The Axiomatic Structure of Empirical Content

In this paper, we provide a formal framework for studying the empirical content of a given theory. We define the falsifiable closure of a theory T with respect to T ′ to be the largest subtheory of T ′ with respect to which T cannot be falsified with a finite data set. We provide a syntactic characterization of the falsifiable closure operator, showing it is equivalent to the theory which is axiomatized by all sentences satisfied by T which are universal negations of conjunctions of atomic formulas (UNCA). We also show that it is the smallest subtheory of T ′ containing T all of whose predictions are empirically falsifiable. Lastly, the falsifiable closure operator has the structure of a topological closure. The ideas here are useful for understanding theories which are well-understood (for example, we describe Afriat’s theorem in our context), but they can also be applied to theories with no known axiomatization. We show, for example, that the theory of multiple selves with a fixed finite set of agents (where multiple selves are aggregated according to a neutral preference aggregation rule satisfying independence of irrelevant alternatives), is a falsifiably complete theory (for any neutral rule satisfying IIA). PRELIMINARY AND INCOMPLETE: PLEASE DO NOT QUOTE OR CITE Date: January 15, 2010. Chambers and Echenique are affiliated with the Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena CA 91125. Shmaya is affiliated with the Kellogg School of Management at Northwestern University, 2001 Sheridan Road, Evanston, IL 60208-2001. Chambers and Echenique acknowledge support from the NSF through grant SES-0751980. Emails: [email protected] (Chambers), [email protected] (Echenique), and [email protected] (Shmaya). 1 2 CHAMBERS, ECHENIQUE, AND SHMAYA