Mathematical Psychology and Psychophysiology

By Stephen Grossberg | Go to book overview

the control of the decision maker, whereas the axioms force a fixed probability on E. For an extended discussion of these matters, see Balch [ 1974], Balch and Fishburn [ 1974], Krantz and Luce [ 1974], and Spohn [ 1977].

A general averaging model of the type assumed by Anderson [ 1974a], [ 1974b] arises from the conditional theory as follows. Let X = {1, 2, . . ., n), E = 2x, Ci, iX, be sets and define
D = {fA∣A ∈ E - {∅}, fA (i) ∈ Ci for iA}.

Observe that if AB = ∅ and fA, gB ∈ D, then, automatically, fAgB ∈ D. Assuming the axioms of the conditional theory, it follows readily that there exist nonnegative weights wi = P({i}) and a real-valued function φ, on Ci such that


preserves the ordering relation ≿. For details, see Luce [ 1981].


CONCLUSIONS

Our understanding of positive concatenation and of conjoint structures is reasonably adequate when the automorphism group is dense and especially so for fundamental unit structures. In contrast, we know very little about the discrete and trivial cases. Undoubtedly, many of these are so irregular as to be of no conceivable scientific interest, but some are clearly of importance, witness the case of probability.

Our understanding of the interplay of solvable conjoint structures having at least one component that is a fundamental unit structure is adequate when they satisfy distributivity, as appears to be the case for classical physics. It is totally unsatisfactory when distributivity does not hold, as arises with velocity in relativistic physics. The problem evidences itself in our inability to relate closely the automorphisms of the conjoint structure and that of the positive concatenation one. Closely related to these problems of dimensional interlocks is the general conceptual issue of meaningfulness and how it should be defined in terms of automorphisms and/or endomorphisms and/or some other invariance concept. Judging by the importance of dimensional analysis, this issue is of rather more than just philosophical interest.

The study of structures with more than one operation, including the case just mentioned, is probably susceptible to considerable generalization, just as fundamental unit structures generalized extensive ones. At the moment, all of the theories lead only to polynomial or metric representations. Almost certainly, more powerful algebraic proof techniques will be required since the existing methods seem to be leading to ever more complex, not easily generalized proofs.


REFERENCES

Adams E. W., R. F. Fagot and R. E. Robinson, 1965. "A theory of appropriate statistics", Psychometrika 30, 99-127.

Anderson N. H., 1974a. "Information integration theory: A brief survey", Contemporary Developments in Mathematical Psychology. II ( D. H. Krantz, R. C. Atkinson, R. D. Luce and P. Suppes, Eds.), Freeman, San Francisco, Calif., pp. 236-305.

_____, 1974b. "Algebraic models in perception", Handbook of Perception, II ( E C. Carterette and M. P. Friedman , Eds.), Academic Press, New York, pp. 215-298.

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