This argument will seem confusing, precisely because it observes the laws of identity being subject to equivocation. If this is kept in mind, the following should make more sense and explain why the argument leads to the following question.
The following Proposition states:
"The Principle of Identity is the Principle of Identity, or The Principle of Non Contradiction is the Principle of Non Contradiction."
The law of identity is a proposition as it expressive both a judgement and assertion of what constitutes identity.
So is the law of non contradiction.
As propositions they are subject to the law of identity respectively.
They are not the same propositions, but both are subject to the laws of identity.
Thus the law of excluded middle requires one to be negated, thus a law being negated.
However, if one is negated then one of the indentity laws is negated.
This leads to excluded middle to be negated as well leaving only one of the laws as true.
Both P and -P are subject to the law of identity as "P=P" and "-P=-P", but one identity is false when excluded middle is applied.
The principle of identity (P), expressed as (P=P), and non contradiction (-P), expressed as (-P=-P) through the law of identity requires P=-P through (P=P)=(-P=-P).
However, non contradiction states P=/-P.
The dualism between identity and noncontradiction is premised on equality and non equality where one law it antithetical to the other.
On one hand P=P necessitates an identity through connection,on the other hand P=/-P offers a negation of identity through an absence of connection.
This connector is equivocation.
Does this mean sometimes the law of identity may be possibly untrue considering "-P", as the law of non contradiction, is subject to it?
The "equality" of identity and the "inequality" of non contradiction are respectively thetical and antithetical, as one statement cancels the other out:
P=P but P cannot equal -P, however -P=-P is necessary if -P is to have an identity.
...thus one principle of identity (P=P) does not equal another (-P=-P).
What are the results of this statement considering both laws exist according to the principle of identity but both are not eachother respectively and only one can be chosen?
To make this simple this proposition simpler the following example can be applied:
"The cat is alive and the cat is dead" is a contradiction.
However if the context is changed:
"The cat is alive and the cat is dead in my memory of January" then the identity of the cat can contradict itself given a change of context.
((C=C)=(-C=-C)) = (M=J)
The law of non contradiction is over ruled, as well as excluded middle by an inherent middle proposition of (M=J)
This necessitates the law of identity as P(x)=-P(x) where x=x but x is undefined considering the law of identity is undefined.
The law of identity is undefined because excluded middle and non contraction are negated...thereby leaving the law of identity as non existent as "=" cannot be defined.
Therefore P(x)=-P(x) cannot exist except as
((P(x))(-P(x))) where ( ) observes all axioms as fundamentally contexts or "sets" of variables.
This necessitates that only P=P as an aristotelian law is left, but that means the law exists only if it relates to itself:
Where P exists as it's own context (P)=(P) considering "=" is undefined except through P.
An example of this can be observed where "equals" as "=" can be replaced with the following meanings with corresponding signs if iPad permits:
Etc. With the "=" sign being represented through a variety of symbolic expressions (cannot copy an paste on ipad).
Thus "=" is subject to equivocation and can be replaced with the variable: P as P(P)P
Thus (=)P(=) where = is P allowing "=" to have an identity.
Thus one identity is a context that is inherently void until another context occurs.
This necessitates, whether I want it or not, that the Principle of identity P=P is void and must be replaced with not a principle (as principles are void) but an assumption:
The Assumption of Inherent Void: (P)