The principle of explosion diverges from correct reasoning

[Flannel Jesus]

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

Wikipedia seems to think it is

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.

That's not too different from what I said is happening here. "the opposite scenario would lead to absurdity" -- is that not pretty much what I said?

[PeteOlcott]

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

Wikipedia seems to think it is

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.

That's not too different from what I said is happening here. "the opposite scenario would lead to absurdity" -- is that not pretty much what I said?

If you end up with a contradiction you know there is a mistake somewhere
thus you already know that beginning with a contradiction is a mistake so
don't do it.

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

By similar reasoning one could presume that a wild Tiger is a milk
cow, and then proceed to milk this tiger. That will probably not end well.

[Flannel Jesus]

Yes, it doesn't end well, it doesn't make sense, you agree, I agree, the person who devised the principle agrees. What is it you're arguing for?

[PeteOlcott]

Yes, it doesn't end well, it doesn't make sense, you agree, I agree, the person who devised the principle agrees. What is it you're arguing for?

So the Principle of Explosion merely answers the question:
What would happen if we rejected reductio ad absurdum and
assumed that contradictions are true?

This could have been stated much more clearly the way that
I stated it above. I will see if Wikipedia accepts my edits.

[Flannel Jesus]

So the Principle of Explosion merely answers the question:
What would happen if we rejected reductio ad absurdum and
assumed that contradictions are true?

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.

[PeteOlcott]

So the Principle of Explosion merely answers the question:
What would happen if we rejected reductio ad absurdum and
assumed that contradictions are true?

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.

I updated the Wikipedia page to say this:

"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} is rejected.
https://en.wikipedia.org/wiki/Principle_of_explosion

[wtf]

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.

[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 am quoting this:

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

[wtf]

I am quoting this:

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

I fail to see how that quote supports your assertion that the POE somehow (how???) entails rejection of the law of non-contradiction.

That you are confused on this point I can well understand, though I don't personally think you should be defacing Wikipedia. But I cannot fathom why @Flannel Jesus, who usually plays a sane person on this and on another forum, is jumping in to support your nonsensical claim.

[PeteOlcott]

I am quoting this:

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

I fail to see how that quote supports your assertion that the POE somehow (how???) entails rejection of the law of non-contradiction.

That you are confused on this point I can well understand, though I don't personally think you should be defacing Wikipedia. But I cannot fathom why @Flannel Jesus, who usually plays a sane person on this and on another forum, is jumping in to support your nonsensical claim.

"a contradiction is true" quoted above entails that the law of non-contradiction has been rejected.

All Wikipedia changes are reviewed and reversed as needed.
Because I directly linked to the source of most of the update I expect that it may be approved.

[wtf]

"a contradiction is true" quoted above entails that the law of non-contradiction has been rejected.

No contradiction is being taken as true. It's a hypothetical.

IF 2 + 2 = 5 THEN I am the Pope.

That is a logically true statement based on the truth table for material implication. The antecedent is FALSE -- do you get that? -- the antecedent is FALSE. Therefore the implication as a whole is true, because any material implication with a false antecedent is true.

There is no claim that 2 + 2 = 5 is true. On the contrary. We note that 2 + 2 = 5 is false. We then look up the rows in the truth table for material implication in which the antecedent is false; and we note that in both cases, namely whether the consequent is true or false, the implication as a whole is true.

See https://en.wikipedia.org/wiki/Material_ ... inference), in particular the truth table:

Code: Select all

P Q  P → Q
= =  =====
T T    T
T F    F
F T    T 
F T    T

Likewise, P ∧ ¬P is false. Let me say that again. P ∧ ¬P is false. Why? Principle of non-contradiction, which is being AFFIRMED.

Therefore the material implication (P ∧ ¬P) → I am the Pope is a valid material implication, whose overall truth value is True.

Why does this simple point elude you?

In material implication, a FALSE antecedent always gives a TRUE material implication. "If 2 + 2 = 5 then I am the Pope." The material implication is true, exactly by virtue of the fact that 2 + 2 = 5 is false.

Nobody is saying P ∧ ¬P is true. Well YOU are saying it, but that's only because you don't understand material implication.

On the contrary, P ∧ ¬P is FALSE FALSE FALSE. That's the principle of explosion. It is NOT a denial of non-contradiction, it's an AFFIRMATION of non-contradiction.

All Wikipedia changes are reviewed and reversed as needed.

I graffitied the wall of a building, because I expect the property owner will go to the expense of having it painted over. So no harm done, right?

Because I directly linked to the source of most of the update I expect that it may be approved.

Like I said, the exact flaw with Wikipedia. "Anyone may edit." That policy giveth, and in this case that policy taketh away.

[PeteOlcott]

No contradiction is being taken as true. It's a hypothetical.

IF 2 + 2 = 5 THEN I am the Pope.

That is a logically true statement based on the truth table for material implication. The antecedent is FALSE -- do you get that? -- the antecedent is FALSE. Therefore the implication as a whole is true, because any material implication with a false antecedent is true.

Within the scope of the Principle of Explosion the law of non-contradiction is rejected.
I tentatively reject implication as not truth preserving.

Along these same lines anything that diverges from relevance logic is also rejected
as a deviation from correct reasoning.

[Agent Smith]

Let's be thorough about the issue, shall we?

A contradiction: p & ~p

Begin!

Ex contradictione sequitur quodlibet. What else do we need?!

[Flannel Jesus]

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.

[Skepdick]

"a contradiction is true" quoted above entails that the law of non-contradiction has been rejected.

No contradiction is being taken as true. It's a hypothetical.

IF 2 + 2 = 5 THEN I am the Pope.

That is a logically true statement based on the truth table for material implication. The antecedent is FALSE -- do you get that? -- the antecedent is FALSE. Therefore the implication as a whole is true, because any material implication with a false antecedent is true.

There is no claim that 2 + 2 = 5 is true. On the contrary. We note that 2 + 2 = 5 is false. We then look up the rows in the truth table for material implication in which the antecedent is false; and we note that in both cases, namely whether the consequent is true or false, the implication as a whole is true.

See https://en.wikipedia.org/wiki/Material_ ... inference), in particular the truth table:

Code: Select all

P Q  P → Q
= =  =====
T T    T
T F    F
F T    T 
F T    T

Likewise, P ∧ ¬P is false. Let me say that again. P ∧ ¬P is false. Why? Principle of non-contradiction, which is being AFFIRMED.

Therefore the material implication (P ∧ ¬P) → I am the Pope is a valid material implication, whose overall truth value is True.

Why does this simple point elude you?

In material implication, a FALSE antecedent always gives a TRUE material implication. "If 2 + 2 = 5 then I am the Pope." The material implication is true, exactly by virtue of the fact that 2 + 2 = 5 is false.

Nobody is saying P ∧ ¬P is true. Well YOU are saying it, but that's only because you don't understand material implication.

On the contrary, P ∧ ¬P is FALSE FALSE FALSE. That's the principle of explosion. It is NOT a denial of non-contradiction, it's an AFFIRMATION of non-contradiction.

All Wikipedia changes are reviewed and reversed as needed.

I graffitied the wall of a building, because I expect the property owner will go to the expense of having it painted over. So no harm done, right?

Because I directly linked to the source of most of the update I expect that it may be approved.

Like I said, the exact flaw with Wikipedia. "Anyone may edit." That policy giveth, and in this case that policy taketh away.

What an unnecessary rant.

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

The reason all you idiot Classical logicians are so confused is because you have IF/THEN without an ELSE.

P ∧ ¬P is an logical expression. It has no value unless evaluated in a value-system; and in some value-systems it evaluates to true.

Code: Select all

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