The New Sudoku Players' Forum

[mith]

Regardless, I will ask one more time:

Do you agree that (16)r4c56 and (16)r4c789 in this puzzle:

a. Can both be false?
b. Are therefore not strongly linked?
c. And thus such a link cannot be used as a strong link in an AIC?

[Ajò Dimonios]

Hi Mith

Regardless, I will ask one more time:

Do you agree that (16)r4c56 and (16)r4c789 in this puzzle:

a. Can both be false?
b. Are therefore not strongly linked?
c. And thus such a link cannot be used as a strong link in an AIC?

Currently each of us builds his AICs not on the current definition of strong inference but by using the concept that if one assumption A is false, another B is true. This construction does not absolutely prove that A or B are true or false but it is useful because the logic of the AIC allows us to solve the puzzle. Once the puzzle is solved I can determine if A and B are both true, both false, A true and B false or B true and A false. So according to the current definition of strong inference, I can only know the truth about A and B after solving the puzzle. This should require each of us to review all strong inferences used and determine whether it was legitimate to use them or not. This is clearly absurd because this would invalidate most of the solutions obtained. For this reason I believe that the definition should be formulated on the method to be used to build a strong inference. We all build a strong inference starting from a hypothesis of falsity of A and verifying the truth of B with correct "true" logic, in this way we are sure to have obtained a strong inference. So according to this method (16) r4c789 = 16r4c56 is a strong inference; if both 1 and 6 in r4c789 are false surely both 1 and 6 in r4c56 are true. The negation of (16) r4c789 means (1 and 6 false; 1 true 6 false; 1 false 6 true), (FF or TF or FT). The first option FF determines that 16r4c56 is TT, so the inference is strong and can be used in the construction of an AIC.

Paolo

[eleven]

The negation of (16) r4c789 means (1 and 6 false; 1 true 6 false; 1 false 6 true), (FF or TF or FT). The first option FF determines that 16r4c56 is TT,

And the second and third ?
The negation of (1|6) r4c789 means (1 and 6 false), no other options here.

Lets formulate your claim this way:
(Cretan) = (liar)
The negation of (Cretan) is no Cretan. One option for non Cretans is to be a liar. So there is a strong inference and we can say, if you are no Cretan, you are a liar.

[mith]

[preview edit]eleven cut straight to the heart of the problem, but to be rigorous...[/pedit]

This construction does not absolutely prove that A or B are true or false but it is useful because the logic of the AIC allows us to solve the puzzle.

The logic of the AIC is only useful if properly constructed. As SpAce already demonstrated, if you are using (16)r4c56 = (16)r4c789 in this puzzle, you can end up with an elimination that is invalid.

So according to the current definition of strong inference, I can only know the truth about A and B after solving the puzzle. This should require each of us to review all strong inferences used and determine whether it was legitimate to use them or not. This is clearly absurd because this would invalidate most of the solutions obtained.

You can only know the truth about A and B individually when you have logically reached a point of placing the relevant digits (or whatever else, depending on what it is that A and B say). But you don't need to solve the puzzle to know whether a strong link is valid or not, and using links without knowing the difference is, indeed, absurd, as it may lead to false conclusions.

So according to this method (16) r4c789 = 16r4c56 is a strong inference; if both 1 and 6 in r4c789 are false surely both 1 and 6 in r4c56 are true. The negation of (16) r4c789 means (1 and 6 false; 1 true 6 false; 1 false 6 true), (FF or TF or FT). The first option FF determines that 16r4c56 is TT, so the inference is strong and can be used in the construction of an AIC.

No, this is simply not correct logically. In order for the material conditional to be true (and therefore have a strong link/inference), all three possible cases of the negation must lead to the same conclusion of TT. If any one of them does not, the material conditional is false. Since the very nature of an AIC is dependent on the truth of alternating conditionals, constructing an AIC with a false conditional is not valid, and doing so anyway could lead to false eliminations.

The first option you listed does indeed have a strong link to (16)r4c56, because it is the negation of (1|6)r4c789. The following is a correct proof using propositional calculus:

Code: Select all

¬[(1|6)r4c789]
 ↔ ¬[1r4c789 ∨ 6r4c789] (definition of (1|6))
 ↔ [¬1r4c789 ∧ ¬6r4c789] (by De Morgan's Theorems)
 ↔ [1r4c56 ∧ 6r4c56] (by remaining candidates)
 ↔ [(16)r4c56] (definition of (16))
therefore
¬[(1|6)r4c789] → [(16)r4c56] (Commutation, Material Equivalence, and Simplification)
 ↔ [(1|6)r4c789] ∨ [(16)r4c56] (Material Implication)
and finally
(1|6)r4c789 = (16)r4c56 (definition of a strong link)

You can't construct the same proof starting from (16)r4c789, because only one of the negation cases implies (16)r4c56. Here's what you get instead:

Code: Select all

¬[(16)r4c789]
 ↔ ¬[1r4c789 ∧ 6r4c789] (definition of (16))
 ↔ [¬1r4c789 ∨ ¬6r4c789] (by De Morgan's Theorems)
 ↔ [1r4c56 ∨ 6r4c56] (by remaining candidates)
 ↔ [(1|6)r4c56] (definition of (1|6))
and ultimately
(16)r4c789 = (1|6)r4c56

Which is itself a valid strong link, but not what you are claiming.

If you want to try to construct a valid logical proof using the rules of propositional calculus and the definition of a strong link/inference to demonstrate that (16)r4c789 = (16)r4c56, be my guest - but both sides are false, so you're not going to be able to. Otherwise, stop making claims that amount to "since (1|6)r4c789 = (16)r4c56, (16)r4c789 = (16)r4c56 can be used as a strong link in an AIC".

[Ajò Dimonios]

Hi Eleven

Eleven wrote:
Ajò Dimonios wrote:
The negation of (16) r4c789 means (1 and 6 false; 1 true 6 false; 1 false 6 true), (FF or TF or FT). The first option FF determines that 16r4c56 is TT,

And the second and third ?
The negation of (1|6) r4c789 means (1 and 6 false), no other options here.

Lets formulate your claim this way:
(Cretan) = (liar)
The negation of (Cretan) is no Cretan. One option for non Cretans is to be a liar. So there is a strong inference and we can say, if you are no Cretan, you are a liar.

We probably don't agree on the definitions, for me (16) r4c789 means (1 and 6) r4c789.
Logical AND operator &: The & operator computes the logical AND of its operands. The result of x & y is true if both x and y evaluate to true. Otherwise, the result is false.
Therefore each of the three options is a negation independently of the other two.
identical result with the OR operator
Logical OR operator |:The | operator computes the logical OR of its operands. The result of x | y is true if either x or y evaluates to true. Otherwise, the result is false.

P.S. while with exclusive OR operator ^
Logical exclusive OR operator ^:The ^ operator computes the logical exclusive OR, also known as the logical XOR, of its operands. The result of x ^ y is true if x evaluates to true and y evaluates to false, or x evaluates to false and y evaluates to true. Otherwise, the result is false.
In this case the negation means only the negation of both or the truth of both is contemplated.

Paolo

[mith]

(16) r4c789 means (1 and 6) r4c789

Everyone agrees on this.

Therefore each of the three options is a negation independently of the other two.

This is ambiguously worded. The negation of A AND B is NOT(A) OR NOT(B).

NOT(A) OR NOT(B) is true for AB having truth values of FF, FT, or TF. It is false for AB having truth values TT.

If we can rule out the TT case, NOT(A) OR NOT(B) is true, and therefore A AND B is false. In that sense, one case (such as TF, or equivalently A AND NOT(B)) negates the original statement A AND B.

The issue is that in order to use an implication like NOT(A AND B) implies (C AND D), all three cases must lead to that conclusion in order for the implication to be true. This is what you seem to be not understanding, or are willfully ignoring. FF implies (C AND D) is not sufficient to show that the whole implication is true. It MUST be true for all three.

We can play semantics all day about the difference between "the negation", "a negation", "negates", etc., or we can stick to sound logical principles. Sound logical principles show that the negation of (16)r4c789 does not imply (16)r4c56. Full stop. You don't appear to even be disputing that.

[eleven]

We probably don't agree on the definitions, for me (16) r4c789 means (1 and 6) r4c789.
Logical AND operator &: The & operator computes the logical AND of its operands. The result of x & y is true if both x and y evaluate to true. Otherwise, the result is false.
Therefore each of the three options is a negation independently of the other two.

Yes.

identical result with the OR operator
Logical OR operator |:The | operator computes the logical OR of its operands. The result of x | y is true if either x or y evaluates to true. Otherwise, the result is false.

Yes (or both are true, not to be ambigous).
So we agree with the definitions.
But what, do you think, is the negation of your or operator, and what do you follow from the other 2 options of the negated and operator - or are they irrelevant for you ?

[Ajò Dimonios]

Hi Eleven

Yes (or both are true, not to be ambigous).
So we agree with the definitions.
But what, do you think, is the negation of your or operator, and what do you follow from the other 2 options of the negated and operator - or are they irrelevant for you ?

I simply adapt to what it means to make the assumption false. We are talking about Boolean logic (true; false).
It is not relevant because it is not used. In these logical operations, subjective interpretations are not contemplated.

Paolo

[eleven]

Subjective is your interpretation of boolean logic. You pick out, what you like, and ignore the others.
Please make a pause and think about it.

[Ajò Dimonios]

we have at least arrived at the same conclusion negating (16) r4c789 and negating (1|6) r4c789 produces the same result.

[eleven]

You are a good bot, congratulations to your creator.

[SpAce]

I know I should just keep eating popcorn and enjoying the show...

nonsense: Show

Currently each of us builds his AICs not on the current definition of strong inference but by using the concept that if one assumption A is false, another B is true. This construction does not absolutely prove that A or B are true or false but it is useful because the logic of the AIC allows us to solve the puzzle. Once the puzzle is solved I can determine if A and B are both true, both false, A true and B false or B true and A false. So according to the current definition of strong inference, I can only know the truth about A and B after solving the puzzle. This should require each of us to review all strong inferences used and determine whether it was legitimate to use them or not. This is clearly absurd because this would invalidate most of the solutions obtained. For this reason I believe that the definition should be formulated on the method to be used to build a strong inference. We all build a strong inference starting from a hypothesis of falsity of A and verifying the truth of B with correct "true" logic, in this way we are sure to have obtained a strong inference. So according to this method (16) r4c789 = 16r4c56 is a strong inference; if both 1 and 6 in r4c789 are false surely both 1 and 6 in r4c56 are true. The negation of (16) r4c789 means (1 and 6 false; 1 true 6 false; 1 false 6 true), (FF or TF or FT). The first option FF determines that 16r4c56 is TT, so the inference is strong and can be used in the construction of an AIC.

There's only one sentence in that whole text (1/11) that is both true and makes sense:

The negation of (16) r4c789 means (1 and 6 false; 1 true 6 false; 1 false 6 true), (FF or TF or FT).

Indeed. Symbolically:

Code: Select all

-(1 & 6) <-> ((-1 & -6) | (1 & -6) | (-1 & 6))

We all agree that those two are logically equivalent, just like you said. The question is, how do you get from that to this:

Code: Select all

-(1 & 6) <-> (-1 & -6)

Since you seem to be such a guru in Boolean logic, can you teach us mere mortals how to perform such a miracle? See, I'm pretty sure all the rest of us think it should be:

Code: Select all

-(1 & 6) <-> (-1 | -6)

[mith]

Let's drop the formal logic for a moment.

Paolo, I have two children. You ask me if they are both girls. I answer no. Can you conclude that they are both boys?

[Ajò Dimonios]

I certainly can't conclude by saying that they are two boys because it could be a girl and a boy. Our case is different in the Eleven AIC we already have the information in the previous step that r5c7 = 1 and r5c9 = 6 therefore the only option that negates 16r4c789 usable in the next inference is FF.

[SpAce]

Arguing with someone like this is akin to solving a puzzle with zero solutions. Perfect facts and logic won't help when the game itself is broken. If it weren't for Hanlon's razor, I'd be certain that this is intentional trolling.