Possible consequences of falsifying the principle of explosion?

[PeteOlcott]

Tarski's proof:
"It is wrong in the sense that it doesn't formalise properly the logic of human reasoning."
When we "formalise properly the logic of human reasoning" we get:
[a connected set of known truths necessarily always derives truth]

With no undecidability, incompleteness or inconsistency.

he 1936 Tarski Undefinability Proof
http://liarparadox.org/Tarski_Proof_275_276.pdf

I don't think this is the problem you seem to believe it is.
The concept of truth which Tarski explicitly admits to in the book your referencing here, is truth as correspondence between a description and the thing described. So, if we accept that we know some thing, then it makes sense to qualify our description of this thing as "true", whatever our description may be. Thus, the description of the thing you know as you know it is necessarily true because true by definition of the word "true". Thus, the concept of truth makes sense.
It seems you've just described the concept of truth, in the context of logic, as you think you know it. We can all do it each in our own way and it's fine.
Yet, deciding on a definition you accept doesn't mean it's free of Tarski proof that truth cannot be defined in a logically coherent way.
So, the proof is in the pudding: Apply Tarski's proof to your definition of truth as a sentence you assert is true. Why exactly would Tarski's proof fail with your sentence given that it applies to all sentences? What's wrong in Tarski's proof?
EB

The 1936 Tarski Undefinability Proof
http://liarparadox.org/Tarski_Proof_275_276.pdf

It is trivial to verify that that the third steps of Tarski's proof:
the symbol 'Pr' which denotes the class of all provable sentences
we denote the class of all true sentences by the symbol 'Tr'
(3) x ∉ Pr if and only if x ∈ Tr // ~Provable(x) ↔ True(x)
simply assumes that Provability diverges from Truth.

[Deductively Sound Formal Proofs]
True Premises Necessarily derive a True Consequence: ◻(True(P) ⊢ True(C))
Provability is the valid inference connection between true premises and consequence.

Thus making the third step of Tarski's proof: ~Provable(x) ↔ True(x) absurd.
Summing up the conclusion of the Tarski Proof we have: ¬◻(True(P) ⊢ True(C))

[Scott Mayers]

Right, so you don't have any example outside cases of a false antecedent.
As I understand it, those cases cannot have any consequence, in mathematics or elsewhere, precisely because the antecedent is false, which makes the argument valid but unsound. Isn't that correct?
Maybe I'm wrong since apparently no one seems willing to answer conclusively my simple question, so, presumably no one knows the answer.
EB

You understand it incorrectly: It has a valid consequence.

https://en.wikipedia.org/wiki/Principle ... esentation
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.)

If living_thing->animal->cat is an office building then it logically
follows that an ice cream cone is the actual creator of the universe.

There is no difference between "P, ¬P ⊢ Q " and P & ~P ⊢ Q

I disagree with that link's assertion as it ignores covering all assumptions in the conclusion. Look at my example argument earlier. The left-hand side indicates the assumptions. The assumptions in the argument in the final conclusion needs to include the given assumptions or is 'incomplete'. In the link's example, it takes the ~P but ignores the P in the final conclusion and thus is incomplete.

For using separate premises P, ~P, this also implies (P & ~P). Prior to recognizing the details needed for closure of a system, these MAY have been considered problems but no longer are when the logic is appropriately done. I'm guessing that many don't follow through properly when studying their logic and miss these details. Then they RE-discover the errors for which they think they've discovered some flaw but misrepresent the proper methods.

[PeteOlcott]

Right, so you don't have any example outside cases of a false antecedent.
As I understand it, those cases cannot have any consequence, in mathematics or elsewhere, precisely because the antecedent is false, which makes the argument valid but unsound. Isn't that correct?
Maybe I'm wrong since apparently no one seems willing to answer conclusively my simple question, so, presumably no one knows the answer.
EB

You understand it incorrectly: It has a valid consequence.

https://en.wikipedia.org/wiki/Principle ... esentation
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.)

If living_thing->animal->cat is an office building then it logically
follows that an ice cream cone is the actual creator of the universe.

There is no difference between "P, ¬P ⊢ Q " and P & ~P ⊢ Q

I disagree with that link's assertion

I had to quit talking to another poster because we disagreed that 1 != 0,
You too are now disagreeing with a known fact.
P ∧ ¬P ⊢ Q is what the principle of explosion is.

[Scott Mayers]

You understand it incorrectly: It has a valid consequence.

https://en.wikipedia.org/wiki/Principle ... esentation
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.)

If living_thing->animal->cat is an office building then it logically
follows that an ice cream cone is the actual creator of the universe.

There is no difference between "P, ¬P ⊢ Q " and P & ~P ⊢ Q

I disagree with that link's assertion

I had to quit talking to another poster because we disagreed that 1 != 0,
You too are now disagreeing with a known fact.
P ∧ ¬P ⊢ Q is what the principle of explosion is.

The "Principle of Explosion" is just the justification for evading CONTRADICTION. It is a negotiated agreement and convention for those developing a system of reasoning for practical reasons to DISALLOW CONTRACTION in binary 'true/false' systems.

I more appropriately expressed what it is above. This is just a colloquial justification, not a logical one, to DEFINE symbolic binary systems to require the apriori limiting condition of disallowing contradiction.

I am NOT denying your simplistic argument concerns by others. I AM competent to argue using the systems in question. I must ask you if YOU are familiar with the form of argument style I presented earlier because I notice whenever I use it, it gets bypassed as though it went over your heads! And that is merely Propositional logic that is fundamental to anyone discussing this. If you lack experience in this, then the apparent confusion of these issues is also going over your heads and it would be I who is wasting time on posters (or, better, 'posers') on logic discussions here.

[PeteOlcott]

There is no difference between "P, ¬P ⊢ Q " and P & ~P ⊢ Q

I disagree with that link's assertion

I had to quit talking to another poster because we disagreed that 1 != 0,
You too are now disagreeing with a known fact.
P ∧ ¬P ⊢ Q is what the principle of explosion is.

The "Principle of Explosion" is just the justification for evading CONTRADICTION. It is a negotiated agreement and convention for those developing a system of reasoning for practical reasons to DISALLOW CONTRACTION in binary 'true/false' systems.

I more appropriately expressed what it is above. This is just a colloquial justification, not a logical one, to DEFINE symbolic binary systems to require the apriori limiting condition of disallowing contradiction.

Ah that makes much more sense. So if one were to further elaborate
something along the lines of relevance logic, one could supersede and
replace the principle of explosion without being denigrated for ignoring
sacred conventions.

[Scott Mayers]

For an example, if P = "I am alive in 2019", with some 'universal' of "Those things alive between 2000 to 2019" AND then if we have

P & ~P,

While 'contradictory' if I lived in this period, we can have an infinite possible Qs. One such Q might be "the universal is NOW between 1800 and 2019."

Here the possibilities are infinite (explosive) but suggest that we can still make a system that makes Q DO something. If we want some kind of system that might be useful, we could have it expand the domain one year more and one year less: New Universe = "Those things alive between X- 1 to X+1."
If it is still true that P & ~P, repeat the process until it is 'true'.

Q can also just BE defined as "Not Real/False" and tells us to 'stop'. This is what lead to the thinking of Turing to define a system that was restricted to 'true/false'. He used our binary NON-contradictory rule (in light of the explanation of explosiveness) but showed that such a system is still incomplete precisely because we CAN have computers based on this that 'hang'. Obviously if it hangs infinitely , we can't make it 'update' to a new universal that stops.

I think reality makes better sense this way. Contradiction to me IS 'force' that compel reality to BE infinite as some truly 'infinite universal machine'. For our practical purposes, we just add another 'logic' that places artificial limits on how many times we update the universal domain.

A human example might be to have some mechanism that can solve some unknown murder. We may never be able to realistically solve some case but KNOW that the greater infinite universe holds the truth. Then we set artificial limits to the overall larger logic we are using to determine this indefinite problem, like a 'statute of limitation' rule in the logic-system of a government that makes laws.

So the reality to the OP's question is that there are an infinity of 'consequences' for falsifying the principle (ie, allow contradiction). This is allowed and is in fact REAL. There is still a 'rationale' to it in a closed sense but it requires an infinite amount of time to actually prove useful when we ourselves are limited.

[Speakpigeon]

The question is whether the inference A ∧ ¬A ⊢ B, which is deemed valid in standard mathematical logic, is used in the proof of theorems in standard mathematics.

That statement does NOT mean anything as it stands. So you need to prove what it could mean first.

LOL
Learn English first, then we might have a conversation.
EB

[Speakpigeon]

Right, so you don't have any example outside cases of a false antecedent.
As I understand it, those cases cannot have any consequence, in mathematics or elsewhere, precisely because the antecedent is false, which makes the argument valid but unsound. Isn't that correct?
Maybe I'm wrong since apparently no one seems willing to answer conclusively my simple question, so, presumably no one knows the answer.

You understand it incorrectly: It has a valid consequence.
https://en.wikipedia.org/wiki/Principle ... esentation
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.)
If living_thing->animal->cat is an office building then it logically
follows that an ice cream cone is the actual creator of the universe.

I understand that all good, thanks.
Maybe I can rephrase my point: Give me an example where p ∧ ¬p ⊢ q is used to prove an inference where a true premise A entails any B (i.e. A ⊢ B).
EB

[PeteOlcott]

Right, so you don't have any example outside cases of a false antecedent.
As I understand it, those cases cannot have any consequence, in mathematics or elsewhere, precisely because the antecedent is false, which makes the argument valid but unsound. Isn't that correct?
Maybe I'm wrong since apparently no one seems willing to answer conclusively my simple question, so, presumably no one knows the answer.

You understand it incorrectly: It has a valid consequence.
https://en.wikipedia.org/wiki/Principle ... esentation
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.)
If living_thing->animal->cat is an office building then it logically
follows that an ice cream cone is the actual creator of the universe.

I understand that all good, thanks.
Maybe I can rephrase my point: Give me an example where p ∧ ¬p ⊢ q is used to prove an inference where a true premise A entails any B (i.e. A ⊢ B).
EB

You have your p,q and A,B conflated so I cannot tell what refers to what.
As soon as you start talking about true premises the Principle of Explosion quits working.

[Speakpigeon]

Maybe I can rephrase my point: Give me an example where p ∧ ¬p ⊢ q is used to prove an inference where a true premise A entails any B (i.e. A ⊢ B).

As soon as you start talking about true premises the Principle of Explosion quits working.

Exactly.
So, how could there be any consequence of the principle of explosion, even for mathematical theorems?
I don't know of any mathematical theorem based on the principle of explosion. Do you?
EB

[PeteOlcott]

Maybe I can rephrase my point: Give me an example where p ∧ ¬p ⊢ q is used to prove an inference where a true premise A entails any B (i.e. A ⊢ B).

As soon as you start talking about true premises the Principle of Explosion quits working.

Exactly.
So, how could there be any consequence of the principle of explosion, even for mathematical theorems?
I don't know of any mathematical theorem based on the principle of explosion. Do you?
EB

The principle of explosion tells the lie that an antecedent unrelated to the consequent logically entails the consequent.

Apparently it was set up this way for practical reasons. In computer science we would call this a kludge.

[Speakpigeon]

Exactly.
So, how could there be any consequence of the principle of explosion, even for mathematical theorems?
I don't know of any mathematical theorem based on the principle of explosion. Do you?

The principle of explosion tells the lie that an antecedent unrelated to the consequent logically entails the consequent.
Apparently it was set up this way for practical reasons. In computer science we would call this a kludge.

Sure, I already said I agree with you on that. I personally researched the reason why mathematicians got there to begin with. It's a pragmatic move. Bad metaphysics but it works all good. It works because there is no known logical truth falsified by ECQ and ECQ was already criticised by most logicians when it was first stated in the 12th century. The reason mathematicians keep it is that it has no known consequence. I also don't believe there is way to show that the truth table of the material implication is wrong even though this is the origin of the paradox. All that this shows is that mathematicians don't really understand Aristotelian logic while at the same time pretending they understand what they are doing. This is what is pathetic. This is what is misleading. Even calling mathematical logic "logic" is misleading. It is mathematics. Geometry is not space. Curved geometry isn't our intuitive sense of space. So, mathematicians should call all their theories something like "material logic", just as there is a logic to a toaster or to a fridge only it's not human logic (and computer logic isn't human logic either). They also insist, most of them at least, that mathematical logic is in line with Aristotelian logic. That is just a pathetic claim. It obviously isn't, essentially for the reasons you give. Whether it's a straightforward lie or evidence they have a very limited intelligence, I don't know, but something is definitely wrong with them.

My question to you is simple: What do you think would have to be the mathematical consequence if mathematicians ever admitted that ECQ was wrong?

For now, you seem to accept the answer is: none whatsoever.
EB

[PeteOlcott]

Exactly.
So, how could there be any consequence of the principle of explosion, even for mathematical theorems?
I don't know of any mathematical theorem based on the principle of explosion. Do you?

The principle of explosion tells the lie that an antecedent unrelated to the consequent logically entails the consequent.
Apparently it was set up this way for practical reasons. In computer science we would call this a kludge.

Sure, I already said I agree with you on that. I personally researched the reason why mathematicians got there to begin with. It's a pragmatic move. Bad metaphysics but it works all good. It works because there is no known logical truth falsified by ECQ and ECQ was already criticised by most logicians when it was first stated in the 12th century. The reason mathematicians keep it is that it has no known consequence. I also don't believe there is way to show that the truth table of the material implication is wrong even though this is the origin of the paradox. All that this shows is that mathematicians don't really understand Aristotelian logic while at the same time pretending they understand what they are doing. This is what is pathetic. This is what is misleading. Even calling mathematical logic "logic" is misleading. It is mathematics. Geometry is not space. Curved geometry isn't our intuitive sense of space. So, mathematicians should call all their theories something like "material logic", just as there is a logic to a toaster or to a fridge only it's not human logic (and computer logic isn't human logic either). They also insist, most of them at least, that mathematical logic is in line with Aristotelian logic. That is just a pathetic claim. It obviously isn't, essentially for the reasons you give. Whether it's a straightforward lie or evidence they have a very limited intelligence, I don't know, but something is definitely wrong with them.

My question to you is simple: What do you think would have to be the mathematical consequence if mathematicians ever admitted that ECQ was wrong?

For now, you seem to accept the answer is: none whatsoever.
EB

"The reason mathematicians keep it is that it has no known consequence."
It sure screws up formalizing natural language using conventional logic as the basis.
In a court of law it would conclude that the witness is telling the truth on the basis that they have lied.

If it was not for screwball stuff like the POE, we could not have gone all these decades
with people believing that Incompleteness and Undefinability was true.

Because of screwball things such as POE every attempt at correcting the error that
lead to the misconceptions of Incompleteness and Undefinability was rejected
out-of-hand as unconventional.

Deductively Sound Formal Proofs
https://www.researchgate.net/publicatio ... mal_Proofs
Now that I corrected this problem, the solution continues to be rejected out-of-hand as unconventional.

[Scott Mayers]

The question is whether the inference A ∧ ¬A ⊢ B, which is deemed valid in standard mathematical logic, is used in the proof of theorems in standard mathematics.

That statement does NOT mean anything as it stands. So you need to prove what it could mean first.

LOL
Learn English first, then we might have a conversation.
EB

What's that supposed to mean?

Given you keep responding similar to me this way in anything I've discussed anything with you, I'm thinking you've got some beef with me? Do you want to let me in on your problem?

[Speakpigeon]

What's that supposed to mean?
Given you keep responding similar to me this way in anything I've discussed anything with you, I'm thinking you've got some beef with me? Do you want to let me in on your problem?

I don't have any problem with you. I do seem to have a problem with most of your comments.
My statement below is perfectly good English and any idiot can understand it.

The question is whether the inference A ∧ ¬A ⊢ B, which is deemed valid in standard mathematical logic, is used in the proof of theorems in standard mathematics.

If you want to claim my statement is meaningless, then it's up to you to prove it is meaningless.
EB