Objective Bayesianism has been criticised for not allowing learning from experience: it is claimed that an agent must give degree of belief ½ to the next raven being black, however many other black ravens have been observed. I argue that this objection can be overcome by appealing to objective Bayesian nets, a formalism for representing objective Bayesian degrees of belief. Under this account, previous observations exert an inductive influence on the next observation. I show how this approach can be used (...) to capture the Johnson-Carnap continuum of inductive methods, as well as the Nix-Paris continuum, and show how inductive influence can be measured. (shrink)

My title is intended to recall Terence Fine's excellent survey, Theories of Probability [1973]. I shall consider some developments that have occurred in the intervening years, and try to place some of the theories he discussed in what is now a slightly longer perspective. Completeness is not something one can reasonably hope to achieve in a journal article, and any selection is bound to reflect a view of what is salient. In a subject as prone to dispute as this, there (...) will inevitably be many who will disagree with any author's views, and I take the opportunity to apologize in advance to all such people for what they will see as the narrowness and distortion of mine. (shrink)

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)

This paper addresses the question whether the past couple of decades of formal research in belief revision offers evidence of a new psychologism in logic. In the first part I examine five potential arguments in favour of this thesis and find them all wanting. In the second part of the paper I argue that belief revision research has climbed up a hierarchy of models for the change of doxastic states that appear to be clearly normative at the bottom, but are (...) more amenable to an empirical-descriptive interpretation on higher levels. I conclude that this observation might offer a foothold for the thesis that there is a new psychologism in logic. (shrink)

Objective Bayesian probability is often defined over rather simple domains, e.g., finite event spaces or propositional languages. This paper investigates the extension of objective Bayesianism to first-order logical languages. It is argued that the objective Bayesian should choose a probability function, from all those that satisfy constraints imposed by background knowledge, that is closest to a particular frequency-induced probability function which generalises the λ = 0 function of Carnap’s continuum of inductive methods.

The problem addressed in this paper is “the main epistemic problem concerning science”, viz. “the explication of how we compare and evaluate theories [...] in the light of the available evidence” (van Fraassen, BC, 1983, Theory comparison and relevant Evidence. In J. Earman (Ed.), Testing scientific theories (pp. 27–42). Minneapolis: University of Minnesota Press). Sections 1– 3 contain the general plausibility-informativeness theory of theory assessment. In a nutshell, the message is (1) that there are two values a theory should exhibit: (...) truth and informativeness—measured respectively by a truth indicator and a strength indicator; (2) that these two values are conflicting in the sense that the former is a decreasing and the latter an increasing function of the logical strength of the theory to be assessed; and (3) that in assessing a given theory by the available data one should weigh between these two conflicting aspects in such a way that any surplus in informativeness succeeds, if the shortfall in plausibility is small enough. Particular accounts of this general theory arise by inserting particular strength indicators and truth indicators. In Section 4 the theory is spelt out for the Bayesian paradigm of subjective probabilities. It is then compared to incremental Bayesian confirmation theory. Section 4 closes by asking whether it is likely to be lovely. Section 5 discusses a few problems of confirmation theory in the light of the present approach. In particular, it is briefly indicated how the present account gives rise to a new analysis of Hempel’s conditions of adequacy for any relation of confirmation (Hempel, CG, 1945, Studies in the logic of comfirmation. Mind, 54, 1–26, 97–121.), differing from the one Carnap gave in § 87 of his Logical foundations of probability (1962, Chicago: University of Chicago Press). Section 6 adresses the question of justification any theory of theory assessment has to face: why should one stick to theories given high assessment values rather than to any other theories? The answer given by the Bayesian version of the account presented in section 4 is that one should accept theories given high assessment values, because, in the medium run, theory assessment almost surely takes one to the most informative among all true theories when presented separating data. The concluding section 7 continues the comparison between the present account and incremental Bayesian confirmation theory. (shrink)

First, a brief historical trace of the developments in confirmation theory leading up to Goodman's infamous "grue" paradox is presented. Then, Goodman's argument is analyzed from both Hempelian and Bayesian perspectives. A guiding analogy is drawn between certain arguments against classical deductive logic, and Goodman's "grue" argument against classical inductive logic. The upshot of this analogy is that the "New Riddle" is not as vexing as many commentators have claimed. Specifically, the analogy reveals an intimate connection between Goodman's problem, and (...) the "problem of old evidence". Several other novel aspects of Goodman's argument are also discussed. (shrink)

We survey a variety of possible explications of the term “Individual Risk.” These in turn are based on a variety of interpretations of “Probability,” including classical, enumerative, frequency, formal, metaphysical, personal, propensity, chance and logical conceptions of probability, which we review and compare. We distinguish between “groupist” and “individualist” understandings of probability, and explore both “group to individual” and “individual to group” approaches to characterising individual risk. Although in the end that concept remains subtle and elusive, some pragmatic suggestions for (...) progress are made. (shrink)

It is generally thought that Carnap’s principle of tolerance cannot be integrated into a coherent overall conception of rationality. The doubts come from many sides, of which two are singled out. This paper argues that both are wrong, and that Carnapian rationality is a viable and perhaps quite interesting program for further development.

According to all traditional theories of confirmation, statements such as “E confirms H” (where H is a hypothesis and E is the evidence that supports H) are a priori. Peter Achinstein has recently challenged this orthodox position. He claims that at least some confirmation statements are empirical. In this paper I criticize this thesis. I first show that Achinstein´s arguments are either flawed or inconclusive. I then argue that there are strong reasons to conceive of all confirmation statements as a (...) priori. I conclude that inductive logic, if possible, must be a priori knowledge. (shrink)

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how recent work on (...) the qualitative part of probabilistic inference leads to a very broad class of 'proto-probability' functions. (shrink)

Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information.

This note argues that, under some circumstances, it is more rational not to behave in accordance with a Bayesian prior than to do so. The starting point is that in the absence of information, choosing a prior is arbitrary. If the prior is to have meaningful implications, it is more rational to admit that one does not have sufficient information to generate a prior than to pretend that one does. This suggests a view of rationality that requires a compromise between (...) internal coherence and justification, similarly to compromises that appear in moral dilemmas. Finally, it is argued that Savage's axioms are more compelling when applied to a naturally given state space than to an analytically constructed one; in the latter case, it may be more rational to violate the axioms than to be Bayesian. (shrink)

Bayesian decision theory is here construed as explicating a particular concept of rational choice and Bayesian probability is taken to be the concept of probability used in that theory. Bayesian probability is usually identified with the agent’s degrees of belief but that interpretation makes Bayesian decision theory a poor explication of the relevant concept of rational choice. A satisfactory conception of Bayesian decision theory is obtained by taking Bayesian probability to be an explicatum for inductive probability given the agent’s evidence.

The Paradox of the Ravens (a.k.a,, The Paradox of Confirmation) is indeed an old chestnut. A great many things have been written and said about this paradox and its implications for the logic of evidential support. The first part of this paper will provide a brief survey of the early history of the paradox. This will include the original formulation of the paradox and the early responses of Hempel, Goodman, and Quine. The second part of the paper will describe attempts (...) to resolve the paradox within a Bayesian framework, and show how to improve upon them. This part begins with a discussion of how probabilistic methods can help to clarify the statement of the paradox itself. And it describes some of the early responses to probabilistic explications. We then inspect the assumptions employed by traditional (canonical) Bayesian approaches to the paradox. These assumptions may appear to be overly strong. So, drawing on weaker assumptions, we formulate a new-and-improved Bayesian confirmation-theoretic resolution of the Paradox of the Ravens. (shrink)

A generalization of Ehrenfest''s urn model is suggested. This will allow usto treat a wide class of stochastic processes describing the changes ofmicroscopic objects. These processes are homogeneous Markov chains. Thegeneralization proposed is presented as an abstract conditional (relative)probability theory. The probability axioms of such a theory and some simpleadditional conditions, yield both transition probabilities and equilibriumdistributions. The resulting theory interpreted in terms of particles andsingle-particle states, leads to the usual formulae of quantum and classicalstatistical mechanics; in terms of chromosomes (...) and allelic types, it allowsthe deduction of many genetical models including the Ewens sampling formula;in terms of agents'' strategies, it gives a justification of the ``herdbehaviour'''' typical of a population of heterogeneous economic agents. (shrink)

We investigate the relative probabilistic support afforded by the combination of two analogies based on possibly different, structural similarity (as opposed to e.g. shared predicates) within the context of Pure Inductive Logic and under the assumption of Language Invariance. We show that whilst repeated analogies grounded on the same structural similarity only strengthen the probabilistic support this need not be the case when combining analogies based on different structural similarities. That is, two analogies may provide less support than each would (...) individually. (shrink)

Das Thema dieses Bandes ist die Frage, ob die Wissenschaftstheorie eine normative Disziplin ist. Zunächst überrascht die Frage, denn für viele Wissenschaftstheoretiker ist die Antwort ein klares „Ja“; sie halten es für einen Allgemeinplatz, dass die Wissenschaftstheorie ein normatives Unternehmen ist. Bei genauerem Hinsehen stellt sich jedoch heraus, dass die Frage unterschiedliche Interpretationen zulässt, die einzeln diskutiert werden müssen. Dies geschieht im ersten Abschnitt. Im zweiten Abschnitt suchen wir nach möglichen Erklärungen dafür, warum die Wissenschaftstheorie bisher bei dem Projekt, eine (...) allseits akzeptable Methodologie wissenschaftlichen Schließens zu formulieren und zu begründen, so wenig Erfolg hatte. Eine mögliche Erklärung für den ausbleibenden Erfolg ist der Partikularismus, wonach Methoden keine allgemeine, sondern nur eine lokale, bereichsabhängige Gültigkeit haben. Im dritten Abschnitt wollen wir an Hand des Bayesianismus zeigen, dass die Methodologie doch nicht so schlecht da steht, wie es zunächst den Anschein hat. Der Bayesianismus ist für viele Wissenschaftstheoretiker der aussichtsreichste Kandidat für eine allgemeine Theorie induktiven Räsonierens. Wir besprechen seine Vorzüge, stellen aber auch dar, welche Konzessionen er an den Partikularismus machen muss. (shrink)

This article discusses the classical problem of zero probability of universal generalizations in Rudolf Carnap’s inductive logic. A correction rule for updating the inductive method on the basis of evidence will be presented. It will be shown that this rule has the effect that infinite streams of uniform evidence assume a non-zero limit probability. Since Carnap’s inductive logic is based on finite domains of individuals, the probability of the corresponding universal quantification changes accordingly. This implies that universal generalizations can receive (...) positive prior and posterior probabilities, even for (countably) infinite domains. (shrink)

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)

Predictions about the future and unrestricted universal generalizations are never logically implied by our observational evidence, which is limited to particular facts in the present and past. Nevertheless, propositions of these and other kinds are often said to be confirmed by observational evidence. A natural place to begin the study of confirmation theory is to consider what it means to say that some evidence E confirms a hypothesis H.

In October 2009 I decided to stop doing philosophy. This meant, in particular, stopping work on the book that I was writing on the nature of probability. At that time, I had no intention of making my unfinished draft available to others. However, I recently noticed how many people are reading the lecture notes and articles on my web site. Since this draft book contains some important improvements on those materials, I decided to make it available to anyone who wants (...) to read it. That is what you have in front of you. The account of Laplace’s theory of probability in Chapter 4 is very different to what I said in my seminar lectures, and also very different to any other account I have seen; it is based on a reading of important texts by Laplace that appear not to have been read by other commentators. The discussion of von Mises’ theory in Chapter 7 is also new, though perhaps less revolutionary. And the final chapter is a new attempt to come to grips with the popular, but amorphous, subjective theory of probability. The material in the other chapters has mostly appeared in previous articles of mine but things are sometimes expressed differently here. I would like to say again that this is an incomplete draft of a book, not the book I would have written if I had decided to finish it. It no doubt contains poor expressions, it may contain some errors or inconsistencies, and it doesn’t cover all the theories that I originally intended to discuss. Apart from this preface, I have done no work on the book since October 2009. (shrink)

I conceive of inductive logic as a project of explication. The explicandum is one of the meanings of the word `probability' in ordinary language; I call it inductive probability and argue that it is logical, in a certain sense. The explicatum is a conditional probability function that is specified by stipulative definition. This conception of inductive logic is close to Carnap's, but common objections to Carnapian inductive logic (the probabilities don't exist, are arbitrary, etc.) do not apply to this conception.

The purpose of this paper is to make a simple observation regarding the Johnson -Carnap continuum of inductive methods. From the outset, a common criticism of this continuum was its failure to permit the confirmation of universal generalizations: that is, if an event has unfailingly occurred in the past, the failure of the continuum to give some weight to the possibility that the event will continue to occur without fail in the future. The Johnson -Carnap continuum is the mathematical consequence (...) of an axiom termed Johnson 's sufficientness postulate, the thesis of this paper is that, properly viewed, the failure of the Johnson -Carnap continuum to confirm universal generalizations is not a deep fact, but rather an immediate consequence of the sufficientness postulate; and that if this postulate is modified in the minimal manner necessary to eliminate such an entailment, then the result is a new continuum that differs from the old one in precisely one respect: it enjoys the desideratum of confirming universal generalizations. (shrink)

The problem of valid induction could be stated as follows: are we justified in accepting a given hypothesis on the basis of observations that frequently confirm it? The present paper argues that this question is relevant for the understanding of Machine Learning, but insufficient. Recent research in inductive reasoning has prompted another, more fundamental question: there is not just one given rule to be tested, there are a large number of possible rules, and many of these are somehow confirmed by (...) the data — how are we to restrict the space of inductive hypotheses and choose effectively some rules that will probably perform well on future examples? We analyze if and how this problem is approached in standard accounts of induction and show the difficulties that are present. Finally, we suggest that the explanation-based learning approach and related methods of knowledge intensive induction could be, if not a solution, at least a tool for solving some of these problems. (shrink)

Theories of truth approximation in terms of truthlikeness almost always deal with approaching deterministic truths, either actual or nomic. This paper deals first with approaching a probabilistic nomic truth, viz. a true probability distribution. It assumes a multinomial probabilistic context, hence with a lawlike true, but usually unknown, probability distribution. We will first show that this true multinomial distribution can be approached by Carnapian inductive probabilities. Next we will deal with the corresponding deterministic nomic truth, that is, the set of (...) conceptually possible outcomes with a positive true probability. We will introduce Hintikkian inductive probabilities, based on a prior distribution over the relevant deterministic nomic theories and on conditional Carnapian inductive probabilities, and first show that they enable again probabilistic approximation of the true distribution. Finally, we will show, in terms of a kind of success theorem, based on Niiniluoto’s estimated distance from the truth, in what sense Hintikkian inductive probabilities enable the probabilistic approximation of the relevant deterministic nomic truth. In sum, the truth approximation perspective on Carnapian and Hintikkian inductive probabilities leads to the unification of the inductive probability field and the field of truth approximation. (shrink)

Objective standards for justification or for being a reason would be desirable, but inductive skepticism tells us that they cannot be presupposed. Rather, we have to start from subjective-relative notions of justification and of being a reason. The paper lays out the strategic options we have given this dilemma. The paper explains the requirements for this subject-relative notion and how they may be satisfied. Then it discusses four quite heterogeneous ways of providing more objective standards, which combine without guaranteeing complete (...) success. (shrink)

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q& R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability (...) calculus for prob(P/Q& R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q& R) 1 A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability—nomic probability—there are principles of "probable probabilities" that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r, s, a) such that if prob(P/Q) = r, prob(P/R) = s, andprob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&R) = Y(r, s, a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities. This has the potential to greatly enhance the usefulness of probabilities in practical application. (shrink)

Single-case and long-run propensity theories are among the main objective interpretations of probability. There have been various objections to these theories, e.g. that it is difficult to explain why propensities should satisfy the probability axioms and, worse, that propensities are at odds with these axioms, that the explication of propensities is circular and accordingly not informative, and that single-case propensities are metaphysical and accordingly non-scientific. We consider various propensity theories of probability and their prospects in light of these objections. We (...) argue that while propensity theories face challenges, these challenges do not undermine their validity as prospective interpretations of probability in science. (shrink)

We examine the idea that similar problems should have similar solutions (to paraphrase van Fraassen’s slogan ‘Problems which are essentially the same must receive essentially the same solution’, see van Fraassen in Laws and symmetry, Oxford Univesity Press, Oxford, 1989, p. 236) in the context of symmetries of sentence algebras within Inductive Logic and conclude that by itself this is too generous a notion upon which to found the rational assignment of probabilities. We also argue that within our formulation of (...) symmetry the paradoxes associated with the so called ‘Principle of Indifference’ collapse, but only to be replaced by genuinely irremediable examples of the same phenomenon. (shrink)

In this paper, I examine the possibility of accounting for the rationality of belief-formation by utilising decision-theoretic considerations. I consider the utilities to be used by such an approach, propose to employ verisimilitude as a measure of cognitive utility, and suggest a natural way of generalising any measure of verisimilitude defined on propositions to partial belief-systems, a generalisation which may enable us to incorporate Popper's insightful notion of verisimilitude within a Bayesian framework. I examine a dilemma generated by the decision-theoretic (...) procedure and consider an adequacy condition (immodesty) designed to ameliorate one of its horns. Finally, I argue in a sceptical vein that no adequate verisimilitude measure can be used decision-theoretically. (shrink)

Summary A survey of recent work in the philosophy of science in Finland, with a bibliography. The main sources of influence emphasized are Eino Kaila (1890–1958) and G. H. von Wright (b. 1916). The main topics covered are: induction and probability; information and explanation; the acceptance and application of theories; the role of auxiliary (theoretical) terms; measurement; general methodology of social and behavioral sciences; finalistic explanation; methodology of sociology and history.

We respond to Morris and Richardson 's claim that Pickering and Chater's arguments about the lack of a relation between cognitive science and folk psychology are flawed. We note that possible controversies about the appropriate uses for the two terms do not affect our arguments. We then address their claim that computational explanation of knowledge-rich processes has proved possible in the domains of problem solving, scientific discovery, and reasoning. We argue that, in all cases, computational explanation is only possible for (...) aspects of those processes that do not make reference to general knowledge. We conclude that consideration of the issues raised by Morris and Richardson reinforces our original claim that there are two fundamentally distinct projects for understanding the mind, one based on justification, and the other on computational explanation, and that these apply to non-overlapping aspects of mental life. (shrink)

Georg Kreisel suggested various ways out of the Gödel incompleteness theorems. His remarks on ways out were somewhat parenthetical, and suggestive. He did not develop them in subsequent papers. One aim of this paper is not to develop those remarks, but to show how the basic idea that they express can be used to reason about the Lucas-Penrose-Putnam arguments that human minds are not finitary computational machines. Another aim is to show how one of Putnam’s two anti-functionalist arguments avoids the (...) logical error in the Lucas-Penrose arguments, extends those arguments, but succumbs to an absurdity. A third aim is to provide a categorization of the Lucas-Penrose-Putnam anti-functionalist arguments. (shrink)

We give an account of some relationships between the principles of Constant and Atom Exchangeability and various generalizations of the Principle of Instantial Relevance within the framework of Inductive Logic. In particular we demonstrate some surprising and somewhat counterintuitive dependencies of these relationships on ostensibly unimportant parameters, such as the number of predicates in the overlying language.

Probability and logic are two branches of mathematics that have important philosophical applications. This article discusses several areas of intersection between them. Several involve the role for probability in giving semantics for logic or the role of logic in governing assignments of probability. Some involve probability over non-classical logic or self-referential sentences.

Additionally, the text shows how to develop computationally feasible methods to mesh with this framework.