1. Peter Abell, `Measurement in Sociology, I, Measurement Systems', Sociology 2, 1-20.

2. We are in possession, of course, of many `low level' empirical generalizations about human behaviour but these are in most cases little more than systematized common sense (see, for instance, B. R. Berelson and G. A. Steiner, Human Behavior; an Inventory of Scientific Findings, New York: Harcourt Brace, 1964). The point stands that these generalizations could have been elaborated and tested without the aid of anything that, even on the most generous interpretation, could be interpreted as theory. Furthermore, these generalizations are almost invariably isolated, not being integrated into a set of related propositions. It is largely to inter-related theoretical generalizations that we look when speaking of law-like propositions.

3. It is often maintained that deterministic propositions which characterize so much of physical science are unlikely in sociology and any propositions we may wish to consider as law-like will be of a probabilistic nature. Statements of the form, `There is more than a random expectation that an A will be a B' (in a specified population) may well be located. But the possibility of attaching a probability number to such a statement that will exhibit stability across different but analogous populations, must necessarily be remote since the populations that sociologists deal with (be they people, roles, institutions, groups, organizations), are always comprised of a `small' number of cases. In fact I would contend that the laws of sociology are unlikely to be probabilistic in nature, much rather they will be structural regularities where the appropriate formalism will be abstract algebra of one sort or another.

4. I had originally intended considering the problems of converting ordinal to metric data in this paper. It turns out, however, to be a problem of such complexity that it must be left for later consideration.

5. It may be objected that colour is a generic concept whereas alienation is not. If, however, the theoretical approach suggested were adopted then alienation itself would most probably turn out to be a generic concept. That is we would locate different types of alienated structure, i.e. structures with different structural properties. Alternatively, `colour' in the argument in the previous paragraph could be replaced by `red' and `redness'.

6. For definitions of elementary structural concepts (graph theoretic) see, for example, Claude Berge (trans. A. Doig), The Theory of Graphs, New York: John Wiley, 1962. For more advanced motions of algebraic topology, see, for example, E. M. Patterson, Topology, London, Oliver and Boyd, 1963.

7. Johan Galtung, International Relations and International Conflicts: A Sociological Approach (mimeo), 1967.

8. Gerhard Lenski, `Status Crystallization: A non-vertical Dimension of Social Status', Amer. Social Rev., 1954, 19, 405-413.

9. Werner Landecker, `Status Crystallization and Class Consciousness', Amier. Social Rev., 1963, 28, 219-229.

10. One reservation however, would be Galtung's assumption of additive scales on and across rank dimensions.

11. This terminology was adopted in paper I.

12. Talcott Parsons and Edward A. Shils (eds.) Toward a General Theory of Action, Cambridge: Harvard University Press, 1951.

13. Mrs. V. West (University of Essex), is in the process of carrying out an analysis of organizational structure from this general standpoint.

14. See, S. F. Nadel, Towards a Theory of Social Structure, London: Cohen and West, 1962. Although Nadel's book is a most stimulating contribution to the theory of social structure, his formalism is of little value. It does not offer any great clarification over and beyond his prose and certainly does not lead to any deductive consequences.

15. Frank Harary and 0. A. Oeser, `A Mathematical Model for Structural Role Theory', Human Relations, 1962, 15, 89-100.

16. Neal Gross, Ward, S., Mason and Alexander, W. McEachem, Explorations in Role Analysis, New York: John Wiley, 1958.

17. This model will be further elaborated in a forthcoming paper: A Formal Model of Social Role.

18. I should like to thank Ron Arkin (Dept. Mathematics, University of Essex), for pointing out the utility of algebraic topology. The presentation of the theory given here is unrigorous. A more rigorous treatment will be given in Formal Models of Social Structure, forthcoming.

19. The requirements for a Euclidian simplex is that (m + 1) vertices are linearly independent.

20. Simplexes generate groups in a number of ways. See any standard text on Algebraic Topology.

21. Functional theory can also be formalized (as indicated in paper I) in terms of the more conventional logic of ternary relations. But the model presented here seems to be better at capturing the structural aspects of fictional analysis. Nagel's brilliant formalization of functionalism (Ernst Nagel, Logic without Metaphysics, Glencoe: Free Press, 1956), which is often adopted as a paradigm in sociology seems to me to be rather inappropriate to sociological problems as it rests upon the assumption of metric measures. Sociology is often concerned with what we might call ordinal functional systems. Ordinal simplexes seem to offer a suitable language for exploring some of the structural aspects of such systems.

22. We have adopted Smelser's model of structural differentiation. (Neil J. Smelser, `Toward a Theory of Modernization', in A. and E. Etzioni (eds.) Social Change, New York: Basic Books, 1964.

23. Peter Winch, The Idea of a Social Science, London: Routledge and Kegan Paul, 1958.

Measurement in Sociology

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