The New Sudoku Players' Forum

[Ajò Dimonios]

d is false

[mith]

Question 15: (¬A OR A) AND (A OR B) AND (B OR ¬B) => (¬A OR ¬B); True or False?
Question 16: What is the definition of (A OR B)?

[Ajò Dimonios]

15 is false
16 16 The logical OR operation returns true if at least one of the elements is true, while it returns false in all other cases.

[mith]

Question 5: What is the definition of a strong inference/link?

Definition 1 : Let A, B be two assumptions. We say that A,B are thicklinked and note A == B when at least one of A,B is true

Question 16: What is the definition of (A OR B)?

16 16 The logical OR operation returns true if at least one of the elements is true, while it returns false in all other cases.

Question 17: So if at least one of A,B is true, A = B. And if at least one of A,B is true, (A OR B) is true. Is that correct?
Question 18: Logically, A = B (A,B are strongly linked) and (A OR B) mean the same thing. True or False?

[Ajò Dimonios]

17 Yes in the current definition
18 Yes

[mith]

Question 19: So if A = B and (A OR B) mean the same thing, they should always have the same truth value, yes?

[Ajò Dimonios]

19 yes

[mith]

Question 20: So, by definition, A = B means the same thing as (A OR B). Does ¬A = ¬B mean the same thing as (¬A OR ¬B)?

[Ajò Dimonios]

Yes. So according to this proof there is no strong link between ¬A and ¬B. But there is a logical link which is not a weak link

[mith]

Hey, that's progress. Just to be clear why there is no strong link:

¬A = A = B = ¬B is short for:
(¬A = A) AND (A = B) AND (B = ¬B)

But as we have just established, the strong link, by definition, is the same as the OR operator. So this is also equivalent:

(¬A OR A) AND (A OR B) AND (B OR ¬B)

Likewise:
(¬A = ¬B) is the same as
(¬A OR ¬B)

But these are just the left side and right side of the implication. Since

(¬A OR A) AND (A OR B) AND (B OR ¬B) => (¬A OR ¬B) is False, ¬A = A = B = ¬B => ¬A = ¬B is also False.

[mith]

In fact, the resulting link is a weak link [edit]in this specific case[/edit]:

¬A = A = B = ¬B => ¬A - ¬B

Why? In order for the left side to be true, A = B must be true. But this means at least one of A,B must be true by definition. Since A and B are not both false, ¬A and ¬B are not both true. At least one of ¬A,¬B is false. Which is the very definition of a weak link.

And to bring it all the way back to the sudoku: (16)r4c789 - (16)r4c56. They cannot both be true, so there is a weak link. They can both be false, so there is no strong link.

[Ajò Dimonios]

It can be shown that in a logical chain alternating strong inference with weak inferences the head and tail of the chain are linked together with a strong bond. In this case a chain of strong inferences alone does not produce the same result. Why?
It is also known that the definition of strong bond also includes in its definition that of the weak link, so these three strong links can be replaced by an alternation of - = -, in this case it is possible to prove that A false implies B false.

[mith]

Because a strong link does not have to be a weak link.
If you had ¬A = A - B = ¬B, this would in fact imply ¬A = ¬B. But A = B does not imply A - B, in general. (As we showed much earlier in the thread T = T, but T is not weakly linked to T. So F = T = T = F does not imply F = F.)

[mith]

In fact, the resulting link is a weak link.

¬A = A = B = ¬B => ¬A - ¬B

To clarify: This is not generally true of chains of strong links. A = B = C = D does not imply any kind of link between A and D. It is only true in the above example with negations. (If A and B are strongly linked, their negations are weakly linked.)

[mith]

It is also known that the definition of strong bond also includes in its definition that of the weak bond

No, this is not correct. The definition of a strong link is "at least one is true". The definition of a weak link is "at least one is false". A strong link does not have to be a weak link, nor does a weak link have to be a strong link. An XOR link would be both a strong and weak link, and would work in the way you want it to.