PeteOlcott wrote: ↑Mon Jun 22, 2020 12:37 am
Eodnhoj7 wrote: ↑Mon Jun 22, 2020 12:28 am
"By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon is not made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true."

A contradiction can be true:

1. All lemons are yellow = true

2. All lemons are not yellow = true

3. All lemons are thus shades of yellow thus contains elements which are not yellow. A multitude of other colors, as not yellow, occur resulting in an infinite variety.

The first two axioms are defined by classical logic.

The last two axioms are my correction to classical logic.

(1) (All-True(Γ) ∧ False(𝒞)) ↔ ¬Valid-Argument(Γ, 𝒞)

(2) (All-True(Γ) ∧ True(𝒞)) ↔ Sound-Argument(Γ, 𝒞)

(3) (¬All-True(Γ) ∧ True(𝒞)) ↔ ¬Valid-Argument(Γ, 𝒞)

(4) (¬All-True(Γ) ∧ False(𝒞)) ↔ Valid-Argument(Γ, 𝒞)

Under my redefinition of classic logic every argument that has contradictory premises

and a True conclusion is invalid per Axiom(3).

First this does not address the argument above as the premises contradict yet result in a true conclusion.

Second, You are starting with contradictory premises though (classical identity laws), thus negating your stance by it's own criteria:

1. "P" is an assumed variable as a point of view of the observer.

2. (P=P) leads to an infinite regress as ((((P=P)=(Q=Q))=(R=R))=(S=S))=....

3. (P=P) has the same premise as the conclusion thus is circular.

Dually each of the laws is subject to the trilemma:

(P=P) is subject to circularity as P is both the premise and conclusion.

(P=/=-P) is subject to infinite regress as -P equates to (R,S,T,...) as variables which are not P

(Pv-P) is subject to assumed assertions as P and -P are strictly taken without proof.

Dually the laws are contradictory if applied to themselves in a circular self referential manner:

((P=P)v(-P=-P)) necessitates under the law of excluded middle one principle of identity exists or the other thus negating the principle of identity into existing in seperate states of either one identity or the other.

(P=P)v(P=/=-P) necessitates that under the law of excluded middle either the law of identity exists or the law of non contradiction. ****If one is false, then P=-P either way. If (P=P) is false then (P=-P) and (P=/=-P) simultaneously. If (P=/=-P) is false then (P=P) and (P=-P) simultaneously

((P=P)=(-P=-P)) necessitates under the law of identity that two opposing values are equal through the law of identity thus negating the law of non contradiction where P cannot equal not P.

((P=P)=/=(-P=-P)) necessitates under the law of non-contradiction that two principles equal through the law of identity are not equal thus the law of identity is not equal to itself.

((P=P)=(-P=-P)) v ((P=P)=/=(-P=-P)) necessitates either the law of identity or the law of non contradiction results, thus negating either the fallacious use of the law of identity or the fallacious use of the law of non-contradiction but not both. Either the law of identity or the law of non contradiction is negated. If the law of non contradiction is negated then the law of identity ceases to exist as P = -P. If the law of identity is negated then the law of non contradiction is negated as P = -P.