The principle of explosion diverges from correct reasoning

[Skepdick]

I stand by what I said.

Stand by it all you want- it's bullshit.

The law of non contradiction is axiomatic in that formalization of logic.

No it isn't. It's just a choice. Is choice axiomatic in that formalization of logic?

Well.. you can choose to accept; or reject the axiom of choice.

The principle of explosion says, "but what if we relaxed that axiom and let a contradiction in?" The result is, if you let a contradiction in, ALL statements are true.

That's not true.

https://en.wikipedia.org/wiki/Paraconsistent_logic

Is there any bit of the above paragraph you disagree with?

Yeah. All of it.

There are systems of logic in which a contradiction entails an explosion.
There are system of logic in which a contradiction doesn't entail an explosion.

Why have you chosen the former?

And then, since we know that not all statements are true, it stands to reason that we cannot let contradictions in.

Of course we can let contradictions in. Just don't let the principle of explosion in.

[Flannel Jesus]

I would replace "reductio ad absurdum" with "the law of non contradiction", although I see why you might treat them as basically interchangeable. But yes, that's what the principle of explosion does.

Are you joking? Why are you encouraging this nonsense? What you said is wrong. The principle of explosion is simply a consequence of the truth table for material implication. It does not falsify or reject reductio or non contradiction in any way.

I stand by what I said.

The law of non contradiction is axiomatic in that formalization of logic. The principle of explosion says, "but what if we relaxed that axiom and let a contradiction in?" The result is, if you let a contradiction in, ALL statements are true.

Is there any bit of the above paragraph you disagree with?

And then, since we know that not all statements are true, it stands to reason that we cannot let contradictions in.

I've been googling around and finding various people explaining the concept in their own words, in ways similar to mine

https://www.quora.com/Is-the-proof-for- ... -reasoning

Mark hasty wrote

This does not establish the fact that a contradiction can be used to prove anything we want. It does establish the fact that any argument which includes a contradiction is a useless argument because it is based on a faulty assumption.

In other words, the principle of explosion is used, in his view, to support the axiom in classical logic that we don't allow in contradictory beliefs.

https://academickids.com/encyclopedia/i ... _explosion

Here's a website for educating children. What does this site say about the principle of explosion?

Besides the general prima facie implausibility of contradictions, this is the primary logical argument for not allowing P ∧ ¬P to be true in a formal system: systems in which any arbitrary formula is a theorem are trivial. Thus explosion justifies the law of noncontradiction.

Pretty much the same thing I'm saying.

https://philosophy.stackexchange.com/qu ... d-absurdum

Mauro explains the logic behind the principle, and then says

Conclusion: contradictions are never true.

I believe I'm interpreting the principle in the standard way it's meant to be interpreted. If you disagree wtf, please let me know where I'm taking a wrong step.

[Skepdick]

In other words, the principle of explosion is used, in his view, to support the axiom in classical logic that we don't allow in contradictory beliefs.

Who are you to "allow" or "deny" contradictory beliefs on behalf of others?

You are making a moral argument about the (im)permissable uses of logic.

please let me know where I'm taking a wrong step.

Sure.

Every time you prescribe to me what I should be doing; should be thinking; should be believing; and what axioms or principles I should accept or reject.

Every time you make a choice on my behalf you are taking a wrong step.

Nihil de nobis, sine nobis

[PeteOlcott]

I updated the Wikipedia page

Good God almighty, this is everything that's wrong with Wikipedia. I note from the Talk page that you edited anonymously. Great work, Pete.

I would replace "reductio ad absurdum" with "the law of non contradiction", although I see why you might treat them as basically interchangeable. But yes, that's what the principle of explosion does.

Are you joking? Why are you encouraging this nonsense? What you said is wrong. The principle of explosion is simply a consequence of the truth table for material implication. It does not falsify or reject reductio or non contradiction in any way.

I stand by what I said.

The law of non contradiction is axiomatic in that formalization of logic. The principle of explosion says, "but what if we relaxed that axiom and let a contradiction in?" The result is, if you let a contradiction in, ALL statements are true.

Is there any bit of the above paragraph you disagree with?

And then, since we know that not all statements are true, it stands to reason that we cannot let contradictions in.

Because I found a source that agrees with you and your explanation makes perfect
sense I updated the Wikipedia page with the paraphrase of your answer. Because
I cited my source the edit may remain making the POE clear to everyone.

The Principle of Explosion is not any actual principle of logic it is merely a
"what if" we ignored the law of non-contradiction hypothetical scenario.

Prior to this explanation I couldn't see how humans could possibly be so
stupid as to not reject the POE the first day it was ever presented.

[wtf]

I believe I'm interpreting the principle in the standard way it's meant to be interpreted. If you disagree wtf, please let me know where I'm taking a wrong step.

I did write a thorough explanation of the topic. The POE is just a restatement or a consequence of the truth table for material implication. IF false THEN anything is a valid implication. It's not saying false is true. It's saying (IF false THEN anything) is true. I can't possibly add to what I've already explained in my earlier more detailed post.

IF p and not-p THEN anything.

That's a valid material implication with a truth value of true.

It's not saying p and not-p is true or rejecting non-contradiction. It's a hypothetical. IF this THEN that.

If 2 + 2 = 5 then I am the Pope.

That does not assert that 2 + 2 = 5. It asserts that IF such an absurdity were true, THEN anything at all would be true. Can you not see that?

[Flannel Jesus]

I believe I'm interpreting the principle in the standard way it's meant to be interpreted. If you disagree wtf, please let me know where I'm taking a wrong step.

I did write a thorough explanation of the topic. The POE is just a restatement or a consequence of the truth table for material implication. IF false THEN anything is a valid implication. It's not saying false is true. It's saying (IF false THEN anything) is true. I can't possibly add to what I've already explained in my earlier more detailed post.

IF p and not-p THEN anything.

That's a valid material implication with a truth value of true.

It's not saying p and not-p is true or rejecting non-contradiction. It's a hypothetical. IF this THEN that.

If 2 + 2 = 5 then I am the Pope.

That does not assert that 2 + 2 = 5. It asserts that IF such an absurdity were true, THEN anything at all would be true. Can you not see that?

None of that disagrees with anything I've said. Can you quote the bit I've said that disagrees with that?

[Skepdick]

I believe I'm interpreting the principle in the standard way it's meant to be interpreted. If you disagree wtf, please let me know where I'm taking a wrong step.

I did write a thorough explanation of the topic. The POE is just a restatement or a consequence of the truth table for material implication. IF false THEN anything is a valid implication. It's not saying false is true. It's saying (IF false THEN anything) is true. I can't possibly add to what I've already explained in my earlier more detailed post.

IF p and not-p THEN anything.

That's a valid material implication with a truth value of true.

It's not saying p and not-p is true or rejecting non-contradiction. It's a hypothetical. IF this THEN that.

If 2 + 2 = 5 then I am the Pope.

That does not assert that 2 + 2 = 5. It asserts that IF such an absurdity were true, THEN anything at all would be true. Can you not see that?

Sigh. Bullshit.

It's not "IF p and not-p THEN anything."

It's "IF p THEN anything" in general. Which happens to include "IF p THEN not-p" in particular.

https://en.wikipedia.org/wiki/Curry%27s_paradox

[wtf]

None of that disagrees with anything I've said. Can you quote the bit I've said that disagrees with that?

You wrote, "I would replace "reductio ad absurdum" with "the law of non contradiction", although I see why you might treat them as basically interchangeable. But yes, that's what the principle of explosion does."

You were agreeing w/Pete that the POE rejects non-contradiction. That's just wrong. On the contrary, POE affirms non-contradiction.

[Flannel Jesus]

None of that disagrees with anything I've said. Can you quote the bit I've said that disagrees with that?

You wrote, "I would replace "reductio ad absurdum" with "the law of non contradiction", although I see why you might treat them as basically interchangeable. But yes, that's what the principle of explosion does."

You were agreeing w/Pete that the POE rejects non-contradiction. That's just wrong. On the contrary, POE affirms non-contradiction.

You have drastically misread.

[Flannel Jesus]

You were right and I was wrong.

"That is to say, the principle of explosion is an argument for the law
of non-contradiction in classical logic, because without it all truth
statements become meaningless."

So the POE has always only been a hypothetical argument to prove
the law of of non-contradiction. That makes sense.

Yes. It's called proof by absurdity, or reductio ad absurdum.

Reductio ad absurdum is a Latin phrase which means "reduction to the absurd". The phrase describes a kind of indirect proof. It is a proof by contradiction, and is a common form of argument. It shows that a statement is true because its denial leads to a contradiction, or a false or absurd result.

So, you believe p and notP cannot both be true simultaneously. How do you prove that? You prove that by assuming that they CAN be true simultaneously and showing that that results in absurdity.

That's all this principle is doing.

Please refer to this post, wtf, if you want to get a bit more context in what I was talking about with pete

[wtf]

Please refer to this post, wtf, if you want to get a bit more context in what I was talking about with pete

I think I've said my piece. I have not read the entire thread. I saw Pete make an error, and then I saw you seemingly agreeing with it. If that is not what you were doing, then I was wrong. IF that is not what you were doing, THEN I was wrong. Material implication again.

ps -- Ok if you happen to have a few spare moments, can you please explain to me what your response to Pete that I quoted is intended to mean? When Pete said that POE is a denial of non-contradiction, and you seemingly agreed, how was I misunderstanding your meaning? Pete at least contextualized his remark by mentioning Relevance logic. Is that what you were talking about?

[PeteOlcott]

"The principle of explosion is a logical rule of inference. According to the rule, from a set of premises in which a sentence A and its negation ¬A are both true (i.e., a contradiction is true), any sentence B may be inferred." [Principle of explosion] Thus the Principle of Explosion shows what happens when Reductio ad absurdum or the Law of noncontradiction are rejected.

[Principle of explosion] linked source of all of the words the precede it.
https://rationalwiki.org/wiki/Principle_of_explosion

[Skepdick]

"The principle of explosion is a logical rule of inference. According to the rule, from a set of premises in which a sentence A and its negation ¬A are both true (i.e., a contradiction is true), any sentence B may be inferred." [Principle of explosion] Thus the Principle of Explosion shows what happens when Reductio ad absurdum or the Law of noncontradiction are rejected.

[Principle of explosion] linked source of all of the words the precede it.
https://rationalwiki.org/wiki/Principle_of_explosion

Olcott, you are fucking idiot. The worst kind of idiot too - an idiot whose self-confidence far outweighs his idiocy.

Have the basic human decency to leave unedited public knowledge that you know nothing about.

Contradiction needs not entail explosion - the two can be decoupled. When they are decoupled there is no longer a need for the "law" of non-contradiction.

https://en.wikipedia.org/wiki/Paraconsistent_logic

Furthermore your inability to tolerate contradiction has a tangible adverse consequence. Non-contradiction imposes unreasonable limits on expressivity!

natural language abounds with directly or indirectly self-referential yet apparently harmless expressions—all of which are excluded from the Tarskian framework." This expressive limitation can be overcome in paraconsistent logic --Solomon Feferman

[PeteOlcott]

Some people don't seem to be able to understand that when someone directly
contradicts themselves that this conclusively proves that they are not telling
the truth. This also applies to expressions of language. When an expression of
language contradicts itself we know that it it not a truth bearer.

The letters I and E in the names of the rules come from
“introduction” and “elimination” respectively...
Here ⊢ is used to point out that the relation of deducibility
holds between premises and the conclusion of a rule instance...
(¬E) φ, ¬φ ⊢ ⊥

https://iep.utm.edu/natural-deduction/#H4

[Skepdick]

Some people don't seem to be able to understand that when someone directly
contradicts themselves that this conclusively proves that they are not telling
the truth.

Bullshit.

Here's an empirical example of P ∧ ¬P being true.

It's true that it's raining ( P )
It's also true that it's NOT raining ( ¬P )

https://www.youtube.com/watch?v=HBFdS1BaoFY