## Simply defining Gödel Incompleteness away V8

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PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Simply defining Gödel Incompleteness away V8

My correction to classical logic:
A valid argument requires that the truth of the conclusion logically
follows from the truth of all of the premises.

If the truth of the conclusion is the same as the truth of the premises
joined together by "and" then the argument is valid otherwise the
argument is invalid.

Mendelson 1997 Introduction to Mathematical Logic page 35: Γ ⊢ 𝒞
Γ = the premises of the proof
𝒞 = the consequence of Γ

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)

This gets rid of the principle-of-explosion.

Only the conclusion of a sound argument counts as true making true and
unprovable impossible refuting Gödel's 1931 Incompleteness Theorem.

Last edited by PeteOlcott on Sun Jun 21, 2020 11:28 pm, edited 1 time in total.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

PeteOlcott wrote: Sun Jun 21, 2020 10:15 pm Mendelson 1997 Introduction to Mathematical Logic page 35: Γ ⊢ 𝒞
Γ = the premises of the proof
𝒞 = the consequence of Γ

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)

Validity is defined in classical logic as follows:
An argument (consisting of premises and a conclusion) is valid if and only if there is
no possible situation in which all the premises are true and the conclusion is false...

All premises and conclusions truth values change as context expands or contracts. All premises and conclusions are thus potentially true and false.

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

Here is what I am adding to that:
and there is no possible situation in which any of the premises are false and the conclusion is true.
(¬All-True(Γ) ∧ True(𝒞)) ↔ ¬Valid-Argument(Γ, 𝒞)
(¬All-True(Γ) ∧ False(𝒞)) ↔ Valid-Argument(Γ, 𝒞)

This gets rid of the principle-of-explosion.

The premises are an assumed starting point, to argue a premise has truth value is to assume it as such. A conclusion does not justify a premise as a true statement without going through a circularity.

The principle of explosion is still valid. The negation of negation results in all possible phenomenon of that negation. The not not cat is cat, the not not all is all.

We keep the convention definition of a sound argument:
A deductive argument is sound if and only if it is both valid, and all of its premises
are actually true. Otherwise, a deductive argument is unsound.
https://www.iep.utm.edu/val-snd/#:~:tex ... ot%20sound.

Only the conclusion of a sound argument counts as true making true and unprovable
impossible refuting Gödel's 1931 Incompleteness Theorem.

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Sun Jun 21, 2020 11:21 pm
PeteOlcott wrote: Sun Jun 21, 2020 10:15 pm Mendelson 1997 Introduction to Mathematical Logic page 35: Γ ⊢ 𝒞
Γ = the premises of the proof
𝒞 = the consequence of Γ

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)

Validity is defined in classical logic as follows:
An argument (consisting of premises and a conclusion) is valid if and only if there is
no possible situation in which all the premises are true and the conclusion is false...

All premises and conclusions truth values change as context expands or contracts. All premises and conclusions are thus potentially true and false.

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

Here is what I am adding to that:
and there is no possible situation in which any of the premises are false and the conclusion is true.
(¬All-True(Γ) ∧ True(𝒞)) ↔ ¬Valid-Argument(Γ, 𝒞)
(¬All-True(Γ) ∧ False(𝒞)) ↔ Valid-Argument(Γ, 𝒞)

This gets rid of the principle-of-explosion.

The premises are an assumed starting point, to argue a premise has truth value is to assume it as such. A conclusion does not justify a premise as a true statement without going through a circularity.

The principle of explosion is still valid. The negation of negation results in all possible phenomenon of that negation. The not not cat is cat, the not not all is all.

We keep the convention definition of a sound argument:
A deductive argument is sound if and only if it is both valid, and all of its premises
are actually true. Otherwise, a deductive argument is unsound.
https://www.iep.utm.edu/val-snd/#:~:tex ... ot%20sound.

Only the conclusion of a sound argument counts as true making true and unprovable
impossible refuting Gödel's 1931 Incompleteness Theorem.

By redefining a valid argument as requiring that the truth of the conclusion is derived from
the truth of all of the premises the principle of explosion ceases to exist.

Every argument where the truth value of the premises joined together by "and" is not
the same as the truth value of the conclusion is redefined to be an invalid argument.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

PeteOlcott wrote: Sun Jun 21, 2020 11:51 pm
Eodnhoj7 wrote: Sun Jun 21, 2020 11:21 pm
PeteOlcott wrote: Sun Jun 21, 2020 10:15 pm Mendelson 1997 Introduction to Mathematical Logic page 35: Γ ⊢ 𝒞
Γ = the premises of the proof
𝒞 = the consequence of Γ

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)

Validity is defined in classical logic as follows:
An argument (consisting of premises and a conclusion) is valid if and only if there is
no possible situation in which all the premises are true and the conclusion is false...

All premises and conclusions truth values change as context expands or contracts. All premises and conclusions are thus potentially true and false.

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

Here is what I am adding to that:
and there is no possible situation in which any of the premises are false and the conclusion is true.
(¬All-True(Γ) ∧ True(𝒞)) ↔ ¬Valid-Argument(Γ, 𝒞)
(¬All-True(Γ) ∧ False(𝒞)) ↔ Valid-Argument(Γ, 𝒞)

This gets rid of the principle-of-explosion.

The premises are an assumed starting point, to argue a premise has truth value is to assume it as such. A conclusion does not justify a premise as a true statement without going through a circularity.

The principle of explosion is still valid. The negation of negation results in all possible phenomenon of that negation. The not not cat is cat, the not not all is all.

We keep the convention definition of a sound argument:
A deductive argument is sound if and only if it is both valid, and all of its premises
are actually true. Otherwise, a deductive argument is unsound.
https://www.iep.utm.edu/val-snd/#:~:tex ... ot%20sound.

Only the conclusion of a sound argument counts as true making true and unprovable
impossible refuting Gödel's 1931 Incompleteness Theorem.

By redefining a valid argument as requiring that the truth of the conclusion is derived from
the truth of all of the premises the principle of explosion ceases to exist.

False.

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.

Two simultaneously different yet true statements can occur simultaneously, resulting in an infinite number of conclusions all of which are valid. Two contradictory true premises are observed and an true conclusion results.

Every argument where the truth value of the premises joined together by "and" is not
the same as the truth value of the conclusion is redefined to be an invalid argument.

Give an example.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Sun Jun 21, 2020 11:58 pm
PeteOlcott wrote: Sun Jun 21, 2020 11:51 pm
Eodnhoj7 wrote: Sun Jun 21, 2020 11:21 pm
By redefining a valid argument as requiring that the truth of the conclusion is derived from
the truth of all of the premises the principle of explosion ceases to exist.

False.

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.

Two simultaneously different yet true statements can occur simultaneously, resulting in an infinite number of conclusions all of which are valid. Two contradictory true premises are observed and an true conclusion results.

Every argument where the truth value of the premises joined together by "and" is not
the same as the truth value of the conclusion is redefined to be an invalid argument.

Give an example.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

PeteOlcott wrote: Mon Jun 22, 2020 12:21 am
Eodnhoj7 wrote: Sun Jun 21, 2020 11:58 pm
PeteOlcott wrote: Sun Jun 21, 2020 11:51 pm

By redefining a valid argument as requiring that the truth of the conclusion is derived from
the truth of all of the premises the principle of explosion ceases to exist.

False.

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.

Two simultaneously different yet true statements can occur simultaneously, resulting in an infinite number of conclusions all of which are valid. Two contradictory true premises are observed and an true conclusion results.

Every argument where the truth value of the premises joined together by "and" is not
the same as the truth value of the conclusion is redefined to be an invalid argument.

Give an example.
"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."

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.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Mon Jun 22, 2020 12:28 am
PeteOlcott wrote: Mon Jun 22, 2020 12:21 am
Eodnhoj7 wrote: Sun Jun 21, 2020 11:58 pm
"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."

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).

Furthermore every argument besides Axiom(2) is unsound, thus untrue.
This makes Gödel's true and unprovable impossible.
Last edited by PeteOlcott on Mon Jun 22, 2020 12:41 am, edited 1 time in total.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

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."

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.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Mon Jun 22, 2020 12:40 am
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."

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:
I add the rule that contradictory premises by themselves make the whole argument invalid.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

PeteOlcott wrote: Mon Jun 22, 2020 12:54 am
Eodnhoj7 wrote: Mon Jun 22, 2020 12:40 am
PeteOlcott wrote: Mon Jun 22, 2020 12:37 am

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:
I add the rule that contradictory premises by themselves make the whole argument invalid.
Not if both have truth values and/or result in a truth value.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Mon Jun 22, 2020 12:55 am
PeteOlcott wrote: Mon Jun 22, 2020 12:54 am
Eodnhoj7 wrote: Mon Jun 22, 2020 12:40 am

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:
I add the rule that contradictory premises by themselves make the whole argument invalid.
Not if both have truth values and/or result in a truth value.
Yes I stipulate it that way. Any argument with contradictory premises is stipulated to be both invalid and unsound.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

PeteOlcott wrote: Mon Jun 22, 2020 12:59 am
Eodnhoj7 wrote: Mon Jun 22, 2020 12:55 am
PeteOlcott wrote: Mon Jun 22, 2020 12:54 am

I add the rule that contradictory premises by themselves make the whole argument invalid.
Not if both have truth values and/or result in a truth value.
Yes I stipulate it that way. Any argument with contradictory premises is stipulated to be both invalid and unsound.
Yet the lemon argument is true. Second the identity laws, which your argument is grounded in, are contradictory.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Mon Jun 22, 2020 1:00 am
PeteOlcott wrote: Mon Jun 22, 2020 12:59 am
Eodnhoj7 wrote: Mon Jun 22, 2020 12:55 am

Not if both have truth values and/or result in a truth value.
Yes I stipulate it that way. Any argument with contradictory premises is stipulated to be both invalid and unsound.
Yet the lemon argument is true.
Cats are goats and goats are dump trucks therefore water is wet.
Has a true conclusion that does not logically follow for its premises
therefore it is both invalid and unsound.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away V8

PeteOlcott wrote: Mon Jun 22, 2020 1:02 am
Eodnhoj7 wrote: Mon Jun 22, 2020 1:00 am
PeteOlcott wrote: Mon Jun 22, 2020 12:59 am

Yes I stipulate it that way. Any argument with contradictory premises is stipulated to be both invalid and unsound.
Yet the lemon argument is true.
Cats are goats and goats are dump trucks therefore water is wet.
Has a true conclusion that does not logically follow for its premises
therefore it is both invalid and unsound.
"Cats" are "goats" through the context of "mammal".
"Goats" are "dump trucks" through the context of "material carrier".
"Water" is "wet" through the context of "fluidity".

All premises equate through a middle context, thus the premises can equate to the conclusion in light of form. The conclusion as equitable through a middle context stems from the same form and function of the premises.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away V8

Eodnhoj7 wrote: Mon Jun 22, 2020 1:10 am
PeteOlcott wrote: Mon Jun 22, 2020 1:02 am
Eodnhoj7 wrote: Mon Jun 22, 2020 1:00 am

Yet the lemon argument is true.
Cats are goats and goats are dump trucks therefore water is wet.
Has a true conclusion that does not logically follow for its premises
therefore it is both invalid and unsound.
"Goats" are "dump trucks" through the context of "material carrier".

All premises equate through a middle context, thus the premises can equate to the conclusion in light of form. The conclusion as equitable through a middle context stems from the same form and function of the premises.
Goats are never dump trucks unless they share an identical set of properties which they do not.
Last edited by PeteOlcott on Mon Jun 22, 2020 1:21 am, edited 1 time in total.