The distance between constituents

Abstract

One thing on which Hilpinen (1977), Niiniluoto (1977), (1978), Tichy (1974), (1976), and I seem to be in agreement is that, at least for a finite language, the problem of verisimilitude, or o? distance from the truth, is in some way reducible to the problem of specifying a satisfactory measure of the distance between constituents for the language under consideration. For every theory A can be represented as a set of constituents (that is, complete consistent theories), namely the set of all those constituents that extend (or entail) A; and the truth Tis itself a single constituent. Thus the distance of A from T is reasonably supposed to be some function of the distances between the various constituents of the language. Exactly which function is most suitable is something on which Hilpinen, Niiniluoto, Tichy, and I are in disagreement; certainly collectively, and even, I suspect, pairwise.1 But that is not my topic here. What I am exclusively concerned with in this paper is the problem of providing a non-arbitrary measure of the distance between any two constituents (and therefore, in particular, between any constituent and T) of a finite language. I maintain that the simple measures proposed by Tichy for sentential languages and by Niiniluoto for monadic predicate languages are seriously arbitrary. Tichy's sentential measure I have discussed on several previous occasions ((1974), Section 6, (1975), Section IV, (1976), Sections 1 and 5), and I return to it here (in Section I) only because it provides an easily comprehensible introduction to the not so easily comprehensible com plexities involved when we turn to the constituents of the predicate calculus. Commenting on Tichy's proposal Niiniluoto writes ((1977), note 19): "He specifies a distance measure between constituents of propositional logic only; this measure is not very plausible, however"; he refers directly to my (1975). Later ((1978), note 10) he goes on to say that "Tichy (1974) develops this suggestion [to define all distances from the truth in terms of distances between constituents] only in the case of propositional logic; his treatment of this case has been effectively criticized by Miller". In Section II of the present paper I intend to establish that a similar criticism is effective against