Synthetic Syllogisms and Analytic Axioms.

[Faust]

This seems to implicitly address my question. But according to you, A → A is not an implication, right? At least you said it is not an inference.

It’s not, but it is written as one - I was quoting you.

[Alpha_Omega]

@Faust

Alphie -

In terms of scientific method and the practical applications of logic, reason is generally accepted to be ‘worth its weight in gold’, so to speak, and to discount its general role in all things reasonable, would be unfortunately beyond my ability.

Well, the “application of logic” and “reason” are synonymous. What I am saying is that axioms are unprovable. This is necessarily so, for they are the starting points of proofs. They are unprovable by definition. In other words, if they were provable, then they wouldn’t be axioms.

A = A, for instance, is considered self-evident. (A → A is not, by the definition of implication itself, which is why implication requires a proof.)

This depends on what you take ‘proof’ to be, and you seems to be taking it to be the construction of formal proof rules, formulæ and theorems.

More generally self-evident is synonomous with self-prooving.

To say that a self-evident truth is ‘unprovable’, is pedantic; by stating the truth itself, you are prooving it to be true by virtue of its self evidence.

Tautologies are self-evident.

If A then A is not considered to be self-evident by formal standards, then by most others (I would imagine) it is very clearly so.

[Sauwelios]

@Faust

Saully - no - I almost cautioned you not to use negative expressions here, but did not.

Your sweating is a sufficient condition of your being thirsty and you being thirsty is a necessary condition of your sweating.

If I am sweating then I am thirsty.

Every time I am sweating I am thirsty.

I am sweating, so I have to be thirsty - have to be - it’s necessary to be thirsty.

Whenever I am sweating…

But other things can make e thirsty, so sweating is not necessary.

So my not sweating is sufficient for me to be thirsty, right? My sweating is not “sufficient” for my being thirsty, but is the necessary accompaniment of my being thirsty.

It is sufficient to not be sweating for me to be thirsty. I don’t need to be sweating to be thirsty.

I’m just trying to understand this word in the context of logic…

[Sauwelios]

@Alpha+Omega

@Faust

Alphie -

In terms of scientific method and the practical applications of logic, reason is generally accepted to be ‘worth its weight in gold’, so to speak, and to discount its general role in all things reasonable, would be unfortunately beyond my ability.

Well, the “application of logic” and “reason” are synonymous. What I am saying is that axioms are unprovable. This is necessarily so, for they are the starting points of proofs. They are unprovable by definition. In other words, if they were provable, then they wouldn’t be axioms.

A = A, for instance, is considered self-evident. (A → A is not, by the definition of implication itself, which is why implication requires a proof.)

This depends on what you take ‘proof’ to be, and you seems to be taking it to be the constuction formal proof rules, formulæ and theorems.

More generally self-evident is synonomous with self-prooving.

Even so, he said it was considered self-evident, not that it was self-evident.

Tautologies are self-evident.

This statement, coming from you, means: “I consider tautologies to be self-evident”. Compare the Declaration of Independence: “We hold these truths to be self-evident”…

[Alpha_Omega]

1. Even so, he said it was considered self-evident, not that it was self-evident.

Tautologies are self-evident.

2. This statement, coming from you, means: “I consider tautologies to be self-evident”. Compare the Declaration of Independence: “We hold these truths to be self-evident”…

1. Lol. It is self-evident. Tautologies are absolute truths. Remember it is you who is asking if A then A is self evident or not, not me.
2. Tautolgies ARE self-evident. The illogic in your comparison of tautology with potentially false starting principles is, although amusing, not worth analysing.

[Sauwelios]

@Alpha+Omega

If A then A is not considered to be self-evident by formal standards, then by most others (I would imagine) it is very clearly so.

I guess it has been predetermined that “A–>B” means “A presupposes B”. B is then a necessary ground for A, but not vice versa.

A
B

~A
B

~A
~B

These are then the only possible cases.

A and B cannot be equal, because something cannot be its own ground.

“A equals A, therefore A equals A” is then invalid. The valid (though partly pre-logical) equivalent would be:

“[Revelation], therefore A equals A.”
“It has been shown to us that A equals A.”
“We have seen* that A equals A.”

*“To see” is videre in Latin, whence “evident”.

[Sauwelios]

@Alpha+Omega

1. Even so, he said it was considered self-evident, not that it was self-evident.

Tautologies are self-evident.

2. This statement, coming from you, means: “I consider tautologies to be self-evident”. Compare the Declaration of Independence: “We hold these truths to be self-evident”…

1. Lol. It is self-evident. Tautologies are absolute truths. Remember it is you who is asking if A then A is self evident or not, not me. :lol:

Not if it is self-evident, but if it (self-)evidently resembles A = A.

[Alpha_Omega]

@Sauwelios

@Alpha+Omega

If A then A is not considered to be self-evident by formal standards, then by most others (I would imagine) it is very clearly so.

I guess it has been predetermined that “A–>B” means “A presupposes B”. B is then a necessary ground for A, but not vice versa.

A
B

~A
B

~A
~B

These are then the only possible cases.

A and B cannot be equal, because something cannot be its own ground.

“A equals A, therefore A equals A” is then invalid. The valid (though partly pre-logical) equivalent would be:

“[Revelation], therefore A equals A.”
“It has been shown to us that A equals A.”
“We have seen* that A equals A.”

*“To see” is videre in Latin, whence “evident”.

… pfft…

Yeah like B is relevant to the truth of the inference from A to itself.

[Alpha_Omega]

1. Even so, he said it was considered self-evident, not that it was self-evident.

Tautologies are self-evident.

2. This statement, coming from you, means: “I consider tautologies to be self-evident”. Compare the Declaration of Independence: “We hold these truths to be self-evident”…

1. Lol. It is self-evident. Tautologies are absolute truths. Remember it is you who is asking if A then A is self evident or not, not me.

Not if it is self-evident, but if it (self-)evidently resembles A = A.

What if was exactly the same thing

[Sauwelios]

@Alpha+Omega

… pfft…

Yeah like B is relevant to the truth of the inference from A to itself.

:lol:

I said “A and B cannot be equal”. If A and B are equal, “A–>B”, “B–>A”, “A–>A”, and “B–>B” are equal. So then the inference from A to itself is the inference from B to itself, or from B to A. So yeah, than B is relevant to its truth: exactly as relevant as A.

[Faust]

Alphie - I’m afraid you’ve got it all backwards.

This depends on what you take ‘proof’ to be, and you seems to be taking it to be the construction of formal proof rules, formulæ and theorems.

More generally self-evident is synonomous with self-prooving.

Well, in the context of formal logic, it’s not.

To say that a self-evident truth is ‘unprovable’, is pedantic; by stating the truth itself, you are prooving it to be true by virtue of its self evidence.

Now i have to wonder if English is your first language.

Tautologies are self-evident.

If they were, we wouldn’t need logic, for the basic purpose of logic is to produce tautologies (at least of a sort) that are not immediately self-evident.

If A then A is not considered to be self-evident by formal standards, then by most others (I would imagine) it is very clearly so.

Yes, you are imagining.

Logic is not like the other fields of philosophy. A lot of logic is settled.

[Sauwelios]

@Alpha+Omega

Not if it is self-evident, but if it (self-)evidently resembles A = A.

:o What if was exactly the same thing :o

:idea:

Faust has denied that from the beginning, and I’m beginning to see why (see above).

[Alpha_Omega]

OK then let me spell it out for you.

[size=200]A ≡ A.[/size]

[Faust]

Saully - in an implication, a and b are not considered equal for the purposes of the implication, even if they turn out to be equal later. But a -->a states that the equality is already known. But this is a problem in syllogistic logic - it is one reason that propositional logic was developed.

[Alpha_Omega]

@Faust

Alphie - I’m afraid you’ve got it all backwards.

This depends on what you take ‘proof’ to be, and you seems to be taking it to be the construction of formal proof rules, formulæ and theorems.

More generally self-evident is synonomous with self-prooving.

1. Well, in the context of formal logic, it’s not.

To say that a self-evident truth is ‘unprovable’, is pedantic; by stating the truth itself, you are prooving it to be true by virtue of its self evidence.

2. Now i have to wonder if English is your first language.

Tautologies are self-evident.

3. If they were, we wouldn’t need logic, for the basic purpose of logic is to produce tautologies (at least of a sort) that are not immediately self-evident.

If A then A is not considered to be self-evident by formal standards, then by most others (I would imagine) it is very clearly so.

4. Yes, you are imagining.

5. Logic is not like the other fields of philosophy. A lot of logic is settled.

You’re just playing games and you know it.

1. Well in the context of wider reality it is and wider reality weigh a lot more than the section that is ‘formal logic’.
2. Indeed
3. ‘at least of a sort’.
5. Yes and it is well settled with me .

[Faust]

I am speaking within the context of formal logic, which is the stated topic of this thread.

[Alpha_Omega]

@Faust

Saully - in an implication, a and b are not considered equal for the purposes of the implication, even if they turn out to be equal later. But a -->a states that the equality is already known. But this is a problem in syllogistic logic - it is one reason that propositional logic was developed.

[Alpha_Omega]

@Faust

I am speaking within the context of formal logic, which is the stated topic of this thread.

The stated topic of this thread is ‘Synthetic Syllogisms and Analytic Axioms’ which leaves a lot open for discussion…

[Alpha_Omega]

Well that was a real good laugh guys thanks, I’m off to bed.

[Sauwelios]

@Sauwelios

@Faust

Saully - no - I almost cautioned you not to use negative expressions here, but did not.

Your sweating is a sufficient condition of your being thirsty and you being thirsty is a necessary condition of your sweating.

If I am sweating then I am thirsty.

Every time I am sweating I am thirsty.

I am sweating, so I have to be thirsty - have to be - it’s necessary to be thirsty.

Whenever I am sweating…

But other things can make e thirsty, so sweating is not necessary.

So my not sweating is sufficient for me to be thirsty, right? My sweating is not “sufficient” for my being thirsty, but is the necessary accompaniment of my being thirsty.

This is not right, of course. It is a “dependant” of my being thirsty. 'Twas late…

It is sufficient to not be sweating for me to be thirsty. I don’t need to be sweating to be thirsty.

I’m just trying to understand this word in the context of logic…

I think I understand it now. “Sufficient” in this context is itself a negative concept: it means as much as “not exclusive”. It does not mean “enough to meet the needs of a situation or a proposed end”, as Merriam-Webster puts it: for there may be other, necessary conditions for my being thirsty.