Possible consequences of falsifying the principle of explosion?

[Speakpigeon]

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...

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.

What surprised me was the realisation that since Russell, nobody in mathematical logic ever claims anything except that his formal logic is logically consistent. That is, they don't claim that their theories are anything beyond consistent theories, which they are. That is, they don't claim that their theories are "correct", i.e. that they would be the correct model of human logic. So, in effect, they're not even wrong and they won't ever be. Pointing out that their theories don't mean anything is just a waste of time.
The real shame is that none of them could come up with anything like a proper method of logic. They keep inventing ever more idiotic theories that don't represent any real logic. Why not try to model proper logic for once?
To date, the only methods of logic that are correct are Aristotle's syllogistic and the Stoics' logic (or Abelard's). Complete the picture with a nice formalisation of these two methods using the form used in mathematical logic and you have the best which has ever been done in the field. And this after 2,500 years of the history of formal logic! Whoa. Really impressive.
Still, perhaps someone found something and nobody noticed... I wouldn't be surprised.
EB

[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...

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.

What surprised me was the realisation that since Russell, nobody in mathematical logic ever claims anything except that his formal logic is logically consistent. That is, they don't claim that their theories are anything beyond consistent theories, which they are. That is, they don't claim that their theories are "correct", i.e. that they would be the correct model of human logic. So, in effect, they're not even wrong and they won't ever be. Pointing out that their theories don't mean anything is just a waste of time.
The real shame is that none of them could come up with anything like a proper method of logic. They keep inventing ever more idiotic theories that don't represent any real logic. Why not try to model proper logic for once?
To date, the only methods of logic that are correct are Aristotle's syllogistic and the Stoics' logic (or Abelard's). Complete the picture with a nice formalisation of these two methods using the form used in mathematical logic and you have the best which has ever been done in the field. And this after 2,500 years of the history of formal logic! Whoa. Really impressive.
Still, perhaps someone found something and nobody noticed... I wouldn't be surprised.
EB

Tarski "proved" that the notion of Truth cannot be formalized for any system as expressive as PA or greater:

I prove that he is incorrect right here by defining the system that he "proved" cannot possibly be defined:

Within the sound deductive inference model True(x) is formalized as:
(a) Axioms stipulated as expressions of language having the semantic
value of Boolean true.

(b) Theorems stipulated as the consequence of conventional formal
mathematical proofs with True(x) premises.

When the formal proofs of mathematical logic are adapted to the
sound deductive inference model all of the undecidable sentences
of other formal systems are decided to be deductively unsound within
this complete and consistent system.

Every formal system having a provability predicate and the above
definitions Axiom and Theorem is complete and consistent.

When Peano Arithmetic is defined to have a provability predicate and
the above definitions of Axiom and Theorem both the Tarski Undefinability
Theorem and Gödel's 1931 Incompleteness Theorem proofs fail.

Copyright 2019 Pete Olcott

[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.

Logical implication
p q p ⇒ q
T T T
T F F
F T T
F F T

I would correct it to conform to the English meaning this way:
Logical implication
p q p ⇒ q
T T T
T F F
F T F
F F F
From falsehood, only falsehood follows.

"it is not the case that p is true and q false".
becomes:
"it is ONLY the case that p is true and q true".

It is ONLY the case that 5 > 3 if 3 < 5.

[wtf]

I would correct it to conform to the English meaning this way:
Logical implication
p q p ⇒ q
T T T
T F F
F T F
F F F

Isn't that what everyone else calls logical AND?

Would you say that

* 2 + 2 = 4 and George Washington is regarded as the father of his country.

and

* 2 + 2 = 4 THEREFORE George Washington is regarded as the father of his country.

are semantically equivalent?

[PeteOlcott]

I would correct it to conform to the English meaning this way:
Logical implication
p q p ⇒ q
T T T
T F F
F T F
F F F

Isn't that what everyone else calls logical AND?

It sure as Hell makes much more sense than saying that every truth is logically
entailed by either falsehood or contradiction.

[wtf]

It sure as Hell makes much more sense than saying that every truth is logically
entailed by either falsehood or contradiction.

Avoiding the question I asked?

[PeteOlcott]

It sure as Hell makes much more sense than saying that every truth is logically
entailed by either falsehood or contradiction.

Avoiding the question I asked?

That question was absurd. I don't answer those.

If you want to make sure things are semantically connected we
have to quit talking about material equivalence and get back to
talking about provability in the sound deductive inference model.

This is what I mean by provability:
http://liarparadox.org/Provable_Mendelson.pdf

This is what I mean by formalizing the sound deductive inference model:
True(Γ) ⊢ C

[wtf]

That question was absurd. I don't answer those.

The question was perfectly sensible. I asked you about the consequences of redefining material implication to be logical conjunction. So that "2 + 2 = 4 AND Washington is regarded as the father of his country" is the same truth connective as "2 + 2 = 4 THEREFORE Washington is regarded as the father of his country." Two true statements.

Clearly you are the one who has stated an absurdity, and now you're running from the obvious challenge. As usual. Since you claim every closed wff is either true or false, what is the truth value of the axiom of choice? You've avoided answering that one as well.

[Speakpigeon]

Logical implication
p q p ⇒ q
T T T
T F F
F T T
F F T

I would correct it to conform to the English meaning this way:
Logical implication
p q p ⇒ q
T T T
T F F
F T F
F F F

Well, that's an idea, but please check if you still have the Modus Tollens and the Transposition. I would expect not.
And some.
And I think we wouldn't want to get rid of any of those...
EB

[PeteOlcott]

Clearly you are the one who has stated an absurdity, and now you're running from the obvious challenge. As usual. Since you claim every closed wff is either true or false, what is the truth value of the axiom of choice? You've avoided answering that one as well.

I am not going to bother to learn every detail of all of mathematics to prove
a single point. I have proved my point categorically thus simultaneously
applying to every category of a question about it.

Within the conventional notion of formal proof: Γ ⊢ C
http://liarparadox.org/Provable_Mendelson.pdf
¬((Γ ⊢ C) ∨ (Γ ⊢ ¬C)) defines ¬Deductively_Sound(Γ, C)

This excludes every undecidable logic sentence of conventional formal proofs
as deductively unsound thus semantically incorrect.

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

[wtf]

I am not going to bother to learn every detail of all of mathematics to prove
a single point.

What you call a "detail" is in fact a counterexample to your thesis. That you can't be bothered to engage with it shows you're not serious. The axiom of choice is the CLASSIC EXAMPLE of a closed wff that is undecidable in ZF.

And now for the second time you've pointedly ignored my example that shows how ridiculous it is for you to redefine material implication as logical conjunction. For the record, and now for the THIRD time:

a) 2 + 2 = 4 AND George Washington is recognized as the father of his country.

b) 2 + 2 = 4 THEREFORE George Washington is recognized as the father of his country.

According to your redefinition of material implication as logical conjunction, these two statements have exactly the same semantic content.

Is that not absurd?

[PeteOlcott]

I am not going to bother to learn every detail of all of mathematics to prove
a single point.

What you call a "detail" is in fact a counterexample to your thesis. That you can't be bothered to engage with it shows you're not serious. The axiom of choice is the CLASSIC EXAMPLE of a closed wff that is undecidable in ZF.

There cannot possibly be a counter-example to my thesis in the
same way that there cannot possibly be an integer X such that X > 5 and X < 5.
Every formal proof based on true premises necessarily has a true consequence.
Counter-examples to stipulated definitions are a categorical impossibility.

Within the conventional notion of formal proof: Γ ⊢ C
http://liarparadox.org/Provable_Mendelson.pdf

Every conventional Mendelson formal proof satisfying this:
¬((Γ ⊢ C) ∨ (Γ ⊢ ¬C)) exactly meets the conventional specification
of the definition of unsound deductive inference.

[wtf]

There cannot possibly be a counter-example to my thesis

I presented one. Several times. But if you don't think the axiom of choice is a counterexample, just tell me whether it's true or false. You'd become famous if you could answer that question. But you can't even be bothered to look it up on Wikipedia to understand why it's a killer counterexample to your entire thesis.

And now you continue to ignore the obvious counterexample to your utterly nonsensical idea to redefine material implication as logical conjunction.

Nobody will take you seriously if you simply ignore every question and challenge.

[PeteOlcott]

There cannot possibly be a counter-example to my thesis

I presented one. Several times. But if you don't think the axiom of choice is a counterexample, just tell me whether it's true or false. You'd become famous if you could answer that question. But you can't even be bothered to look it up on Wikipedia to understand why it's a killer counterexample to your entire thesis.

And now you continue to ignore the obvious counterexample to your utterly nonsensical idea to redefine material implication as logical conjunction.

Nobody will take you seriously if you simply ignore every question and challenge.

The axiom of choice seems obviously correct.

I spent a very long time carefully composing a reply and the system erased it.
What I am stating is the whole idea of stipulated definitions are irrefutable.
When 2 + 3 = 5 is defined to be true, you can't say wait wait I have a counter-example.
This is semantic tautology, not quite the same thing as logical tautology.

https://www.britannica.com/topic/tautology
Tautology, in logic, a statement so framed that it cannot be denied
without inconsistency. Thus, “All humans are mammals”
prevents any rebuttal that humans are not mammals.

Combining the conventional notion of {formal proofs of mathematical logic}
with the conventional notion of {sound deductive inference} necessarily
creates {sound deductive formal proofs of mathematical logic} which does
indeed necessarily reject some expressions of language as deductively unsound.
All of this is a matter of definition, thus not subject to any counter-examples.

Any possible counter-example to the contrary would be exactly the same thing
as arguing against any stipulated definition such as: “All humans are mammals”.

[PeteOlcott]

I am not going to bother to learn every detail of all of mathematics to prove
a single point.

What you call a "detail" is in fact a counterexample to your thesis. That you can't be bothered to engage with it shows you're not serious. The axiom of choice is the CLASSIC EXAMPLE of a closed wff that is undecidable in ZF.

There cannot possibly be a counter-example to my thesis in the
same way that there cannot possibly be an integer X such that X > 5 and X < 5.
Every formal proof based on true premises necessarily has a true consequence.
Counter-examples to stipulated definitions are a categorical impossibility.

Within the conventional notion of formal proof: Γ ⊢ C
http://liarparadox.org/Provable_Mendelson.pdf

Every conventional Mendelson formal proof satisfying this:
¬((Γ ⊢ C) ∨ (Γ ⊢ ¬C)) exactly meets the conventional specification
of the definition of unsound deductive inference.

Whoops I forgot a step:
¬(True(Γ) ∧ ((Γ ⊢ C) ∨ (Γ ⊢ ¬C))) exactly meets the conventional specification