XPost: comp.theory, sci.math

On 12/14/23 10:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
 can be inferred from it; this is known as deductive explosion." https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

Nope, it establishes that some Dogs are not Dogs. That is a FULL
"semantic" reasoning from the premises.

This comes because the cats that are the "Some Cats" in (b), MUST BE, by
(a) Dogs, so we can conclude that Those Dogs are Not Dogs.

In other words, it proves the system is inconsistant.

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XPost: comp.theory, sci.math

On 12/14/23 7:14 PM, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
  can be inferred from it; this is known as deductive explosion."
https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

Whereas the intersection of the sets specified by
(a) and (b) is the empty set, thus derives no conclusion.

But logic doesn't take the intersetion of the premises, but, in one
sense, the Union.

Or, are you saying that it implies that it is describing a world with no
cats or dogs?

But that would violate the clear meaning of the word "Some", which
implies existance.

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XPost: sci.logic, comp.theory, sci.math

On 2023-12-14 17:14, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
  can be inferred from it; this is known as deductive explosion."
https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

No. It doesn't say that. Given a contradiction (I'll use A & ¬A), the principle of explosion says that for any statement X, "A & ¬A therefore
X" is a *valid* argument.

To *prove* a statement, the statement needs to appear as the conclusion
to a *sound* argument (being valid is necessary but not sufficient), and
the principle of explosion does *not* claim that your hypothetical
argument is sound.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

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XPost: comp.theory, sci.math

On 12/14/23 11:20 PM, olcott wrote:

On 12/14/2023 9:56 PM, André G. Isaak wrote:

On 2023-12-14 17:14, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
  can be inferred from it; this is known as deductive explosion."
https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

No. It doesn't say that. Given a contradiction (I'll use A & ¬A), the
principle of explosion says that for any statement X, "A & ¬A
therefore X" is a *valid* argument.

*Which is itself conventionally defined incorrectly*
The correct way that valid should be defined is that the
conclusion is a necessary consequence of all of its premises.

Which it is, according to the rules of the logic system. You are just
showing your lack of understanding.

Any system which claims to be non-contradictory in logc form, that has a
pair of statements that are contradictory, is just broken. The Principle
of Explosion just makes the breakage total,

This eliminates the Principle of Explosion before it
even gets started.

Nope, it proves that you don't understand what you are talking about.

Truth is established by having a set (possibly infinite) of valid steps
from the initial truthmakers of the system to the statement.

A Proof is just a finite listing of one possible set of those links,
thus anything that can be proven, must be true.

Yes, if you limit the forms of links that can be used as steps, you can
make some things not provable, but this MIGHT also reduce what is
actually true in the system.

To *prove* a statement, the statement needs to appear as the
conclusion to a *sound* argument (being valid is necessary but not
sufficient), and the principle of explosion does *not* claim that your
hypothetical argument is sound.

André

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XPost: comp.theory, sci.math

On 12/14/23 11:20 PM, olcott wrote:

On 12/14/2023 9:56 PM, André G. Isaak wrote:

On 2023-12-14 17:14, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
  can be inferred from it; this is known as deductive explosion."
https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

No. It doesn't say that. Given a contradiction (I'll use A & ¬A), the
principle of explosion says that for any statement X, "A & ¬A
therefore X" is a *valid* argument.

*Which is itself conventionally defined incorrectly*
The correct way that valid should be defined is that the
conclusion is a necessary consequence of all of its premises.

And they are.

Note, ANY system that starts with a contradiction in it is just "broken"
and "necessary consequence" isn't really defined.

Your problem is you don't understand the nature of the proof of the
principle of explosion.

It isn't removed by use of "meaning", as a system that allows the
contradiction in the first place has already broken the definition of "meaning", but is removed by weaking the logic system to restrict how
broad of a circle the break can infest.

This eliminates the Principle of Explosion before it
even gets started.

Nope, the logic system was broken as soon as the truthmakers that
allowed the derivation of the contradiction were added to the system.

To *prove* a statement, the statement needs to appear as the
conclusion to a *sound* argument (being valid is necessary but not
sufficient), and the principle of explosion does *not* claim that your
hypothetical argument is sound.

André

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XPost: sci.logic, comp.theory, sci.math

On 12/15/23 05:20, olcott wrote:

On 12/14/2023 9:56 PM, André G. Isaak wrote:

On 2023-12-14 17:14, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
  can be inferred from it; this is known as deductive explosion."
https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

No. It doesn't say that. Given a contradiction (I'll use A & ¬A), the
principle of explosion says that for any statement X, "A & ¬A
therefore X" is a *valid* argument.

*Which is itself conventionally defined incorrectly*
The correct way that valid should be defined is that the
conclusion is a necessary consequence of all of its premises.

This eliminates the Principle of Explosion before it
even gets started.

To *prove* a statement, the statement needs to appear as the
conclusion to a *sound* argument (being valid is necessary but not
sufficient), and the principle of explosion does *not* claim that your
hypothetical argument is sound.

André

"The moon is made from green cheese" is a necessary consequence of "all
cats are dogs" and "some cats are not dogs". Or can you imagine a world
where all cats are dogs and some cats are not dogs, but the moon isn't
made from green cheese?

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XPost: comp.theory, sci.math

On 12/17/23 12:11 PM, olcott wrote:

On 12/17/2023 2:17 AM, immibis wrote:

On 12/15/23 05:20, olcott wrote:

On 12/14/2023 9:56 PM, André G. Isaak wrote:

On 2023-12-14 17:14, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
  can be inferred from it; this is known as deductive explosion." >>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

No. It doesn't say that. Given a contradiction (I'll use A & ¬A),
the principle of explosion says that for any statement X, "A & ¬A
therefore X" is a *valid* argument.

*Which is itself conventionally defined incorrectly*
The correct way that valid should be defined is that the
conclusion is a necessary consequence of all of its premises.

This eliminates the Principle of Explosion before it
even gets started.

To *prove* a statement, the statement needs to appear as the
conclusion to a *sound* argument (being valid is necessary but not
sufficient), and the principle of explosion does *not* claim that
your hypothetical argument is sound.

André

"The moon is made from green cheese" is a necessary consequence of
"all cats are dogs" and "some cats are not dogs". Or can you imagine a
world where all cats are dogs and some cats are not dogs, but the moon
isn't made from green cheese?

It is not true that anything is semantically entailed by any
contradiction. When the Principle of explosion says that everything is syntactically entailed by a contradiction the POE is a liar that denies
the law of non-contradiction. For analytical truth coherence is the
measure.

Just shows you don't understand how semantic logic actually works.

The Principle of Explosion says that, for a logic system with certain
logical operations, that are normally included in logic, once you have a contradiction provable in the system, you can prove any statement from it.

Yes, there are systems with weakened logic system that this does not
apply to, but such system can not prove as many true statements themselves.

It is also a fact, that ANY logic system, which claims to have logic
that is non-contradictory, that can prove a contradiction, is no longer
a sound logic system, as at least one of its truth makers must not be
actually true.

So, in one sense you are right, give the statements shown to be true in
a system that (a) All Cats are Dogs, and (b) Some Cats are not Dog, yes,
we can conclude that the FULL logic system shows the NULL set, as
nothing in the set can be believed.

If that is your goal, to assert that it is impossible to know if
anything is actually true, and thus it is just as valid to claim any
stateement we want as true, you have succeeded with your logic system.

That seems to be just the opposite of what you have claimed to be trying
to do, so you are just at total failure.

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XPost: sci.logic, comp.theory, sci.math

On 12/17/23 18:11, olcott wrote:

On 12/17/2023 2:17 AM, immibis wrote:

"The moon is made from green cheese" is a necessary consequence of
"all cats are dogs" and "some cats are not dogs". Or can you imagine a
world where all cats are dogs and some cats are not dogs, but the moon
isn't made from green cheese?

It is not true that anything is semantically entailed by any
contradiction. When the Principle of explosion says that everything is syntactically entailed by a contradiction the POE is a liar that denies
the law of non-contradiction. For analytical truth coherence is the
measure.

Can you imagine a world where all cats are dogs and some cats are not
dogs, but the moon isn't made from green cheese?

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XPost: sci.logic, comp.theory, sci.math

On 12/19/23 04:02, olcott wrote:

On 12/18/2023 11:37 AM, immibis wrote:

On 12/17/23 18:11, olcott wrote:

On 12/17/2023 2:17 AM, immibis wrote:

"The moon is made from green cheese" is a necessary consequence of
"all cats are dogs" and "some cats are not dogs". Or can you imagine
a world where all cats are dogs and some cats are not dogs, but the
moon isn't made from green cheese?

It is not true that anything is semantically entailed by any
contradiction. When the Principle of explosion says that everything is
syntactically entailed by a contradiction the POE is a liar that denies
the law of non-contradiction. For analytical truth coherence is the
measure.

Can you imagine a world where all cats are dogs and some cats are not
dogs, but the moon isn't made from green cheese?

That would be incoherent: The coherence theory of truth applies to the analytical body of knowledge.

I've never heard of these two, and they seem to be fully immersed in philosophy, not computer science or mathematical logic.

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XPost: sci.logic, comp.theory, sci.math

On 12/19/23 16:22, olcott wrote:

A deductive argument is said to be valid if and only if it takes a form
that makes it impossible for the premises to be true and the conclusion nevertheless to be false. https://iep.utm.edu/val-snd/

On that basis we can conclude that this sentence is valid:
"Kittens are 15 story office buildings therefore water is H2O."

When we redefine value to be a conclusion must be a necessary
consequence of all of its premises then the above nonsense
sentence is not valid.

What is a necessary consequence?

A consequence is said to be necessary if and only if it takes a form
that makes it impossible for the antecedents to be true and the
consequence nevertheless to be false...

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XPost: comp.theory, sci.math

On 12/19/23 12:05 PM, olcott wrote:

On 12/19/2023 9:55 AM, immibis wrote:

On 12/19/23 16:22, olcott wrote:

A deductive argument is said to be valid if and only if it takes a form
that makes it impossible for the premises to be true and the conclusion
nevertheless to be false. https://iep.utm.edu/val-snd/

On that basis we can conclude that this sentence is valid:
"Kittens are 15 story office buildings therefore water is H2O."

When we redefine value to be a conclusion must be a necessary
consequence of all of its premises then the above nonsense
sentence is not valid.

What is a necessary consequence?

◊ means possibly
◻ means necessarily
¬ means not
◊P means ¬◻¬P
◻P means ¬◊¬P

A---B---A ◻ B
t---t-----t
t---f-----f
f---?-----? When A is false then we know nothing about B

In other words, your system of logic can not assign a validity to an implication.

Note, your "conclusion" actually comes out of the normal definition of implication, since A->B is true for A being false and B being either
True or False, then we know nothing about B.

Note, for YOUR "truth Table" if we know that A -> B is a true sttement,
then we can not determine that A is false from knowing that B is false.

You have lost the relationship that A -> B alse means that ~B -> ~A

A consequence is said to be necessary if and only if it takes a form
that makes it impossible for the antecedents to be true and the
consequence nevertheless to be false...

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XPost: comp.theory, sci.math

On 12/19/23 10:22 AM, olcott wrote:

On 12/19/2023 7:34 AM, immibis wrote:

On 12/19/23 04:02, olcott wrote:

On 12/18/2023 11:37 AM, immibis wrote:

On 12/17/23 18:11, olcott wrote:

On 12/17/2023 2:17 AM, immibis wrote:

"The moon is made from green cheese" is a necessary consequence of >>>>>> "all cats are dogs" and "some cats are not dogs". Or can you
imagine a world where all cats are dogs and some cats are not
dogs, but the moon isn't made from green cheese?

It is not true that anything is semantically entailed by any
contradiction. When the Principle of explosion says that everything is >>>>> syntactically entailed by a contradiction the POE is a liar that
denies
the law of non-contradiction. For analytical truth coherence is the
measure.

Can you imagine a world where all cats are dogs and some cats are
not dogs, but the moon isn't made from green cheese?

That would be incoherent: The coherence theory of truth applies to
the analytical body of knowledge.

I've never heard of these two, and they seem to be fully immersed in
philosophy, not computer science or mathematical logic.

Without Philosophy logic has no basis. The basis that logic does have is incoherent because they got the philosophy wrong.

Nope, Without logic, Philosophy has no basis.

A deductive argument is said to be valid if and only if it takes a form
that makes it impossible for the premises to be true and the conclusion nevertheless to be false. https://iep.utm.edu/val-snd/

On that basis we can conclude that this sentence is valid:
"Kittens are 15 story office buildings therefore water is H2O."

Yes. Can you show it to NOT be valid?

Is there a case where we have Kittens as 15 story office buildings and
NOT have water as H2O?

Your problem is you don't understand how logic works, and thus you don't
really understand philosophy.

When we redefine value to be a conclusion must be a necessary
consequence of all of its premises then the above nonsense
sentence is not valid.

And thus such a system is incorrect, as it makes an impossible to be
false statement not to be not true.

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XPost: comp.theory, sci.math

On 12/19/23 12:26 PM, olcott wrote:

On 12/19/2023 9:55 AM, immibis wrote:

On 12/19/23 16:22, olcott wrote:

A deductive argument is said to be valid if and only if it takes a form
that makes it impossible for the premises to be true and the conclusion
nevertheless to be false. https://iep.utm.edu/val-snd/

On that basis we can conclude that this sentence is valid:
"Kittens are 15 story office buildings therefore water is H2O."

When we redefine value to be a conclusion must be a necessary
consequence of all of its premises then the above nonsense
sentence is not valid.

What is a necessary consequence?

A consequence is said to be necessary if and only if it takes a form
that makes it impossible for the antecedents to be true and the
consequence nevertheless to be false...

*This may be a more exactly precise way to say what I mean*
My correction to the notion of a valid argument means that the
truth of the conclusion depends on the truth all of the premises.

If any premise is false or irrelevant then the conclusion is not proved.
(a) I go outside
(b) I am unprotected from the rain
(c) then I get wet.

(a) I go outside
(b) I eat a popsicle
(c) Do I get wet? impossible to tell.

Which means, for standard logic, your second set (where (c) makes an
actual statement about getting wet) is just a false implication an not
valid.

A & B -> C is true ONLY if any time A and B are True then C is also True.

So, a implication like

If (a) I go outside, and (b) I eat a popsicle, then (c) I get wet is
just a false implication, as there are cases where (a) and (b) are true
but (c) isn't.

Somehow you don't seem to understand that not all implications that can
be stated are true.

Note, just because ONE time I went outside and ate a popsicle I got wet,
does NOT prove that implication, as to prove it you need to be able to
look at ALL POSSIBLE cases.

But, I guess since you think proof by example is valid, I guess that
shows your problem with implication,

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XPost: sci.logic, comp.theory, sci.math

On 12/19/23 18:26, olcott wrote:

*This may be a more exactly precise way to say what I mean*
My correction to the notion of a valid argument means that the
truth of the conclusion depends on the truth all of the premises.

If any premise is false or irrelevant then the conclusion is not proved.
(a) I go outside
(b) I am unprotected from the rain
(c) then I get wet.

(a) I go outside
(b) I eat a popsicle
(c) Do I get wet? impossible to tell.

Alright so the moon being blue is a necessary consequence of me being
wet and not wet. If I'm wet and not wet, this proves the moon is blue,
we can tell that, so it's a necessary consequence.

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