This review represents an attempt to draw together two kernels of an idea. First, how is the correspondence between number and quantity defined--extensionally or intensionally? Second, what does this definition have to do with causality? I entertain the possibility that causal relations can be understood in terms of the idea that intension implies extension, such that intension (meaning or definition) calls out or defines extension (referents or examples) in one of three senses.The first and strongest sense in which intension could imply extension is that intension is the way of recognizing an extension--that is, whether an extension instantiates an intension or not. The latter interpretation is consistent with measurement by way of operational definitions, in which a variable is completely defined by the operations or measurements used to recognize it.
The second sense in which intension could imply extension is in terms of intensional measurement based on frames of reference. A frame of reference is a generalization of Kant's philosophical approach ("We do not derive our laws from nature but impose our laws upon nature"). Numerical frames of reference, in particular, are crucial to Suppes' (1970) probabilistic theory of causality as logical implication, which is a generalization of the principle that intension implies extension to the probabilistic case. Suppes' theory starts with a probabilistic frame of reference--a probabilistic intension--and then derives extensions by logical implication. If probability is intensive, then this implies an epistemic, as opposed to aleatory, conception of probability--probabilities are parasitic on the state of our knowledge, they do not exist independently in nature but rather are imposed by our minds upon nature. Probabilistic concepts, although not derived from nature, nevertheless establish a sort of correspondence with a qualitative natural world, and when this correspondence meets certain criteria, it takes the form of causality.
The third sense in which intension could imply extension is in terms of extensional measurement of ratio quantities. In this vein, Mulaik (1986) has shown that it is possible to obtain probabilistic causal relations as asymmetric functional relations. Thus, intension still implies extension, but the intension is a function, say, of person ability and item difficulty, and the extension is the probability of a given response. Thus, probability of a given response is the effect of some cause expressed as a function of person and item parameters, but this relation is asymmetric--person and item parameters are not and cannot be the effect of the probability of a given response. If probability is extensive, as it seems to be on this conception, then this implies an aleatory, as opposed to epistemic, conception of probability--probabilities exist in nature, they are not parasitic on the state of our knowledge, and probability and error must be measurable in ratio units. Thus, in this sense, Rasch measurement (Rasch, 1980) is the discovery of constants of nature.
If factors are real measurements of quantitative attributes, then going back to Spearman's (1904) definition of factor analysis as true score plus error, this implies that true scores and errors must be quantitative attributes. What if error is not a quantitative attribute? Error is simply what we did not measure, where measure is used in the operational sense of applying a rule to reality rather than in the sense of fundamental measurement--in the former sense, it is permissible to say, "Error is what we did not intend to measure," because our intentions are definitive. What if our intentions were not definitive, as on the traditional theory of measurement? Then we would have to observe error in ratio units, we could not rely on a definition of error as what we did not attempt to measure or were not interested in measuring.
A frame of reference evidently exists in the mind of the scientist. Or perhaps it exists in the form of scholarly journal articles in university libraries. Lakatos (1970) distinguished the material world (World 1) from the world of consciousness (World 2) and the world of objective knowledge (World 3). Coherence competence derives from a mirroring in one's mind (World 2) of laws such as the probability calculus (World 3). Correspondence competence derives from a mirroring in one's mind (World 2) of relations in the material world (World 1) (Hammond, 1996). Conceptions based on the notion that intension implies extension, such as those above, concern correspondence competence.
Both the traditional theory of measurement (e.g., as represented by Mulaik) and the representational theory of measurement (e.g., as represented by Suppes) are correspondence theories--they have to do with the correspondence between intensions and extensions--but their approaches to establishing the correspondence are opposite. The traditional theory begins with numbers, extensionally defined (think of definition by pointing to examples) and attempts to derive not only concepts (intensions) but quantitative concepts. Russell (1920/1993), a representationalist, preferred to define extensions in terms of intensions--he defined numbers in terms of their definitions or concepts. From the standpoint of mathematical logic, this may have been a very fruitful thing to do, but from the standpoint of measurement, perhaps not. It does not represent measurement as practiced in Euclidean geometry or Newtonian physics, for example.
Hammond, K. R. (1996). Human judgment and social policy: Irreducible uncertainty, inevitable error, unavoidable injustice. New York: Oxford University Press. References
Lakatos, I. (1970). Falsification and the methodology of scientific research programmes. In I. Lakatos & A. Musgrave (Eds.), Criticism and the growth of knowledge (pp. 91-196). New York: Cambridge University Press.
Mulaik, S. A. (1986). Toward a synthesis of deterministic and probabilistic formulations of causal relations by the functional relation concept. Philosophy of Science, 53, 313-332.
Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests. Chicago: University of Chicago Press.
Russell, B. (1993). Introduction to mathematical philosophy. New York: Dover. (Original work published 1920)
Spearman, C. (1904). "General intelligence," objectively determined and measured. American Journal of Psychology, 15, 201-293.
Suppes, P. (1970). A probabilistic theory of causality. Amsterdam: North-Holland.
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