Invariance and Structural Dependence

This essay is about the scientific use of dependence and independence and other related concepts. There are many notions of dependence and independence used in science but only some of them will be studied here. One important notion of independence which I shall not deal with is the notion of independent events in probability theory. The notion of causal dependence is another important notion I shall leave out of account. The subject matter is rather the notions of dependence and independence which are central in most applications of mathematics to science.

This chapter is devoted to, on the one hand, a brief presentation of the standard logical and mathematical terminology and theory which will be frequently used in the rest of this book, and on the other, to proofs of some results within elementary set theory and algebra which will be used in later chapters. It is intended to function mainly as a kind of “dictionary” which the reader can consult when necessary. Section 2.1 and the beginning of sections 2.7 and 2.9 contain the basic apparatus for Part 2 and should therefore be at least cursorily perused before proceeding further. But the remaining sections can wait until they are referred to. Even though most terms which will be used are defined, a familiarity with elementary set theory and logic as presented in for example Suppes (1957) or Stoll (1963) is presupposed.

As we pointed out in chapter 1, structural dependence is a relation between aspects, and aspects will be represented here by systems of relationals. In this chapter we will introduce the notion of a relational in detail. A relational is a representation of the ordinary language notion of relation and we shall use it in connections with aspects, rather than the usual set-theoretical representation of a relation as a set of ordered n-tuples. Formally a relational is a function which takes sets as arguments and sets of ordered n-tuples as values. The values of a relational are thus relations in the set-theoretical sense. The basic idea is that the value of the relational for a set is the extension on this set of the relation which the relational represents.

In this chapter we shall introduce three main concepts we are going to use in the study of structural dependence, viz. subordination, uncorrelation and derivation. Subordination is one end of the “dependence-scale”—complete determination—and uncorrelation the other end—complete undetermination. (To be exact, there are as we shall see in later chapters many “dependence-scales” and subordination and uncorrelation are the endpoints of one of them.) We start off in section 4.1 with a presentation of the core of subordination, that is isomorphism preservation. In that section we also introduce the notion of a transition, which will be one of our basic tools in the rest of the investigation. Section 4.2 is devoted to subordination and its relation to definability while uncorrelation is treated in section 4.3. In section 4.4 we show that R* is not generally subordinate to R. What is meant by equality, =, for relationals is discussed in section 4.5. There we also introduce the notion of a decision method for relationals. Decision methods are also the topic of section 4.6, now applied to derivations of one relational from another. That a relational is derivable from another turns out to be another kind of dependence than isomorphism preservation and its relatives. In section 4.7 the notions of stability, introduced in chapter 3 as properties of relationals, will be applied to transitions and derivations.

In this chapter we shall apply some of the concepts presented in earlier chapters to a well-known subject, social choice theory. Social choice theory is rather closely related to group decision theory, although the exact character of the relation is hard to state. What is said here seems equally applicable to both subjects. We use the terminology from social choice theory essentially in accordance with Arrow (1951, 1963), Sen (1970) and Luce & Raiffa (1957). Our purpose is primarily to exemplify what has been said in earlier sections and not to conduct an analysis of the problem of social choice.

Homomorphic representations of relational structures into systems of real numbers is often held to be the kernel of measurement theory. Scales and measures are usually regarded as essentially such homomorphic representations. Let us say, as is customary in measurement theory, that the function h on A into B is a homomorphism on (or of) the relational structure 〈A,ρ〉 into 〈B,σ〉 iff for all x₁,...,x_n ∈ A, . A homomorphic representation of a relational into a structure is a more complicated notion, and a measure or scale for a relational will be a homomorphic representation of a special kind.

In this chapter we shall study transitions between relational systems. Transitions were first introduced in section 4.1. We are interested in transitions as a device for studying dependence, and this will determine the form and content of the chapter. In section 7.1 we generalize the notion of finitary system of relationals from section 3.4 so that a system of relationals may consist of infinitely many relationals, and give a short recapitulation of central notions such as range of definition, extension and characteristic class, domain stability etc. The investigation of transitions is begun in section 7.2, where we study relationships between properties of transitions and subordination. In section 7.3 we conduct a similar investigation for correlation rather than subordination.

Subordination, sub, is a relation between systems of relational whose formal properties will be more closely examined in this chapter. We will see that it generates a lattice in a natural way, and one of our main tasks is to find out what kind of lattice. This is done in sections 8.1 and 8.2, where we introduce two new concepts, the partial order which sub generates and the notion of the rank of a relational system. In section 8.3 we will devote some effort to investigating what role the pair corr and uncorr, i.e. the relations correlation and uncorrelation, play in the lattice generated by sub. A generalization of the notion of a rank is studied in section 8.4. Already in chapter 1 I said that sub expresses the idea of equality preservation applied to structures. In section 8.5 this suggestion will be made more precise and we will see that uncorr expresses the idea of realizability of structures. It is tempting to regard sub, corr and uncorr themselves as relational, but I shall not pursue this line of thought here.

A relational is a kind of generalization of a relation in the ordinary sense of a set of ordered n-tuples. One might therefore ask if the notion of an isomorphism for a relation in the ordinary sense can be generalized to relationals. In section 9.2 we will see how this can be done. Given this notion of an isomorphism for a relational it is a straightforward matter to say what is meant by an automorphism for a relational. As a preliminary to the study of isomorphisms for relationals, the notion of a mapping is introduced in section 9.1. In section 9.3 we study invariance under the set of automorphisms for a relational and see how this notion is related to subordination. The idea of stability can be applied in a natural way to automorphisms for relationals, and invariance under stable automorphisms will be studied in section 9.4 It turns out that invariance under stable automorphisms gives rise to a notion of dependence of different character to that of subordination because it is not domainwise. We shall be primarily interested in a notion which is somewhat stronger than invariance under stable automorphisms, to be called global subordination, by contrast with the usual local subordination. (The notion of global subordination and the theorems 9.4.4 and 9.4.5 were suggested to me by Sten Lindström.)