Possible consequences of falsifying the principle of explosion?

[Speakpigeon]

"The reason mathematicians keep it is that it has no known consequence."
It sure screws up formalizing natural language using conventional logic as the basis.
In a court of law it would conclude that the witness is telling the truth on the basis that they have lied.
If it was not for screwball stuff like the POE, we could not have gone all these decades
with people believing that Incompleteness and Undefinability was true.
Because of screwball things such as POE every attempt at correcting the error that lead to the misconceptions of Incompleteness and Undefinability was rejected out-of-hand as unconventional.

I don't have the time now to look at these two theorems. They certainly look fishy to me but I don't know that they rely on ECQ at all. If they do, then they're wrong.

"Deductively Sound Formal Proofs
https://www.researchgate.net/publicatio ... mal_Proofs
Now that I corrected this problem, the solution continues to be rejected out-of-hand as unconventional.

Did you try any of the big journals of logic?
EB

[PeteOlcott]

"Deductively Sound Formal Proofs
https://www.researchgate.net/publicatio ... mal_Proofs
Now that I corrected this problem, the solution continues to be rejected out-of-hand as unconventional.

Did you try any of the big journals of logic?
EB

I have tried academic journals and they rejected what I said out-of-hand mostly without review.
So I decided to keep making my words clearer and more conventional until they can be understood
and agreed to. I think that I may be at that point now.

Anyone that understands both formal proof from mathematical logic and the sound deductive
inference model should be able to understand what I say and agree that it is correct.

Logik and I got stuck when he refused to agree that ¬(1 = 0).
I called this a psychotic break from reality later on to realize that this is the view of radical skepticism.

This view is apparently held by people that "believe in" Quine's view that analytical truth is circular.
This view of circularity seems quite nutty to me when I realize through my own insights that

Analytical (or self-evident) truth is expressions of language that can be verified as completely true
entirely based on the meaning of their words. Copyright 2012 Pete Olcott.

Deductively Sound Formal Proofs
https://www.researchgate.net/publicatio ... mal_Proofs

[Scott Mayers]

What's that supposed to mean?
Given you keep responding similar to me this way in anything I've discussed anything with you, I'm thinking you've got some beef with me? Do you want to let me in on your problem?

I don't have any problem with you. I do seem to have a problem with most of your comments.
My statement below is perfectly good English and any idiot can understand it.

The question is whether the inference A ∧ ¬A ⊢ B, which is deemed valid in standard mathematical logic, is used in the proof of theorems in standard mathematics.

If you want to claim my statement is meaningless, then it's up to you to prove it is meaningless.
EB

Your language is insulting and inappropriate. I KNOW what I'm talking about BEYOND your own apparent background and don't appeal to being dismissed for accent I did not intend against you.

What is "meaningless" is NOT you personally, but the statement since "B" is the conclusion of a sequent that doesn't use what is IN the premises. For propositional logic, the conclusion MUST at least contain the terms within its premises and so that statement is INCOMPLETE as it stands. That is, "B" is not even dependent upon the given proposition and cannot be 'valid'.

The Wikipedia page also linked by you or Pete is incorrect for even using that. It ignores that all complete Propositional logic axioms include a rule that outlaws contradiction.

[Speakpigeon]

I have tried academic journals and they rejected what I said out-of-hand mostly without review. So I decided to keep making my words clearer and more conventional until they can be understood and agreed to. I think that I may be at that point now.

I think academia is now a business and most journals are part of the business of academia. These people have a career to take care of. They are all very busy competing against each other and the game is to not rock the boat you're in, even when you realise there's something no quite right. It is Kripke I think who stated explicitly that material implication was not exactly correct but that now it was too late! Too much work had been done and it was inconceivable to start from a new foundation.

I see all alternative logics, like relevance logics, constructive etc. as symptoms of this decision to keep the original material implication regardless of its obvious meaninglessness. But instead of trying to put the building on proper ground, they sort of build their own floor apart from the other mathematicians, with the resulting and extraordinary cacophony.
Personally, I'm quite sure standard classical (1st order) logic is the least wrong of them all, that is, the one which is closest to the logic of human reasoning.
I don't believe that apart from ECQ and EFQ there is any tautology of 1st order logic that contradicts our intuition or even the small list of historical logical truths people have discovered since Aristotle, and this also helps explain why mathematicians have no incentive to look at the foundation of their system. The only motivation is the discomfort we all experience when considering ECQ and EFQ. And I don't know if these two are of any consequence.

What's puzzling, though, is that you find really many people successfully building all sorts of idiotic non-standard logics, but not one to put standard logic right. It may even be that ECQ and EFQ help mainstream mathematicians get rid of the not-so-bright among them by diverting them towards non-standard logics and ultimately irrelevant research. People who can't stand working with ECQ and EFQ on their shoulder are more likely a bit "unstable". And if you remember it's all essentially a business, you'll understand why it is very convenient to get rid of the potentially "unstable". Except that now the non-standard is sort of thriving on their own logorrhea.

Anyone that understands both formal proof from mathematical logic and the sound deductive
inference model should be able to understand what I say and agree that it is correct.

Possibly, but the question is whether the question has any importance at all if ECQ and EFQ don't have any consequence.

Logik and I got stuck when he refused to agree that ¬(1 = 0).

Strictly speaking, "=" is not a logic connector, so ¬(1 = 0) can be true depending on your mathematical assumptions.
That being said, Logik doesn't seem to understand logic at all. The few time he tried to justify his claims, his justification was logically incoherent. And you can't have any sensible conversation with him. I did warn you, if you remember.

I called this a psychotic break from reality later on to realize that this is the view of radical skepticism.

He doesn't have a coherent view, so I wouldn't use any name for it at all. It's just incoherent.

This view is apparently held by people that "believe in" Quine's view that analytical truth is circular. This view of circularity seems quite nutty to me when I realize through my own insights that
Analytical (or self-evident) truth is expressions of language that can be verified as completely true entirely based on the meaning of their words. Copyright 2012 Pete Olcott.

It depends what you mean precisely by analytical truth. This view is essentially the realisation that dictionaries don't tell you anything but definitions. As if these people really believed truth was to be found in a dictionary. So, I think a lot of mathematicians and logicians are truly idiots and sometimes that realise what they believe is wrong and write books about truth being circular. Such is life.

Still, if you were able to find even one 1st order tautology that's obviously wrong (outside ECQ and EFQ), you would have a stronger position because science and engineering would definitely pay attention. Because money.

But, as I said, I don't believe there's any.
EB

[Speakpigeon]

Your language is insulting and inappropriate. I KNOW what I'm talking about BEYOND your own apparent background and don't appeal to being dismissed for accent I did not intend against you.

It's up to you to make sure your comments and replies are in good English, mean something and are relevant. Short of that, you'll get the harsh end of the stick.

What is "meaningless" is NOT you personally, but the statement since "B" is the conclusion of a sequent that doesn't use what is IN the premises. For propositional logic, the conclusion MUST at least contain the terms within its premises and so that statement is INCOMPLETE as it stands. That is, "B" is not even dependent upon the given proposition and cannot be 'valid'.

Oh, but you do understand what the statement mean since you now explain why you think it's wrong! And yet, you keep insisting it's meaningless! That's a contradiction, you know?

The Wikipedia page also linked by you or Pete is incorrect for even using that.

Possibly, but nobody is going to believe you on trust. You would need either to explain yourself or provide a link explaining your point. If you can't bother, that's fine, I'm not expecting you to be able to articulate your position properly or anybody to do it for you.

It ignores that all complete Propositional logic axioms include a rule that outlaws contradiction.

That is idiotic. Contradictions are not "outlawed". They are just false. And False implies any B until you prove otherwise. Can you?
EB

[PeteOlcott]

Anyone that understands both formal proof from mathematical logic and the sound deductive
inference model should be able to understand what I say and agree that it is correct.

Possibly, but the question is whether the question has any importance at all if ECQ and EFQ don't have any consequence.

This notion of sound deductive inference:

Within the sound deductive inference model True(x) is defined as:
(1) Any expression of language having the semantic value of Boolean True.
(2) The consequence of any formal mathematical proof where every element of its set of premises is True.
This makes the truth value of the consequence necessarily true.

Would decide that both: ECQ and EFQ are unsound.

What we get for free with this solution is that every undecidable logic sentence
of mathematical logic would also be decided to be unsound thus semantically incorrect.

Instead of undecidable logic sentences "proving" incompleteness of formal systems they
are merely rejected as derived from unsound deduction.

[wtf]

Instead of undecidable logic sentences "proving" incompleteness of formal systems they
are merely rejected as derived from unsound deduction.

That doesn't make any sense. If a sufficiently interesting system is consistent, it must necessarily contain closed wffs that can neither be proven nor disproven. They are formally undecidable or formal unentscheidbare in the original.

On another note, I am concerned that our friend Logik has disappeared. I don't think he would have lost interest ... I hope nothing more serious has occurred, but online one can never know.

[PeteOlcott]

Instead of undecidable logic sentences "proving" incompleteness of formal systems they
are merely rejected as derived from unsound deduction.

That doesn't make any sense. If a sufficiently interesting system is consistent, it must necessarily contain closed wffs that can neither be proven nor disproven. They are formally undecidable or formal unentscheidbare in the original.

On another note, I am concerned that our friend Logik has disappeared. I don't think he would have lost interest ... I hope nothing more serious has occurred, but online one can never know.

(1) I spent a lot of years on the 1931 Incompleteness Theorem.
My definition of [deductively sound formal systems] makes undecidable sentences impossible.
They are decided true, false, or semantically incorrect. Since this is nothing more than
making formal systems conform to the deductive inference model it is already accepted as correct.

(2) When logik kept insisting that (1 = 0) is true I got very frustrated and called this a psychotic
break from reality. I hadn't considered the possibility that a statement like this could have
adverse psychological effects.

[Scott Mayers]

Your language is insulting and inappropriate. I KNOW what I'm talking about BEYOND your own apparent background and don't appeal to being dismissed for accent I did not intend against you.

It's up to you to make sure your comments and replies are in good English, mean something and are relevant. Short of that, you'll get the harsh end of the stick.

What is "meaningless" is NOT you personally, but the statement since "B" is the conclusion of a sequent that doesn't use what is IN the premises. For propositional logic, the conclusion MUST at least contain the terms within its premises and so that statement is INCOMPLETE as it stands. That is, "B" is not even dependent upon the given proposition and cannot be 'valid'.

Oh, but you do understand what the statement mean since you now explain why you think it's wrong! And yet, you keep insisting it's meaningless! That's a contradiction, you know?

The Wikipedia page also linked by you or Pete is incorrect for even using that.

Possibly, but nobody is going to believe you on trust. You would need either to explain yourself or provide a link explaining your point. If you can't bother, that's fine, I'm not expecting you to be able to articulate your position properly or anybody to do it for you.

It ignores that all complete Propositional logic axioms include a rule that outlaws contradiction.

That is idiotic. Contradictions are not "outlawed". They are just false. And False implies any B until you prove otherwise. Can you?
EB

I'll leave you guys to your own thinking discussion.I disagree with your take on it and HAVE explained it sufficiently. I don't know your background history to know what you are biased on and can't be bothered when I'm already far beyond this. Good luck.

[Speakpigeon]

This notion of sound deductive inference:

Within the sound deductive inference model True(x) is defined as:
(1) Any expression of language having the semantic value of Boolean True.
(2) The consequence of any formal mathematical proof where every element of its set of premises is True.
This makes the truth value of the consequence necessarily true.

Would decide that both: ECQ and EFQ are unsound.

What we get for free with this solution is that every undecidable logic sentence
of mathematical logic would also be decided to be unsound thus semantically incorrect.

Instead of undecidable logic sentences "proving" incompleteness of formal systems they
are merely rejected as derived from unsound deduction.

I broadly agree with that but I would need to look at the detail of both Tarski 's and Gödel's proofs. So, pending that, I have to suspend my judgement.
Of course, on the bright side, Tarski 's and Gödel's proofs only apply to standard mathematical logic. So, assuming your system is correct, you would need to identify some conjecture you could prove with your system that standard mathematical logic could not due to Tarski 's and Gödel's proofs.
Yeah, I know, sounds like a lot of work...
EB

[Speakpigeon]

The Wikipedia page also linked by you or Pete is incorrect for even using that.

Possibly, but nobody is going to believe you on trust. You would need either to explain yourself or provide a link explaining your point. If you can't bother, that's fine, I'm not expecting you to be able to articulate your position properly or anybody to do it for you.

I'll leave you guys to your own thinking discussion.I disagree with your take on it and HAVE explained it sufficiently. I don't know your background history to know what you are biased on and can't be bothered when I'm already far beyond this. Good luck.

Sorry for that, but as I said, nobody is going to believe you on trust. You would need either to explain yourself or provide a link explaining your point. If you can't bother, that's fine, I'm not expecting you to be able to articulate your position properly or anybody to do it for you.
Still, I understand you. It was so much easier for you to claim without justification that what I said didn't make sense than to justify your claim.
Too bad, I would have been interested if you were up to proving the Wiki page wrong.
EB

[PeteOlcott]

The question is whether the inference A ∧ ¬A ⊢ B, which is deemed valid in standard mathematical logic, is used in the proof of theorems in standard mathematics.

If you want to claim my statement is meaningless, then it's up to you to prove it is meaningless.
EB

https://en.wikipedia.org/wiki/Colorless ... _furiously is not meaningless its meaning is incoherent.
Likewise with this: A ∧ ¬A ⊢ B

If Batman is both Godzilla and not Godzilla then it is proven that the creator of the universe is an ice cream sandwich.

It makes more sense to have the consequent of EFQ and EFC simply be Boolean false:
(A ∧ ¬A ) ⊢ False
False ⊢ False

It seems quite nutty that they resolved is the way that they did.

[PeteOlcott]

What would be the consequences for mathematics of falsifying the principle of explosion resulting from the truth-table of the material implication?

According to the truth-table of the material implication, for any implication, if the antecedent is false, including if it is a contradiction, then the implication is valid. This is true whatever the consequent might be, and whether it is true or false.

Given that the principle only affects validity, not soundness, I would expect no consequence at all. Is that correct do you think?
EB

EDIT
By falsifying, I mean proving false, i.e. proving that there isn't any "explosion" to begin with, i.e. proving true A ∧ ¬A ⊬ B.

https://en.wikipedia.org/wiki/Paradoxes ... mplication

[PeteOlcott]

"Deductively Sound Formal Proofs
https://www.researchgate.net/publicatio ... mal_Proofs
Now that I corrected this problem, the solution continues to be rejected out-of-hand as unconventional.

Did you try any of the big journals of logic?
EB

On many different forums what-so-ever I say is rejected out-of-hand merely on the basis
that it is already known with complete certainty that what I am trying to do is totally impossible.

The Tarski and Gödel theorems are accepted as the word of God and anyone speaking against
them commits the sin of blasphemy and is thus dismissed are a crank.

[PeteOlcott]

I broadly agree with that but I would need to look at the detail of both Tarski 's and Gödel's proofs. So, pending that, I have to suspend my judgement.
Of course, on the bright side, Tarski 's and Gödel's proofs only apply to standard mathematical logic. So, assuming your system is correct, you would need to identify some conjecture you could prove with your system that standard mathematical logic could not due to Tarski 's and Gödel's proofs.
Yeah, I know, sounds like a lot of work...
EB

Mathematical logic is currently considered to be the ultimate foundation of the notion of truth.

Because of Tarski and Gödel truth conditional semantics is "known" with certainty to be either
incomplete or inconsistent thus putting a huge damper on the possibility of any meaningful
success of any AI projects.

Through the understanding that all analytical truth is simply the tautological connections
between concepts that are validated through deduction the broken system of
mathematical logic is corrected.