David H. Krantz, working at the University of Michigan and later Columbia University, was the lead author and driving force behind the monumental three-volume Foundations of Measurement (with Luce, Suppes, & Tversky, 1971, 1989, 1990). This work provided the most comprehensive axiomatic treatment of measurement ever produced, establishing the conditions under which empirical observations can be faithfully represented by numerical structures.
Representational Measurement Theory
a >=_e b iff phi(a) >= phi(b)
phi(a o b) = phi(a) + phi(b)
Representation theorem: When does phi exist?
Uniqueness theorem: How unique is phi?
Krantz and his co-authors formalized measurement as the construction of a homomorphism from an empirical relational system to a numerical relational system. For any proposed measurement, two questions must be answered: the representation theorem (under what axioms does a valid numerical assignment exist?) and the uniqueness theorem (how much freedom is there in the assignment, determining the scale type?). These results provided the rigorous mathematical foundation for Stevens' scale taxonomy and for all quantitative work in psychology.
Krantz, together with Tversky, extended the Luce-Tukey conjoint measurement theory to a broader class of polynomial combination rules. Their work showed that independence tests on multi-attribute stimuli can determine whether the underlying measurement structure is additive, multiplicative, or some more complex polynomial form -- providing empirically testable conditions for the meaningfulness of quantitative models.
Contributions Beyond Measurement
Krantz also contributed to color science (developing a Grassmann-structure theory of color matching), decision theory, and health psychology. His later career included influential work on the psychology of risk perception and health behavior, applying the rigorous theoretical standards of mathematical psychology to applied problems.
Legacy and Impact
Foundations of Measurement remains the definitive reference on the axiomatic basis of psychological and physical measurement. Its framework shapes how psychologists, economists, and decision theorists think about what their numbers mean. Krantz's combination of mathematical depth and psychological insight made representational measurement theory not just technically rigorous but genuinely useful for understanding the structure of psychological phenomena.