Information theory and statistical mechanics.

Bertrand's random-chord paradox purports to illustrate the inconsistency of the principle of indifference when applied to problems in which the number of possible cases is infinite. This paper shows that Bertrand's original problem is vaguely posed, but demonstrates that clearly stated variations lead to different, but theoretically and empirically self-consistent solutions. The resolution of the paradox lies in appreciating how different geometric entities, represented by uniformly distributed random variables, give rise to respectively different nonuniform distributions of random chords, and hence (...) to different probabilities. The principle of indifference appears consistently applicable to infinite sets provided that problems can be formulated unambiguously. (shrink)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

In this paper I shall venture into an area with which I am not very familiar and in which I feel far from confident; namely into phenomenology. My main motive is not to get away from standard, boring, methodological questions like those of induction and demarcation; but the conviction that a phenomenological account of the empirical basis forms a necessary complement to Popper's falsificationism. According to the latter, a scientific theory is a synthetic and universal, hence unverifiable proposition. In fact, (...) in order to be technologically useful, a scientific hypothesis must refer to future states-of-affairs; it ought therefore to remain unverified. But in order to be empirical, a theory must bear some kind of relation to factual statements. According to Popper, such a relation can only be one of potential conflict. Thus a theory T will be termed scientific if and only if T is logically incompatible with a so-called basic statement b, where b is both empirically verifiable and empirically falsifiable. In other words: T is scientific if it entails ¬b; where b, hence also ¬b, is an empirically decidable proposition. (shrink)

Van Fraassen has recently argued that empiricism can be construed as a stance, involving commitments, attitudes, values, and goals, in addition to beliefs and opinions. But this characterisation emerges from his recognition that to be an empiricist can not be to believe, or decide to commit to belief in, a foundational proposition, without removing any basis for a non-dogmatic empiricist critique of other philosophical approaches, such as materialism. However, noticeable by its absence in Van Fraassen's discussions is any mention of (...) Bartley's ‘pancritical rationalism’, for Bartley offers a cohesive argument that genuine dogmatism lies precisely in the act of commitment to an idea. The consequence of denying this, he thinks, is an opening of the floodgates to irrationalism: if to rely on reasoned argument in decision-making is fundamentally an act of faith, then there is a tu quoque – “I simply have a different faith” – that may be employed by those who wish to shield their views from criticism. This raises the following question: why should it be any less dogmatic to adopt particular commitments, attitudes, values, and goals, rather than a particular belief or opinion, come what may? And if Bartley is right that there is only one non-dogmatic attitude – the critical attitude – then why might this not be adopted by an empiricist, a materialist, a metaphysician, or anyone else? (shrink)

It is a common view that the axioms of probability can be derived from the following assumptions: probabilities reflect degrees of belief, degrees of belief can be measured as betting quotients; and a rational agent must select betting quotients that are coherent. In this paper, I argue that a consideration of reasonable betting behaviour, with respect to the alleged derivation of the first axiom of probability, suggests that and are incorrect. In particular, I show how a rational agent might assign (...) a ‘probability’ of zero to an event which she is sure will occur. (shrink)

How sensitive should you be to the testimony of others? You saw the car that caused an accident going through traffic lights on the red; or so you thought. Should you revise your belief on discovering that the majority of bystanders, equally well-equipped, equally well-positioned and equally impartial, reported that it went through on the green? Or take another case. You believe that intelligent design is the best explanation for the order of the living universe. Should you revise that belief (...) on finding that most other people, or at least most who by your own lights are as intelligent, informed and impartial as yourself, believe that evolutionary theory offers the better account? Should you do this, in particular, if your own personal sense of where the evidence points – like your own vivid memory of the car going through on the red – remains firmly on the side of intelligent design? Assume, to take a third case, that there is a matter of fact about whether abortion is right or wrong. You believe that it is wrong, having a firm picture of it as an act on a par with murder. Should you revise that belief on discovering that among those whom you regard as equally intelligent, informed and impartial, most believe that abortion is not wrong, or at least not wrong in the way that murder is wrong? Should you do this, in particular, if your own personal sense of abortion remains unchanged; it still seems to you to be a grievous wrong? Should you put aside your own sense of things as mistaken, in the way you might put aside your imagined memory of the car going through on the red, and decide to go along with the majority view? (shrink)

Many statistical problems, including some of the most important for physical applications, have long been regarded as underdetermined from the standpoint of a strict frequency definition of probability; yet they may appear wellposed or even overdetermined by the principles of maximum entropy and transformation groups. Furthermore, the distributions found by these methods turn out to have a definite frequency correspondence; the distribution obtained by invariance under a transformation group is by far the most likely to be observed experimentally, in the (...) sense that it requires by far the least “skill.” These properties are illustrated by analyzing the famous Bertrand paradox. On the viewpoint advocated here, Bertrand's problem turns out to be well posed after all, and the unique solution has been verified experimentally. We conclude that probability theory has a wider range of useful applications than would be supposed from the standpoint of the usual frequency definitions. (shrink)

This paper introduces what is called the intersubjective interpretation of the probability calculus. Intersubjective probabilities are related to subjective probabilities, and the paper begins with a particular formulation of the familiar Dutch Book argument. This argument is then extended, in Section 3, to social groups, and this enables the concept of intersubjective probability to be introduced in Section 4. It is then argued that the intersubjective interpretation is the appropriate one for the probabilities which appear in confirmation theory whether of (...) a Bayesian or a Popperian variety. The final section of the paper states and tries to answer an objection due to Putnam. (shrink)

This paper is a critique of Contessa’s . First, I show that Popper in The Logic of Scientific Discovery argues against the view that the logical probability of a hypothesis is identical to its degree of confirmation , rather than against Bayesianism. Second, I explain that his argument to this effect does not depend on the assumption that ‘the universe is infinite’. Third, and finally, I refine Popper’s case by developing an argument which requires only that some universal laws have (...) a logical probability of zero relative to any finite evidence, and providing an example concerning Newtonian mechanics. (shrink)

This paper is a critique of Contessa’s. First, I show that Popper in The Logic of Scientific Discovery argues against the view that the logical probability of a hypothesis is identical to its degree of confirmation, rather than against Bayesianism. Second, I explain that his argument to this effect does not depend on the assumption that ‘the universe is infinite’. Third, and finally, I refine Popper’s case by developing an argument which requires only that some universal laws have a logical (...) probability of zero relative to any finite evidence, and providing an example concerning Newtonian mechanics. (shrink)

In the appendices to his Logic of Scientific Discovery, Karl Popper presents a number of arguments in favour of the thesis that the logical probability of any universal law in an infinite universe must be zero. According to Popper, from this it follows that any attempt to apply a Bayesian approach to the confirmation of scientific laws is a non-starter—if the prior probability of any hypothesis h is zero = 0), it follows from Bayes Theorem that p = 0 for (...) any possible e. In this essay, I will discuss what I deem to be Popper’s main argument in support of this thesis. I shall argue that the general validity of two of the three assumptions on which the argument is based can be challenged. (shrink)