In Whitehead and Russell's PM, does not identity imply existance?

Question

At the end of ✳96.48, $ \sim(w=\overset{\smile}{R}‘max_R‘J_R‘x)$ is chosen over $ w\neq\overset{\smile}{R}‘max_R‘J_R‘x$, on account of the latter's implication of existence. But ✳13.02 states that they are the same.

✳13.02 $ x\neq y .=.\sim(x=y) $ Df

I wonder what I have missed. Where does PM say identity has anything to do with existence? Thanks,

Answer 1

After a sleepless week I've decide to ask to Prof.Gregory Landini (Iowa University), my preferred "interpreter" of Russell's logic.

Here is his kind answer; I'll give it verbatim :

This is a nice passage that illustrates something I've been arguing for years -- viz., definintions formed with individual variables cannot be applied to definite descriptions (because definite descriptions are not genuine terms). I didn't know it and find is wholly enjoyable that you found it and brought it to my attention.

According to ✳30.01 $R‘y = (\iota x)(xRy)$ Df; thus, in our formula, "$\overset{\smile}{R}‘ ...$" is a definite description of a class and not a class term. Thus there is an issue of scope of the description.

In order to apply the definition ✳13.02 [i.e. $x≠y.=.∼(x=y)$ Df] one must first eliminate the definite description. Note as well that when scope marker are dropped, smallest (most secondary scope possible) is intended.

The general form of the case at hand is this:

$w \ne \iota aFa =_{df} [\iota aFa]( w \ne \iota aFa)$.

This is the smallest scope possible -- because definition ✳13.02 is framed with individual variables and thus cannot apply until after the definite description has been eliminated.

In contrast,

$\lnot(w = \iota aFa) =_{df} \lnot [\iota aFa](w = \iota aFa)$

The two scopes are not equivalent.

Added

This is explained by the "paradigmatic" case regarding "the King of France" and the management of scope in definite descriptions.

See Principia page 70 :

$\quad \quad \quad \quad \quad [(\iota x) (\phi x)]. \lnot \psi(\iota x)(\phi x)$

will mean $\quad \quad (\exists c): \phi x . \equiv_x. x = c: \lnot \psi c$,

while $\quad \quad \lnot \{ [(\iota x) (\phi x)]. \psi(\iota x)(\phi x) \}$

will mean $\quad \quad \lnot \{ (\exists c): \phi x . \equiv_x. x = c: \psi c \}$.

Here again, when $(\iota x) (\phi x)$ does not exist, the first is false and the second true.

Thus, in ✳96.48, $\lnot(w=\overset{\smile}{R}‘ ...)$ corresponds to the second case, in which $\overset{\smile}{R}‘...$ does not exists.

Answer 2

W&R's comment above is not clear to me.

According to logical rules, an abbreviation is only a "symbol": it cannot alter the theorems provable in the system.

Thus, if $x \ne y$ is (as usual) an abbreviation for $\lnot (x=y)$, we have no possibility of having different "sets of consequences".

I think that the comment is related to ✳13.19 : $\vdash ∃y(y=x)$, which is a correct logic rule, and to the "missing" semantics of PM.

In "modern" first-order logic, the law : $\vdash ∃y(y=x)$ is (universally) valid because we assume that every universe of discourse (i.e.every domain of interpretation) is not empty. Thus, in every interpretation, we have at least one object, which is for sure "equal to itself".

The above law is usually derived from the identity axiom : $x=x$ through the $\exists$-introduction rule :

from $\varphi(t)$, infer $\exists x \varphi(x)$.

If we apply the rule to $y \ne x$ (which is exactly : $\lnot (y=x)$), we can infer $∃y (y \ne x)$, but now we do not obtain a logical law.

The formula $∃y (y \ne x)$ is not (universally) valid because it is false in every domain with at least two objects, exactly like $y \ne x$.

I think that in ✳96.48 the free variable $w$ into $\sim(w=\overset{\smile}{R}‘max_R‘J_R‘x)$ must be read as implicitly universally quantified.

If so, the sub-formula is equivalent to $∀w \sim(w=\overset{\smile}{R}‘max_R‘J_R‘x)$ i.e.to $\sim \exists w (w=\overset{\smile}{R}‘max_R‘J_R‘x)$.

A possible reading of the annotation can be this: W&R want to avoid the misconception related to the above rule.

If we apply it to the formula $w\neq\overset{\smile}{R}‘max_R‘J_R‘x$ we can infer $\exists w (w\neq\overset{\smile}{R}‘max_R‘J_R‘x)$, which is not $\sim \exists w (w=\overset{\smile}{R}‘max_R‘J_R‘x)$.