full facts. In this work, Jon Williamson embarks on a (...) quest to find a general, reasonable, applicable inductive logic (GRAIL), all the while examining why pioneers such as Ludwig Wittgenstein and Rudolf Carnap did not entirely succeed in this task. (shrink)

Inductive logic admits a variety of semantics (Haenni et al., 2011, Part 1). This paper develops semantics based on the norms of Bayesian epistemology (Williamson, 2010, Chapter 7). §1 introduces the semantics and then, in §2, the paper explores methods for drawing inferences in the resulting logic and compares the methods of this paper with the methods of Barnett and Paris (2008). §3 then evaluates this Bayesian inductive logic in the light of four traditional critiques of inductive (...) logic, arguing (i) that it is language independent in a key sense, (ii) that it admits connections with the Principle of Indifference but these connections do not lead to paradox, (iii) that it can capture the phenomenon of learning from experience, and (iv) that while the logic advocates scepticism with regard to some universal hypotheses, such scepticism is not problematic from the point of view of scientific theorising. (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)

This book revives inductive logic by bringing out the underlying epistemology. The resulting structural reliabilist theory propounds the view that justification supervenes on syntactic and semantic properties of sentences as justification-bearers. It is claimed to set up a genuine alternative to the prevailing theories of justification. Kawalec substantiates this claim by confronting structural reliabilism with a number of epistemological problems. While the book is addressed to both professionals and students of philosophical logic, probability, epistemology, and philosophy of science, (...) it also surveys ideas central to the development of philosophy in the 20th century. It will be a valuable companion to multifarious graduate and postgraduate courses. (shrink)

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)

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.

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)

Philosophers of science have recently been urged by Arthur Fine to collaborate with physicists and with other scientists in constructing scientific theories.2 What I am proposing is a collaboration at the other pole of scientific activity; the pole of experiment.I consider this effort to be part of a tendency within philosophy to naturalize epistemology. The banner of naturalistic epistemology has attracted such men as Quine and Goldman. I consider the effort as one small part of that program which involves not (...) only theoretical integration of at least portions of the two corpuses of knowledge, but the employment of methodologies which are common to them both.The theory of inductive logic today is in stasis. A number of highly ingenious and very sophisticated theories of inference vie for primacy. As in the case of many philosophical theses, no closure seems in sight. (shrink)

Inductive logic generalizes the idea of logical entailment and provides standards for the evaluation of non-conclusive arguments. A main application of inductive logic is the generalization of observational data to theoretical models. In the empirical sciences, the mathematical theory of statistics addresses the same problem. This paper argues that there is no separable purely logical aspect of statistical inference in a variety of complex problems. Instead, statistical practice is often motivated by decision-theoretic considerations and resembles empirical science.

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

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.

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)

The paper explores the handling of singular analogy in quantitative inductive logics. It concentrates on two analogical patterns coextensive with the traditional argument from analogy: perfect and imperfect analogy. Each is examined within Carnap’s λ-continuum, Carnap’s and Stegmüller’s λ-η continuum, Carnap’s Basic System, Hintikka’s α-λ continuum, and Hintikka’s and Niiniluoto’s K-dimensional system. Itis argued that these logics handle perfect analogies with ease, and that imperfect analogies, while unmanageable in some logics, are quite manageable in others. The paper concludes with a (...) modification of the K-dimensional system that synthesizes independent proposals by Kuipers and Niiniluoto. (shrink)

Carnap's Inductive Logic, like most philosophical discussions of induction, is designed for the case of independent trials. To take account of periodicities, and more generally of order, the account must be extended. From both a physical and a probabilistic point of view, the first and fundamental step is to extend Carnap's inductive logic to the case of finite Markov chains. Kuipers (1988) and Martin (1967) suggest a natural way in which this can be done. The probabilistic character (...) of Carnapian inductive logic(s) for Markov chains and their relationship to Carnap's inductive logic(s) is discussed at various levels of Bayesian analysis. (shrink)

The resolving of the main problem of quantum mechanics about how a quantum leap and a smooth motion can be uniformly described resolves also the problem of how a distribution of reliable data and a sequence of deductive conclusions can be uniformly described by means of a relevant wave function “Ψdata”.

Logic is the study of the quality of arguments. An argument consists of a set of premises and a conclusion. The quality of an argument depends on at least two factors: the truth of the premises, and the strength with which the premises confirm the conclusion. The truth of the premises is a contingent factor that depends on the state of the world. The strength with which the premises confirm the conclusion is supposed to be independent of the state (...) of the world. Logic is only concerned with this second, logical factor of the quality of arguments. (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)

I argue, in opposition, to the traditional approach that systematic psychological inquiry of a type frequently practiced by people like Edwards, Kahneman and Tversky, and Schum is relevant to the choice of the best inductive logic. In the paper, I present some provisional arguments against the traditional view and sketch some of the relevant evidence. This effort is made with the aim of aiding the development of a naturalistic epistemology.

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)

Although we often see references to Carnap’s inductive logic even in modern literatures, seemingly its confusing style has long obstructed its correct understanding. So instead of Carnap, in this paper, I devote myself to its necessary and sufficient commentary. In the beginning part (Sections 2-5), I explain why Carnap began the study of inductive logic and how he related it with our thought on probability (Sections 2-4). Therein, I trace Carnap’s thought back to Wittgenstein’s Tractatus as well (Section (...) 5). In the succeeding sections, I attempt the simplest exhibition of Carnap’s earlier system, where his original thought was thoroughly provided. For this purpose, minor concepts to which researchers have not paid attention are highlighted, for example, m-function (Section 8), in-correlation (Section 10), C-correlate (Section 10), statistical distribution (Section 12), and fitting sequence (Section 17). The climax of this paper is the proof of theorem (56). Through this theorem, we will be able to overview Carnap’s whole system. (shrink)