¹ The study of Communist voting can be formulated at two levels of analysis of which only one will be of concern here. Several studies have examined the determinants of the aggregate strength of the Communist movement under different socio-economic conditions. Communist strength has been said to depend upon the degree of industrialization, upon legitimacy and efficiency of a government, degree of massification of a society, extent of anti-industrial sentiments prevailing among the population, and the gap between economic and social development. It has been frequently suggested that the strength of the Communist movement is curvilinearily related to the socio-economic factors. Soares hypothesized that historically and cross-nationally this relationship may well fit a second degree curve. Ulam has observed that the reactions to industrial society which generate support for radical movements take place during the early stages of industrialization. Kornhauser pointed out that the discontinuities emerging during the early stages of industrialization favor mass politics, Communism included. Recently, two studies have reached the conclusion that the relationship between economic development and Communism is curvilinear. Using cross-national data, the authors of these studies observed that the highest degree of Communist support occurs at the intermediate stages of economic development. Although these two studies do not agree as to exactly when the peak is likely to occur, they share the view that in both highly industrial and highly traditional societies Communist strength is low. However, our concern is strictly limited to the explanation of the voting behavior of individuals under given socio-economic conditions. We are not concerned here with the explanation of the aggregate strength of the Communist movement under changing conditions. See Soares, G. A. D., “Desenvolvimento Economico e Radicalismo Politico: O teste de uma hipotese (Chile),” America Latino, 5 (1962)Google Scholar; Ulam, Adam, The Unfinished Revolution (New York: Random House, 1960)Google Scholar; Kornhauser, William, The Politics of Mass Society (London: Routledge and Kegan Paul, 1960)Google Scholar; Marsh, R. M. and Paris, W. L. “Modernization and Communism: A Retest of Lipset's Hypotheses,” American Sociological Review, 30 (1965)CrossRefGoogle Scholar; Benjamin, R. W. and Kautsky, J. H., “Communism and Economic Development,” this Review, 62 (1968)Google Scholar.

² Soares, Glaucio and Hamblin, Robert L., “Socio-Economic Variables and Voting for the Radical Left: Chile, 1952,” this Review, 61 (1967)Google Scholar. A new study by W. D. Burnham and J. D. Sprague follows the general problems raised by Soares and Hamblin and discusses in general terms the relative merits of the linear and power functions. Burnham, W. D. and Sprague, J. D., “Additive and Multiplicative Models of the Voting Universe: The Case of Pennsylvania, 1960–68,” this Review, 64 (1970)Google Scholar. Voting models based on interaction were developed earlier by Boudon, Raymond, L'Analyse Mathématique de Faits Sociaux, (Paris: Plon, 1967), pp. 183–196 Google Scholar, and by Coleman, J. S., Introduction to Mathematical Sociology (New York: Free Press, 1964)Google Scholar, Chapter 11, section 5.

³ Soares and Hamblin, op. cit., p. 1064.

⁴ For a discussion of some criteria for evaluating theories see Hempel, C. G., Aspects of Scientific Explanation and Other Essays in the Philosophy of Science (New York: Free Press, 1965)Google Scholar; see particularly the title essay.

⁵ An excellent discussion of the invalidity of the Lockean assumptions is contained in Carr, E. H., The New Society (London: Macmillan, 1951)Google Scholar.

⁶ The literature on what is variously called “structural,” “compositional,” or “contextual” effects is quite extensive and growing rapidly, but we cannot review it here. For general statements of the problems of contextual influences see Kendal, P. L. and Lazersfeld, B. F. (eds.), Studies in Scope and Method of the ‘American Soldier’ (Glencoe: Free Press, 1955)Google Scholar; Blau, P. M., “Formal Organizations: Dimensions of Analysis,” American Journal of Sociology, 58 (1957)Google Scholar; and Boudon, Raymond, “Prioprietés individuelles et prioprietés collectives,” Revue française de sociologie, 4 (1963)Google Scholar. Techniques for identification of contextual effects have been developed by Davis, J. A. et al., “Technique for Analyzing the Effects of Group Composition,” American Sociological Review, 26 (1961)CrossRefGoogle Scholar, and were further discussed and modified by Tannenbaum, A. S. and Bachman, J. G., “Structural versus Individual Effects,” American Journal of Sociology, 69 (1964)CrossRefGoogle Scholar, and by Valkonen, Papani, “Individual Effects in Ecological Research,” in Dogan, Mattei and Rokkan, Stein (eds.), Quantitative Ecological Analysis in the Social Sciences (Cambridge, Mass.: MIT Press, 1969)Google Scholar. Several other articles in this volume also discuss the problem from a theoretical perspective, particularly the articles by Erik Allardt and by Erwin Scheuch. Applications of contextual formulations to the study of political behavior can be found in Ennis, P. H., “The Contextual Dimension in Voting,” in McPhee, W. N. and Glaser, W. A., (eds.) Public Opinion and Congressional Elections, (Glencoe: Free Press, 1962)Google Scholar and in Putnam, R. D., “Political Attitudes and the Local Community,” this Review, 60 (1966)Google Scholar. This list is incomplete and concerns only the texts which treat the problem from the theoretical rather than from the data analysis point of view. Several additional discussions of contextual effects can also be found in the literature concerning the problem of “ecological inferences.”

⁷ According to Dahrendorf, the main difference between the stratificational and the Marxist concept of class is that the first approach treats classes as characteristics of continuous distributions while the latter views them in nominal terms. We follow here the second usage of this concept. See Dahrendorf, Ralf, Class and Class Conflict in Industrial Society (Stanford, Calif.: Stanford University Press, 1959)Google Scholar.

⁸ Note that one of the simplifying assumptions of this theory is that there is only one party which appeals to the particular group. This theory will not explain the distribution of working class vote among the reformist Socialist, traditional Communist, and revolutionary left wing parties.

⁹ A mathematically minded reader will recognize Y′ as the first derivative of Y with regard to X.

¹⁰ Soares, G. A. D., “Desarrollo Economico y Radicalismo Politico,” in Kahl, J. A. (ed.), La. Industrialization en America Latina (Mexico, D.F., 1965)Google Scholar.

¹¹ This discussion of the concept of “class-consciousness” should not be treated as an attempt to find “what Marx really meant” Arguments about correct interpretation can rarely be solved, even after a careful analysis of the content. The definition introduced here is greatly arbitrary, although we hope that it is consonant with Marx's methodology.

¹² Among empirical analyses of class voting see Alford, R. R., “Class Voting in the Anglo-American Political Systems,” and several other articles in Lipset, S. M. and Rokkhan, Stein (eds.), Party Systems and Voter Alignments: Cross-National Perspectives, (New York: Free Press, 1967)Google Scholar.

¹³ Erik Allardt and Pertti Pesonnen, “Cleavages in Finnish Politics,” in Stein Rokkan and S. M. Lipset, op. cit.

¹⁴ Actually a part of class consciousness may be independent of the social context: it may be a result of internalization of norms. The we would have

Y′ = 2pX + b,

where b represents the strength of the “core” component of consciousness, and

Y = pX² + bX + a

This is the general form of the model.

¹⁵ See in particular Gramsci, Antonio, The Modern Prince (New York: International Publishers, 1957)Google Scholar; and Il Materialismo Storico e La Filosofia Di Benerdetto Croce, (Torino: Einaudi, 1949)Google Scholar.

¹⁶ Since Y_x, = Y – (Contribution of non-workers) = ae^px — ae^po = a (e^px — 1).

¹⁷ However, note that even this simple model has one non-intuitive consequence. Class-consciousness depends upon a, the extent to which others vote for the left party when no workers are present This model implies that, with p given, the effectiveness of interaction is higher in a congenial atmosphere in which non-workers tend to vote for the left party. This consequence is contrary to the situations contained in the Marxist model discussed below.

¹⁸ See Bartholomew, D. J., Stochastic Models for Social Processes (New York: Wiley, 1967)Google Scholar, Chapter 8.

¹⁹ This interpretation of c makes it theoretically awkward. Clearly, values of c > 1.00 must be rejected on a priori grounds. But also rejected should be those models which do not exhibit limiting behavior in the 0 < X < 1 interval. In such cases the exponential rather than the logistic model is appropriate.

²⁰ It must be noted that the p which appears in the models in which class-consciousness, Y!, depends upon Y cannot be interpreted as a probability in its original form. This is due to the fact that we are operating on proportions rather than on the absolute numbers. Strictly speaking we should have written,

Thus, if p is to be interpreted as a probability its value must be 100 times lower than the estimated value of this parameter.

The mathematical treatment of the logistic function is based on Czechowski, Tadeusz, Rachunek rozniczkowy i calkowy, (PWN, Waraszawa, 1966)Google Scholar. The solution of the equation 4.1. in terms of 4.2. is given by Allen, R. G. D., Mathematical Analysis for Economists (London: Macmillan, 1962), p. 419 Google Scholar, example 3.

²¹ This model is based on Boudon, op. cit., pp. 183–184. Boudon's text, to which we have repeatedly referred here, is fascinating both in its theoretical richness and its treatment of logical conditions imposed on a theory. However, Boudon's method of developing the contextual models is based on direct substitution of a contextual hypothesis in place of the regression coefficient in a linear model. This method yields results identical to ours when the effect of context is made dependent upon X but cannot be used to treat hypotheses of dependence upon Y.

²² J. S. Coleman, op. cit., Chap. 16.

²³ This model greatly oversimplifies Marx's theory. We are merely attempting to represent formally the situation concerning the influence of class conflict on class consciousness. However, the assumption that class conflict is linear on the proportion of workers represents at best a Marxist theory in its crudest formulations. Marxist theory of class consciousness is so complex that even a half-way attempt to formalize it would greatly exceed the limitations of space, not to speak of mathematics which we know.

²⁴ See, for example, Rokkan's data on class voting in Norway. Stein Rokkan, “Geography, Religion, and Social Class: Crosscutting Cleavages in Norwegian Politics,” in Lipset and Rokkan, op. cit., pp. 430–431.

²⁵ In order to obtain an explicit solution for Y, one must solve two first order differential equations given by the hypotheses concerning the rates of change of left vote among workers and non-workers:

Since Y is unknown, the solution becomes very complex. Generally, it seems that the way to proceed is to formulate a second order differential equation for Y′ ′i−x and then work backwards by substitutions. However, we have not attempted to obtain the exact solution, and the results discussed here are purely intuitive.

²⁶ Mattei Dogan, “Political Cleavage and Social Stratification in France and Italy,” in Rokkan and Lipset, op. cit.

²⁷ Robert T. McKenzie and Allan Silver, “The Delicate Experiment: Industralism, Conservatism, and Working Class Tories in England,” in Rokkan and Lipset, op. cit.

²⁸ Sartori, Giovanni, “The Sociology of Parties, A Critical Review,” in Party Systems, Party Organizations, and the Politics of New Masses (Berlin: Committee on Political Sociology of the I.S.A., 1968)Google Scholar.

²⁹ Note that the proportion of the labor force in industry and mining is treated as a measure of the relative size of the working class, not as an indicator of the level of development, as in the Soares and Hamblin analysis.

³⁰ Arauco is the center of the Chilean coal mining and has long traditions of left-wing radicalism. Atacama is a province dominated by copper mining. Both Arauco and Atacama have an exceedingly high percentage of trade union members among the working class, but it seems that the influence of the unions operates in different directions in these two provinces. We suspect that different unions operate in the two provinces, bu we do not have this information.

³¹ The logistic curve was fitted by an iterative procedure and thus the solution obtained may not be optimal. An analytical procedure for estimation is discussed in Johnston, J., Econometric Methods (New York: McGraw-Hill, 1963), pp. 49–50 Google Scholar, and in Tintner, G., Econometrics (New York: Wiley, 1965), pp. 208–211 Google Scholar.

³² Tingsten, Herbert, Political Behavior (Totawa, N.J.: The Bedminister Press, 1963), pp. 178–179 Google Scholar.

³³ Bush, R. R. and Mosteller, Frederick, “A Comparison of Eight Models,” in Lazersfeld, P. F. and Henry, N. W. (eds.), Readings in Mathematical Social Sciences (Cambridge, Mass.: MIT Press, 1966)Google Scholar. This is an interesting article, since it provides a concrete example of the meta-theoretical issues discussed here. The authors introduce the article by arguing that “In the testing of a scientific model or theory, one rarely has a general measure of goodness-of-fit, a universal yardstick, by which one accepts or rejects the model. Indeed, science does not and should not work this way; a theory is kept until a better one is found. One way that science does work is by comparing two or more theories to determine their relative merits in handling relevant data.” Page 335.

³⁴ We cannot do better at this point than refer to William Feller's observations concerning the “law of logistic growth.” Having cited the extensive theoretical, empirical, and practical preoccupations with the logistic functions applied to the analysis of growth, Feller concludes: “The only trouble with the theory is that not only the logistic distributions can be fitted to the same material with the same or better goodness-of-fit. In this competition the logistic distribution plays no distinguished role whatever; most contradictory theoretical models can be supported by the same observational material.” Feller, William, An Introduction to Probability Theory and Its Applications, Volume II, (New York: Wiley, 1966), p. 52 Google Scholar.

³⁵ It is interesting to note that the problems of form of prediction are nearly completely absent from the otherwise instructive new book on theory construction by Blalock, H. M. Jr., Theory Construction from Verbal to Mathematical Formulations, (Englewood Cliffs, N.J.: Prentice-Hall, 1969)Google Scholar. “Propositional inventories,” even from the best literature, are no substitute for theory.