Inconsistent Theories Metatheoretically Prove Trivialism

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paulemok
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Inconsistent Theories Metatheoretically Prove Trivialism

Post by paulemok »

Definition of Inconsistent Theory. An inconsistent theory is a theory in which some contradiction exists.

Let T be an inconsistent theory and let p be a proposition. Since T is inconsistent, by definition of inconsistent theory, some contradiction exists in T. So, by ex contradictione quodlibet, the following two propositions are true in T.

1. p
2. It is not true that p.

So, both in T and out of T, the following two propositions are true.

3. p is true in T.
4. It is not true that "p is true in T."

Since (4) is the negation of (3), some contradiction exists both in and out of T. Thus, by ex contradictione quodlibet, trivialism is true in and out of T. Therefore, trivialism is true.

Paul E. Mokrzecki
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Namelesss »

paulemok wrote: April 21st, 2018, 10:07 pm Definition of Inconsistent Theory. An inconsistent theory is a theory in which some contradiction exists.

Let T be an inconsistent theory and let p be a proposition. Since T is inconsistent, by definition of inconsistent theory, some contradiction exists in T. So, by ex contradictione quodlibet, the following two propositions are true in T.

1. p
2. It is not true that p.

So, both in T and out of T, the following two propositions are true.

3. p is true in T.
4. It is not true that "p is true in T."

Since (4) is the negation of (3), some contradiction exists both in and out of T. Thus, by ex contradictione quodlibet, trivialism is true in and out of T. Therefore, trivialism is true.

Paul E. Mokrzecki
Which is why formal logic has nothing to do with Reality!
Like the magician waving his hands, telling us that there's nothing up his sleeve...
It proves(?) that false is true, and vice versa.
Mental masturbation, at best.
A theory with inconsistencies and contradictions is a failed theory. Real simple. History demonstrates this!
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Eduk »

Trivialism seems well named.
Unknown means unknown.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Mosesquine »

It seems that the inconsistent theory T is wrong. I mean, it is not the case that trivialism is proved to be right, but merely the inconsistent theory T is wrong.
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The Beast
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by The Beast »

In his Ethics and of the mind, Spinoza said:
A1: “The essence of man does not involve necessary existence, that is, from the order of Nature it can happen equally that this or that man does exist or that he does not exist.”
At the Boolean level virtualism will be hard to prove. However, if all propositions are true, it only means this: a proposition with fiction and the intangible consistency of the unknown... Or that you can only know the half of it and choose virtualism. It is in the imaginary logic that we find the plurivalent truth tables. Is there a mind independent truth? Reality arises from the breaking of symmetry and who is to say that there is only one truth or no truth or all that exist have a little piece of it a little not. The composition of truth evolves as there is no constant. So there is a mind independent truth as not truth, maybe a plurivalent truth table that evolves to truth as proven by the breaking of symmetry.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by paulemok »

I was considering the epistemic possibility, with respect to all I know, that no inconsistent theories exist. However, there's strong evidence that some inconsistent theories do exist.

Example 1. A theory E has the following two postulates.

Postulate 1. Today is Monday.
Postulate 2. It is not true that today is Monday.

Since Postulate 2 is the negation of Postulate 1, some contradiction exists in E. Therefore, by definition of inconsistent theory, E is inconsistent. Nonetheless, it's clear E exists. This concludes the example.


I was also considering some definitions of theory that involve a theory being a set of propositions. I'm skeptical of such definitions of theory because it does not seem that a theory is merely a mathematical set. I feel that when a theory is defined to be a set of propositions, some of the features that a theory naturally has are stripped from the concept of theory. An inconsistent theory would then be an inconsistent set. And the concept of an inconsistent set may not make much sense.

Example 2. For this example, a theory is defined as the union of a nonempty set of postulates and the set of all theorems of the set of postulates. D is an inconsistent theory. The following two propositions are elements of D.

5. p
6. It is not true that p.

Two traditional claims against "the existence of some contradiction in a metatheory of D" may then be that "it is not true that '(5) is not an element of D'" and "it is not true that '(6) is not an element of D.'" However, since (5) is an element of D, (6) is not an element of D. And since (6) is an element of D, (5) is not an element of D. So, in a metatheory of D, the following two propositions are true.

7. (5) is an element of D and (5) is not an element of D.
8. (6) is an element of D and (6) is not an element of D.

Since (7) and (8) are both contradictions, some contradiction exists. Therefore, by ex contradictione quodlibet, trivialism is true. This concludes the example.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by JamesOfSeattle »

My study of formal logic is limited to high school level, but I feel pretty comfortable saying that the problem with the OP is that an inconsistent theory tells you nothing about truth. Thus, these statements:
So, both in T and out of T, the following two propositions are true.

3. p is true in T.
4. It is not true that "p is true in T."
are completely wrong. It is true that “p” is in T, but “p” is not true in T. Nothing is true in T, because T is inconsistent.

*
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paulemok
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by paulemok »

JamesOfSeattle wrote: April 24th, 2018, 7:41 pmNothing is true in T, because T is inconsistent.
That is both true and false. Since T is inconsistent, by definition of inconsistent theory, some contradiction exists in T. So, by ex contradictione quodlibet, every proposition is true in T. Also, since every proposition is true in T, the proposition "no proposition is true in T" is true in T. So, by simplification, no proposition is true in T.

Since the claim that I quoted at the beginning of this post is true and false, some contradiction exists. So, by ex contradictione quodlibet, the proposition "trivialism is true" is true. Therefore, by simplification, trivialism is true.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Namelesss »

paulemok wrote: April 24th, 2018, 2:02 pm I was considering the epistemic possibility, with respect to all I know, that no inconsistent theories exist. However, there's strong evidence that some inconsistent theories do exist.
"Every kind of partial and transitory disequilibrium must perforce contribute towards the great equilibrium of the whole.." - Rene' Guenon

That could as well translate that '"Every kind of partial and transitory disequilibrium/inconsistency must perforce contribute towards the great equilibrium/consistency (Balance) of the whole.."
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Namelesss »

Ultimately, there is no difference between the 'consistent' and the 'inconsistent'!
They refer to One Reality, they merely evidence mutually arising opposite Perspectives.
All 'meaning' (consistent/incon...) exist in the thought/eye of the beholder, not in perceived 'objects'.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Mosesquine »

paulemok wrote: April 24th, 2018, 2:02 pm I was considering the epistemic possibility, with respect to all I know, that no inconsistent theories exist. However, there's strong evidence that some inconsistent theories do exist.

Example 1. A theory E has the following two postulates.

Postulate 1. Today is Monday.
Postulate 2. It is not true that today is Monday.

Since Postulate 2 is the negation of Postulate 1, some contradiction exists in E. Therefore, by definition of inconsistent theory, E is inconsistent. Nonetheless, it's clear E exists. This concludes the example.


The sentence "some contradiction exists in a theory" means the same as the sentence "the theory is wrong". I think that the sentence "some inconsistent theories do exist" demonstrates a trivial matter. I can make the theory P as follows:

The Theory P:
Postulate 1. The thing that is-paulemok is not a creature with torso.
Postulate 2. The thing that is-paulemok is a creature with torso.

Suppose that Postulate 1 is true, but Postulate 2 is false. Then, we can make the modified theory P-2 as follows:

The Theory P-2:
Postulate 1. The thing that is-paulemok is not a creature with torso.
Postulate 2. It is not the case that the thing that is-paulemok is a creature with torso.

Postulate 1 of the theory P-2 is logically equivalent to Postulate 2 of the theory P-2. We shift from the inconsistent theory P to the consistent theory P-2.
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paulemok
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by paulemok »

Namelesss wrote: April 24th, 2018, 10:14 pmUltimately, there is no difference between the 'consistent' and the 'inconsistent'!
That's a good way of describing the relationship between the two.
Mosesquine wrote: April 25th, 2018, 8:11 amThe sentence "some contradiction exists in a theory" means the same as the sentence "the theory is wrong".
The proposition "some contradiction exists in a theory" is logically equivalent to the proposition "the theory is right and wrong."

Proof. Assume some contradiction exists in a theory. Name the theory T. Then, just as in my original post, the following two propositions are true in T, by ex contradictione quodlibet.

9. p
10. It is not true that p.

So, in a metatheory of T, the following two propositions are true.

11. p is true in T.
12. It is not true that "p is true in T."

Since (12) is the negation of (11), some contradiction exists. Therefore, by ex contradictione quodlibet, T is right and wrong. Discharge the assumption.

To prove the converse, assume a theory is right and wrong. Name the theory T. Since T is right and wrong, by definition of wrong, T is right and not right. So, some contradiction exists. By ex contradictione quodlibet, some contradiction exists in T. Discharge the assumption. This concludes the proof.
Mosesquine wrote: April 25th, 2018, 8:11 amWe shift from the inconsistent theory P to the consistent theory P-2.
I'm not quite sure what the purpose of making theories P and P-2 was.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by Mosesquine »

paulemok wrote: April 25th, 2018, 3:57 pm
Namelesss wrote: April 24th, 2018, 10:14 pmUltimately, there is no difference between the 'consistent' and the 'inconsistent'!
That's a good way of describing the relationship between the two.
Mosesquine wrote: April 25th, 2018, 8:11 amThe sentence "some contradiction exists in a theory" means the same as the sentence "the theory is wrong".
The proposition "some contradiction exists in a theory" is logically equivalent to the proposition "the theory is right and wrong."

Proof. Assume some contradiction exists in a theory. Name the theory T. Then, just as in my original post, the following two propositions are true in T, by ex contradictione quodlibet.

9. p
10. It is not true that p.

So, in a metatheory of T, the following two propositions are true.

11. p is true in T.
12. It is not true that "p is true in T."

Since (12) is the negation of (11), some contradiction exists. Therefore, by ex contradictione quodlibet, T is right and wrong. Discharge the assumption.

To prove the converse, assume a theory is right and wrong. Name the theory T. Since T is right and wrong, by definition of wrong, T is right and not right. So, some contradiction exists. By ex contradictione quodlibet, some contradiction exists in T. Discharge the assumption. This concludes the proof.
Mosesquine wrote: April 25th, 2018, 8:11 amWe shift from the inconsistent theory P to the consistent theory P-2.
I'm not quite sure what the purpose of making theories P and P-2 was.

I think that the trick is an assumption that 'p' and 'not-p' are true at the same time. Your hypothetic theory T contains the propositions as follows:

(1) p
(2) It is not the case that p

You are irresponsibly assuming that (1) and (2) above are true at the same time, but logical contradictions like (1) and (2) above are not allowed in traditional logic. Your whole argument is a failure, unless you prove that contradictions can be established.
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by The Beast »

It is possible to use the Sheffer stroke in the system. P | not p.
Then if there is a continuity in a dynamic system, that is a unit of time (unit map) defined by the smallest possible unit: (p ∆ p1) is also true ∴ (not p1 ∆ not p) … since (p | not p) it is possible that:
(p ∆ p(n+1)) ∘ (not p(n))
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Re: Inconsistent Theories Metatheoretically Prove Trivialism

Post by paulemok »

Mosesquine wrote: April 26th, 2018, 5:16 amI think that the trick is an assumption that 'p' and 'not-p' are true at the same time. Your hypothetic theory T contains the propositions as follows:

(1) p
(2) It is not the case that p

You are irresponsibly assuming that (1) and (2) above are true at the same time, but logical contradictions like (1) and (2) above are not allowed in traditional logic.
There is no assumption that "(1) is true in T." There is no assumption that "(2) is true in T." The propositions "(1) is true in T" and "(2) is true in T" are logical consequences.
Mosesquine wrote: April 26th, 2018, 5:16 amYour whole argument is a failure, unless you prove that contradictions can be established.
The very argument we are discussing proves that contradictions can be established.
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