On the general theory of meaningful representation

The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received adequate justification: the assumption that a proposition invariant under the appropriate group is therefore meaningful, and the assumption that representations should be unique up to a transformation of the appropriate group. A general theory of representational meaningfulness is offered, based on a semantic and syntactic analysis of representational propositions. Two neglected features of representational propositions are formalized and made use of: (a) that such propositions are induced by more general propositions defined for other structures than the one being represented, and (b) that the true purpose of representation is the application of the theory of the representing system to the represented system. On the basis of these developments, justifications are offered for the two problematic assumptions made by the existing theories.Material from this paper was presented at a conference on meaningfulness in the theory of measurement held at New York University in December 1984, hosted by J. C. Falmagne. I would like to thank Patrick Suppes for arranging my invitation to this conference, and David Krantz, R. Duncan Luce, and Fred Roberts for helpful comments. I would also like to thank an anonymous referee for extremely detailed and helpful comments and suggestions, the most important of which are acknowledged in footnotes.