Possible consequences of falsifying the principle of explosion?

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Speakpigeon
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Possible consequences of falsifying the principle of explosion?

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.
Last edited by Speakpigeon on Wed May 01, 2019 5:38 pm, edited 2 times in total.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Possible consequences of falsifying the principle of explosion?

Speakpigeon wrote:
Wed May 01, 2019 12:22 pm
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
I went through this with many hundreds of messages.
https://plato.stanford.edu/entries/logic-relevance/ is one way to break the POE.
Another way is sound deductive inference that requires all premises to be true.

It is a quite nutty idea that logic allows logical consequence to follow from a contradiction,
it is almost like concluding that someone it telling the truth on the basis that they are a liar.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Possible consequences of falsifying the principle of explosion?

PeteOlcott wrote:
Wed May 01, 2019 1:37 pm
Another way is sound deductive inference that requires all premises to be true.
P1. If it rains the ground will be wet.
P2. It rains.
C. The ground is wet.

Empirical counter-example (a.k.a FALSIFICATION) : https://www.youtube.com/watch?v=XjXvkIzUTtk

Examining the argument + evidence holistically, what you have is a sound-and-valid argument with an incorrect conclusion. Oops?

https://en.wikipedia.org/wiki/Is_Logic_Empirical%3F
What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In his seminal paper "Two Dogmas of Empiricism," the logician and philosopher W. V. Quine argued that all beliefs are in principle subject to revision in the face of empirical data, including the so-called analytic propositions. Thus the laws of logic, being paradigmatic cases of analytic propositions, are not immune to revision.
The "laws" of logic are provisional - they are subject to revision.

Please stop peddling your dogma. Logic is just a tool. There is no "truth" hiding there - look somewhere else.

Scott Mayers
Posts: 1391
Joined: Wed Jul 08, 2015 1:53 am

Re: Possible consequences of falsifying the principle of explosion?

Speakpigeon wrote:
Wed May 01, 2019 12:22 pm
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
One of the first rules of Propositional Calculus is one often called, "OR Introduction". It says that given ANY assumed proposition, we can assert that that proposition OR some other arbitrary one is 'true'. This is an incidental truth in that the given proposition is assumed 'true' and so the statement is true BY ACCIDENT.

This is related to the same concern of material implication given any argument is itself an 'implication' of its input premises to its conclusion. As such, the implication HAS to be equivalently thought of as identical in FORM to any logic's input-to-output standard of VALIDITY:

That IF the inputs ARE 'true', then the outputs MUST be 'true' to define a "valid" form. So reduce this to a simple argument one premise, P.

If P is 'input', there are two possible separable types of conclusions using only P. We have either that P is 'output' or it is not (which may mean that there is either no conclusion or some other one indeterminately). We can then say we have two forms possible:

(1)..... ..P.
...........P

or

(2).......P.
.........~P

This simplifies the problem and demonstrates the definition of what we expect of 'validity' alone. Here, IF the premise was NOT true, then it wouldn't matter what the conclusion is because we only attend to the arguments defined WHEN the input premises are assumed 'true'.

This should help show that implication rules are not at odds here. If you wanted to ADD the (2) as a possible form in some logic, then you might define this possibility as the 'Negation of Validity' where the (1) is the 'Position of Validity'. See how this can still enable inclusion if you WANT to cover all possibilities?

I don't think "exploding" to an infinity of possibilities is faulty because any non-P is understood to be the infinity of all that is not-P.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Possible consequences of falsifying the principle of explosion?

Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
Speakpigeon wrote:
Wed May 01, 2019 12:22 pm
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
One of the first rules of Propositional Calculus is one often called, "OR Introduction". It says that given ANY assumed proposition, we can assert that that proposition OR some other arbitrary one is 'true'. This is an incidental truth in that the given proposition is assumed 'true' and so the statement is true BY ACCIDENT.

This is related to the same concern of material implication given any argument is itself an 'implication' of its input premises to its conclusion. As such, the implication HAS to be equivalently thought of as identical in FORM to any logic's input-to-output standard of VALIDITY:

That IF the inputs ARE 'true', then the outputs MUST be 'true' to define a "valid" form. So reduce this to a simple argument one premise, P.

If P is 'input', there are two possible separable types of conclusions using only P. We have either that P is 'output' or it is not (which may mean that there is either no conclusion or some other one indeterminately). We can then say we have two forms possible:

(1)..... ..P.
...........P

or

(2).......P.
.........~P

This simplifies the problem and demonstrates the definition of what we expect of 'validity' alone. Here, IF the premise was NOT true, then it wouldn't matter what the conclusion is because we only attend to the arguments defined WHEN the input premises are assumed 'true'.

This should help show that implication rules are not at odds here. If you wanted to ADD the (2) as a possible form in some logic, then you might define this possibility as the 'Negation of Validity' where the (1) is the 'Position of Validity'. See how this can still enable inclusion if you WANT to cover all possibilities?

I don't think "exploding" to an infinity of possibilities is faulty because any non-P is understood to be the infinity of all that is not-P.
https://en.wikipedia.org/wiki/Principle ... esentation
P, ¬P ⊢ Q

If cats are dogs and cats are not dogs then an ice cream cone is the creator of the universe.

P, ¬P ⊢ Q

https://plato.stanford.edu/entries/logic-relevance/
Relevance logicians claim that what is unsettling about these
so-called paradoxes is that in each of them the antecedent
seems irrelevant to the consequent.

Speakpigeon
Posts: 957
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

Re: Possible consequences of falsifying the principle of explosion?

PeteOlcott wrote:
Wed May 01, 2019 1:37 pm
I went through this with many hundreds of messages.
https://plato.stanford.edu/entries/logic-relevance/ is one way to break the POE.
Another way is sound deductive inference that requires all premises to be true.
I'm not asking how to prove the principe wrong but what would be the consequences of it being proved wrong.
PeteOlcott wrote:
Wed May 01, 2019 1:37 pm
It is a quite nutty idea that logic allows logical consequence to follow from a contradiction,
it is almost like concluding that someone it telling the truth on the basis that they are a liar.
I agree it's a nutty idea but as I understand it, mathematicians adopted it because, first, they couldn't find any better and, second, the principle has no adverse consequence precisely because it only affects validity, not soundness. Don't you agree with that?
EB

Speakpigeon
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Location: Paris, France, EU

Re: Possible consequences of falsifying the principle of explosion?

PeteOlcott wrote:
Wed May 01, 2019 3:03 pm
Relevance logicians claim that what is unsettling about these so-called paradoxes is that in each of them the antecedent seems irrelevant to the consequent.
It just shows "logicians" don't understand logic, not that there is any real problem, i.e. any real consequence for mathematics and mathematical theorems.
EB

Speakpigeon
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Location: Paris, France, EU

Re: Possible consequences of falsifying the principle of explosion?

Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
One of the first rules of Propositional Calculus is one often called, "OR Introduction". It says that given ANY assumed proposition, we can assert that that proposition OR some other arbitrary one is 'true'. This is an incidental truth in that the given proposition is assumed 'true' and so the statement is true BY ACCIDENT.

This is somewhat confusing... I don't know if you're trying to explain there is a problem?
Me, I don't think there is any. In the valid inference A ⊢ A ∨ B, the proposition B is not "assumed" true, "by accident" or otherwise, and it is not inferred as true either. Do you agree?
Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
This is related to the same concern of material implication given any argument is itself an 'implication' of its input premises to its conclusion. As such, the implication HAS to be equivalently thought of as identical in FORM to any logic's input-to-output standard of VALIDITY:

That IF the inputs ARE 'true', then the outputs MUST be 'true' to define a "valid" form. So reduce this to a simple argument one premise, P.

If P is 'input', there are two possible separable types of conclusions using only P. We have either that P is 'output' or it is not (which may mean that there is either no conclusion or some other one indeterminately). We can then say we have two forms possible:

(1)..... ..P.
...........P

or

(2).......P.
.........~P

This simplifies the problem and demonstrates the definition of what we expect of 'validity' alone. Here, IF the premise was NOT true, then it wouldn't matter what the conclusion is because we only attend to the arguments defined WHEN the input premises are assumed 'true'.
Does this lean you agree that the principle of explosion is of no consequence at all?
Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
This should help show that implication rules are not at odds here. If you wanted to ADD the (2) as a possible form in some logic, then you might define this possibility as the 'Negation of Validity' where the (1) is the 'Position of Validity'. See how this can still enable inclusion if you WANT to cover all possibilities?

I don't think "exploding" to an infinity of possibilities is faulty because any non-P is understood to be the infinity of all that is not-P.
???
Sorry, I don't understand your point here.
It should be remembered that it is mathematicians themselves who call it a "paradox", the paradox of material implication. So, this was an admission that it was an unfortunate situation at the very least.
EB

Speakpigeon
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Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

Re: Possible consequences of falsifying the principle of explosion?

I agree mathematical logic is wrong. It is wrong in the sense that it doesn't formalise properly the logic of human reasoning. However, it seems we have to put up with this situation because nobody can offer a better alternative. Further, if the problem has no impact on mathematics then there is in fact no problem. If you disagree with this, please explain what bad consequence on mathematics you think there are.
EB

Scott Mayers
Posts: 1391
Joined: Wed Jul 08, 2015 1:53 am

Re: Possible consequences of falsifying the principle of explosion?

Speakpigeon wrote:
Wed May 01, 2019 5:08 pm
Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
One of the first rules of Propositional Calculus is one often called, "OR Introduction". It says that given ANY assumed proposition, we can assert that that proposition OR some other arbitrary one is 'true'. This is an incidental truth in that the given proposition is assumed 'true' and so the statement is true BY ACCIDENT.

This is somewhat confusing... I don't know if you're trying to explain there is a problem?
Me, I don't think there is any. In the valid inference A ⊢ A ∨ B, the proposition B is not "assumed" true, "by accident" or otherwise, and it is not inferred as true either. Do you agree?
You are just rephrasing me in a different way. I just don't see a paradox at all because B, in your expression IS the accidental factor of the expression (A v B). The expression here is NOT actually about A nor B in isolation but the relationship between the two. As such, the statement will always enable ANY meaning of B without a problem. It relates to the problem of implication in that when the antecedent is NOT true, the consequent can be anything also. The 'validity' definition only concerns itself with the condition THAT the antecedent is true.
Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
This is related to the same concern of material implication given any argument is itself an 'implication' of its input premises to its conclusion. As such, the implication HAS to be equivalently thought of as identical in FORM to any logic's input-to-output standard of VALIDITY:

That IF the inputs ARE 'true', then the outputs MUST be 'true' to define a "valid" form. So reduce this to a simple argument one premise, P.

If P is 'input', there are two possible separable types of conclusions using only P. We have either that P is 'output' or it is not (which may mean that there is either no conclusion or some other one indeterminately). We can then say we have two forms possible:

(1)..... ..P.
...........P

or

(2).......P.
.........~P

This simplifies the problem and demonstrates the definition of what we expect of 'validity' alone. Here, IF the premise was NOT true, then it wouldn't matter what the conclusion is because we only attend to the arguments defined WHEN the input premises are assumed 'true'.
Does this [m]ean you agree that the principle of explosion is of no consequence at all?
Yes. "Explosion" usually refers to contradictions and there is no contradiction (no paradox). But if you are interpreting the 'explosion' to be of any possibility, you need to prove where the contradiction exists first.
Speakpigeon wrote:
Scott Mayers wrote:
Wed May 01, 2019 2:47 pm
This should help show that implication rules are not at odds here. If you wanted to ADD the (2) as a possible form in some logic, then you might define this possibility as the 'Negation of Validity' where the (1) is the 'Position of Validity'. See how this can still enable inclusion if you WANT to cover all possibilities?

I don't think "exploding" to an infinity of possibilities is faulty because any non-P is understood to be the infinity of all that is not-P.
???
Sorry, I don't understand your point here.
It should be remembered that it is mathematicians themselves who call it a "paradox", the paradox of material implication. So, this was an admission that it was an unfortunate situation at the very least.
EB
SOME call it a paradox for it being unresolved in their opinion. This is noted in most texts to be fair about the controversies of others but not usually of those writing the texts themselves.

I'm not sure what concern you might have for this thread and so would need a better example that you could express of what is contradicting here.

If you already agree but pointing out that it isn't an actual contradiction, I agree and my post is in support of that point.

Scott Mayers
Posts: 1391
Joined: Wed Jul 08, 2015 1:53 am

Re: Possible consequences of falsifying the principle of explosion?

The actual paradox of material implication is precisely stating that GIVEN any P, you can conclude some arbitrary Q implies P.

P ⊢ Q ⭢ P

Here, the Q is arbitrary in the similar way as to the OR-introduction rule. As long as P still follows from P, some Q may also be added as

P ⊢ P v Q

Since P is always true even where Q MAY be false, then if follows that IF Q were to be true, P happens to already be true.

EDIT: And it is only paradoxical if you interpret the meaning of the statements as related to each other.

Letting Q = ~P, we get

from
[theorem]: P ⊢ Q ⭢ P,

P ⊢ (~P) ⭢ P

And if I were to have this used in some argument, it SEEMS odd when we go

1......(1)P.....................[assumption]
.......(2)P ⊢ ~P ⭢ P.........[theorem/non-assumption]
1......(3)~P ⭢ P ............[1,2 Modus Ponendo Ponens]
4......(4)~P...................[assumption]
1,4...(5) P & ~P..............[3,4 Modus Ponendo Ponens]

The last line is contradictory ONLY based upon the assumption THAT ~P. Propositional Calculus adds another rule, to repair this with a rule against contradiction called, Reductio-ad-Absurdum (RAA). Continuing from above,

1.....(6)~~P..................[4,5 RAA]
1.....(7)P....................[6 Double Negation]

Thus we end only on the original assumption.
[I don't know how familiar or not you are with this but only I only add in case you already do.]

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Possible consequences of falsifying the principle of explosion?

Logik wrote:
Wed May 01, 2019 2:16 pm
PeteOlcott wrote:
Wed May 01, 2019 1:37 pm
Another way is sound deductive inference that requires all premises to be true.
P1. If it rains the ground will be wet.
P2. It rains.
C. The ground is wet.

Empirical counter-example (a.k.a FALSIFICATION) : https://www.youtube.com/watch?v=XjXvkIzUTtk

Examining the argument + evidence holistically, what you have is a sound-and-valid argument with an incorrect conclusion. Oops?

https://en.wikipedia.org/wiki/Is_Logic_Empirical%3F
What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In his seminal paper "Two Dogmas of Empiricism," the logician and philosopher W. V. Quine argued that all beliefs are in principle subject to revision in the face of empirical data, including the so-called analytic propositions. Thus the laws of logic, being paradigmatic cases of analytic propositions, are not immune to revision.
The "laws" of logic are provisional - they are subject to revision.

Please stop peddling your dogma. Logic is just a tool. There is no "truth" hiding there - look somewhere else.
The only actual truth that exists is merely the tautological connections between names.
One cannot eat a bowl of integers.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Possible consequences of falsifying the principle of explosion?

Speakpigeon wrote:
Wed May 01, 2019 4:48 pm
PeteOlcott wrote:
Wed May 01, 2019 1:37 pm
I went through this with many hundreds of messages.
https://plato.stanford.edu/entries/logic-relevance/ is one way to break the POE.
Another way is sound deductive inference that requires all premises to be true.
I'm not asking how to prove the principe wrong but what would be the consequences of it being proved wrong.
PeteOlcott wrote:
Wed May 01, 2019 1:37 pm
It is a quite nutty idea that logic allows logical consequence to follow from a contradiction,
it is almost like concluding that someone it telling the truth on the basis that they are a liar.
I agree it's a nutty idea but as I understand it, mathematicians adopted it because, first, they couldn't find any better and, second, the principle has no adverse consequence precisely because it only affects validity, not soundness. Don't you agree with that?
EB
Well put !

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Possible consequences of falsifying the principle of explosion?

Speakpigeon wrote:
Wed May 01, 2019 4:53 pm
PeteOlcott wrote:
Wed May 01, 2019 3:03 pm
Relevance logicians claim that what is unsettling about these so-called paradoxes is that in each of them the antecedent seems irrelevant to the consequent.
It just shows "logicians" don't understand logic, not that there is any real problem, i.e. any real consequence for mathematics and mathematical theorems.
EB
I think that it formalizes the concept of logical entailment incorrectly.
It seems to show cause-and-effect between semantically unrelated things
when none actually exists because they are semantically unrelated.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Possible consequences of falsifying the principle of explosion?

Speakpigeon wrote:
Wed May 01, 2019 5:15 pm
I agree mathematical logic is wrong. It is wrong in the sense that it doesn't formalise properly the logic of human reasoning. However, it seems we have to put up with this situation because nobody can offer a better alternative. Further, if the problem has no impact on mathematics then there is in fact no problem. If you disagree with this, please explain what bad consequence on mathematics you think there are.
EB
"It is wrong in the sense that it doesn't formalise properly the logic of human reasoning."
This screws all kind of things up such as "proving" Tarski Undefinability when Truth <is>
defined as simply as this:

If we simply construe Axioms as expressions of language having the semantic property of
Boolean true this would anchor the syntax of formal proofs to the semantics of Boolean values.

Now we have: [Deductively Sound Formal Proofs] True(x) ↔ (⊢x)
[a connected sequence of inference from axioms to a true consequence].
AKA the same universal truth predicate that Tarski "proved" was impossible.

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