A Verisimilar Ordering of Theories Phrased in a Propositional Language

Abstract

Popper introduced the concept of verisimilitude in the early sixties; Miller and Tichy deflated his definition in the early seventies. These facts, and the ensuing debate on verisimilitude, have been well chronicled in the pages of this Journal, as a look at, for example, Oddie [1981] and Urbach [19831 will show. Various attempts have been made to construct an acceptable definition of verisimilitude; these have mostly centred around the idea of distance from the truth. But this shared approach has not led to any substantial agreement. Moreover, of late there has been growing pessimism concerning the very possibility of success. This is due mostly to an argument first raised in Miller [1974], which says that our intuitions concerning verisimilitude are dependent on the language in which theories are expressed. In this paper we do not tread any of the well-worn paths: this is not another de novo investigation of the issues involved in verisimilitude. It is rather the serendipitous application to this concept of a certain construction applied to algebraic structures. Namely, to any structure there corresponds its power structure, essentially built up by taking the power relation of each relation in that structure. And for any relation R between elements of a set A, its power relation R+ relates subsets of A in a way dependent on R. It turns out that there is a natural power relation to be found between formulae of a propositional language. We offer this relation for consideration as a verisimilar ordering of theories phrased in a proposition language. Our approach, then, if we are to be charged with one, is to think of verisimilitude as an ordering relation between theories, and to present a model of this relation. We agree that an acceptable definition of verisimilitude should lead to an ordering of theories phrased in a first-order language. Nevertheless, for a start, to illustrate the ideas involved, and because the subject