Definitions are not Free!

[Willem F Esterhuyse]

In Logic we can distinguish wrong from right definitions: "A -> B defined = ~A OR B" is right, while "A -> B defined = A OR ~B" is wrong. Other definitions are similar such as definitions according to De Morgan's Law. Now I wonder if the same applies to Mathematics?

[studiot]

25 minutes ago, Willem F Esterhuyse said:

In Logic we can distinguish wrong from right definitions: "A -> B defined = ~A OR B" is right, while "A -> B defined = A OR ~B" is wrong. Other definitions are similar such as definitions according to De Morgan's Law.

 

1)  Have you ever heard of fuzzy logic?

2)  Even in first order logic it is not always possible to conclude that a statement is right or wrong, have you ever heard of the liar paradox ?

 

26 minutes ago, Willem F Esterhuyse said:

Now I wonder if the same applies to Mathematics?

 

Apart from the mathematical versions of the above examples, what about statistics and probability ?

[dimreepr]

40 minutes ago, Willem F Esterhuyse said:

In Logic we can distinguish wrong from right definitions: "A -> B defined = ~A OR B" is right, while "A -> B defined = A OR ~B" is wrong. Other definitions are similar such as definitions according to De Morgan's Law. Now I wonder if the same applies to Mathematics?

Only if you're a logic gate, but I can't be sure...

[studiot]

12 minutes ago, dimreepr said:

Only if you're a logic gate, but I can't be sure...

In general I would agree with you but how about tristate logic gates ?

https://en.wikipedia.org/wiki/Three-state_logic

Edited December 14, 2022 by studiot

[dimreepr]

3 minutes ago, studiot said:

In general I would agree with you but how about tristate logic gates ?

https://en.wikipedia.org/wiki/Three-state_logic

I just wondered if he got the joke?

[Willem F Esterhuyse]

On 12/14/2022 at 3:00 PM, studiot said:

Have you ever heard of fuzzy logic?

Yes I heard about it but I didn't read papers about it yet.

On 12/14/2022 at 3:00 PM, studiot said:

have you ever heard of the liar paradox

Yes. It does not contribute to the topic.

[studiot]

33 minutes ago, Willem F Esterhuyse said:

Yes. It does not contribute to the topic.

1)Why not ?

It is an example (there are many) where your claim that either A is true or not A is true is violated.

 

2) What about the answer I gave to your actual question ?

[Willem F Esterhuyse]

On 12/15/2022 at 5:09 PM, studiot said:

On 12/15/2022 at 4:33 PM, Willem F Esterhuyse said:

Yes. It does not contribute to the topic.

1)Why not ?

It does not contribute to the question of right and wrong definitions.

On 12/15/2022 at 5:09 PM, studiot said:

2) What about the answer I gave to your actual question ?

You just gave two examples in Logic, didn't answer the question.

[Genady]

On 12/14/2022 at 8:31 AM, Willem F Esterhuyse said:

In Logic we can distinguish wrong from right definitions

How? Your examples didn't make it clear to me.

[Willem F Esterhuyse]

The definitions follow proven Laws.

[Genady]

6 minutes ago, Willem F Esterhuyse said:

The definitions follow proven Laws.

How are they proven without definitions?

[Willem F Esterhuyse]

16 minutes ago, Genady said:

How are they proven without definitions?

They are proven using the Axioms (intuitively true rules).

[Genady]

8 minutes ago, Willem F Esterhuyse said:

They are proven using the Axioms (intuitively true rules).

So, you mean in your OP, "we can intuitively distinguish intuitively wrong from intuitively right definitions"?

Also, if they follow from axioms, then they are theorems rather than definitions.

[Willem F Esterhuyse]

On 12/31/2022 at 3:03 PM, Genady said:

So, you mean in your OP, "we can intuitively distinguish intuitively wrong from intuitively right definitions"?

No I mean they follow from the Axioms or not (but you can state it that way). Yes they are theorems.

However, the specific example: "A -> B define = Ã OR B" follows from intuition (the symbols of this intuition are not given).

Edited January 2, 2023 by Willem F Esterhuyse

[Genady]

25 minutes ago, Willem F Esterhuyse said:

"A -> B define = Ã OR B" follows from intuition

How do you know that it follows from intuition?

Whose intuition?

How does a procedure of "following from intuition" work?

[Willem F Esterhuyse]

3 minutes ago, Genady said:

How do you know that it follows from intuition?

I verified it in my own mind.

4 minutes ago, Genady said:

Whose intuition?

George Boole.

5 minutes ago, Genady said:

How does a procedure of "following from intuition" work?

You basically form the premise in your mind and then see/feel the conclusion following. It must use hidden symbols.

[Genady]

1 minute ago, Willem F Esterhuyse said:

I verified it in my own mind.

George Boole.

You basically form the premise in your mind and then see/feel the conclusion following. It must use hidden symbols.

There is no "intuition" in this text.

[Willem F Esterhuyse]

"intuition" is in the quoted questions. The fact that the answers do not contain "intuition" does not invalidate it.

[Genady]

1 minute ago, Willem F Esterhuyse said:

The fact that the answers do not contain "intuition" does not invalidate it.

Yes, it does. Otherwise, it does not "follow from intuition."

[Willem F Esterhuyse]

If I answered in terms of "intuition" the answer would be circular! It does not follow from intuition but from memories in my mind.

Edited January 2, 2023 by Willem F Esterhuyse

[Genady]

4 minutes ago, Willem F Esterhuyse said:

If I answered in terms of "intuition" the answer would be circular! It does not follow from intuition but from memories in my mind.

A procedure of following from intuition has to include intuition on one or more steps. Just like a procedure of following from axioms has to include the axioms on one or more steps.

[dimreepr]

20 minutes ago, Willem F Esterhuyse said:

I verified it in my own mind.

George Boole.

You basically form the premise in your mind and then see/feel the conclusion following. It must use hidden symbols.

If the symbols are hidden, how do you know they are there?

Besides, what's George Boole got to do with your own verification? 

[Willem F Esterhuyse]

1 minute ago, dimreepr said:

If the symbols are hidden, how do you know they are there?

I actually saw the symbols the second time I verified it. The first time I went on feeling.

5 minutes ago, dimreepr said:

Besides, what's George Boole got to do with your own verification? 

I read his writings and found the definition there.

5 minutes ago, Genady said:

A procedure of following from intuition has to include intuition on one or more steps.

No it only needs to contain specific intuitions not the word "intuition" itself and the answer does not follow from intuition but from memories of mental operations.

[Genady]

1 minute ago, Willem F Esterhuyse said:

it only needs to contain specific intuitions not the word "intuition" itself

Which specific intuition does it contain?

 

2 minutes ago, Willem F Esterhuyse said:

the answer does not follow from intuition

Then what does?

[Willem F Esterhuyse]

1 minute ago, Genady said:

Which specific intuition does it contain?

Some basic symbols (un-externalizable, because I have to believe in order to externalize them and if a part externalized does not fit in a system I will disbelieve it and scratch it out).

4 minutes ago, Genady said:

Then what does?

That is just stupid: read the whole sentence.