The principle of explosion diverges from correct reasoning

[PeteOlcott]

All of these years I simply relied on a bad source, here is what the Principle of Explosion really is
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.
https://rationalwiki.org/wiki/Principle_of_explosion

In symbolic logic, the principle of explosion can be expressed schematically in the following way:

P, ¬P ⊢ Q For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.

Step --- Proposition --- Derivation
1 -------- P ------------ Assumption
2 ------- ¬P ------------ Assumption
3 -------- P ∨ Q ------- Disjunction introduction (1)
4 -------- Q ------------ Disjunctive syllogism (3,2)
https://en.wikipedia.org/wiki/Principle ... sion#Proof

Truth table for P ∧ ¬P
P----¬P----P ∧ ¬P
T-----F------F
F-----T------F

With arithmetic we know that we must perform the operations in a
specific order or we get the wrong answer. (2 * 3) + (4 * 5) = (6 + 20)
We resolve the inner operations before proceeding. If we don't do this
same thing in logic we get absurd results.

When the text says P, ¬P ⊢ Q it failed to resolve P ∧ ¬P ⊢ False
(according to the above truth table) before moving to the next step.
This is a key aspect of the absurd results. It is not a logic error. It is
the divergence of logic from correct reasoning. When two premises
contradict each other (in correct reasoning) they resolve to false.

When we resolve P ∨ ¬P to False then it becomes clear that Disjunction
introduction cannot be performed (False ∨ Q) ⊢ Q transforms the unknown
truth value of Q to true. Thus Disjunction introduction is not a truth
preserving operation in this case.

[Skepdick]

In symbolic logic, the principle of explosion can be expressed schematically in the following way:

P, ¬P ⊢ Q For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.

Step --- Proposition --- Derivation
1 -------- P ------------ Assumption
2 ------- ¬P ------------ Assumption
3 -------- P ∨ Q ------- Disjunction introduction (1)
4 -------- Q ------------ Disjunctive syllogism (3,2)
https://en.wikipedia.org/wiki/Principle ... sion#Proof

Truth table for P ∧ ¬P
P----¬P----P ∧ ¬P
T-----F------F
F-----T------F

With arithmetic we know that we must perform the operations in a
specific order or we get the wrong answer. (2 * 3) + (4 * 5) = (6 + 20)
We resolve the inner operations before proceeding. If we don't do this
same thing in logic we get absurd results.

When the text says P, ¬P ⊢ Q it failed to resolve P ∧ ¬P ⊢ False
(according to the above truth table) before moving to the next step.
This is a key aspect of the absurd results. It is not a logic error. It is
the divergence of logic from correct reasoning. When two premises
contradict each other (in correct reasoning) they resolve to false.

When we resolve P ∨ ¬P to False then it becomes clear that Disjunction
introduction cannot be performed (False ∨ Q) ⊢ Q transforms the unknown
truth value of Q to true. Thus Disjunction introduction is not a truth
preserving operation in this case.

Olcott, you are a fucking idiot.

You don't even understand the difference between truth-preserving and property-preserving transformations.

(False ∨ Q) ⊢ Q is property-preserving.

It preserves Q whether Q is true; or false.

[Flannel Jesus]

I feel that you've misunderstood the point of the principle of explosion. You don't need to come here to prove that it diverges from correct reasoning. The reason it was invented in the first place is to demonstrate that "P and notP" diverges from correct reasoning.

[PeteOlcott]

I feel that you've misunderstood the point of the principle of explosion. You don't need to come here to prove that it diverges from correct reasoning. The reason it was invented in the first place is to demonstrate that "P and notP" diverges from correct reasoning.

You may understand that P ∧ ¬P ⊬ Q because P ∧ ¬P ⊢ False,
yet the Principle of Explosion does not seem to understand this.

[Flannel Jesus]

I feel that you've misunderstood the point of the principle of explosion. You don't need to come here to prove that it diverges from correct reasoning. The reason it was invented in the first place is to demonstrate that "P and notP" diverges from correct reasoning.

You may understand that P ∧ ¬P ⊬ Q because P ∧ ¬P ⊢ False,
yet the Principle of Explosion does not seem to understand this.

The principle itself doesn't understand this, you're right. The person who devised the principle, however, does. The reason he devised the principle was to show that, if you loosen up the laws of logic to allow P ∧ ¬P, nonsense ensues.

He understands that P ∧ ¬P is necessarily false, it's all a thought experiment, "... but what if it wasn't false?" The answer to "what if it wasn't false?" is "Then all other statements are true", at least according to the principle.

Do you get that it's a thought experiment, and that the man who devised it doesn't actually think P ∧ ¬P can be true?

[PeteOlcott]

I feel that you've misunderstood the point of the principle of explosion. You don't need to come here to prove that it diverges from correct reasoning. The reason it was invented in the first place is to demonstrate that "P and notP" diverges from correct reasoning.

You may understand that P ∧ ¬P ⊬ Q because P ∧ ¬P ⊢ False,
yet the Principle of Explosion does not seem to understand this.

The principle itself doesn't understand this, you're right. The person who devised the principle, however, does. The reason he devised the principle was to show that, if you loosen up the laws of logic to allow P ∧ ¬P, nonsense ensues.

He understands that P ∧ ¬P is necessarily false, it's all a thought experiment, "... but what if it wasn't false?" The answer to "what if it wasn't false?" is "Then all other statements are true", at least according to the principle.

Do you get that it's a thought experiment, and that the man who devised it doesn't actually think P ∧ ¬P can be true?

I know that there are several cases where logic diverges from correct reasoning.

When analytically true means
(a) Finite strings are stipulated to have the semantic property of Boolean true.

Only by stipulating relations between finite strings do finite strings acquire semantic meaning otherwise they remain utterly meaningless. All of these stipulated relations are stipulated to have the semantic property of Boolean true. This makes these finite strings tautologies that we know must be true.

(b) Finite strings are semantically deduced from the above set

Tarski Undefinability and Gödel 1931 incompleteness cannot occur

[Flannel Jesus]

I'm not getting the impression that you understand what I've said...

[Skepdick]

I feel that you've misunderstood the point of the principle of explosion. You don't need to come here to prove that it diverges from correct reasoning. The reason it was invented in the first place is to demonstrate that "P and notP" diverges from correct reasoning.

I feel like you have no idea what principles are.

You can accept or reject the principle of explosion.

There are logical consequences of each choice. Neither of those consequences are "correct" or "incorrect".

That's just your opinion.

[Skepdick]

P ∧ ¬P ⊢ False,

Olcott, you are a fucking idiot!

The value of "P ∧ ¬P " depends on the logic you are using!

Code: Select all

❯ pry -r./universe.rb
[1] pry(main)> p and not p
=> true

[PeteOlcott]

I'm not getting the impression that you understand what I've said...

I understand what you said. It seems to me that no one else understands it that way.

On a tangential subject:
Everyone else simply takes it for granted that no consistent truth predicate can
be defined simply because Tarski said so.

[Agent Smith]

Kindly review yer argument ... I sense an illegal move! Not unexpected of course, given the territory ... hic sunt leones/dracones.


By the way, true that contradictions aren't logical errors as is obvious from the principle of explosion. What are they then? Again, mon ami, hic sunt dracones/leones. Apologies if this upsets ya in any way but it is what it is, oui?

[PeteOlcott]

I feel that you've misunderstood the point of the principle of explosion. You don't need to come here to prove that it diverges from correct reasoning. The reason it was invented in the first place is to demonstrate that "P and notP" diverges from correct reasoning.

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.

[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.

[Skepdick]

You prove that by assuming that they CAN be true simultaneously and showing that that results in absurdity.

Why is it an "absurdity" exactly?

A Qubit is both 0 and not-0 simultaneously. It's a perfectly coherent superposition.

[PeteOlcott]

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.

Yet that is not the way that reductio ad absurdum works.

a method of proving the falsity of a premise by showing
that its logical consequence is absurd or contradictory.

If you end up with a contradiction then you know that your premise is false.
It is incorrect to start with a contradiction all you get is this: P ∧ ¬P ⊢ False