Hume: Philosophy of Probability

The Heythrop Journal, Volume 63, Issue 4, Page 547-552, July 2022.

This paper examines Hume’s formulations and uses of the conceivability principle and the inconceivability principle. In Hume’s works, we identify different versions of CP and ICP, including proper CP, proper ICP, the weak versions of CP and ICP, the epistemic versions of CP and ICP, and show that Hume not only expresses ICP, but also really maintains it. Assuming an axiomatic characterization of modalities, we argue that if there is a sharp distinction between levels of modalities, then Hume’s conceivability arguments (...) do not hold. But, in a rather different way, we also argue that if Hume’s conceivability arguments hold, then there should be no distinction between levels of modalities. Finally, we argue that after Hume, there are lots of endeavors in logic and philosophy to distinguish different levels of modalities, and to accept new concepts of necessity other than logical necessity. (shrink)

Classical statistical mechanics posits probabilities for various events to occur, and these probabilities seem to be objective chances. This does not seem to sit well with the fact that the theory’s time evolution is deterministic. We argue that the tension between the two is only apparent. We present a theory of Humean objective chance and show that chances thus understood are compatible with underlying determinism and provide an interpretation of the probabilities we find in Boltzmannian statistical mechanics.

In an illuminating article, Claus Beisbart argues that the recently-popular thesis that the probabilities of statistical mechanics (SM) are Best System chances runs into a serious obstacle: there is no one axiomatization of SM that is robustly best, as judged by the theoretical virtues of simplicity, strength, and fit. Beisbart takes this 'no clear winner' result to imply that the probabilities yielded by the competing axiomatizations simply fail to count as Best System chances. In this reply, we express sympathy for (...) the 'no clear winner' thesis. However, we argue that an importantly different moral should be drawn from this. We contend that the implication for Humean chances is not that there are no SM chances, but rather that SM chances fail to be sharp. (shrink)

John Arbuthnot's celebrated but flawed paper in the Philosophical Transactions of 1711-12 is a philosophically and historically plausible target of Hume's probability theory. Arbuthnot argues for providential design rather than chance as a cause of the annual birth ratio, and the paper was championed as a successful extension of the new calculations of the value of wagers in games of chance to wagers about natural and social phenomena. Arbuthnot replaces the earlier anti-Epicurean notion of chance with the equiprobability assumption of (...) Huygens's mathematics of games of chance, and misrepresents the birth ratio data to rule out chance in favour of design. The probability sections of Hume's Treatise taken together correct the equiprobability assumption and its extension to other kinds of phenomena in the estimation of wagers or expectations about particular events. Hume's probability theory demonstrates the flaw in this version of the design argument. (shrink)

Hume appeals to different kinds of certainties and necessities in the Treatise. He contrasts the certainty that arises from intuition and demonstrative reasoning with the certainty that arises from causal reasoning. He denies that the causal maxim is absolutely or metaphysically necessary, but he nonetheless takes the causal maxim and ‘proofs’ to be necessary. The focus of this paper is the certainty and necessity involved in Hume’s concept of knowledge. I defend the view that intuitive certainty, in particular, is certainty (...) of the invariability or necessity of relations between ideas. Against David Owen and Helen Beebee, I argue that the certainty involved in intuition depends on the activity of the mind. I argue, further, that understanding this activity helps us understand more clearly one of Hume’s most important theses, namely that experience is the source of a distinct kind of certainty and of necessity. (shrink)

This paper defends a Bayesian approach to confirming a miracle against Jordan Howard Sobel’s recent novel interpretation of Hume’s criticisms. In his book, ’Logic and Theism’, Sobel offers an intriguing and original way to apply Hume’s criticisms against the possibility of having sufficient evidence to confirm a miracle. The key idea behind Sobel’s approach is to employ infinitesimal probabilities to neutralize the cumulative effects of positive evidence for any miracle. This paper aims to undermine Sobel’s use of infinitesimal probabilities to (...) block a Bayesian approach to confirming a miracle. (shrink)

The theorem of convergence of opinions is an important theorem in the subjective theory of probability.It demonstrates that the subjectivity of a prior probability will be substituted with the objectivity of a posterior probability as evidences increase.The theorem of convergence of opinions is regarded as the dynamic principle of rationality concerning the subjective probability,and therefore is used to resolve Hume's problem,i.e.,the problem of inductive rationality.However,Hacking convincingly argues that the theorem of convergence of opinions is not about the convergence of a (...) posterior probability Pre(h),but about the convergence of a conditional probability Pr(h/e).The subjective theory of probability implies a tacit acceptance of an equation that is Pre(h)=Pr(h/e),usually called "the rule of conditionalisation".Thus,the problem of rationality of induction is converted into one concerning the rule of conditionalisation.In this paper a new principle of rationality(the principle of minimum number of initial probabilities)is put forward,which,in combination with the concept of "local rationality",offers a justification for the rule of conditionalisation. (shrink)

This paper contends that the first argument of Hume's "Of scepticism with regard to reason" entails that humans have no knowledge as Hume understands knowledge. In defending this claim, we also see how Hume's argument anticipates an important aspect of an extremely influential 20th century development: the collapse of the analytic/synthetic distinction.

We must rethink our assessment of Hume’s theory of probabilistic inference. Hume scholars have traditionally dismissed his naturalistic explanation of how we make inferences under conditions of uncertainty; however, psychological experiments and computer models from cognitive science provide substantial support for Hume’s account. Hume’s theory of probabilistic inference is far from obsolete or outdated; on the contrary, it stands at the leading edge of our contemporary science of the mind.

The topic and methods of David Hume’s "Of Miracles" resemble his historiographical more than his philosophical works. Unfortunately, Hume and his critics and apologists have shared the prescientific, indeed ahistorical, limitations of Hume’s original historical investigations. I demonstrate the advantages of the critical methodological approach to testimonies, developed initially by German biblical critics in the late eighteenth century, to a priori discussions of miracles. Any future discussion of miracles and Hume must use the critical method to improve the quality and (...) relevance of the debate. (edited). (shrink)

The most important theories in fundamental physics, quantum mechanics and statistical mechanics, posit objective probabilities or chances. As important as chance is there is little agreement about what it is. The usual “interpretations of probability” give very different accounts of chance and there is disagreement concerning which, if any, is capable of accounting for its role in physics. David Lewis has contributed enormously to improving this situation. In his classic paper “A Subjectivist's Guide to Objective Chance” he described a framework (...) for representing single case objective chances, showed how they are connected to subjective credences, and sketched a novel account what they are within his Humean account of scientific laws. Here I will describe these contributions and add a little to them. (shrink)

Marc Lange has criticized my assertion that relative to a Bayesian conception of inductive reasoning, Hume's argument for inductive scepticism cannot be run. I reply that the way in which Lange suggests one should run the Humean argument in a Bayesian framework ignores the fact that in Bayesian models of learning from experience, the domain of an agent's probability measure is exogenously determined. I also show that Lange is incorrect to equate probability distributions which 'support inductive inferences' with probability distributions (...) which assign probability to contingent propositions/events. (shrink)

Analyses of the argument from design in Hume's Dialogues concerning Natural Religion have generally treated that argument as an example of reasoning by analogy. In this paper I examine whether it is in accord with Hume's thinking about the argument to subsume the version of it given in the Dialogues under the model of probabilistic reasoning offered by Bayes's theorem. Wesley Salmon attempted this project in 1978. In related projects, David Owen as well as Philip Dawid and Donald Gillies have (...) more recently attempted to construct Bayesian analyses of Hume's argument concerning testimony in "Of Miracles.". (shrink)

The foremost advocate of Baconian probability, L. J. Cohen, has credited Hume for being the first to explicitly recognize that there is an important kind of probability which does not fit into the framework afforded by the calculus of chance, a recognition that is evident in Hume's distinction between analogical probability and probabilities arising from chance or cause. This essay defends Hume's account of the credibility of testimony, including his notorious argument against the credibility of testimony to miracles, in light (...) of this insight. (shrink)

Comienzo este artículo mostrando que las teorías neohumeanas de la causalidad probabilista basadas en la noción de relevancia estadlstica (como la teoria de Suppes, 1970) se encuentran con múltiples e insuperables dificultades. Luego analizo brevemente algunas versiones de la causalidad probabilista que relativizan o prescinden de dicha noción: la de Cartwright, que postula la existencia de capacidades causales, y las de Salmon y Dowe, quienes, aunque se proponen no abandonar el suelo humeano, creen necesario introducir una ontología de propensiones. Y (...) concluyo que el análisis de estas versiones demuestra que la causalidad probabilista constituye un nuevo y serio obstáculo para el enfoque humeano o neohumeano de la causalidad.In this paper I first show that the neohumean theories of probabilistic causality based on the notion of statistical relevance (as that of Suppes, 1970) run into many and unsolvable difficulties. Then I briefty analyze some accounts of probabilistic causality which relativize or avoid this notion: the Cartwright’s account, claiming the existence of causal capacities, and those of Salmon and Dowe, though trying to remain on a Humean ground, believe that the introduction of an ontology of propensities is required. I finally conclude that the analysis of these accounts shows that probabilistic causality constitutes a new and serious obstacle to the Humean or neohumean view of causality. (shrink)

Hume's argument concerning miracles is interpreted by making approximations to terms in Bayes's theorem. This formulation is then used to analyse the impact of multiple testimony. Individual testimonies which are ‘non-miraculous’ in Hume's sense can in principle be accumulated to yield a high probability both for the occurrence of a single miracle and for the occurrence of at least one of a set of miracles. Conditions are given under which testimony for miracles may provide support for the existence of God.

This paper presents a new account of Hume’s “probability of causes”. There are two main results attained in this investigation. The first, and perhaps the most significant, is that Hume developed – albeit informally – an essentially sound system of probabilistic inductive logic that turns out to be a powerful forerunner of Carnap’s systems. The Humean set of principles include, along with rules that turn out to be new for us, well known Carnapian principles, such as the axioms of semiregularity, (...) symmetry with respect to individuals (exchangeability), predictive irrelevance and positive instantial relevance. The second result is that Hume developed an original conception of probability, which is subjective in character, although it differs from contemporary personalistic views because it includes constraints that are additional to simple consistency and do not vary between different persons. The final section is a response to Gower’s thesis, by which Hume’s probability of causes is essentially non-Bayesian in character. It is argued that, on closer examination, Gower’s reading of the relevant passages is untenable and that, on the contrary, they are in accordance with the Bayesian reconstruction presented in this paper. (shrink)

Bayesian analyses are prominent among recent and allegedly novel interpretations of Hume’s argument against the justified belief in miracles. However, since there is no consensus on just what Hume’s argument is any Bayesian analysis will beg crucial issues of interpretation. Apart from independent philosophical arguments—arguments that would undermine the relevance of a Bayesian analysis to the question of the credibility of reports of the miraculous—no such analysis can, in principle, prove that no testimony can (or cannot) establish the credibility of (...) a miracle. Bayesian analyses of Hume’s argument are not analyses of Hume’s argument at all—but superfluous representations of it. (shrink)

At the beginning of his section “Of Miracles,” Hume mentions an argument of Dr. Tillotson. The doctrine of “the real presence” seems contradicted by our senses. We see a piece of bread, but are asked to believe it consists in the substance of the body of Christ.

Recent attempts to cast Hume’s argument against miracles in a Bayesian form are examined. It is shown how the Bayesian apparatus does serve to clarify the structure and substance of Hume’s argument. But the apparatus does not underwrite Hume’s various claims, such as that no testimony serves to establish the credibility of a miracle; indeed, the Bayesian analysis reveals various conditions under which it would be reasonable to reject the more interesting of Hume’s claims.

Hume’s main argument against rational belief in miracles might seem to rule out rational belief in other antecedently improbable occurrences as well--for example, a certain person’s having won the lottery. Dorothy Coleman has recently defended Hume against the lottery counterexample, invoking Hume’s distinction between probability of chances and probability of causes. I argue that Coleman’s defence fails.

THIS PAPER ANSWERS RECENT CRITICISMS OF HUME’S SKEPTICISM WITH REGARD TO MIRACLES BY THOSE WHO ARGUE THAT THERE ARE COUNTEREXAMPLES, ILLUSTRATED BY LOTTERIES, TO HUME’S ACCOUNT OF HOW THE TRUTH OF REPORTS ABOUT IMPROBABLE EVENTS MUST BE EVALUATED. THE AUTHOR FIRST SHOWS THAT THESE ARGUMENTS ARE ANALOGOUS TO BUTLER’S CRITICISM OF HUME’S PREDECESSORS IN THE DEBATE ABOUT MIRACLES. IT IS THEN ARGUED THAT EACH OF THESE CRITICISMS COLLAPSES THE DISTINCTION BETWEEN PROBABILITIES PERTAINING TO EVENTS QUA UNIQUE OCCURRENCES AND PROBABILITIES PERTAINING (...) TO EVENTS QUA TOKENS INSTANTIATING EVENT-TYPES. (shrink)

Hume’s celebrated argument concerning miracles, and an 18th century criticism of it put forward by Richard Price, is here interpreted in terms of the modern controversy over the base-rate fallacy. When considering to what degree we should trust a witness, should we or should we not take into account the prior probability of the event reported? The reliability of the witness (’Pr’(says e/e)) is distinguished from the credibility of the testimony (’Pr’(e/says e)), and it is argued that Hume, as a (...) good proto-Bayesian, argued that the credibility of the testimony should be calculated in terms of both the reliability of the witness and the prior probability of the event reported. (shrink)

A BAYESIAN ARTICULATION OF HUME’S VIEWS IS OFFERED BASED ON A FORM OF THE BAYES-LAPLACE THEOREM THAT IS SUPERFICIALLY LIKE A FORMULA OF CONDORCET’S. INFINITESIMAL PROBABILITIES ARE EMPLOYED FOR MIRACLES AGAINST WHICH THERE ARE ’PROOFS’ THAT ARE NOT OPPOSED BY ’PROOFS’. OBJECTIONS MADE BY RICHARD PRICE ARE DEALT WITH, AND RECENT EXPERIMENTS CONDUCTED BY AMOS TVERSKY AND DANIEL KAHNEMAN ARE CONSIDERED IN WHICH PERSONS TEND TO DISCOUNT PRIOR IMPROBABILITIES WHEN ASSESSING REPORTS OF WITNESSES.

The bulk of this book is the second series of John Dewey Lectures, delivered by Professor Ayer in April 1970. To this, Ayer has added a criticism of Roy Harred’s purported refutation of Hume and a chapter about "non-truth-functional" conditionals that rounds out the lectures. Leaving Harred aside, this book provides an elegant, concise, and up-to-date introduction to the problem of induction and related issues concerning probability. Hume is here vindicated. Beginning by giving what may be the best, updated paraphrase (...) of Hume’s negative argument about causal judgments, Ayer explicates the subsumed atomicity, the denial of natural necessity, and the claimed vacuity or circularity of introducing the "uniformity of nature." Ayer goes on to distinguish a priori or logical probabilities, statistical or frequency judgments, and credibility or ground floor inductive judgments, and he summarily argues that the admissibility of the first two forms of judgment, properly understood, provides no real justification for credibility judgments, which, though not defined as subjective, turn out to need some support from our decision to project and therefore entrench certain predicates, or take as lawlike certain generalizations, rather than others. Ayer here includes a very clear explication and criticism of Carnap’s way out in the Logical Concept of Probability, and elsewhere. Ayer acutely summarizes and discusses the Hempel and Goodman paradoxes, taking a position substantially in agreement with Quine and Goodman, and welds this discussion into a Humean conclusion: "In a certain sense cases are what we choose them to be. We do not decide what facts habitually go together but we do decide what combinations are to be imaginatively projected. The despised savages who beat gongs at solar eclipses to summon back the sun are not making any factual error. It is a true generalization that whenever they beat the gongs the sun does shine again, and if they always keep up the ceremony, it is also a true generalization that the sun comes out again only when they beat the gongs. If we despise them, it is because they tell a fictional story about what would happen if they did not beat the gongs, which we do not accept. They see what goes on as well as we do; it is just that we have a different and, we think, a better idea of the way the world works." To say all this is just, of course, to summarize and refurbish Hume’s position on causality. But to do that has two very real values now: 1) It underlines and exposes the epistemic basis of the position taken by Hume’s heirs; 2) It makes clear what are the departures required if one is to differ in substance from Hume and Ayer.—J. L. (shrink)

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

The overall aim of this thesis is to understand Hume’s famous argument concerning induction, and to appraise its success in establishing its conclusion. The thesis accordingly falls into two main parts, the first being concerned with analysis and interpretation of the argument itself, and the second with investigation of possible responses to it. Naturally the argument’s interpretation strongly constrains the range of possible replies, and indeed the results of Part I indicate that the only kind of strategy which stands much (...) prospect of defeating Hume’s argument is one based on a priori probabilistic reasoning – hence the overwhelming majority of Part II is devoted to a thorough investigation of this approach. (shrink)

The centrepiece of Earman’s provocatively titled book Hume’s Abject Failure: The Argument against Miracles is a probabilistic interpretation of Hume’s famous ‘maxim’ concerning the credibility of miracle reports, followed by a trenchant critique of the maxim when thus interpreted. He argues that the first part of this maxim, once its obscurity is removed, is simply trivial, while the second part is nonsensical. His subsequent discussion culminates with a forthright challenge to any would-be defender of Hume to ‘point to some thesis (...) which is both philosophically interesting and which Hume has made plausible’. My main aim here is to answer this challenge, by demonstrating a preferable interpretation of Hume’s maxim, according to which its first half is both plausible and non-trivial, while its second half sketches a useful, albeit approximate, corollary. I conclude by contesting Earman’s negative views on the originality and philosophical significance of Hume’s justly famous essay. (shrink)