Online research in philosophy

In the following we propose a variant of the frequency interpretation of probability of Richard von Mises; one of our aims is to address recent criticisms that have been formulated against this interpretation. Following von Mises, we will argue that (objective) probability can only be defined for events that can be repeated in similar conditions, and that exhibit ‘frequency stabilization’. The central idea of the present article is that the mentioned ‘conditions’ should be well-defined and ‘partitioned’. More precisely, we will divide probabilistic systems into object, environment, and probing subsystem, and show that such partitioning allows to solve problems. By the same token we will be able to derive a definition of what ‘similar events’ are – a problematic concept in traditional interpretations. Our general conclusion will be that the probability of an event or system is only defined if all subsystems that compose the latter are defined – in a slogan: probability is composed.

Two experiments were conducted to investigate the roles of covariation and of causality in people's readiness to believe a conditional. The experiments used a probabilistic truth-table task (Oberauer & Wilhelm, 2003) in which people estimated the probability of a conditional given information about the frequency distribution of truth-table cases. For one group of people, belief in the conditional was determined by the conditional probability of the consequent, given the antecedent, whereas for another group it depended on the probability of the conjunction of antecedent and consequent. There was little evidence that covariation, expressed as the probabilistic contrast or as the pCI rule (White, 2003), influences belief in the conditional. The explicit presence of a causal link between antecedent and consequent in a context story had a weak positive effect on belief in a conditional when the frequency distribution of relevant cases was held constant.

A popular conception of probability for many years now has been the relative frequency interpretation, made famous by the work of Reichenbach and von Mises, and more recently by Salmon and others. The frequency view has played important roles of various sorts in virtually every area in epistemology and the philosophy of science, including explanation, causation, the justification of induction, the nature of laws and lawlike statements, and so on. A major attraction of the frequency conception has been its claim to be a strictly empirical view. In this paper we argue that on prima facie grounds the frequency view violates some of our deepest intuitions regarding the notions of probability and possibility.

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.

Quantum mechanics may be formulated as Sensible Quantum Mechanics (SQM) so that it contains nothing probabilistic except conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued awareness operators. Ratios of the measures for these sets of perceptions can be interpreted as frequency- type probabilities for many actually existing sets. These probabilities gener- ally cannot be given by the ordinary quantum “probabilities” for a single set of alternatives. Probabilism, or ascribing probabilities to unconscious aspects of the world, may be seen to be an aesthemamorphic myth.

John Leslie's Doomsday argument uses the frequency interpretation of probability to argue that the end of the universe is closer than we might have thought. Oh well - all the worse for the frequency interpretation.

The modus ponens (A -> B, A :. B) is, along with modus tollens and the two logically not valid counterparts denying the antecedent (A -> B, ¬A :. ¬B) and affirming the consequent, the argument form that was most often investigated in the psychology of human reasoning. The present contribution reports the results of three experiments on the probabilistic versions of modus ponens and denying the antecedent. In probability logic these arguments lead to conclusions with imprecise probabilities. In the modus ponens tasks the participants inferred probabilities that agreed much better with the coherent normative values than in the denying the antecedent tasks, a result that mirrors results found with the classical argument versions. For modus ponens a surprisingly high number of lower and upper probabilities agreed perfectly with the conjugacy property (upper probabilities equal one complements of the lower probabilities). When the probabilities of the premises are imprecise the participants do not ignore irrelevant (“silent”) boundary probabilities. The results show that human mental probability logic is close to predictions derived from probability logic for the most elementary argument form, but has considerable difficulties with the more complex forms involving negations.

The aim of the paper is to draw a connection between a semantical theory of conditional statements and the theory of conditional probability. First, the probability calculus is interpreted as a semantics for truth functional logic. Absolute probabilities are treated as degrees of rational belief. Conditional probabilities are explicitly defined in terms of absolute probabilities in the familiar way. Second, the probability calculus is extended in order to provide an interpretation for counterfactual probabilities--conditional probabilities where the condition has zero probability. Third, conditional propositions are introduced as propositions whose absolute probability is equal to the conditional probability of the consequent on the antecedent. An axiom system for this conditional connective is recovered from the probabilistic definition. Finally, the primary semantics for this axiom system, presented elsewhere, is related to the probabilistic interpretation.

I sketch a new objective interpretation of probability, called "mechanistic probability", and more specifically what I call "far-flung frequency (FFF) mechanistic probability". FFF mechanistic probability is defined in terms of facts about the causal structure of devices and certain sets of collections of frequencies in the actual world. The relevant kind of causal structure is a generalization of what Strevens (2003) calls microconstancy. Though defined partly in terms of frequencies, FFF mechanistic probability avoids many drawbacks of well-known frequency theories. It at least partly explains stable frequencies, which will usually be close to the values of corresponding mechanistic probabilities; FFF mechanistic probability thus satisfies what in my view is a core desideratum for any objective interpretation. However, FFF mechanistic probabilities are not single case probabilities, and FFF mechanistic probability explains stable frequencies directly rather than by inference from combinations of single case probabilities.

Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in infinite sets. This interpretation vindicates part of what the logical interpretation of probability aimed to establish.