On inductive relations
C. H. Langford · Bulletin of the American Mathematical Society · 1927
Introduction,Difficulties, connected with the occurrence of reflexive fallacies, appear when we attempt to give straightforward definitions of inductive series.These difficulties arise inevitably from the fact that, if a series is to be inductive, in the ordinary sense, properties of all orders must be transmitted in the series, and it would seem that no proposition, nor any finite set of propositions, can assert that properties of all orders are transmitted.! In the discussion which follows, we shall confine attention to the definition of a particular type of inductive series, namely the order-type co.This restriction is made for the sake of definiteness and simplicity, and it does not entail any loss of generality in the points we shall wish to illustrate.In a note in Mind for 1923, Dr. J. E. McTaggart holds that there is a sense in which the dictum, No proposition can be about itself, is false ;$ and he maintains, on the contrary, that propositions in intension can be about themselves in the sense that they are applicable to themselves; and that distinctions of type are transcended in the case of these propositions.I think it just possible that this distinction is an important one, and wish to inquire what can be made of it by way of overcoming certain difficulties in the theory of types.McTaggart's view requires that we make a sharp distinction between modal and non-modal propositions, and that non-modal or material propositions be strictly subject to differences of type, in contrast to modal propositions.