George Boolos, 1940-1996

Boolos Memorial Service

October 5, 1996
Eulogy by Richard Cartwright

Newspaper notices of George’s death described him as “philosopher and logician”. I think he would not have been altogether happy with the description: accurate, no doubt, but faintly redundant–a little like describing someone as “mathematician and algebraist” or “linguist and grammarian”. Though he would have understood those who find logic hard to place, he himself thought it belonged to philosophy. Logicians are as often as not mathematicians, sometimes engineers, or economists. Well, so much for academic affiliations. The subject was started by Aristotle, and to this day its central topics are centrally philosophical. The study of logical implication and associated concepts has benefitted enormously from the application of mathematical techniques, but it has not thereby ceased to be part of philosophy.

Logic and what for lack of a better term may be called philosophy of logic are distinct. Of course, anyone who goes in for the latter had better know something about the former; but just as a physicist need know nothing of philosophy of science, so a logician can do his work without regard to the philosophical questions it raises. But that was not George’s way. Indeed, in much of his work–his most characteristic and memorable work, I think–logic and philosophy of logic are intimately connected: one cannot pull them apart without serious loss. Not surprisingly, this is especially so in his work on the question of the nature of logic, a question that concerned him throughout his professional career.

Professor Quine has said:

Logic, like any science, has as its business the pursuit of truth. What are true are certain statements; and the pursuit of truth is the endeavor to sort out the true statements from the others, which are false. (Methods of Logic, p. 1)

Not, of course, that logic is the indiscriminate pursuit of truths of any and all sorts. Hence the question, What sort or sorts of truths are sought within the science of logic? The question divides into two. One concerns boundaries: are there lines of demarcation separating logic from arithmetic, set theory, chemistry, zoology, and so on; and if there are, how exactly and on what basis are they to be drawn? The other has to do, not with the extent of logic, but with the nature of the truths found or sought within what is indisputably the province of logic. George dealt with both questions, but I think his treatment of the second shows him at his most inventive.

Plato, in the Republic, has Socrates say to Glaucon: “It is clear that one thing cannot act in opposite ways or be in opposite states at the same time and in the same part of itself in relation to the same other things” (436b). Glaucon assents, of course; but what is he assenting to? It has the ring of a principle of logic, but what exactly is it? Aristotle put it this way: “For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible” (Metaphysics IV, 1005b18-21). An improvement, I suppose, but what are these “things” that may or may not “hold good” of this or that? Expositions of elementary logic typically avoid the question: we are in effect given, as a so-called theorem: “Nothing is both such-and-such and not such-and-such”, and told that we may put anything grammatical for ‘such-and-such’ and thereby obtain a truth. A strange theorem, however; for, being a mere schema, it is not even a truth.

A natural thought is that Aristotle’s talk of “things” may be replaced with talk of attributes–or, as some prefer, of classes, which are said to have the advantage of a clear criterion of identity: a class A is the same as a class B just in case A and B have the same members. Thus, fixing on classes, Aristotle’s formulation would give way to: “Whatever class you take, nothing both is and isn’t a member of it”. This is surely true. But does it have the wanted power? The schema, “Nothing is both such-and-such and not such-and-such”, was to yield a truth no matter what predicative phrase was put for ‘such-and-such’. The suggested class-principle will yield those truths only if there are enough classes–only if, that is, corresponding to every predicative phrase there is a class the members of which are precisely those objects to which the phrase correctly applies. That there are in that sense enough classes may seem undeniable. As George wrote in his classic “The Iterative Conception of Set”:

How could there not be a [class] of just those things to which any given predicate applied? Isn’t anything to which a predicate applies similar to all other things to which it applies in precisely the respect that it applies to them; and how could there fail to be a [class] of things similar to one another in that respect? Wouldn’t it be extremely implausible to say, of any particular predicate one might consider, that there weren’t two kinds of things it determined, namely a kind of thing of which it is true, and a kind of thing of which it is not true? And why should one not take these kinds of things to be [classes]? Aren’t kinds [classes]?

And yet, as of course he knew, Russell had discovered that the general principle is false, in fact self-contradictory. For take the predicate ‘nonselfmembered’. Certainly, nothing is both nonselfmembered and also not nonselfmembered. So by the general principle some class–R, say–has for its members all and only the objects that are nonselfmembered. Is R one of them? Even if it is, it isn’t; so it isn’t. And even if it isn’t, it is; and so it is. So since nothing is both nonselfmembered and not nonselfmembered, there is no such class as R.

If this were an isolated case, maybe we could ignore it; but it is just one of infinitely many. And if it concerned classes specifically, we might hope for consistency by switching to, say, attributes. But that would help none; for just as no class has as its members just the nonselfmembered objects, so no attribute has as its instances the objects that are nonselfinstantiated.

What is to be done? One strategy is to renounce direct generalization on the instances of the schema and to take as the relevant truth of logic a statement about the schema, to the effect that each exemplification of it is true. To follow this strategy appears to sacrifice the view, implicitly Frege’s and explicitly Russell’s– and, I think, favored by George, that (as Russell put it) “logic is concerned with the real world just as truly as zoology” (Introduction to Mathematical Philosophy, p. 169). A closely related strategy, one that tempted George for a time, is to take the heading “Theorem” above e.g. “Nothing is both such-and-such and not such-and- such” as signalling simply affirmation of, or at least commitment to, each particular case–a little like writing down a dozen or so instances and then writing ‘etc.’. Each such affirmation concerns the real world, but unfortunately not, to complete Russell’s remark, “with its more abstract and general features”. The wanted generality of truths of logic is lost.

It was in the early 1980s, I think, that George first hit on another strategy. In a pre-publication version of “Nominalist Platonism”, he describes its origin this way:

Frege’s definition of “x is an ancestor of y” is: x is in every class that contains y’s parents and also contains the parents of any member. Some years ago, I was discussing this definition with Peter Unger, who asked me, “Do you mean to tell me that because I believe that Napoleon was not one of my ancestors, I am committed to such philosophically dubious entities as classes?” At the time, I didn’t know what to say about the point Unger had raised. I thought–I think nearly everyone would have thought the same thing–that one couldn’t simply throw out the definition, whose logical utility and fruitfulness had been established beyond doubt, for such a crazy reason. But I certainly didn’t have a good answer to Unger’s question. I think I have one now.

The answer was, in a word, “No”. Napoleon is not one of Unger’s ancestors just in case either no one is a parent of Unger or there are some people such that (i) each of Unger’s parents is one of them, (ii) each parent of any one of them is one of them, and (iii) Napoleon is not one of them. Here talk of classes is avoided by using plural nouns and pronouns and the locution ‘is one of’: ‘Napoleon is not a member of it‘ gives way to ‘Napoleon is not one of them‘. To return to our original example: “Whatever class you take, nothing both is and isn’t a member of it” is replaced by “Given any objects whatever, nothing both is and isn’t one of them”. There you have a truth of the science of logic.

I have not meant to imply that George was an enemy of classes. He did think that logic ought to be, in David Lewis’s phrase, “ontologically innocent”–to which he might have added, twinkle-in-eye, “and ought implies can“. And he did think that the theory of classes (“set theory”, as it is known) becomes mythology somewhere in the region of the smaller large cardinals: see his delightful “Must We Believe in Set Theory?”. But he had no opposition to classes as such, nor for that matter to abtract entities generally. Recall after all the title, “Nominalist Platonism“.

Neither have I meant to imply that George’s philosophical work, or his philosophical interests, were limited to the topic I’ve talked about. I have alluded to his work on the foundations of set theory; and this was closely connected with his important research on Frege’s philosophy of mathematics, research he once described–tongue half in cheek–as a foray into history of science. And his philosophical interests, indeed his intellectual interests, were far wider than his bibliography suggests. Hints occasionally appear in his published works: references to Leibniz, of course, to Berkeley and Hume, to Austin and Austen, and even to the Eleatic Stranger. (I suspect this last eluded a good many readers.) But one cannot tell from those works that he enjoyed Chaucer, untranslated, or that he was a regular reader of Philosophy and Public Affairs.

Friendships nourished George’s interests. His many friends will miss him.