Online research in philosophy

This paper sketches a concept of higher-level objective probability (“short-run mechanistic probability”, SRMP) inspired partly by a style of explanation of relative frequencies known as the “method of arbitrary functions”. SRMP has the potential to fill the need for a theory of objective probability which has wide application at higher levels and which gives probability causal connections to observed relative frequency (without making it equivalent to relative frequency). Though this approach provides probabilities on a space of event types, it does not provide probabilities for outcomes on particular trials. This allows SRMP to coexist with lower-level probabilities which do govern individual trials.

Oaksford and Chater (1994) proposed to analyse the Wason selection task as an inductive instead of a deductive task. Applying Bayesian statistics, they concluded that the cards that participants tend to select are those with the highest expected information gain. Therefore, their choices seem rational from the perspective of optimal data selection. We tested a central prediction from the theory in three experiments: card selection frequencies should be sensitive to the subjective probability of occurrence for individual cards. In Experiment 1, expected frequencies of the p- and the q-card were manipulated independently by concepts referring to large vs. small sets. Although the manipulation had an effect on card selection frequencies, there was only a weak correlation between the predicted and the observed patterns. In the second experiment, relative frequencies of individual cards were manipulated more directly by explicit frequency information. In addition, participants estimated probabilities for the four logical cases and of the conditional statement itself. The experimental manipulations strongly affected the probability estimates, but were completely unrelated to card selections. This result was replicated in a third experiment. We conclude that our data provide little support for optimal data selection theory.

When applied to a family of sets, the term differentiation designates a measure of the totality of those members which appear in only one of the sets. This basic set theoretic concept involves the formation of intersections, unions, and complements of sets. However, populations as special kinds of sets may share types, but they do not share the carriers of these types; intersections of different populations are thus always empty. The resulting conceptual dilemma is resolved by considering the joint representation of members of different populations that have the same type; populations then intersect with respect to joint representation of types. Two forms of representation reflect relative and absolute characteristics of differentiation by accounting for the distributions of types as relative frequencies within populations (as is commonly done) and as absolute frequencies (including effects of population sizes on differentiation), respectively. Corresponding classes of differentiation measures are developed, and existing measures are discussed in relation to these classes. In particular, the affinity of the measurement of distances between populations and the special case of differentiation of two-population families is examined in order to distinguish between the notions of distance and differentiation.

A standard view of probability and statistics centers on distributions and hypothesis testing. To solve a real problem, say in the spread of disease, one chooses a “model”, a distribution or process that is believed from tradition or intuition to be appropriate to the class of problems in question. One uses data to estimate the parameters of the model, and then delivers the resulting exactly specified model to the customer for use in prediction and classification. As a gateway to these mysteries, the combinatorics of dice and coins are recommended; the energetic youth who invest heavily in the calculation of relative frequencies will be inclined to protect their investment through faith in the frequentist philosophy that probabilities are all really relative frequencies. Those with a taste for foundational questions are referred to measure theory, an excursion from which few return. That picture, standardised by Fisher and Neyman in the 1930s, has proved in many ways remarkably serviceable. It is especially reasonable where it is known that the data are generated by a physical process that conforms to the model. It is not so useful where the data is a large and little understood mess, as is typical in, for example, insurance data being investigated for fraud. Nor is it suitable where one has several speculations about possible models and wishes to compare them, or..

This paper deals with the problem of vindicating a particular type of inductive rule, a rule to govern inferences from observed frequencies to limits of relative frequencies. Reichenbach's rule of induction is defended. By application of two conditions, normalizing conditions and a criterion of linguistic invariance, it is argued that alternative rules lead to contradiction. It is then argued that the rule of induction does not lead to contradiction when suitable restrictions are placed upon the predicates admitted. Goodman's grue-bleen paradox is considered, and an attempt to resolve it is offered. Finally, Reichenbach's pragmatic argument, hinging on convergence properties, is applied.

One finds intertwined with ideas at the core of evolutionary theory claims about frequencies in counterfactual and infinitely large populations of organisms, as well as in sets of populations of organisms. One also finds claims about frequencies in counterfactual and infinitely large populations—of events—at the core of an answer to a question concerning the foundations of evolutionary theory. The question is this: To what do the numerical probabilities found throughout evolutionary theory correspond? The answer in question says that evolutionary probabilities are “hypothetical frequencies” (including what are sometimes called “long-run frequencies” and “long-run propensities”). In this paper, I review two arguments against hypothetical frequencies. The arguments have implications for the interpretation of evolutionary probabilities, but more importantly, they seem to raise problems for biologists’ claims about frequencies in counterfactual or infinite populations of organisms and sets of populations of organisms. I argue that when properly understood, claims about frequencies in large and infinite populations of organisms and sets of populations are not threatened by the arguments. Seeing why gives us a clearer understanding of the nature of counterfactual and infinite population claims and probability in evolutionary theory.

Biologists often define evolution as a change in allele frequencies. Consideration of the evolution of the pocket mouse will show that it is possible to have evolution without any change in the allele frequencies in a population (through change in the genotype frequencies). The implications of this for genic selectionism are then discussed. Sober and Lewontin (1982) have constructed an example to demonstrate the blindness of genic selectionism in certain cases. Sterelny and Kitcher (1988) offer a defense against these arguments which assumes a conventionalist approach to populations. The example considered here will be shown to offer a more plausible and far-reaching argument against the view that alleles can always be seen as the units of selection.

A law about frequencies would be a law of nature that imposes a constraint on one or more (actual, global) frequencies. On any of the leading philosophical approaches to laws of nature, there could be laws about frequencies. Hypotheses that posit laws about frequencies turn out to behave very similarly to hypotheses that posit corresponding laws about probabilities or chances -- they make the same predictions, provide similar explanations, and are confirmed or disconfirmed by empirical evidence in the same ways. This makes it interesting to consider the possibility of interpreting probabilistic laws from scientific theories as laws about frequencies. This is surprising proposal, but I argue that the resulting view (which I call 'nomic frequentism') is able to overcome all of the standard objections to frequentist interpretation of objective probabilities.

Bayesians take “definite” or “single-case” probabilities to be basic. Definite probabilities attach to closed formulas or propositions. We write them here using small caps: PROB(P) and PROB(P/Q). Most objective probability theories begin instead with “indefinite” or “general” probabilities (sometimes called “statistical probabilities”). Indefinite probabilities attach to open formulas or propositions. We write indefinite probabilities using lower case “prob” and free variables: prob(Bx/Ax). The indefinite probability of an A being a B is not about any particular A, but rather about the property of being an A. In this respect, its logical form is the same as that of relative frequencies. For instance, we might talk about the probability of a human baby being female. That probability is about human babies in general — not about individuals. If we examine a baby and determine conclusively that she is female, then the definite probability of her being female is 1, but that does not alter the indefinite probability of human babies in general being female. Most objective approaches to probability tie probabilities to relative frequencies in some way, and the resulting probabilities have the same logical form as the relative frequencies. That is, they are indefinite probabilities. The simplest theories identify indefinite probabilities with relative frequencies.3 It is often objected that such “finite frequency theories” are inadequate because our probability judgments often diverge from relative frequencies. For example, we can talk about a coin being fair (and so the indefinite probability of a flip landing heads is 0.5) even when it is flipped only once and then destroyed (in which case the relative frequency is either 1 or 0). For understanding such indefinite probabilities, it has been suggested that we need a notion of probability that talks about possible instances of properties as well as actual instances..