The only symbolic logic used in this book is a small part of propositional logic, also called sentential logic or Boolean logic. In this appendix, I review the relevant part of this simple area of logic and clarify some notation and terminology. This appendix is not an introduction to logic; various important fine points and distinctions are not be mentioned. But I hope this will suffice as an introduction to the basic ideas in the elementary part of logic used in this book, and as a clarification of the logical terminology and symbols used in this book.

The basic entities of the formal propositional calculus are usually called the propositions and the propositional connectives (and the language of propositional logic usually includes punctuation marks, usually parentheses, that are used to avoid ambiguity of grouping when “propositions” are “connected” in complex ways).

In this book, it is factors (or properties, or types) that play the role of the so-called propositions of propositional logic. The abstract and formal propositional calculus can be interpreted as applying to propositions in a number of ways in which the term “proposition” could be understood. For example, we could think of propositions as sentences (which may be understood as concrete linguistic entities such as utterances or inscriptions). Or we could think of them as statements (understood in such a way that many sentences can all be used to “make” the same statement, and the same sentence, if used in different contexts, would make different statements).

The first main qualification of the basic probability-increase idea of probabilistic causation, explained in Chapter 1, is the relativity of the causal relation to a given token population, considered to be of a given (appropriate) kind that the population exemplifies. The second main qualification of the basic probability-increase idea, to be explored in this chapter, involves the possibility of what has been called “spurious correlation.” Of course, what is meant by saying that a factor X raises the probability of a factor X is that Pr(Y/X) > Pr(Y) – equivalently, Pr(Y/X) > Pr(Y/∼X) Another way of expressing this relation is to say that Y is positively probabilistically correlated with X. It is famous that “correlation is no proof of causation,” and it is also true that causation does imply correlation. The possibility of spurious correlation is one reason why.

In this book, I will actually explore in detail three general ways in which probability increase may fail to coincide with causation, and I will show how the probability-increase idea of causation should be adjusted to accommodate these three possibilities. After briefly describing the three possibilities below, this chapter will concentrate on one of them, the one called “spurious correlation.” The other two will be dealt with in subsequent chapters.

One simple way to see that probability increase does not imply causation is to notice that the relation of positive correlation is symmetric. If X raises the probability of Y, then Y raises the probability of X.

For the examples of spurious correlation discussed in Chapter 2, it sufficed to hold fixed all (independent) positive, negative, and mixed causes of the candidate effect factor, in order for the probability-increase idea to deliver the right answers about what caused what. For these examples, only factors that were causally relevant to the candidate effect factor needed to be held fixed. In this chapter, I will argue that other kinds of factors, which may be causally irrelevant to (neutral for) the effect factor in question, must be held fixed as well, if the probability-increase theory is to deliver the right answers in other kinds of cases.

For example, if the right answer in Dupré's example, discussed in Chapter 2, is that smoking has a mixed causal role (not positive, negative, or neutral) for lung cancer, then it will be necessary to hold fixed the factor of that rare physiological condition. Otherwise, causal relevance would go by average probabilistic impact of smoking on lung cancer, across the presence and absence of that condition, and this cannot give the correct answer of mixed causal relevance. However, as noted in Chapter 2 and explained more fully in this chapter, that physiological condition need not itself be a positive, negative, or mixed cause of lung cancer.

At the beginning of Chapter 2, the possibility of there being such factors as that physiological condition in Dupré's example was called the problem of causal interaction.

We turn finally to the role of time in the theory of proper tylevel probabilistic causation. This will be dealt with in Sections 5.1 and 5.2, and this will complete the theory of property level probabilistic causation offered in this book. Section 5.3 offers some comparisons and contrasts between this theory and several others.

First, recall the problem that, at the beginning of Chapter 2, I called “the problem of temporal priority of the cause to the effect.” This problem arises from the fact that probabilistic correlation is symmetric. Leaving aside qualifications having to do with causal background contexts, if a factor X is a genuine probabilistic cause of a factor Y in a population, then X raises the probability of Y in that population. This implies that Y raises the probability of X in the same population. But we cannot infer that Y is a cause of X in the population, for while correlation is symmetric, causation is not.

Second, if we agree that property-level probabilistic causation is asymmetric, then we may want to capture this by saying that one factor can only be a cause of “later” factors, and that it can only be caused by “earlier” factors. But what is it for one factor itself–that is, one event type or one property – to be earlier or later than another? How can we make sense of the idea that such abstract things as factors (or types, or properties) enter into temporal relations among themselves?

Type-level probabilistic causation is sometimes called “population-level” probabilistic causation and sometimes “property-level” probabilistic causation. And the items that enter into type-level probabilistic causal relations are called “factors,” or “properties,” or “event types.” The basic idea in the theory of type-level probabilistic causation is that causes raise the probabilities of their effects. A factor C is a property-level probabilistic cause of – or a positive causal factor for – a factor E, if the probability of E is higher in the presence of C than it is in the absence of C. And C is causally negative or causally neutral for E if the presence of C lowers or leaves unchanged the probability of JS, respectively. But this basic idea needs several clarifications and qualifications.

In this chapter, I explain the importance of the idea of a population to type-level probabilistic causal connection. I argue that type-level probabilistic causation is a relation among four things: a cause factor, an effect factor, a token population within which the first is some kind of cause of the second, and, finally, a kind (of population) that is associated with the given token population. Subsequent chapters reveal how important relativity to populations is for the versatility of the probabilistic theory and how it renders the theory immune to a number of criticisms that have recently been advanced.

Of course, some kind of clarification of the idea of probability is in order.

In this appendix, I will present some of the basic ideas of the mathematical theory of probability. As in the case of Appendix 1, this will not be a comprehensive or detailed survey; it is only intended to introduce the basic formal probability concepts and rules used in this book, and to clarify the terminology and notation used in this book. Here I will discuss only the abstract and formal calculus of probability; in Chapter 1, the question of interpretation is addressed.

A probability function, Pr, is any function (or rule of association) that assigns to (or associates with) each element X of some Boolean algebra B (see Appendix 1) a real number, Pr(X), in accordance with the following three conditions:

For all X and Y in B,

1. Pr(X) 0;

2. Pr(X) = 1, if X is a tautology (that is, if X is logically true, or X = 1 in B);

3. Pr(X∨Y) = Pr(X) + Pr(Y), if X&Y is a contradiction (that is, if X&Y is logically false, or X&Y = 0 in B).

These three conditions are the probability axioms, also called “the Kolmogorov axioms” (for Kolmogorov 1933). A function Pr that satisfies the axioms, relative to an algebra B, is said to be a probability function on B – that is, with “domain” B (that is, the set of propositions of B) and range the closed interval [0,1]. In what follows, reference to an assumed algebra B will be implicit.

My interest in probabilistic causality arose naturally from my earlier interest, as a graduate student and after, in the area of the philosophical foundations of decision theory. I was especially interested in the decision-theoretical puzzle known as Newcomb's paradox and the idea of causal decision theory (Eells 1982). Causal decision theory was designed to accommodate the fact that the evidential, or “average” probabilistic, significance of one factor or event for another need not coincide with the causal significance of the first factor or event for the other – a fact vividly illustrated by the Newcomb problem. Causal decision theory involves ideas and techniques quite similar to the ideas and techniques involved in untangling and understanding the relations between probabilistic and causal significance in the theory of probabilistic causality.

Most of the recent philosophical literature in this area has seemed to concentrate on what I call here type-level probabilistic causation, though some authors have either noted or developed theories of what I call here token-level probabilistic causation. The first five chapters of this book are about type-level probabilistic causation. The last, very long, chapter is on token-level probabilistic causation. It is probably Chapter 6, which gives a new theory of token-level probabilistic causation, that contains the most novel proposals of this book.

In the past 30 years or so, philosophers have become increasingly interested in developing and understanding probabilistic conceptions of causality – conceptions of causality according to which causes need not necessitate their effects, but only, to put it very roughly, raise the probabilities of their effects. This philosophical project is of interest not only because the problem of the nature of causation is itself so central in philosophy, and not only because of the nature of causation as well as physical indeterminism in current scientific theory. The theory of probabilistic causation also has applications in other philosophical problems, such as the nature of scientific explanation and the nature of probabilistic laws in a variety of sciences, as well as the character of rational decision. And the theory has applications in these areas whether or not determinism is assumed. In this book, however, very little is said about such applications. I focus on the theory of probabilistic causation itself.

In philosophy, the development of the probabilistic view of causality owes much to the work of I. J. Good (1961–2), Patrick Suppes (1970), Wesley Salmon (1971, 1978, 1984), and Nancy Cartwright (1979) (as well as others). In this book, I articulate and defend a conception of probabilistic causation that owes much to, but differs in important details from, the work of these and other authors. I also examine and appraise several alternatives to the ideas advanced here.

In Chapter 2, a spurious correlation of a factor Y with a factor X was characterized as a situation in which, because of separate causes of Y, the degree of correlation of Y with X is different from the degree of causal significance of X for Y. The possibility of a spurious correlation of Y with X was diagnosed as arising when there are factors Z that are correlated with X and that are positive, negative, or mixed causes of Y, independently of X, where the correlation in question may be unconditional or conditional on other such factors Z. But as noted in Chapter 2, not all cases in which X is correlated with separate causes of Y give rise to a spurious correlation: The separate causes must be causally independent of X. In Chapter 3, we saw that when a factor X interacts, with respect to a factor Y, with a factor Z that is causally independent of X, then we should say that X is causally mixed for Y. But as noted in Chapter 3, not all cases in which a factor X interacts with a factor Z in the production of a factor Y are cases of mixed causal relevance of X for Y. Again, Z must be causally independent of X.

As explained in the introduction, the relation I am calling “token causation” is a relation between two actually occurring, concrete, token events, while type-level causation relates abstract entities called “properties,” “types,” or “factors.” In the preceding chapters, I used upper case italicized letters to represent factors. Now we need to refer to token events, and I will use lower case italicized letters, x, y, z, and so on, for this purpose. As explained more fully below, the relation I wish to analyze in this chapter, in terms of probability relations, is roughly this (where x is of type X and y is of type Y): x's being of type X caused (atemporally) y's being of type Y. Another way of putting it is as follows. Where x takes place at time and place <tx,sx> and y takes place at time and place <ty,sy>, the relation I wish to analyze is this: things’ being X at <tx,sx caused things to be Y at <ty, sy.

The basic idea in the probabilistic theory of type level causation was that causes raise the probability of their effects. We saw that this idea needed several qualifications. The possibilities of spurious correlation and causal interaction had to be accommodated, and it was necessary to build into the theory the requirement that causes precede their effects in time.

Abstract

Necessary conditions are given for the Hermite–Fejér interpolation polynomials based at the zeros of orthogonal polynomials to converge in weighted Lp spaces at the Jackson rate. These conditions are known to be sufficient in the case of the generalized Jacobi polynomials.

Introduction

The first detailed study of weighted mean convergence of Hermite–Fejér interpolation based at the zeros of orthogonal polynomials was accomplished in [13] and [14], where it was shown that some of the most delicate problems associated with mean convergence of Hermite–Fejér interpolation can be approached through the general theory of orthogonal polynomials; in particular, a distinguished role is played by Christoffel functions. As opposed to Lagrange interpolation operators, Hermite–Fejér interpolation operators are not projectors, and thus in general the rate of convergence cannot be expected to equal the rate of the best approximation. Nevertheless, Jackson rates can be obtained.

Unaware of the general theory in [13] and [14] and of a variety of technical tools developed in [6], [9] and [10] (see [11] for a survey), A. K. Varma & J. Prasad in [22] investigated mean convergence of Hermite–Fejér interpolation in a particular case, namely in the case of interpolation based at the zeros of the Chebyshev polynomials. Subsequently, P. Vértesi & Y. Xu [23] wrote a paper dealing with the case of generalized Jacobi polynomials. However, their results left a significant gap between the necessary and the sufficient conditions.