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Grelling on Dependence Grelling on dependence Jouko Va¨an¨ anen¨ ∗ Abstract Kurt Grelling wrote a paper in 1939 presenting various concepts of de- pendence. The paper remained unpublished, but deserves to be read today. Many of the ideas of the paper have been subsequently reinvented but one concept, which we call G-dependence, is still genuinely new, and that is the main topic of this paper. We isolate some basic properties of G-dependence and pose the question of finding simple axioms for it. 1 Introduction In his 1939 [11] the mathematician and logician Kurt Grelling2 developed, in co- operation with Paul Oppenheim3, a completely abstract theory of dependence and independence with apparently no connection to algebra or probability theory, al- though a study of dependence was emerging in these fields too. His starting point seems to have been so-called Gestalt Theory [23], but his prime example was the earthly way in which commercial price depends on supply and demand. This paper will present an overview of Grelling’s theory. Department of Mathematics and Statistics, University of Helsinki, Finland · Institute for Logic, Language and Computation, University of Amsterdam, The Netherlands ∗ Research partially supported by grant 40734 of the Academy of Finland. 2 Kurt Grelling was born in Berlin in 1886. He studied in Gottingen¨ with David Hilbert as his supervisor, graduating in 1910 with a thesis on set theory. Already in 1908 he published a paper with L. Nelson on Russell’s paradox, in which they introduced what is today known as the Grelling- Nelson paradox. He was subsequently influential in philosophy in the so-called Berlin Circle. He perished in Auschwitz in 1942. 3 Paul Oppenheim was born in Frankfurt am Main in 1885, emigrated from Germany to Belgium in 1933, and then to USA in 1939. He contributed to philosophy, especially Gestalt Theory, and to philosophy of science, working with Grelling and also with Carl Gustav Hempel (1905-1997). 1 2 Jouko Va¨an¨ anen¨ The concepts of linear and algebraic dependence have a long history. They were known early in the 19th century in the methodology of solving systems of equations. Likewise, the concept of independence of events was at the heart of probability the- ory from the moment of its conception. In the thirties van den Waerden [25] and Whitney [26] pointed out similarities in the properties of linear and algebraic de- pendence and suggested an abstract concept which covers both, without assuming any background algebraic operations. This concept is nowadays known as the con- cept of a matroid or a pregeometry. Also in the thirties, Kolmogoroff [20] gave an axiomatic basis for probability theory, including an exact formulation of indepen- dence of events, with events being essentially just abstract sets. Apart from algebra, probability theory and logic, the concepts of dependence and independence were, of course, used throughout the centuries of development of ex- perimental science. In physics Galileo argued already in the 16th century that the time of descent of two falling bodies is dependent on the height from where they are dropped but independent of their weight. In the 18th century in philosophy Hume developed the theory of causality, continued by Mill in the 19th century; and causal- ity is intimately connected with dependence: an effect depends in a natural sense on its cause, and on the other hand, what is independent of the effect cannot be its cause either. In biology Mendel argued in 1866 that what are now called alleles of genes of plants are totally dependent on the alleles of the parents, but alleles of dif- ferent genes are inherited independently of each other. When we come to the early 19th century the concept of independence was widely used in physics, chemistry, biology, statistics, and other fields. In quantum physics the concept of independence of events received notoriety from the EPR-phenomenon [5], according to which two however distant entangled particles apparently depend on each other in an instanta- neous cause-effect sense, although they are in the sense of classical physics totally independent of each other. The entanglement phenomenon was proved with a math- ematical argument in 1964 by Bell [2], and later also experimentally demonstrated. In an unrelated development in computer science in the seventies Codd [4] in- troduced the concept of functional dependence in a relational database. This is, as we argue below, essentially equivalent to Grelling’s concept of dependence. Many variations of functional dependence were introduced in database theory including the concept of independence (under the name ‘embedded multivalued dependence’ [6]). This concept is a special case of the probability theoretic concept of indepen- dence. We use team semantics [24] as a universal approach covering all the individual cases of dependence. Notation: If A is a set, P(A) is the power-set of A. We use x, y, z and so on for finite set. Then xi, y j, zk and so on refer to individual elements of x, y, z, respectively. If x and y are sets, then xy is a shorthand notation for the union x [ y. If x ⊆ I and s : I ! M is a function, then sx is the restriction of s to the domain x. Grelling on dependence 3 Fig. 1 A gestalt phenomenon. 2 Dependence Grelling’s interest in dependence arose from Gestalt theory. Gestalt theory goes back to the essay On ‘Gestalt Qualities’ by the philosopher Christian von Ehrenfels, pub- lished in 1890. This theory started in philosophy and then moved to psychology. An example of a Gestalt concept is the phenomenon that our perception “fills” the missing parts in Figure 1. The missing parts are in a sense functionally dependent on—or in the closure of—the visible parts. Instead of a visual image we could con- sider a piece of music or poetry. Grelling started to develop a mathematical theory of such dependence. Grelling states, in his own notation, a definition equivalent to the following: Definition 1 (Grelling [11]). Let I be a set of functions g of the same variable x (or more generally, a finite string of variables). The closure cl(j) of j ⊆ I consists of those functions f in I for which the following holds: for any x1 and x2, if each function g in j satisfies g(x1) = g(x2), then f (x1) = f (x2). We say that y depends (only) on j if y ⊆ cl(j). Typically g would be a magnitude with x as a parameter. A typical parameter would be time. We can think of Grelling’s functions as columns of a table of data and his arguments x as indexes of the rows of the table (see Figure 2), although Grelling does not refer explicitly to a table of data as his model. x f1 f2 ··· fm a1 f1(a1) f2(a1) ::: fm(a1) . ak f1(ak) f2(ak) ::: fm(ak) . Fig. 2 Grelling’s setup 4 Jouko Va¨an¨ anen¨ The closure operation cl of Definition 1 satisfies, as Grelling observes, the axioms C1 x ⊆ cl(x). C2 If x ⊆ y, then cl(x) ⊆ cl(y). C3 cl(cl(x)) ⊆ cl(x). Any function P(I) ! P(I) as above is nowadays called a closure operation. If I is a vector space, then cl(x) can be taken to be the linear span [x] of x. Now dependence means the same as linear dependence. Likewise, in an algebraically closed field we can let cl(x) be the algebraic closure of x, and then dependence means algebraic dependence. If we have a topology on I, we can let cl(x) be the topological closure of x, and dependence means being in the set or on the boundary of the set. For an example from model theory, suppose M is an infinite first order structure. An element a of M is algebraic over x ⊆ M if there is a first order formula (with identity) f(y;z1;:::;zm) and elements b1;:::;bm of M such that the set A = fa 2 M : M j= f(a;b1;:::;bm)g is finite and a 2 A. We get a closure operation cl on M be letting cl(x) be the set of elements of M which are algebraic over x. If G is a finite graph we can let cl(x) be the set of points which are on a closed walk (a sequence of vertices starting and ending at the same vertex, with each two consecutive vertices in the sequence connected by an edge in the graph) including x. In this case dependence on x means being within a closed walk from some elements of x. If I is the set of attributes (fields) of a database, then cl(x) can be taken to consist of those attributes yi for which the attributes in x functionally determine yi, i.e. if we take any two tuples from the database and they agree about the attributes in x they also agree about the attribute yi. Dependence in this case means the same as functional dependence in database theory. A classical closure operation in logic is logical consequence: Suppose I is the set of all first order sentences in a given vocabulary. Let cl(x) be the set of sentences that logically follow from the sentences of x. Then cl is a closure operation on I. Dependence in this structure means the same as logical consequence. For an example from recursion theory, suppose I is uncountable the set of all subsets of N. For a (typically finite) subset x of I let cl(x) consist of those (countably many, if x is countable) subsets of N which can be computed with some finite number of oracles from x.
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