How to explain ✳43.3 and ✳43.31 in Whitehead and Russell's PM?

Question

Take ✳43.3 for example, I presume

$ P = R |Q $ where R is fixed.

$ R| $ is the relation between $R|Q$ and $Q$, ie. $ R| = \hat{P} \hat{Q} \{ P = R|Q \} $

$Ɑ‘R|= \hat{Q}\{ E! R|‘Q \}$

Given that $R$ is "father to daughter," then $ Ɑ‘R| $ is the class of relations whose referents are women, e.g. {"mother to son," "sister to brother," "daughter to father," "aunt to nephew," ... }

When $Q$ is "mother to son," $P$ is "grand father to grandson." Therefore P does not belong to $Ɑ‘R|$, which contradicts ✳43.3. Please point out what is wrong with my reasoning.

There is no doubt regarding the converse domain and field of $R|$

$Ɑ‘R|= \hat{Q}\{(∃T) T=R|Q \} $

$C‘R|=\hat{S}\{(∃T) S=R|T .∨. T=R|S\}$

So exactly what $P$ stands for in each of these numbers?

Answer 1

I think that we must "read" $R|$ as an "operation" which takes as "input" a relation $S$ and produces as "output" their composition : $R|S$.

If we use a "dummy" symbol $Comp_R(x)$ defining a mapping form "the set of all relations" into itself, we have that :

$Comp_R(S) = R|S$.

The basic properties stated into *43 are quite "obvious". Consider :

*43.11 : $\vdash R|‘Q = R|Q$.

If we remember that $R‘y=(\iota x)xRy$, i.e.$R‘$ is a function which, from input $y$ produces as output the unique $x$ such that $xRy$, the proposition says that the "operator" $R|‘$ applied to the relation $Q$ give us as result the "composition" $R|Q$ (as expected).

If so, we can read

*43.3 : $\vdash (P). P \in \mathbb D ‘R|$ (I've used $\mathbb D$ in place of the "inverted-D", wich I'm not able to type)

as stating that we can "compose" $R$ with every relation $P$ (i.e. the "operator" $R|$ can be applied to every relation $P$ to produce $R|P$).

The same consideration applies to *43.301 and .302.

Answer 2

One way to avoid contradiction is to assume $P$ stands for different things in different numbers. The summary for ✳43 used $ P $ for both referent and relatum of $R|$.

Assume $P$ in ✳43.3 stands for one of the relata of $(R|)$, thus $P$ is any relation that can be argument to $R|‘Q$. Then ✳43.3 states that all relations either belong to the converse domain of $R|$ or are incapable of being argument to $R|‘Q$. In other words, the converse domain of $R|$ encompasses the entire type defined by the descriptive function $(R|‘Q)$ where |R| is fixed and $Q$ is the argument.

In ✳43.31 $P$ stands for a relation of relations. $P$'s converse domain overlaps with the converse domain of $(R|)$. Since whenever two types overlap they also coincide, the type of $P$'s relata is the same as the type defined by the descriptive function $(R|‘Q)$ where $R$ is fixed and $Q$ is the argument. As 43.3 shows, the converse domain of $R|$ encompasses the entire type defined by the descriptive function $(R|‘Q)$, it follows that the converse domain of P belongs to $(Ɑ‘R|)$