Gordon Brittan

The logic of Simpson’s paradox

There are three distinct questions associated with Simpson’s paradox. (i) Why or in what sense is Simpson’s paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal (and questionable) account of the first two questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”

Algebra and Intuition

A great deal of excellent work on Kant’s philosophy of mathematics has been done in the recent past, much of it facilitated by a deep knowledge of the traditional criticisms made of his philosophy and by contemporary developments in logic and set theory. In particular, I have in mind papers by Michael Friedman, Jaakko Hintikka, Charles Parsons, Carl Posy, Manley Thompson, and J. Michael Young.1 Any new discussion of the topics they address should be able to presuppose some acquaintance with them. Their net effect has been to make Kant’s position, in its general outlines, philosophically credible as well as historically interesting, perhaps especially as concerns arithmetic. Moreover, with the exception of Posy, whose commentary on Kant takes place against a background provided by Brouwer and the Dutch Intuitionists, there is a surprising degree of consensus among them. There remain important differences, of course, and a great deal of disagreement concerning the details. But in this paper I will be more interested in the similarities, and will in any case spend an inadequate amount of time on the details.

Two Dogmas of Strong Objective Bayesianism

We introduce a distinction, unnoticed in the literature, between four varieties of objective Bayesianism. What we call ‘strong objective Bayesianism’ is characterized by two claims, that all scientific inference is ‘logical’ and that, given the same background information two agents will ascribe a unique probability to their priors. We think that neither of these claims can be sustained; in this sense, they are ‘dogmatic’. The first fails to recognize that some scientific inference, in particular that concerning evidential relations, is not (in the appropriate sense) logical, the second fails to provide a non‐question‐begging account of ‘same background information’. We urge that a suitably objective Bayesian account of scientific inference does not require either of the claims. Finally, we argue that Bayesianism needs to be fine‐grained in the same way that Bayesians fine‐grain their beliefs.

Acceptibility, Evidence, and Severity

The notion of a severe test has played an important methodological role in the history of science. But it has not until recently been analyzed in any detail. We develop a generally Bayesian analysis of the notion, compare it with Deborah Mayo’s error-statistical approach by way of sample diagnostic tests in the medical sciences, and consider various objections to both. At the core of our analysis is a distinction between evidence and confirmation or belief. These notions must be kept separate if mistakes are to be avoided; combined in the right way, they provide an adequate understanding of severity.Those who think that the weight of the evidence always enables you to choose between hypotheses “ignore one of the factors (the prior probability) altogether, and treat the other (the likelihood) as though it ...meant something other than it actually does. This is the same mistake as is made by someone who has scruples about measuring the arms of a balance (having only a tape measure at his disposal ...), but is willing to assert that the heavier load will always tilt the balance (thereby implicitly assuming, although without admitting it, that the arms are of equal length!). (Bruno de Finetti, Theory of Probability)2

Non-Bayesian Accounts of Evidence: Howson’s Counterexample Countered

ABSTRACT There is a debate in Bayesian confirmation theory between subjective and non-subjective accounts of evidence. Colin Howson has provided a counterexample to our non-subjective account of evidence: the counterexample refers to a case in which there is strong evidence for a hypothesis, but the hypothesis is highly implausible. In this article, we contend that, by supposing that strong evidence for a hypothesis makes the hypothesis more believable, Howson conflates the distinction between confirmation and evidence. We demonstrate that Howson’s counterexample fails for a different pair of hypotheses.

Determinism, Determination, and Objectivity in Modern Physics

Kants case for the objectivity of at least some of our experience is more threatened by the indeterminate than the indeterministic character of modern physics. Indeterminancy is a complex notion. It can be understood, ultimately, in terms of the failure of a “separability” principle, that objects can be individuated only with respect to non-vanishing spatial-temporal intervals. Its failure seems to follow from the fact that it is indispensable to the derivation of Bells Theorem and that the conclusion of the Theorem is incompatible with well-established empirical results. But Kants case for objectivity depends on it. The result is unsettling.

Algebra, Constructibility, and the Indeterminate

1. There is little to be gained, I believe, in trying to puzzle through the classic philosophical texts of the 17th and 18th centuries which have to do in important ways with on-going developments in mathematics and the various sciences without also having some knowledge of those developments. Lacking this sort of historical, although at times rather technical, knowledge, commentators often miss the point of the texts. Perhaps more important, they fail to bring out their full sophistication. The main figures of the 17th and 18th centuries simply knew a great deal more about mathematics and the various sciences than do the majority of their present commentators.

A Subjective Bayesian Surrogate for Evidence

We contend that Bayesian accounts of evidence are inadequate, and that in this sense a complete theory of hypothesis testing must go beyond belief adjustment. Some prominent Bayesians disagree. To make our case, we will discuss and then provide reasons for rejecting the accounts of David Christensen, James Joyce, and Alan Hajek. The main theme and final conclusions are straightforward: first, that no purely subjective account of evidence, in terms of belief alone, is adequate and second, that evidence is a comparative notion, applicable only when two hypotheses are confronted with the same data, as has been suggested in the literature on “crucial experiments” from Francis Bacon on.

Descartes’ Argument from Dreaming and the Problem of Underdetermination

Very possibly the most famously intractable epistemological conundrum in the history of modern western philosophy is Descartes’ argument from dreaming. It seems to support in an irrefutable way a radical scepticism about the existence of a physical world existing independent of our sense-experience. But this argument as well as those we discussed in the last chapter and many others of the same kind rest on a conflation of evidence and confirmation: since the paradoxical or sceptical hypothesis has as much “evidence” going for it as the conventional or commonly accepted hypothesis, it is equally well supported by the data and there is nothing to choose between them. By this time, however, we understand very well that data that fail to discriminate hypotheses do not constitute “evidence” for any of them, i.e., that “data” and “evidence” are not interchangeable notions, that it does not follow from the fact that there is strong evidence for a hypothesis against one or more of its competitors that it is therefore highly confirmed, and that it does not follow from the fact that a hypothesis is highly confirmed that there is strong evidence for it against its rivals.

Bayesian and Evidential Paradigms

The first step is to distinguish two questions: 1. Given the data, what should we believe, and to what degree? 2. What kind of evidence do the data provide for a hypothesis H 1 as against an alternative hypothesis H 2 , and how much? We call the first the “confirmation”, the second the “evidence” question. Many different answers to each have been given. In order to make the distinction between them as intuitive and precise as possible, we answer the first in a Bayesian way: a hypothesis is confirmed to the extent that the data raise the probability that it is true. We answer the second question in a Likelihoodist way, that is, data constitute evidence for a hypothesis as against any of its rivals to the extent that they are more likely on it than on them. These two simple ideas are very different, but both can be made precise, and each has a great deal of explanatory power. At the same time, they enforce corollary distinctions between “data” and “evidence”, and between different ways in which the concept of “probability” is to be interpreted. An Appendix explains how our likelihoodist account of evidence deals with composite hypotheses.