In Whitehead and Russell's PM, are overlapping ranges of significance necessarily identical?

Question

In Principia Mathematica summary of ✳63

In virtue of ✳20.8, we have $\vdash : \phi a ∨ \sim\phi a . ⊃ . \hat{x}(\phi x \vee \sim \phi x ) =t‘a$

i.e. if "$\phi a$" is significant, then the range of significance of the function $\phi \hat{z} $ is the type of $a$. It follows that two ranges of significance which overlap are identical, and two different ranges of significance have no member in common.

Given $T$ is the relation of team mate, then $\overset{→}{T}$ is the relation of team to team member. For example:

Jordan is Pippen's teammate .≡. (Jordan $.T.$ Pippen),

Chicago Bulls is Jordan's team .≡. (Chicago Bulls) $\overset{→}{T} $ (Jordan)

Chicago Bulls is Pippen's team .≡. (Chicago Bulls) $\overset{→}{T}$ ( Pippen)

$\overset{→}{T} = \hat{\alpha}\hat{y} \{ \alpha = \hat{x}(xTy) \}$

Thus the field of $\overset{→}{T}$ is a class whose members include two types: $C‘\overset{→}{T}$ ={Chicago Bulls, Jordan, Pippen, ..., Lakers, Bryant, O'Neal... }

1. let $\phi(x) = x$ is a basketball player;
2. let $ \psi(\alpha)=\alpha$ is a team;
3. let $\chi(t) = t $ is a member of $C‘\overset{→}{T}$

In the case of #1, $\phi(Jordan)$ is true because Jordan is a basketball player; $\phi(Chicago Bulls)$ is meaningless;

In the case of #2. $\psi(Chicago Bulls)$ is true, but $\psi(Jordan)$ meaningless.

In the case of #3. Both Jordan and Chicago Bulls are members of $C‘\overset{→}{T}$, thus both $\chi(Jordan)$ and $\chi(ChicagoBulls)$ are true and significant.

Both $\phi{\hat{x}}$'s type and $\psi \hat{\alpha}$'s type overlap with $\chi(\hat{t})$'s type, but none are identical with any other. Thus it appears that two ranges of significance which overlap are not necessarily identical. So, what is wrong with my reasoning?

I can't emphasize enough of this: At this stage I have no desire to find faults in PM. All I want is to understand PM to the best of my ability.

Answer 1

The statement of $*20.8$ is preceded [see Alfred North Whitehead & Bertrand Russell, Principia Mathematica to 56 (2nd ed - 1927), page 198] by a comment that relates the proposition to the theory of relative types.

In the same paragraph we have that :

The "type" of any object $x$ will be defined in $*63$ as the class of terms either identical with $x$ or not identical with $x$. We may define the "type of the arguments to $\phi x$" as the class of arguments $x$ for which "$\phi \hat{z}$" is significant, i.e. the class $\hat{x}(\phi x \lor \sim \phi x$).

Simplifying a lot, we must think at the "universe" as made of "levels" (i.e.types) : the "lowest" is that containing the individuals; the next one is the level of classes of individuals, and so on.

The "range of significance" of a formula $\phi x$ (with $x$ free) must be a type.

Following your example, if $t_0$ is the type of basketball players, and if we assume that a team is a class of basketball players, the type of teams will be $t_1$.

Thus, regarding your example :

given $T$ is the relation of team mate, then $\overset{→}{T}$ is the relation of team to team member

I think that we have some "restriction" about the ensuing definition :

$\overset{→}{T} = \hat{\alpha}\hat{y} \{ \alpha = \hat{x}(xTy) \}$,

because I think that we cannot have

"the class whose members include two types".

According to $*32.01$, the $T$-predecessors of $y$ : $\overset{→}{T}‘y = \hat{x}(xTy)$ is, in modern notation : $\{x : xTy \}$.

If we put $\alpha = \overset{→}{T}‘y$, this is a class; if we assume $y := Jordan$, this is the class of teammates of Jordan, i.e. the team of Chicago Bulls.

Being a class, it is not of the same type of Jordan; thus we cannot "predicate" of it the same "predicates" that applies to Jordan.

The relation of "membership" (in modern notation $\in$) is not in PM a relation between individuals (or objects of the same type).

It must be defined according to the "criteria" of significance; i.e. $x \in \hat{z} \phi z$ (that I fear is not well-formed in PM language) iff $\phi x$.

All the crux of the matter with the "horrible" PM's theory of types is that you cannot have $\hat{z} \phi z$ at the "same level" of $x$ : if $x$ is of type $t_i$, then every class to which $x$ belongs must be of type at least $t_{i+1}$.

I think taht the mistake is in the "use" you have done of $\overset{→}{T}$; in the relation $T$, both arguments ($x$ and $y$) are players :

$Jordan .T. Pippen$.

When using $\overset{→}{T}$, we have :

$Chicago Bulls = \hat{x}(xT Jordan)$

i.e.

$Chicago Bulls = \overset{→}{T}‘ Jordan$;

but we cannot have :

$Chicago Bulls .\overset{→}{T}. Jordan$,

because the two "relata" are not of the same type [see Alfred North Whitehead & Bertrand Russell, Principia Mathematica to 56 (2nd ed - 1927), page 248].