BR TO RALPH G. HAWTREY, 23 SEPT. 1908
BRACERS 55814. ALS(X). Churchill College, Cambridge
Proofread by K. Blackwell

Bagley Wood,1
Sp. 23. ’08

My dear Hawtrey

I have left your letters2 rather long unanswered, because I was too much absorbed by the job I had in hand to give them due thought.

Your proof of $\mu , \nu \; \varepsilon \; \text{NC induct} \, . \supset \, . \mu + \nu \; \varepsilon \; \text{NC induct}$ appears to me to be valid, assuming that the difficulties about identity have been disposed of.

With regard to identity, I cannot think you are right. I suggest the following conundrums. (1). I gather you think that “$x = y$” means that the meanings of $x$ and $y$ are identical. What does this mean? (2). Conversely, if identity only has to do with symbols, so has diversity; thus to say that $x$ and $y$ are diverse will mean that they have diverse meanings. I suppose you might reply that you do not mean that $x$ and $y$ are to have “the same meaning” when they are identical, but that there is an object which they both mean. That, perhaps, disposes of the circle in definition. But then how do you deal with such propsa as “Scott is the author of Waverley”?b The meanings of “Scott” and “the author of Waverley” are not the same. There is implied a proposition “Nobody other than Scott wrote Waverley”, i.e. $x \, \ne \, \text{Scott} \, . \supset _x \, . \sim$ {$x$ wrote Waverley}. I cannot believe that diversity, as here employed, means merely something to do with the symbols. Speaking as a philosopher rather than a mathematician, I should say that “diversity” is an ultimate notion, and that identity may perhaps be correctly regarded as the negation of diversity. In any case, I think that if the definition of $x = y$ as “$\phi ! x \, . \supset _\phi \, . \phi ! y$” is abandoned, it will be necessary to introduce either identity or diversity as a primitive idea. If you can deal with “descriptions” otherwise, I should be grateful for an indication of your plan. The points concerned are those dismissed in my article “On Denoting”.3

To come to the axiom of reducibility: Can you prove that if a number $\mu$ possesses all first-order inductive properties, it also possesses all 2nd-order inductive properties? I cannot. On the other hand, supposing this cannot be proved (without the axiom of reducibility), can you bring yourself to believe that there are numbers which are inductive for (say) properties of the first order but not for properties of the 2nd? And if so, what order of inductiveness belongs to the numbers of every-day life, i.e. those expressible in the usual decimal notation? I believe myself that the axiom of reducibility is true, but that, with a better philosophy, it would not appear as an axiom. But I only think so because of the oddity of the results of supposing it false.

Your earlier letter on classes4 is very interesting. To take your illustration, namely “all the $\alpha$’s together form a square”, where $\alpha$ is the class consisting of the four sides of a square, it is of course possible to form a statement about the members of $\alpha$ which is equivalent to this; in fact, Euclid’s definition is virtually such a statement. But it certainly seems as if all statements as to geometrical form, tho’ reducible by violence to statements about the points or lines involved, were more naturally regarded as statements about the class. To say (e.g.) that a certain area is a circle is, I admit, primâ facie to say something about the area, and not about its constituent points. But we know that the circle can be defined as the points whose distance from a given point does not exceed a given distance. And so with “all the $\alpha$’s together form a square”: it can be interpreted as

$\alpha \; \varepsilon \; 4 : \, .  \,x \; \varepsilon \; \alpha \, . \supset _x \, : (\exists y, z) \, . y \ne z . \; \text{angle} \; \widehat{xy}, \; \text{angle} \; \widehat{xz} \; \varepsilon \; \text{right angle} :$
$(\exists w) \, . \, w \ne x \, . \, w \; \text{ is parallel to} \; x.$

I should not prefer this interpretation but for the reflexive fallacies. But I do not think you are right in saying they can be avoided by avoiding reflexive fallacies in the defining propositional functions. I do not see how we are to avoid regarding “$\alpha \; \varepsilon \; \alpha$” as significant, if there are such things as classes; and if so, $\hat{\alpha} (\alpha \sim \varepsilon \; \alpha)$ lands us in the old difficulties. But perhaps I have missed some point over this.

It seems to me that if extensional classes are admitted, the axiom of reducibility must be true. You raised a question as to whether “$x \; \varepsilon \; \alpha$” is predicative. But what the axiom of reducibility is really required to assert is not that all functions are reducible to predicative functions, but merely that all functions are reducible to such-and-such a type. That is, “$\phi ! x$” stands for any function of such-and-such a type. Now it does seem obvious that when the types of $x$ and $\alpha$ are given, “$x \; \varepsilon \; \alpha$” must be of one determinate type, if there are such things as extensional classes. The axiom

$$(\exists \alpha) : \phi x \, . \, \equiv_x \, . \, x \; \varepsilon \; \alpha$$

has exactly the same symbolic effect as the axiom of reducibility. It is, however, a greater assumption, since it implies the axiom of reducibility, but not vice versa.

What you say against intensional classes is, so far as I understand, my own view. But as regards the extensional class, I still need conversion.

For the present, I am going on writing out the later parts of the book,5 as they cannot well be affected by these points, and I am anxious to have the whole thing written out somehow as soon as possible. I like to hear from you, because the points take time to think over. After Xmas I expect to have pretty well finished the later parts, so far as they are my job. Then I shall settle down to think about types etc. for some time. But till then, I shall not definitely work at these questions.

Yrs ever
Bertrand Russell.

Typeset by A. Duncan 05/02/2019; proofread against a photocopy of a photocopy, K. Blackwell 2019/02/10.

• 1

  [document] The letter was edited from

• 2

  your letters Hawtrey to BR, ? and ?.

• 3

  “On Denoting”.Mind n.s. 14 (Oct. 1905): 479–93 (B&R C05.05); 16 in Papers 4.

• 4

  Your earlier letter on classes Hawtrey to BR, ?.

• 5

  the bookPrincipia Mathematica.

Textual Notes

• a

  props Propositions.

• b

  Waverley Italics added editorially.