This article is an attempt to use the insights of objective probability theory to solve the Sleeping Beauty problem. The approach is to develop a partial theory of direct inference and then apply that partial theory to the problem. One of the crucial components of the partial theory is the thesis that expected indefinite probabilities provide a reliable basis for direct inference. The article relies heavily on recent work by Paul D. Thorn to defend that thesis. (...) The article’s primary conclusion is that Beauty can by way of a justifiable direct inference from a statement of expected indefinite probability reach the conclusion that the epistemic probability that the relevant coin toss lands heads is 1/3. The article also provides an account of why the self-locating information that Beauty acquires on Monday is evidentially relevant to the question of whether the coin toss lands heads or tails. (shrink)

John D. Norton’s “Material Theory of Induction” has been one of the most intriguing recent additions to the philosophy of induction. Norton’s account appears to be a notably natural account of actual inductive practices, although his theory has attracted considerable criticism. I detail several novel issues for his theory but argue that supplementing the Material Theory with a theory of direct inference could address these problems. I argue that if this combination is possible, a stronger theory of inductive (...) reasoning emerges, which has a more propitious answer to the Problem of Induction. (shrink)

It is well known that there are, at least, two sorts of cases where one should not prefer a direct inference based on a narrower reference class, in particular: cases where the narrower reference class is gerrymandered, and cases where one lacks an evidential basis for forming a precise-valued frequency judgment for the narrower reference class. I here propose (1) that the preceding exceptions exhaust the circumstances where one should not prefer direct inference based on a (...) narrower reference class, and (2) that minimal frequency information for a narrower (non-gerrymandered) reference class is sufficient to yield the defeat of a direct inference for a broader reference class. By the application of a method for inferring relatively informative expected frequencies, I argue that the latter claim does not result in an overly incredulous approach to direct inference. The method introduced here permits one to infer a relatively informative expected frequency for a reference class R', given frequency information for a superset of R' and/or frequency information for a sample drawn from R'. (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)

The article responds to some of the points raised by B. van Fraassen concerning probability kinematics and direct inference within the framework of the approach to the revision of probability judgment proposed by Levi in The Enterprise of Knowledge. In particular, the critical importance of the question of direct inference is emphasized and explained.

A group of philosophers led by the late John Pollock has applied a method of reasoning about probability, known as direct inference and governed by a constraint known as Reichenbach's principle, to argue in support of ‘thirdism’ concerning the Sleeping Beauty Problem. A subsequent debate has ensued about whether their argument constitutes a legitimate application of direct inference. Here I defend the argument against two extant objections charging illegitimacy. One objection can be overcome via a natural (...) and plausible definition, given here, of the binary relation ‘logically stronger than’ between two properties that can obtain even when the respective properties differ from one another in ‘arity’; given this definition, the Pollock group's argument conforms to Reichenbach's principle. Another objection prompts a certain refinement of Reichenbach's principle that is independently well-motivated. My defense of the Pollock group's argument has epistemological import beyond the Sleeping Beauty problem, because it both widens and sharpens the applicability of direct inference as a method for inferring single-case epistemic probabilities on the basis of general information of a probabilistic or statistical nature. (shrink)

It would be difficult to overestimate the influence Hume’s problem of induction exercises on contemporary epistemology. At the same time, the problem of induction has not perceptibly slowed the progress of mathematics and science. This ironic state of affairs, immortalized by C. D. Broad’s description of induction as “the glory of science” and “the scandal of philosophy,” ought in all fairness to give both sides some pause. And on occasion, it does: the mathematicians stop to concede that Hume has not (...) yet been answered, the scientists worry about randomization of experiments, and inductive skeptics stir uneasily in their chairs at the mention of certain mathematical theorems that seem palpably to have bearing on the problem. (shrink)

There are two uses of randomization in efforts to control systematic bias in experimental design: (a) Alchemical uses seek to convert unavoidable systematic errors into random errors. (b) Hygienic uses seek to reduce the prospect of the experimenter's involvement with the implementation of the experiment contributing to bias. A few remarks are made at the end of the paper about the hygienic use of randomization as a preventative against sticky fingers. The bulk of the discussion addresses the alchemical applications. The (...) thesis is that attitudes towards the cogency of Fisher's alchemical use of randomization ought to depend on views concerning statistical deduction or direct inference. (shrink)

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New results in the theory of nomic probability have led to a theory of probable probabilities, which licenses defeasible inferences between probabilities that are not validated by the probability calculus. Among these are classical principles of direct inference together with some new more general principles that greatly strengthen direct inference and make it much more useful.

Direct inferences identify certain probabilistic credences or confirmation-function-likelihoods with values of objective chances or relative frequencies. The best known version of a direct inference principle is David Lewis’s Principal Principle. Certain kinds of statements undermine direct inferences. Lewis calls such statements inadmissible. We show that on any Bayesian account of direct inference several kinds of intuitively innocent statements turn out to be inadmissible. This may pose a significant challenge to Bayesian accounts of direct (...) inference. We suggest some ways in which these challenges may be addressed. (shrink)

The article begins by describing two longstanding problems associated with direct inference. One problem concerns the role of uninformative frequency statements in inferring probabilities by direct inference. A second problem concerns the role of frequency statements with gerrymandered reference classes. I show that past approaches to the problem associated with uninformative frequency statements yield the wrong conclusions in some cases. I propose a modification of Kyburg’s approach to the problem that yields the right conclusions. Past theories (...) of direct inference have postponed treatment of the problem associated with gerrymandered reference classes by appealing to an unexplicated notion of projectability . I address the lacuna in past theories by introducing criteria for being a relevant statistic . The prescription that only relevant statistics play a role in direct inference corresponds to the sort of projectability constraints envisioned by past theories. (shrink)

One argument for the thirder position on the Sleeping Beauty problem rests on direct inference from objective probabilities. In this paper, I consider a particularly clear version of this argument by John Pollock and his colleagues (The Oscar Seminar 2008). I argue that such a direct inference is defeated by the fact that Beauty has an equally good reason to conclude on the basis of direct inference that the probability of heads is 1/2. Hence, (...) neither thirders nor halfers can find direct support in an appeal to objective probabilities. (shrink)

I here aim to show that a particular approach to the problem of induction, which I will call “induction by direct inference”, comfortably handles Goodman’s problem of induction. I begin the article by describing induction by direct inference. After introducing induction by direct inference, I briefly introduce the Goodman problem, and explain why it is, prima facie, an obstacle to the proposed approach. I then show how one may address the Goodman problem, assuming one (...) adopts induction by direct inference as an approach to the problem of induction. In particular, I show that a relatively standard treatment of what some have called the “Reference Class problem” addresses the Goodman Problem. Indeed, plausible and relatively standard principles of direct inference yield the conclusion that the Goodman inference (involving the grue predicate) is defeated, so it is unnecessary to invoke considerations of ‘projectibility’ in order to address the Goodman problem. I conclude the article by discussing the generality of the proposed approach, in dealing with variants of Goodman’s example. (shrink)

In, Levi has laid out the issues involving chances, frequencies, and direct inference with admirable precision. Nevertheless, puzzles remain. The chief puzzle to which I wish to draw attention is this: Under certain circumstances, we can combine knowledge of chances and knowledge of frequencies to yield new knowledge of chances. If Petersen is “drawn at random” from among Swedes, and we also know that the proportion of Protestants among Swedes is 0.9, then we can say that the chance (...) that Petersen is a Protestant is 0.9. But if we apply this principle generally, we are led to generally trivial results: direct inference yields probabilities constrained only to lie in a broad interval like [0, 1]. On the other hand, if we can't always combine knowledge of chances and knowledge of frequencies to get new knowledge of chances, or if this knowledge can be overridden by other considerations, how do we know when we can usefully apply direct inference? (shrink)

According to an infinite frequency principle, it is rational, under certain conditions, to set your credence in an outcome to the limiting frequency of that outcome if the experiment were repeated indefinitely. I argue that most infinite frequency principles are undesirable in at least one of the following ways: accepting the principle would lead you to accept bets with sure losses, the principle gives no guidance in the case of deterministic experiments like coin tosses and the principle relies on a (...) metaphysical property, ‘chanciness’, whose necessary and sufficient conditions are unknown. I show that a frequency principle that is based on the principal principle suffers from problems related to the definition of ‘chance’ or ‘chanciness’, which could lead to all three of the above problems. I introduce a version of the infinite frequency principle that does not rely on a notion of chance or chanciness and does not suffer from any of these problems. (shrink)

In this paper we discuss the ``admissibility troubles'' for Bayesian accounts of direct inference proposed in, which concern the existence of surprising, unintuitive defeaters even for mundane cases of direct inference. We first show that one could reasonably suspect that the source of these troubles was informal talk about higher-order probabilities: for cardinality-related reasons, classical probability spaces abound in defeaters for direct inference. We proceed to discuss the issues in the context of the rigorous (...) framework of Higher Probability Spaces. However, we show that the issues persist; we prove a few facts which pertain both to classical probability spaces and to HOPs, in our opinion capturing the essence of the problem. In effect we strengthen the message from the admissibility troubles: they arise not only for approaches using classical probability spaces---which are thus necessarily informal about metaprobabilistic phenomena like agents having credences in propositions about chances---but also for at least one respectable framework specifically tailored for rigorous discussion of higher-order probabilities. (shrink)

In Chapter I of his celebrated Foundations of Probability, A. N. Kolmogorov proposed an axiomatic treatment of the mathematical theory of probability—the approach that assimilated probability theory into measure theory. Kolmogorov followed his statement of the axioms with an account of how “we apply the theory of probability to the actual world of experiments.”.

The conflict of narrowness and precision in direct inference occurs if a body of evidence contains estimates for frequencies in a certain reference class and less precise estimates for frequencies in a narrower reference class. To develop a solution to this conflict, I draw on ideas developed by Paul Thorn and John Pollock. First, I argue that Kyburg and Teng’s solution to the conflict of narrowness and precision leads to unreasonable direct inference probabilities. I then show (...) that Thorn’s recent solution to the conflict leads to unreasonable direct inference probabilities. Based on my analysis of Thorn’s approach, I propose a natural distribution for a Bayesian analysis of the data directly obtained from studying members of the narrowest reference class. (shrink)

In this paper we dispel the supposed ``admissibility troubles'' for Bayesian accounts of direct inference proposed by Wallmann and Hawthorne, which concern the existence of surprising, unintuitive defeaters even for mundane cases of direct inference. We show that if one follows the majority of authors in the field in using classical probability spaces unimbued with any additional structure, one should expect similar phenomena to arise and should consider them unproblematic in themselves: defeaters abound! We then show (...) that the framework of Higher Probability Spaces allows the natural modelling of the discussed cases which produces no troubles of this kind. (shrink)

In science, it sometimes occurs that an event is directly observed, and on other occasions that it is not directly observed but one can make the unambiguous inference that it has occurred. Is there any difference concerning the analysis of data arising from these two situations? In this note we show that there is such a difference in one case arising frequently in genetics. The difference derives from the fact that the ability to make the unambiguous inference arises (...) only from a restricted form of data. (shrink)

While there is virtually a consensus among contemporary philosophers of perception that some form of direct realism is true, there is less than complete agreement about whether normal, direct perceptions involve mental inferences in any sense. In taking another look at this recurrent question, my aim is twofold: first, to examine some of the arguments and evidences that have been offered in favor of inferences and to see if they can be accommodated within the direct realist framework, (...) and second, to attempt to clarify and defend the insight of direct realism that normal perceptions are noninferential. (shrink)

The extent to which visual processing can proceed in the visual hierarchy without awareness determines the magnitude of perceptual delay. Increasing data demonstrate that primary visual cortex (V1) is involved in consciousness, constraining the magnitude of visual delay. This makes it possible that visual delay is actually within the optimal lengths to allow sufficient computation; thus it might be unnecessary to compensate for visual delay.

A main message from the causal modelling literature in the last several decades is that under some plausible assumptions, there can be statistically consistent procedures for inferring (features of) the causal structure of a set of random variables from observational data. But whether we can control the error probabilities with a finite sample size depends on the kind of consistency the procedures can achieve. It has been shown that in general, under the standard causal Markov and Faithfulness assumptions, the procedures (...) can only be pointwise but not uniformly consistent without substantial background knowledge. This implies the impossibility of choosing a finite sample size to control the worst case error probabilities. In this paper, I consider the simpler task of inferring causal directions when the skeleton of the causal structure is known, and establish a similarly negative result concerning the possibility of controlling error probabilities. Although the result is negative in form, it has an interesting positive implication for causal discovery methods. (shrink)

The attempt to relate distinctions in perceptual theory to different physiological systems leads to numerous exceptions and inconsistencies. A more promising approach to the reconciliation of constructivist theory and direct perception is to recognize that perception does involve inference, as the constructivists insist, but that inference is a process in logic that does not require unconscious reasoning and need be no more thought-like than resonance.

Two opposing tendencies in the philosophy of language go by the names of ‘referentialism’ and ‘inferentialism’ respectively. In the crudest version of the contrast, the referentialist account of meaning gives centre stage to the referential semantics for a language, which is then used to explain the inference rules for the language, perhaps as those which preserve truth on that semantics (since a referential semantics for a language determines the truth-conditions of its sentences). By contrast, the inferentialist account of meaning (...) gives centre stage to the inference rules for the language, which are then used to explain its referential semantics, perhaps as the semantics on which the rules preserve truth. On pain of circularity, we cannot combine both directions of explanation. (shrink)

This article compares inference to the best explanation with Bayes’s rule in a social setting, specifically, in the context of a variant of the Hegselmann–Krause model in which agents not only update their belief states on the basis of evidence they receive directly from the world, but also take into account the belief states of their fellow agents. So far, the update rules mentioned have been studied only in an individualistic setting, and it is known that in such a (...) setting both have their strengths as well as their weaknesses. It is shown here that in a social setting, inference to the best explanation outperforms Bayes’s rule according to every desirable criterion. 1 What Is Inference to the Best Explanation?2 Judging the Rules—By Which Lights?3 From an Individualistic to a Social Perspective 3.1 The Hegselmann–Krause model 3.2 A probabilistic extension of the Hegselmann–Krause model 3.3 Simulations4 Results and Discussion5 Interpretation6 Conclusion. (shrink)

There are several approaches implementing reasoning based on conditional knowledge bases, one of the most popular being System Z (Pearl, Proceedings of the 3rd conference on theoretical aspects of reasoning about knowledge, TARK ’90, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, pp. 121–135, 1990). We look at ranking functions (Spohn, The Laws of Belief: Ranking Theory and Its Philosophical Applications, Oxford University Press, Oxford, 2012) in general, conditional structures and c-representations (Kern-Isberner, Conditionals in Nonmonotonic Reasoning and Belief Revision: Considering (...) Conditionals as Agents, vol. 2087 of LNCS, Springer, Berlin, 2001) in order to examine the reasoning strength of the different approaches by learning which of the known calculi of nonmonotonic reasoning (System P and R) and Direct Inference are applicable to these inference relations. Furthermore we use the recently proposed Enforcement-postulate (Kern-Isberner and Krümpelmann, Proceedings of the 22nd international joint conference on artificial intelligence, vol. 2, IJCAI’11, AAAI Press, pp. 937–942, 2011) to show dependencies between these approaches. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...) necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)