# Does the Law Of Excluded Middle Apply to the Principle Of Identity and Non Contradiction? [closed]

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).

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.

(C=C)=/(-C=-C)

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:

(P=P)

((P=P)=(P=P))

....

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:

Is

Therefore

Because

If then

Tends to

Therefore/because

Necessitates

Implies

Infers

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.

(P)=(P)

(=)P(=)

Therefore

((P)P)

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)

• I do not see any "fallacy" in the above sentences... and I do not see any special meaning also. – Mauro ALLEGRANZA Aug 27 '19 at 18:36
• I am not clear about what you are asking. – Mark Andrews Aug 27 '19 at 19:16
• @Conifold Perhaps that's what Eodnhoj7 is asking? – Geremia Aug 27 '19 at 21:02
• If you get unclear answers or comments telling you that what you write hardly makes sense, then you should ask yourself if the post really is as clear as you might think. It is your responsibility to make yourself understood beyond equivocation. – Philip Klöcking Aug 28 '19 at 21:41
• Michael's answer does a decent job in telling you why your conclusions are based on a wrong usage of signs, not insight. You can reject that and reiterate the same problematic argument time and again, it does not change anything. – Philip Klöcking Aug 29 '19 at 6:09

The Principle of Identity is the Principle of Identity, or The Principle of Non Contradiction is the Principle of Non Contradiction.

It is not right to say, of the propositions making up this disjunction, that "the law of excluded middle requires one to be negated".

I'm not exactly sure what is causing the confusion, but it seems to me that you think the law of excluded middle is the following:

In a proposition of the form "p or q" exactly one of the disjuncts is true.

That is not a logical law. Maybe you are confusing it with the similar, but (classically) true:

In a proposition of the form "p or not-p" exactly one of the disjuncts is true.

So for

The Principle of Identity is the Principle of Identity, or The Principle of Identity is not the Principle of Identity.

it is right to say that one disjunct must be false.

Edit:
Another guess at what is causing the confusion: You might think that the equivalence of a=a and b=b means that a=b. This might seem plausible if one confuses identity with equivalence, using the same sign for both: a=a=b=b. But the middle «=» should be a propositional connective. So the relevant form is really «a=a if and only if b=b». Cats are cats if and only if dogs are dogs. It does not follow that cats are dogs. Nothing changes if «a» and «b» stand for propositions instead of animals.

Second edit:
Let’s agree that «The law of identity is a proposition». It is the proposition that, for all individuals a, a is a. Let’s signify the proposition with «LI». We can also agree that the law of noncontradiction is a proposition. It is the proposition that, for all propositions p, it is not the case that both p and not-p. Let’s give it the propositional sign «NC».

You say that both propositions are subject to the law of identity. I can also grant that, but remember: The letters that flank the equality sign are names of individuals, so if we plug our propositional sign «LI» into the position of such a name, we are mentioning the proposition. LI=LI can be rendered intuitively as

[The proposition which says that, for all individuals a, a is a] is [the proposition which says that for all individuals a, a is a].

(The brackets are only there to parse the texts for ease of reading.)

And NC=NC:

[The proposition which says that, for all propositions p, it is not the case that both p and not-p] is [the proposition which says that, for all propositions p, it is not the case that both p and not-p].

Since both of these resulting propositions are trivially true, they are equivalent. We can express that equivalence like this: «LI=LI if and only if NC=NC», or like this «(LI=LI)<->(NC=NC)». But remember the difference between «=» and «<->». In particular, note that «(LI=LI)<->(NC=NC)» does not say the same thing as (LI=LI)=(NC=NC). Moreover, the equivalence does not justify the equality, and neither does the law of identity (LI). The proposition (LI=LI)=(NC=NC) is of the form a=b, not a=a. This is so even in the case where «a» and «b» stand for equivalent propositions.

It might be difficult to grasp the difference between the occurrence of a proposition as an individual and its occurrence as a proposition. It might help to see that the propositional connectives do not apply to individuals, even if that individual happens to be a proposition. So while «a=a» is grammatical «a=it is not the case that a» is not, because «it is not the case that …» cannot operate on «a». The negation operator does not operate on the positions that flank the equality sign. This is easy to see when "a" stands for an animal:

[A cat] is [it is not the case that a cat].

But the point holds equally well when «a» stands for a proposition:

[The proposition that cats are cats] is [it is not the case that the proposition that cats are cats].

I hope you see that «it is not the case that the proposition that cats are cats» is ungrammatical, and thus that "p=p" only mentions, and does not assert "p".

• The principle of identity is grounded in equality. The principle of non contradiction inequality. Both symbols are the opposition of the other and are undefined except through P and -P which are in opposition. One propostion, equality and the other non equality, observes one as the negation of the other, thus excluded middle necessitates one proposition as true and the other false...but this negates one of the laws in doing so. – Eodnhoj7 Aug 27 '19 at 21:16
• The principle of noncontradiction says of a proposition and its negation that both cannot be true. The principle of identity says of every individual that it is identical to itself. How is one the negation of the other? – Michael Amundsen Aug 27 '19 at 21:26
• Edited question to make it clearer. One negates the other as P must exist through P=P and -P must exist through -P=-P, but one identity cannot equal the other yet they require the same law to define them as having an identity. One identity is true and the other is false as equality and inequality negate eachother, one law of identity is not equal to another law of identity. – Eodnhoj7 Aug 27 '19 at 21:36
• I edited my answer to add something that might address what you are saying here. – Michael Amundsen Aug 27 '19 at 22:27
• – Michael Amundsen Aug 28 '19 at 16:26