In Whitehead and Russell's PM, is there a typo in ✳72.23?

Question

It seems that where $\gamma$ appears at the end of line 1, 4, 5, there should be a $z$ instead, i.e.

$✳72.23\hspace{10pt} \vdash : R,S \in 1 \rightarrow Cls .\supset. R‘‘S‘‘\gamma=\hat{x}\{(\exists z) . z \in \gamma . x=R‘S‘ z \} $

Dem.

$\hspace{41pt} \vdash . ✳37.33 . \supset \vdash . R‘‘S‘‘\gamma = (R|S)‘‘\gamma \hspace{130pt}(1)$

$\hspace{41pt} \vdash. ✳71.25.4 . \supset \vdash : Hp . \supset . (R|S)‘‘\gamma=\hat{x}\{(\exists z). z \in \gamma . x=(R|S)‘z \} $

$\hspace{41pt}[✳72.2]\hspace{107pt} =\hat{x}\{(\exists z). z \in \gamma . x=R‘S‘z \} \hspace{13pt}(2) $

$\hspace{41pt}\vdash. (1).(2).\supset \vdash . Prop$

The typos are the same on 1st ed Merchant Books, 1910 and 2nd ed Cambridge at the University Press, 1963. Neither listed them on ERRATA.

There are obvious typos at the early part of the book, but I am not sure about myself now as I was half a book ago. If you see what I see, please give me a confirmation.

Thanks,

Answer 1

I think you are right.

The "syntax" of $R‘‘\gamma$ is in *37.01. :

$R‘‘β = \hat x \{ ∃y.y∈β.xRy \}$ Df,

and according to *37.6. $R‘‘β= \hat x \{ ∃y.y∈β.x=R‘y \}$.

In the second line of the proof of *72.23. W&R introduces $(R|S)‘‘\gamma$ and this must be defined in term of $z ∈ \gamma. x =(R|S)‘z$.

See also Florian Cajori, A history of mathematical notations (1928 - Dover reprint), page 310 of II vol :

$R‘y = (\iota x) (xRy)$ Df.

The inverted comma may be read "of": $R‘y$ is read "the $R$ of $y$"

i.e. if $R$ is the father of, then $R‘y$= the father of $y$.

And :

$R‘‘\alpha$ is the class of terms $x$ which have the relation $R$ to some member of a class $\alpha$. Thus, if $R$ is inhabiting and $\alpha$ is towns, then $R‘‘\alpha=$ inhabitants of towns.

Thus the syntax of $R‘y$ "ask for" an individual variable, while that of $R‘‘\alpha$ "ask for" a class variable.