An absolute-difference measurement space is a pair (X, e) where the real-valued function e on X2 satisfies conditions which are shown in the paper to be necessary and sufficient for its representability by the absolute distance on the real line. A positive-difference measurement space is a pair (X, l), where the real-valued function l on X2 satisfies conditions necessary and sufficient for its representability by positive distances on the real line. The conditions imposed on e and l make these functions extensive measurements of proximity and dominance, the two basic predicates of social enquiry. Another way of treating these conditions is to translate them to the formal language of multivalued logic. The translation is easy and the sentences obtained have plausible intuitive meanings such as reflexivity, symmetry, and transitivity. The two sets of conditions thus become formal theories of proximity and dominance. Our difference measurement spaces are relational structures for the multi-valued logic and models of the two formal theories. Thus proximity and dominance are considered dichotomous in principle and the multiple truth-values represent degrees of error. We suggest adopting multivalued logic as a framework within which the problem of measurement error can be treated together with the formal axiomatization of social and phychological theories.
Bibliographical noteFunding Information:
Section 2 and parts of Section 5 of this paper are based on Chapter I of my doctoral dissertation. I wish to thank my Oxford supervisor, Dana Scott, for his help in the writing of my dissertation. Other sections benefited from stimulating discussions I held, at the London School of Economics, with Clyde Coombs and Larry Jones. During the writing of this paper I was partly supported by Grant A4007 of the Canadian National Research Council.
ASJC Scopus subject areas
- Psychology (all)
- Applied Mathematics