THE THOUGHTS recorded in this monograph were first set down in three separate papers in the academic year 1954-55. Since then there have been more revisions than I like to admit, in view of the always difficult and, I fear, often obscure character of the argument. I found at a late stage that it was necessary to expand into a full chapter what had been a brief introduction to the main theory. That, the initial chapter, is exceedingly abstract and general, and probably the most difficult part of the book.
While my most recent thinking on these abstract matters confirms the general course of speculation in chapter i, all has not yet been placed upon a secure enough footing, nor is it sufficiently general in scope of application. But none of this could be remedied short of increasing the size of the book by half. As these efforts would shed little if any further light on the problem of universals, I decided to forgo them, having resort, faute de mieux, to a series of rather long notes which I hope will give the sense of the necessary additions and emendations.
These speculations arose out of my endeavors to answer three seemingly unrelated questions. First: What accounts for our quickness to agree with Locke and Hume that some ideas are simple and others complex, and precisely what distinction do we have in the back of our minds when we use the terms "simple" and "complex"? Second: What is the operating principle behind our use of inverted commas as a device for referring to expressions? Third: What can it mean to say that the natural numbers are denumerably infinite, they being the ultimate standard for determining that sets are denumerably infinite? These questions, as I pursued them, were transfigured into the apparently more general forms: What is a property? What is an expression? What is a natural number? (The second question, while interesting and difficult, receives but passing attention, as an example. Each of my answers to the other two questions fills a chapter.) The common character of these queries suggests a common approach. I always ask, What is peculiar and characteristic in referring to . . . (properties, expressions, numbers)? I probe the metaphysics of the referent by picking at the logic of the reference, and in doing so follow a procedure as old as Plato.
Clearly, I am seeking localized solutions to that most ancient problem, so-called, of universals. Before casting off into my technical exposi