*Principle of explosion*
https://en.wikipedia.org/wiki/Principle_of_explosion#Proof

<quote>
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)
</quote>

*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
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) this resolves 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.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 5/21/23 10:52 PM, olcott wrote:
> *Principle of explosion*
> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>
> <quote>
> 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

Not really an "Assumption", but a proven given.

The problem with working with oversimpified explanations.

> 2 ------- ¬P ------------ Assumption

Not really an "Assumption", but a proven given.

> 3 -------- P ∨ Q -------- Disjunction introduction (1)
> 4 -------- Q ------------ Disjunctive syllogism (3,2)
> </quote>
>
> *Truth table for P ∧ ¬P*
> P----¬P----P ∧ ¬P
> T-----F------F
> F-----T------F

So?

There is no requirement to check a given resul with all rules defined in
the system.

>
> 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

Where is that statement in the expression?

Also, we know that True ∧ ¬True ⊢ True,

So which rule gets priority.

Note, it is the "Assumptions" that "violate" the rule, so the
assumptions are the source of the issue.

> 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) this resolves to false.

No, it is the divergance of the assumptions from correct logic. The key
is that the principle of explosion points out that if we can prove a
contradiction, our system has inconsistent givens.

>
> 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.
>

It isn't that the principle of eplosion diverges from correct reasoning,
the principle of explosion just points out that a system that has
allowed the proving of a contradiction has deviated from "Correctness"
by the fact that it can prove a contradiction, and a system that can
prove one contradiction can prove many.

You are just showing how little you understand how logic works.

On 5/22/23 7:49 AM, Richard Damon wrote:
> On 5/21/23 10:52 PM, olcott wrote:
>> *Principle of explosion*
>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>
>> <quote>
>> 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
>
> Not really an "Assumption", but a proven given.
>
> The problem with working with oversimpified explanations.
>
>> 2 ------- ¬P ------------ Assumption
>
>
> Not really an "Assumption", but a proven given.
>
>> 3 -------- P ∨ Q -------- Disjunction introduction (1)
>> 4 -------- Q ------------ Disjunctive syllogism (3,2)
>> </quote>
>>
>> *Truth table for P ∧ ¬P*
>> P----¬P----P ∧ ¬P
>> T-----F------F
>> F-----T------F
>
> So?
>
> There is no requirement to check a given resul with all rules defined in
> the system.
>
>>
>> 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
>
> Where is that statement in the expression?
>
> Also, we know that True ∧ ¬True ⊢ True,
>
> So which rule gets priority.
>
>
> Note, it is the "Assumptions" that "violate" the rule, so the
> assumptions are the source of the issue.
>
>
>> 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) this resolves to false.
>
> No, it is the divergance of the assumptions from correct logic. The key
> is that the principle of explosion points out that if we can prove a
> contradiction, our system has inconsistent givens.
>
>>
>> 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.
>>
>
> It isn't that the principle of eplosion diverges from correct reasoning,
> the principle of explosion just points out that a system that has
> allowed the proving of a contradiction has deviated from "Correctness"
> by the fact that it can prove a contradiction, and a system that can
> prove one contradiction can prove many.
>
> You are just showing how little you understand how logic works.
>

As an addition, since it is your adding the assumption that H can call
an input non-halting just because it determines that no variation of H
can correctly simulate that input to a final state, and this rules leads
us to being able to actually prove that D(D) is both Halting (by running
it) and Non-Halting (by asking H) says that you added assumption makes
the system inconsistent, and thus your assumption is a violation of
correct reasoning.

On 5/22/2023 6:49 AM, Richard Damon wrote:
> On 5/21/23 10:52 PM, olcott wrote:
>> *Principle of explosion*
>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>
>> <quote>
>> 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
>
> Not really an "Assumption", but a proven given.
>

So you don't understand that the premises to a proof are only
assumptions? A proof is always based on the hypothetical assumption that
its premises are true. The intent is to see if a conclusion can be drawn
by applying truth preserving operations to these premises. If this is
the case then we know that the conclusion logically follows from its
premises, we don't know that the conclusion is true unless it is also
stipulated that the premises are true.

> The problem with working with oversimpified explanations.
>
>> 2 ------- ¬P ------------ Assumption
>
>
> Not really an "Assumption", but a proven given.
>
>> 3 -------- P ∨ Q -------- Disjunction introduction (1)
>> 4 -------- Q ------------ Disjunctive syllogism (3,2)
>> </quote>
>>
>> *Truth table for P ∧ ¬P*
>> P----¬P----P ∧ ¬P
>> T-----F------F
>> F-----T------F
>
> So?
>
> There is no requirement to check a given resul with all rules defined in
> the system.
>
>>
>> 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
>
> Where is that statement in the expression?
>

I am saying that by not first resolving P ∧ ¬P to false according to
the truth table of P ∧ ¬P an absurd conclusion is drawn. This is where
logic diverges from correct reasoning.

> Also, we know that True ∧ ¬True ⊢ True,
>

We are not dealing with True ∧ ¬True.
We are dealing with P ∧ ¬P ⊢ False.

> So which rule gets priority.
>
>
> Note, it is the "Assumptions" that "violate" the rule, so the
> assumptions are the source of the issue.
>

Easily resolved when P ∧ ¬P ⊢ False is resolved before proceeding.

>> 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) this resolves to false.
>
> No, it is the divergance of the assumptions from correct logic. The key
> is that the principle of explosion points out that if we can prove a
> contradiction, our system has inconsistent givens.
>

Yet we cannot prove a contradiction when when P ∧ ¬P ⊢ False is
resolved before proceeding.

>>
>> 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.
>>
>
> It isn't that the principle of eplosion diverges from correct reasoning,
> the principle of explosion just points out that a system that has
> allowed the proving of a contradiction has deviated from "Correctness"
> by the fact that it can prove a contradiction, and a system that can
> prove one contradiction can prove many.
>
> You are just showing how little you understand how logic works.
>

I am showing how little logic knows about correct reasoning.
We know from its truth table that P ∧ ¬P ⊢ False, thus the
short-circuit that proves a contradiction violates the truth
table of P ∧ ¬P.

We also know that Disjunction introduction to False breaks the
truth preserving requirements by transforming the unknown truth
value of Q to true.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 5/22/23 10:12 AM, olcott wrote:
> On 5/22/2023 6:49 AM, Richard Damon wrote:
>> On 5/21/23 10:52 PM, olcott wrote:
>>> *Principle of explosion*
>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>
>>> <quote>
>>> 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
>>
>> Not really an "Assumption", but a proven given.
>>
>
> So you don't understand that the premises to a proof are only
> assumptions? A proof is always based on the hypothetical assumption that
> its premises are true. The intent is to see if a conclusion can be drawn
> by applying truth preserving operations to these premises. If this is
> the case then we know that the conclusion logically follows from its
> premises, we don't know that the conclusion is true unless it is also
> stipulated that the premises are true.

No, most of the times, they are proven statements in the system. Yes,
you can start a proof starting with assuptions and get an answer that is
conditional on the assumption being true, but most of the time, your
input premises are known to be true, so you aren't working on a possible
dead end.

For the principle of explosion, they are statements of pre-condition,
the principle of explosion only applies *IF* you can establish that
there is some condition P, such that you can show that P and ~P are both
true.

Thus, they are "assumptions", but pre-conditions for the results.

Yes, you perform the operations of the proof starting with them as
"assumptions", but then they become the pre-condition of the theory, and
to be applied, they must be proven facts.

>
>> The problem with working with oversimpified explanations.
>>
>>> 2 ------- ¬P ------------ Assumption
>>
>>
>> Not really an "Assumption", but a proven given.
>>
>>> 3 -------- P ∨ Q -------- Disjunction introduction (1)
>>> 4 -------- Q ------------ Disjunctive syllogism (3,2)
>>> </quote>
>>>
>>> *Truth table for P ∧ ¬P*
>>> P----¬P----P ∧ ¬P
>>> T-----F------F
>>> F-----T------F
>>
>> So?
>>
>> There is no requirement to check a given resul with all rules defined
>> in the system.
>>
>>>
>>> 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
>>
>> Where is that statement in the expression?
>>
>
> I am saying that by not first resolving P ∧ ¬P to false according to
> the truth table of P ∧ ¬P an absurd conclusion is drawn. This is where
> logic diverges from correct reasoning.

So, do you check ALL your premises to ALL other possible rules?

If so, why don't you see that H must be wrong, because even without
doing your argument about what any H can do, since we have the
definition of your H, and it is establish fact, just by running it, that
H(D,D) returns 0, and it is established fact that by running D(D) it
Halts, that H must be returning the wrong answer, since that is what it
is supposed to be answering.

ANSWER: You haven't check the established rules, and thus don't see that
your "assumption" that H can answer non-halting because no H can
simulate its input to a final state leads to an inconsistant logic system.

>
>> Also, we know that True ∧ ¬True ⊢ True,
>>
>
> We are not dealing with True ∧ ¬True.
> We are dealing with P ∧ ¬P ⊢ False.

No, we are dealing with a logic arguement that begins:

Assume P,
Assume ¬P

Therefore, when we evaluate P ∧ ¬P we know the values and can substitute
them in. And thus your rule becomes True ⊢ False

In fact, once we have the two assumptions, we can show the system in
inconsistant, and the principle of explosion is just showing the
possible effect of that.

>
>> So which rule gets priority.
>>
>>
>> Note, it is the "Assumptions" that "violate" the rule, so the
>> assumptions are the source of the issue.
>>
>
> Easily resolved when P ∧ ¬P ⊢ False is resolved before proceeding.

But you don't need to do that, and in fact, it is superfluous, as the
definition of NOT is defied already.

>
>>> 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) this resolves to false.
>>
>> No, it is the divergance of the assumptions from correct logic. The
>> key is that the principle of explosion points out that if we can prove
>> a contradiction, our system has inconsistent givens.
>>
>
> Yet we cannot prove a contradiction when when P ∧ ¬P ⊢ False is
> resolved before proceeding.

So, you don't understand how logic works, proving your ignorance.

We HAD the contradition once we had P and ~P as premises.

>
>>>
>>> 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.
>>>
>>
>> It isn't that the principle of eplosion diverges from correct
>> reasoning, the principle of explosion just points out that a system
>> that has allowed the proving of a contradiction has deviated from
>> "Correctness" by the fact that it can prove a contradiction, and a
>> system that can prove one contradiction can prove many.
>>
>> You are just showing how little you understand how logic works.
>>
>
> I am showing how little logic knows about correct reasoning.
> We know from its truth table that P ∧ ¬P ⊢ False, thus the
> short-circuit that proves a contradiction violates the truth
> table of P ∧ ¬P.

No, you are showing how little you know about logic.

You concept of "Correct Reasoning" seems doomed to failure because it
doesn't understand how logic works.

>
> We also know that Disjunction introduction to False breaks the
> truth preserving requirements by transforming the unknown truth
> value of Q to true.
>

So, if A and B are know Truth Bearers, and A ∨ B is proven to be a True
expression, and then A is proven to be False, what is B?

On 5/22/2023 5:58 PM, Richard Damon wrote:
> On 5/22/23 10:12 AM, olcott wrote:
>> On 5/22/2023 6:49 AM, Richard Damon wrote:
>>> On 5/21/23 10:52 PM, olcott wrote:
>>>> *Principle of explosion*
>>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>>
>>>> <quote>
>>>> 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
>>>
>>> Not really an "Assumption", but a proven given.
>>>
>>
>> So you don't understand that the premises to a proof are only
>> assumptions? A proof is always based on the hypothetical assumption that
>> its premises are true. The intent is to see if a conclusion can be drawn
>> by applying truth preserving operations to these premises. If this is
>> the case then we know that the conclusion logically follows from its
>> premises, we don't know that the conclusion is true unless it is also
>> stipulated that the premises are true.
>
> No, most of the times, they are proven statements in the system. Yes,
> you can start a proof starting with assuptions and get an answer that is
> conditional on the assumption being true, but most of the time, your
> input premises are known to be true, so you aren't working on a possible
> dead end.
>

Do You know the Mendelson Text?
According to Mendelson only theorems have premises known to be true.

> For the principle of explosion, they are statements of pre-condition,
> the principle of explosion only applies *IF* you can establish that
> there is some condition P, such that you can show that P and ~P are both
> true.
>

That is the same thing as saying if horses were office buildings then
you could wash their windows.

It is a truism that when you assume contradictory premises that they
only derive False. When you try this semantically it is obvious.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 5/22/23 8:48 PM, olcott wrote:
> On 5/22/2023 5:58 PM, Richard Damon wrote:
>> On 5/22/23 10:12 AM, olcott wrote:
>>> On 5/22/2023 6:49 AM, Richard Damon wrote:
>>>> On 5/21/23 10:52 PM, olcott wrote:
>>>>> *Principle of explosion*
>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>>>
>>>>> <quote>
>>>>> 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
>>>>
>>>> Not really an "Assumption", but a proven given.
>>>>
>>>
>>> So you don't understand that the premises to a proof are only
>>> assumptions? A proof is always based on the hypothetical assumption that
>>> its premises are true. The intent is to see if a conclusion can be drawn
>>> by applying truth preserving operations to these premises. If this is
>>> the case then we know that the conclusion logically follows from its
>>> premises, we don't know that the conclusion is true unless it is also
>>> stipulated that the premises are true.
>>
>> No, most of the times, they are proven statements in the system. Yes,
>> you can start a proof starting with assuptions and get an answer that
>> is conditional on the assumption being true, but most of the time,
>> your input premises are known to be true, so you aren't working on a
>> possible dead end.
>>
>
> Do You know the Mendelson Text?
> According to Mendelson only theorems have premises known to be true.

Right, or CONDITIONS under which they hold. For instance, Godel's
incompleteness proof only holds for systems that support the needed
properties of the Natural Numbers, and which are consistant.

The principle of Explosion holds for system that system that support the
Disjunctive Syllogism (or some other similar operations) AND can prove
two contradictory premises.

>
>> For the principle of explosion, they are statements of pre-condition,
>> the principle of explosion only applies *IF* you can establish that
>> there is some condition P, such that you can show that P and ~P are
>> both true.
>>
>
> That is the same thing as saying if horses were office buildings then
> you could wash their windows.
>

You would first need some axioms that establish that horses have windows.

Note, "Formal Systems" don't need to model the physical universe.

Remember, if wishes were horses then beggars would ride.

> It is a truism that when you assume contradictory premises that they
> only derive False. When you try this semantically it is obvious.
>

Right, but the principle of explosion isn't based on ASSUMING
contradictory premises, but on being able to PROVE contradictory
premises, which is an indication that you axioms are inconsistent.

Like it appears your "Correct Reasoning" is, as somehow it seems to be
saying that a Halting Computation can be corrected described as
Non-Halting, amoung other contradictions.

On 5/22/2023 8:09 PM, Richard Damon wrote:
> On 5/22/23 8:48 PM, olcott wrote:
>> On 5/22/2023 5:58 PM, Richard Damon wrote:
>>> On 5/22/23 10:12 AM, olcott wrote:
>>>> On 5/22/2023 6:49 AM, Richard Damon wrote:
>>>>> On 5/21/23 10:52 PM, olcott wrote:
>>>>>> *Principle of explosion*
>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>>>>
>>>>>> <quote>
>>>>>> 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
>>>>>
>>>>> Not really an "Assumption", but a proven given.
>>>>>
>>>>
>>>> So you don't understand that the premises to a proof are only
>>>> assumptions? A proof is always based on the hypothetical assumption
>>>> that
>>>> its premises are true. The intent is to see if a conclusion can be
>>>> drawn
>>>> by applying truth preserving operations to these premises. If this is
>>>> the case then we know that the conclusion logically follows from its
>>>> premises, we don't know that the conclusion is true unless it is also
>>>> stipulated that the premises are true.
>>>
>>> No, most of the times, they are proven statements in the system. Yes,
>>> you can start a proof starting with assuptions and get an answer that
>>> is conditional on the assumption being true, but most of the time,
>>> your input premises are known to be true, so you aren't working on a
>>> possible dead end.
>>>
>>
>> Do You know the Mendelson Text?
>> According to Mendelson only theorems have premises known to be true.
>
> Right, or CONDITIONS under which they hold. For instance, Godel's
> incompleteness proof only holds for systems that support the needed
> properties of the Natural Numbers, and which are consistant.
>
> The principle of Explosion holds for system that system that support the
> Disjunctive Syllogism (or some other similar operations) AND can prove
> two contradictory premises.
>
So assuming that P ∧ ¬P ⊢ True (which it does not) then the principle
of explosion is entailed.

"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."

We already knew that the law of of non-contradiction is necessarily true
so the POE is redundant.

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

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 5/22/23 9:22 PM, olcott wrote:
> On 5/22/2023 8:09 PM, Richard Damon wrote:
>> On 5/22/23 8:48 PM, olcott wrote:
>>> On 5/22/2023 5:58 PM, Richard Damon wrote:
>>>> On 5/22/23 10:12 AM, olcott wrote:
>>>>> On 5/22/2023 6:49 AM, Richard Damon wrote:
>>>>>> On 5/21/23 10:52 PM, olcott wrote:
>>>>>>> *Principle of explosion*
>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>>>>>
>>>>>>> <quote>
>>>>>>> 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
>>>>>>
>>>>>> Not really an "Assumption", but a proven given.
>>>>>>
>>>>>
>>>>> So you don't understand that the premises to a proof are only
>>>>> assumptions? A proof is always based on the hypothetical assumption
>>>>> that
>>>>> its premises are true. The intent is to see if a conclusion can be
>>>>> drawn
>>>>> by applying truth preserving operations to these premises. If this is
>>>>> the case then we know that the conclusion logically follows from its
>>>>> premises, we don't know that the conclusion is true unless it is also
>>>>> stipulated that the premises are true.
>>>>
>>>> No, most of the times, they are proven statements in the system.
>>>> Yes, you can start a proof starting with assuptions and get an
>>>> answer that is conditional on the assumption being true, but most of
>>>> the time, your input premises are known to be true, so you aren't
>>>> working on a possible dead end.
>>>>
>>>
>>> Do You know the Mendelson Text?
>>> According to Mendelson only theorems have premises known to be true.
>>
>> Right, or CONDITIONS under which they hold. For instance, Godel's
>> incompleteness proof only holds for systems that support the needed
>> properties of the Natural Numbers, and which are consistant.
>>
>> The principle of Explosion holds for system that system that support
>> the Disjunctive Syllogism (or some other similar operations) AND can
>> prove two contradictory premises.
>>
> So assuming that P ∧ ¬P ⊢ True (which it does not) then the principle
> of explosion is entailed.

Sort of, you don't seem to understand that logical terms:

The PRECONDITION to use the Principle of Explosion is that a system can
establish both P and ¬P. You can't say that it can't happen, because
people playing with system accedentally create conditions that allow
such a statement to be proven.

>
> "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."

The Principle of Explosion shows that a classical logic system can not
allow a contradiction, as if it does, it makes truth within the system
meaningless.

>
> We already knew that the law of of non-contradiction is necessarily true
> so the POE is redundant.

No, it shows the effect of violating the "law of non-contradiction".

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

It isn't just hypothetical, because systems exist that violate the "lot
of non-contradiction", and it explains why we need to discard such
systems, like yours.

On 5/22/2023 9:02 PM, Richard Damon wrote:
> On 5/22/23 9:22 PM, olcott wrote:
>> On 5/22/2023 8:09 PM, Richard Damon wrote:
>>> On 5/22/23 8:48 PM, olcott wrote:
>>>> On 5/22/2023 5:58 PM, Richard Damon wrote:
>>>>> On 5/22/23 10:12 AM, olcott wrote:
>>>>>> On 5/22/2023 6:49 AM, Richard Damon wrote:
>>>>>>> On 5/21/23 10:52 PM, olcott wrote:
>>>>>>>> *Principle of explosion*
>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>>>>>>
>>>>>>>> <quote>
>>>>>>>> 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
>>>>>>>
>>>>>>> Not really an "Assumption", but a proven given.
>>>>>>>
>>>>>>
>>>>>> So you don't understand that the premises to a proof are only
>>>>>> assumptions? A proof is always based on the hypothetical
>>>>>> assumption that
>>>>>> its premises are true. The intent is to see if a conclusion can be
>>>>>> drawn
>>>>>> by applying truth preserving operations to these premises. If this is
>>>>>> the case then we know that the conclusion logically follows from its
>>>>>> premises, we don't know that the conclusion is true unless it is also
>>>>>> stipulated that the premises are true.
>>>>>
>>>>> No, most of the times, they are proven statements in the system.
>>>>> Yes, you can start a proof starting with assuptions and get an
>>>>> answer that is conditional on the assumption being true, but most
>>>>> of the time, your input premises are known to be true, so you
>>>>> aren't working on a possible dead end.
>>>>>
>>>>
>>>> Do You know the Mendelson Text?
>>>> According to Mendelson only theorems have premises known to be true.
>>>
>>> Right, or CONDITIONS under which they hold. For instance, Godel's
>>> incompleteness proof only holds for systems that support the needed
>>> properties of the Natural Numbers, and which are consistant.
>>>
>>> The principle of Explosion holds for system that system that support
>>> the Disjunctive Syllogism (or some other similar operations) AND can
>>> prove two contradictory premises.
>>>
>> So assuming that P ∧ ¬P ⊢ True (which it does not) then the principle
>> of explosion is entailed.
>
> Sort of, you don't seem to understand that logical terms:
>
> The PRECONDITION to use the Principle of Explosion is that a system can
> establish both P and ¬P. You can't say that it can't happen, because
> people playing with system accedentally create conditions that allow
> such a statement to be proven.
>
>>
>> "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."
>
> The Principle of Explosion shows that a classical logic system can not
> allow a contradiction, as if it does, it makes truth within the system
> meaningless.
>
>>
>> We already knew that the law of of non-contradiction is necessarily
>> true so the POE is redundant.
>
> No, it shows the effect of violating the "law of non-contradiction".
>

All these years I thought that it was being asserted that the POE is
true and that everything is actually proven from a contradiction.

All the time they were never saying that, they were only saying what
happens when we hypothetically assume that a contradiction is true.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 5/22/23 10:17 PM, olcott wrote:
> On 5/22/2023 9:02 PM, Richard Damon wrote:
>> On 5/22/23 9:22 PM, olcott wrote:
>>> On 5/22/2023 8:09 PM, Richard Damon wrote:
>>>> On 5/22/23 8:48 PM, olcott wrote:
>>>>> On 5/22/2023 5:58 PM, Richard Damon wrote:
>>>>>> On 5/22/23 10:12 AM, olcott wrote:
>>>>>>> On 5/22/2023 6:49 AM, Richard Damon wrote:
>>>>>>>> On 5/21/23 10:52 PM, olcott wrote:
>>>>>>>>> *Principle of explosion*
>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion#Proof
>>>>>>>>>
>>>>>>>>> <quote>
>>>>>>>>> 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
>>>>>>>>
>>>>>>>> Not really an "Assumption", but a proven given.
>>>>>>>>
>>>>>>>
>>>>>>> So you don't understand that the premises to a proof are only
>>>>>>> assumptions? A proof is always based on the hypothetical
>>>>>>> assumption that
>>>>>>> its premises are true. The intent is to see if a conclusion can
>>>>>>> be drawn
>>>>>>> by applying truth preserving operations to these premises. If
>>>>>>> this is
>>>>>>> the case then we know that the conclusion logically follows from its
>>>>>>> premises, we don't know that the conclusion is true unless it is
>>>>>>> also
>>>>>>> stipulated that the premises are true.
>>>>>>
>>>>>> No, most of the times, they are proven statements in the system.
>>>>>> Yes, you can start a proof starting with assuptions and get an
>>>>>> answer that is conditional on the assumption being true, but most
>>>>>> of the time, your input premises are known to be true, so you
>>>>>> aren't working on a possible dead end.
>>>>>>
>>>>>
>>>>> Do You know the Mendelson Text?
>>>>> According to Mendelson only theorems have premises known to be true.
>>>>
>>>> Right, or CONDITIONS under which they hold. For instance, Godel's
>>>> incompleteness proof only holds for systems that support the needed
>>>> properties of the Natural Numbers, and which are consistant.
>>>>
>>>> The principle of Explosion holds for system that system that support
>>>> the Disjunctive Syllogism (or some other similar operations) AND can
>>>> prove two contradictory premises.
>>>>
>>> So assuming that P ∧ ¬P ⊢ True (which it does not) then the principle
>>> of explosion is entailed.
>>
>> Sort of, you don't seem to understand that logical terms:
>>
>> The PRECONDITION to use the Principle of Explosion is that a system
>> can establish both P and ¬P. You can't say that it can't happen,
>> because people playing with system accedentally create conditions that
>> allow such a statement to be proven.
>>
>>>
>>> "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."
>>
>> The Principle of Explosion shows that a classical logic system can not
>> allow a contradiction, as if it does, it makes truth within the system
>> meaningless.
>>
>>>
>>> We already knew that the law of of non-contradiction is necessarily
>>> true so the POE is redundant.
>>
>> No, it shows the effect of violating the "law of non-contradiction".
>>
>
> All these years I  thought that it was being asserted that the POE is
> true and that everything is actually proven from a contradiction.
>
> All the time they were never saying that, they were only saying what
> happens when we hypothetically assume that a contradiction is true.
>

You do understand that a proof can only happen from a true premise?

The principle of explosion says that if you ACTUALLY HAVE a
contradiction that is proven in the system, that the system blows up.

I guess this shows who little you understand of how logic actually works.

If you just "assume" a contradiction to be true, yes, you can prove that
based on that assumption, you can show anything true, but that is preety
worthless, as we should know the contradiction isn't true.

The issue is what happens when you find a contradiction in a logic
system, and why you really need to trace back and find what assumption /
axiom in the system causes that to be able to be proven.

For instance, this is what happened to naive set theory, and the work
done to remove the problem by changing fundamental rules on how you can
define sets.

On 5/21/2023 9:52 PM, olcott wrote:

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

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 5/22/23 11:18 PM, olcott wrote:
> On 5/21/2023 9:52 PM, olcott wrote:
>
> 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
>
>

So, the idiot can learn something.

MAybe some day you will come to a similar realization about all the
other things you think you know.

On Tuesday, May 23, 2023 at 4:47:14 AM UTC-7, Richard Damon wrote:
> On 5/22/23 11:18 PM, olcott wrote:
> > On 5/21/2023 9:52 PM, olcott wrote:
> >
> > 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
> >
> >
> So, the idiot can learn something.
>
> MAybe some day you will come to a similar realization about all the
> other things you think you know.

"You people are dopes."