BR TO RALPH G. HAWTREY, 5 SEPT. 1908
BRACERS 55818. ALS(X), Churchill College, Cambridge
Proofread by K. Blackwell

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Bagley Wood,1
5 Sep. ’08

Dear Hawtrey

Your letter2 did not strike me as insulting, except to the axiom of reducibility, which can’t hit back. Unfortunately the letter you wrote me about types in the spring is at Cranmer Rd,3 and as Whitehead is in Switzerland I can’t get it at present. I would accept almost any theory which would get rid of the axiom of reducibility, but I certainly did not realize that your theory would do so. I can’t remember it with sufficient distinctness to feel certainty about it, but I cannot well believe that it does get over the need of that axiom. If you are not too busy, would you mind telling me how you meet such points as the following:

Whenever a Df involves an apparent function-variable, as most Dfsa do, all props in which this Df is used surpass, in type, whatever limit may have been previously laid down. Take e.g.

$$x = y \, . = \: : (\phi) : \phi ! x \, . \supset \, . \phi ! y \quad \text{Df}$$

Suppose $\phi$ is confined by convention to functions of not more than the 30th order. Then $x = y$ is a function of the 31st order. Hence whatever has been previously proved for all $\phi$’s may be false for $x = y$. It is true that particular props can be proved by the methods of $\ast 10$; e.g. to prove $x = y \, . \supset \, . x = y$, we have

$ \begin{align} & \vdash \: : \, . \phi ! x \, . \supset \, . \phi ! y \, : \, \supset \: : \phi ! x \, . \supset \, . \phi ! y \\ \text{whence} \qquad & \vdash \: : \, . (\phi) : \, . \phi ! x \, . \supset \, . \phi ! y \, : \, \supset \: : \phi ! x \, . \supset \, . \phi ! y \quad & \text{by} \, \ast \! 10 \! \cdot \! 11 \\ \text{whence} \qquad & \vdash \: : \, . (\phi) : \phi ! x \, . \supset \, . \phi ! y \, : \, \supset \: : (\phi) : \phi ! x \, . \supset \, . \phi ! y \quad & \text{by} \, \ast \! 10 \! \cdot \! 27 \end{align}$

But we shall never be able to substitute $x = y$ for a variable $\phi !x$. Thus e.g. we shall find ourselves in difficulties with the class $\iota `x$, which is defined as $$\hat{y} (y = x).$$ The point is that, however far up the hierarchy of types we have previously gone, $x = y$ goes higher.

If the axiom of reducibility is false, there will be orders of equality, according to the type of $\phi$ in “$\phi ! x \, . \operatorname*{\supset}_{\phi} \, . \phi !y$”. The higher the type of $\phi$, the nearer $x$ and $y$ approach to identity; but we cannot get a Df of identity this way, since obviously where there is identity, we must have $\phi x \equiv \phi y$ whatever the type of $\phi$. Hence if we have no axiom of reducibility, $x = y$ will have to be a primitive idea, and such props as $$x = x, \quad x = y \, . \, \supset \, . \, y = x, \quad x = y \, . \, y = z \, . \, \supset \, . \, x = z$$ will have to be axioms.

(But see p. 4)b

Consider again the Df of finite cardinals, or inductive cardinals, as I call them. We have

$ \text{NC induct} = \hat{\nu} \{ \phi ! O : \phi ! \mu \, . \, \operatorname*{\supset}_{\mu} \, . \phi ! (\mu + 1) : \operatorname*{\supset}_{\phi} \, . \phi ! \nu \} \quad \text{Df}$

Thus whatever the type of $\phi$ may be, the type of “$\nu \; \varepsilon \; \text{NC induct}$” is higher. There will thus be orders of finitude: there will be the numbers which are inductive with respect to 1st-order functions — this will be the largest class that can be called inductive — then those inductive with regard to 2nd-order functions, and so on. Whatever order of functions we take, anything involving “$\text{NC induct}$” is of higher order, and therefore must not be a value of $\phi$ in our Df. You will find, I think, that elementary arithmetic becomes impossible under these circumstances.

Take e.g. $\qquad \mu, \nu \; \varepsilon \; \text{NC induct} \, . \, \supset \, . \, \mu + \nu \; \varepsilon \; \text{NC induct.}$

In order to prove this, it is necessary that $\phi ! (\mu + \alpha)$ should be of the same type with respect to $\alpha$ ($\mu$ being kept fixed) as $\phi ! \alpha$ is. But in fact $\phi ! ( \mu + \alpha )$ is of higher type. Hence $\mu + \nu$ can only be proved to have a lower order of inductiveness than $\mu$ and $\nu$ have. As with identity, so here, it will be necessary to take inductiveness as a primitive idea. The same necessity will recur at every stage in the work, wherever we have, otherwise, a Df involving a function as apparent variable.

Unless I have forgotten the essential point of your theory, you have overlooked the following fact. When an expression contains an apparent function, it is useless to make the function of a very high type, since the expression always goes one better; yet it is constantly necessary to be able to draw inferences as to such expressions from statements about all functions of the order (whatever it is) of the apparent function in question.

The theory of identity could be treated as follows (in your theory):

Pic $\qquad x = y$, meaning “$x$ is identical with $y$”.

Observation. “$x = y$” is a first-order function; hence, since $x = x$, it follows that we shall have “$x = y$” if $y$ has all 1st-order properties which $x$ has, i.e.

$$\vdash \: : \, . \, \phi ! x \, . \operatorname*{\supset}_{\phi} \, . \phi ! y \, : \, \supset \, . \, x = y$$

This may be taken as a Pp.d We require also the Pp

$$\vdash \: : \, . \, x = y \, . \, \supset \: : \phi x \, . \, \supset \, . \, \phi y$$

From these two Pp’s it follows that

$$\vdash \: : \, . \, x = y \, . \, \equiv \: : \phi ! x \, . \operatorname*{\supset}_{\phi} \, . \phi ! y$$

Thence the theory proceeds as before.

The above is set out in your interest. It is, I suppose, possible to conduct mathematics in this way, by having Pi’s in place of Dfs, and having Pp’s which are equivalent to instances of the axiom of reducibility. Or we might simply take

$$x = y \, . \, = \, . \, ( \phi ) \, . \, \phi ! x \, \supset \, \phi ! y \quad \text{Df}$$

as before, and put

$$\vdash \: : \, (\exists \psi ) : x = y \, . \operatorname*{\equiv}_{x, y} \, . \, \psi ! (x, y) \quad \text{Pp}$$

The Pp would then amount to a confession that our Df committed the naturalistic fallacy: it would state that there is a notion “identity”, which, however, we do not find it necessary to introduce explicitly. But I think at some point the difficulties will become intolerable within the axiom of reducibility.

If you can possibly find time, please answer this, and let me know how you dispense with the axiom of reducibility in such cases as $NC \; induct$. My A. J. article4 was given to the editor 14 months ago, and I felt I couldn’t re-write it in proof. There is nothing in it of whose falsehood I feel convinced.

Yrs ever
B. Russell.

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Typeset by A. Duncan; proofread against a photocopy of a photocopy, K. Blackwell 24/01/2019.

• 1

  [document] The letter was edited from

• 2

  Your letter Hawtrey to BR, ?.

• 3

  Cranmer Rd Where the Whiteheads lived in Cambridge.

• 4

  My A. J. article “Mathematical Logic as Based on the Theory of Types”, The American Journal of Mathematics 30 (July 1908): 222–62 (B&R C08.05); 22 in Papers 5.

Textual Notes

• a

  Dfs Definitions.

• b

  (But see p. 4) P. 4 begins “Unless I have forgotten the essential point”.

• c

  Pi Primitive idea.

• d

  Pp Primitive proposition.