## Six statements. How many are true?

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Cradarc
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### Six statements. How many are true?

A. This statement is true if and only if the next statement is true.
B. This statement is false if at least two of the statements below this statement are true.
C. All the statements above this statement is true if none of the statements below this statement is true.
D. There are exactly four true statements out of all of these statements.
E. Statement D is false if and only if statement F is true.
F. Statement A is false.

How many of the above statements are true?

My Answer:
Spoiler:
3
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Carlington
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### Re: Six statements. How many are true?

I have very little knowledge of logic, but here's my attempt with working:

Spoiler:
Assume A is true.
Since A is true iff B is true, B must be true. (1)
B can be rewritten as "This statement is true only if less than two of the statements below this statement are true."
Since B is true, a maximum of one out of C, D, E, and F can be true. (2)
If C is true, then A, B and C are true, and D, E, and F are false, and we have 3 true statements. (3)
If D is true, then at least one other statement out of C, E, and F must be true - but this means two statements below B are true, which is a contradiction with (2)
If E is true, either D or F must also be true, which, if F is true, leads to a contradiction with our assumption that A is true, and if E is D is true, leads to the contradiction with (2) from earlier - and in any case, E leads directly to this same contradiction.
If F is true then our assumption is false.
So, assuming A is true, exactly three statements are true: A, B, and C.

I don't think this is a complete solution, as I haven't followed the paths that result in discarding our assumption when we hit a contradiction.
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qetzal
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### Re: Six statements. How many are true?

I'm having a major difficulty with this puzzle.

Statement A is "This statement is true if and only if the next statement is true." I don't think you can determine that A is true just by showing (or assuming) that B is true. If that were the intent of the puzzle, then A should have been worded simply "The next statement is true." Instead, it seems to me that you have to show that it's true that A is true IFF B is true. IOW, you have to show that A is true if B is true and that A is false if B is false.

B seems even harder to parse. Note that it doesn't just say "At least two of the statements below this statement are true." That would be straightforward. Instead, it says "This statement is false if at least two of the statements below this statement are true." Suppose at least 2 of C-F really were true. Wouldn't that make B equivalent to "This statement is false?"

I can't decide if the puzzle was written this way intentionally or not, mainly because I'm not good enough to figure out if there's non-contradictory solution.

HonoreDB
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### Re: Six statements. How many are true?

This is fairly simple if you disallow statements from being reducible to "this statement is false", a statement that can be considered neither false nor true. Assuming that,

Spoiler:
A forces B to be true, regardless of what B says. If B were false, A would be asserting that it's false, which we've disallowed. So B is true.

Also, if D,E, or F were true, that would automatically make C true, since "x if y" is equivalent to "not y or x". But then that would mean that two statements below B were true, so B would be asserting it was false. So D,E, and F are all false.

Since F is false, A is true. Since A and B are true, C is true. So the only possible consistent state is A,B,C true, D,E,F false. We can confirm that this is consistent by checking that D and E are indeed false: D says there are four true statements, which is false. E says that D and F have different truth values, which is false. So this is indeed consistent, and the answer is 3.

qetzal
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### Re: Six statements. How many are true?

Except, if "This statement is false" is neither true nor false, then

Spoiler:
I don't see how B can be true. For B to be true, it would have to be the case that if at least 2 of C-F were true, then B would be false. But you've already stipulated that "This statement is false" is neither true nor false. So, if at least 2 of C-F are true, then B is neither true nor false, which means it's not false (as it asserts), so I think B must actually be false.

But I agree with you that if B is false, then that implies that A simplifies to "A is false."

So I'm still stuck wondering if this puzzle is just poorly written, or if I'm over-interpreting, or missing something, or what.

Cradarc
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### Re: Six statements. How many are true?

I did differently than you guys.
There are 2^6 different ways to assign true/false to each of the statements. The goal is to determine which of those 64 different combinations are valid. Upon doing that, you must find how many "true" are in such combinations. I don't think you can go in assuming only one valid combination exists.

Spoiler:
I focused on statement F because it seems the least convoluted.

Suppose D-F are all false.
Because F is false, A is true. If A is true, then B is true.
Thus, if D-F are all false, then A and B are both true.
We just derived statement C without any assumptions*, thus C is absolutely true.
*The assumption about D-F all false is part of the assertion in Statement C.

So C is true in the same sense that "Shroedinger's cat is alive if the cyanide didn't trigger" is true. This is independent of whether or not the cyanide triggered or if the cat is dead or alive.

Suppose F is false.
Because F is false, A is true. If A is true, B is true.
We also know C is true from the prior deduction.
The possible solutions at this point are A,B,C,?D,?E,~F.
If E is true, D has to be true because F is false. However that will contradict D.
If E is false, D can be true or false and stay consistent with itself.
However B will be inconsistent if D is true, so D must be false.
Thus, given F is false, the solution would be A,B,C,~D,~E,~F.

Suppose F is true.
Because F is true, A is false.
We know C is true from the prior deduction.
Because C and F are both true, B must be false or it will contradict itself.
The possible solutions at this point are ~A,~B,C,?D,?E,F.
If D is true, E must be true to keep D consistent.
However E will be inconsistent if D and F are both true, so D must be false.
If D is false and F is true, E must be true.
Thus, given F is true, the solution would be ~A,~B,C,~D,E,F.

F has to be either true or false (otherwise this problem would have no correct answer).
We just considered both cases, F is false and F is true. In both cases that led to a solution with three truths. Thus, the answer is 3.
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Qaanol
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### Re: Six statements. How many are true?

Spoiler:
Suppose all statements have well-defined truth values.

If A is true, then so is B.
But if A is false, then B must be true.
So B is true.

Since B is true, therefore at most 1 of C–F can be true.
This means D is false, because at most A, B, and one of the other statements could be true.

If E were true then so would be F, contradicting B.
So E is false.

Because D is false, E being false implies F is false.
Since F is false, A is true.

Now A and B are true, while D, E, and F are false.
It follows that C is true.

Hence {A, B, C} are all true, and {D, E, F} are all false.

I have not analyzed what might happen if some statements are allowed to have non-well-defined truth values.
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phlip
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### Re: Six statements. How many are true?

Qaanol's logic is how I solved this one too... but for completeness:
Spoiler:

Code: Select all

`def imp(a,c): return not a or cdef count(*a): return len(filter(None, a))for A in [True,False]: for B in [True,False]:  for C in [True,False]:   for D in [True,False]:    for E in [True,False]:     for F in [True,False]:      if (A == (A == B) and          B == imp(count(C,D,E,F) >= 2, not B) and          C == imp(not D and not E and not F, A and B) and          D == (count(A,B,C,D,E,F) == 4) and          E == (not D == F) and          F == (not A)):       print [A,B,C,D,E,F]`

Code: Select all

`[True, True, True, False, False, False]`
qetzal wrote:
Spoiler:
I don't see how B can be true. For B to be true, it would have to be the case that if at least 2 of C-F were true, then B would be false. But you've already stipulated that "This statement is false" is neither true nor false. So, if at least 2 of C-F are true, then B is neither true nor false, which means it's not false (as it asserts), so I think B must actually be false.

Spoiler:
What you're missing is that you have two assumptions leading to a contradiction - if B is true, and at least 2 of C-F are true, then B contradicts itself. But we've decided that B has to be true (because otherwise A is a contradiction) so we have to deduce that our other assumption is the one that's incorrect - that is, at most 1 of C-F are true.

Cradarc wrote:
Spoiler:
Suppose F is true.
Because F is true, A is false.
We know C is true from the prior deduction.
Because C and F are both true, B must be false or it will contradict itself.
The possible solutions at this point are ~A,~B,C,?D,?E,F.
If D is true, E must be true to keep D consistent.
However E will be inconsistent if D and F are both true, so D must be false.
If D is false and F is true, E must be true.
Thus, given F is true, the solution would be ~A,~B,C,~D,E,F.

Spoiler:
This solution doesn't satisfy A or B, though... as you're saying they're false, but they would both be true with that assignment of truth-values.

Code: Select all

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Cradarc
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### Re: Six statements. How many are true?

@Phlip
Spoiler:
phlip wrote:This solution doesn't satisfy A or B, though... as you're saying they're false, but they would both be true with that assignment of truth-values.

Yes, but they aren't assigned true values, so it's still consistent. Consider this claim:
"I am sleeping if and only if I am lying on a bed."
If I am not lying on a bed and I am not sleeping, does that make the above claim true? Possibly, but it doesn't have to be.

When I initially assumed F is true, I isolated a certain set of possibilities. None of those possibilities allow A or B to be true. Others started by assuming A is true, and then concluding F is false. You can argue F could be true, but that would be purposely contradicting your assumption. Same concept.
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phlip
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### Re: Six statements. How many are true?

Cradarc wrote:
Spoiler:
Yes, but they aren't assigned true values, so it's still consistent. Consider this claim:
"I am sleeping if and only if I am lying on a bed."
If I am not lying on a bed and I am not sleeping, does that make the above claim true? Possibly, but it doesn't have to be.

Spoiler:
That's... not how propositional logic works. If you're not sleeping and not lying in a bed, then "I am sleeping if and only if I am lying on a bed" is a true statement.

Similarly, if A is false and B is false, then "A is true iff B is true" is a true statement, which contradicts the previous statement that A is false.

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Yat
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### Re: Six statements. How many are true?

Of course I solved tis with the classic method, but considering how some of the statements were designed, I wanted to try something else.

Spoiler:
I consider the nature of a statement depends only on how this statement matches reality, and not on what other statements say about it.
For example "This statement is true if and only if two plus two equals seventeen" is neither true nor false because either case would lead to a contradiction. In that sense, B does not have to be true just because A seems to leave only this option, anymore than saying "This statement is true if and only if I will receive a twelve billion dollars transfer on my bank account in the next twenty seconds" will make me rich. This is just an inconsistent statement.

So if B is not true, then A is inconsistent. But what happens if B is true ? There is no more contradiction, but we have no way of knowing wether A is true or false. So I'm pushing further, saying that in this case, it is either true or false, such as "this statement is true". It is and undecidable statement.

As I said earlier, an external statement about this can't make A true or false, therefore F is false, because A is either undecidable or inconsistent and therefore can't be true.
As F is false, E is strictly equivalent to "Statement D is not false" and as A is not true, C is strictly equivalent to "At least one of the statements below is true"

As A and F are not true, D needs B, C and E to be true to be undecidable and therefore not false. But if C and E were true, then B would state "This statement is false", which would be an inconsistent statement, not a true one. Therefore, B, C and E can't be all true, so D is false, and so is E.

At this point, it is obvious that C is false, so the condition in B is false, which makes B true. Therefore, A is undecidable and we're done.

A. Undecidable.
B. True.
C. False.
D. False.
E. False.
F. False.

One statement is true.

SPACKlick
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### Re: Six statements. How many are true?

My analysis was based simply on a logic tree. But I agree with most of the above methods
Spoiler:
The only consistent truth vales are {A,B,C} = T, {D,E,F} = F

I categorised the statements (with 0 = F and 1 =T as)

A=If A=B then 1 else 0
B=If C+D+E+F > 1 then 0 else 1
C=If D+E+F > 0 then 1 else If A+B =2 then 1 else 0
D=If A+B+C+D+E+F=4 then 1 else 0
E=If D=F then 0 else 1
F=1-A

If A is true then B is True
If A and B are True then C is True
If B is true and C is true then D E and F are false

If A is False then B is True
If A is False then F is True
If B is true then only one of C, D E and F can be true, and F is true so C, D and E are false
If F Is True, C is true
C is both false and True Contradiction
If F is true and D is false then E is true.
E is both false and true, Contradiction

snowyowl
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### Re: Six statements. How many are true?

Edit: Whoops, misread statement E as "Statement D is false if and only if statement F is false". Interestingly, the puzzle still has a unique solution, so I include it for completeness' sake.

Quaanol has it right.

Spoiler:
In my modified version of the puzzle, A, B and E are true.

A: "A if and only if B".
If A is true, then B is true iff A is true, so B is true.
If A is false, then B is true iff A is false, so B is true.
So B is definitely true.

B: "Not B if and only if (at least two of C, D, E and F are true)".
B is true, so at most one of C, D, E and F is true.

D is false: A, B, and up to one other statement is true, so there cannot be four true statements.

E: "D is false if and only if F is false"
We know D is false, so E is true if and only if F is false.

So exactly one of E and F is true, which means that C cannot be true (or there would be two of C, D, E and F which are true).

C: "All the statements above this statement are true if none of the statements below this statement are true".
In propositional logic, the only way this can be false is if all the statements above it are true and some of the statements below it are true.
Hence, A is true.

Hence, F is false.

Hence, E is true.

Conclusion: A, B, not C, not D, E, not F.

And we're done.
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Who
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### Re: Six statements. How many are true?

Spoiler:
A: "A iff B", or, "B iff A":
There are two cases: (I think it's called the law of the excluded middle or something)
A is true:
B is also true. (Modus Ponens)
A is false:
Thus ~(A iff B), since A is false that's ~(False iff B) thus ~~B thus B.
Thus, B.
B: "At least 2 statements below B -> ~B"
or
"B -> <2 statements below B"
B is true by A, thus <2 statements below B are true.
C: Since B is true it can be ignored, thus C="~D & ~E & ~F -> A", the negation of which would be " ~A & (~D & ~E & ~F)", which requires ~A and ~F both being true, which can't happen, thus C is true.
Everything else is false by B. (We don't even have to look at D or E)
Because F is false, A is true.
Thus, the answer is 3 statements are true. (A, B, and C).

When we look at D and E because we're bored we get:
Assuming the solution, D can be false if we want by itself.
E says "D ≠ F", since E is false then D must equal F, which is great because they're the same and both false.
And then F is false.

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### Re: Six statements. How many are true?

Cradarc wrote:@Phlip
Spoiler:
phlip wrote:This solution doesn't satisfy A or B, though... as you're saying they're false, but they would both be true with that assignment of truth-values.

Yes, but they aren't assigned true values, so it's still consistent. Consider this claim:
"I am sleeping if and only if I am lying on a bed."
If I am not lying on a bed and I am not sleeping, does that make the above claim true? Possibly, but it doesn't have to be.

When I initially assumed F is true, I isolated a certain set of possibilities. None of those possibilities allow A or B to be true. Others started by assuming A is true, and then concluding F is false. You can argue F could be true, but that would be purposely contradicting your assumption. Same concept.

Spoiler:
The problem is that you have three different statements instead of two (where one is self-referential.)

A: I am sleeping iff I am lying on a bed.
B: I am sleeping.
C: I am lying on a bed.

That means that A is equivalent to "B iff C", and in that case you are correct. C can be false, and B can still be either false (making A true) or true (making A false) and there is no contradiction.

In this case though, A is equivalent to "A iff B" so it's more like you are saying "This statement is true if and only if I am lying on a bed." It is not possible in propositional logic for you to not be lying on a bed. It's only as possible as saying "this statement is false" which is exactly the sort of contradiction that makes finding a solution possible in the first place.

Examining your first post, your second option - F being true - can't occur because then A and B are both false, making A true, leading to a contradiction.

There is only one solution.

EDIT: After reading Phlip's reply I realize that I may have misunderstood your criticism. I guess the real issue is confusing how language works with how logic works. When I say "I am sleeping if and only if I am lying in bed," that might not be a logical statement. In logic, "A if and only if B" is equivalent to "(A and B) or (~A and ~B)" and so the fact that you are doing neither means that the statement is true. In language, you could be describing your life, during which you can (and probably have) run the gamut of the truth table with regard to those statements. Logically, though, the statement "A iff B" would be true or false at different times, depending on the truth of A or B.

SPACKlick
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### Re: Six statements. How many are true?

Not sure if it's appropriate to post here or if I should start a new thread but this puzzle really got me thinking of self referential tests. There's a 20 question version here but a simpler 5 question version is

1) The answer that is not an answer to any other question is
A) A B) B C) C D) D

2) The only question with the same answer as this one is
A) 5 B) 1 C) 3 D) 4

3) The answer to question number 5 is
A) B B) D C) A D) C

4) The first question whose answer is A is
A) 2 B) 3 C) 4 D) 5

5) The answer to number 3 is
A) C B) B C) D D) A

And the answer is... to come later

phlip
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### Re: Six statements. How many are true?

SPACKlick wrote:Not sure if it's appropriate to post here or if I should start a new thread but this puzzle really got me thinking of self referential tests. There's a 20 question version here

Ah yes, that classic. We've had a thread about that one before.

Code: Select all

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douglasm
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### Re: Six statements. How many are true?

SPACKlick wrote:Not sure if it's appropriate to post here or if I should start a new thread but this puzzle really got me thinking of self referential tests. There's a 20 question version here but a simpler 5 question version is

1) The answer that is not an answer to any other question is
A) A B) B C) C D) D

2) The only question with the same answer as this one is
A) 5 B) 1 C) 3 D) 4

3) The answer to question number 5 is
A) B B) D C) A D) C

4) The first question whose answer is A is
A) 2 B) 3 C) 4 D) 5

5) The answer to number 3 is
A) C B) B C) D D) A

And the answer is... to come later

I'm getting two possible solutions to that.
Spoiler:
3 and 5 make a nice pair to start with, referencing each other. For an answer in one to be valid, the other has to have the reverse answer. The only pairs there that match are 3C/5A and 3D/5C.

Now let's look at question 4. Answer A quickly leads to a contradiction with question 2. Answer B is invalid because question 3 can't be A as per above. Answer C is a self reference contradiction. Thus question 4 must be answered D.

4D being a correct answer means the 3C/5A pair must be the correct one.

There are now questions with answers of A, C, and D. That leaves only B as a possibility for question 1.

So far we have:
1B
2?
3C
4D
5A

Now to look at question 2. Answer A would contradict question 4. Answer B would contradict question 1. Questions 1 and 4 don't say anything about other answers, and questions 3 and 5 say nothing about other questions. Thus, in question 2, answers C and D only have to worry about contradicting themselves - and both of them are consistent.

Final answer:
1: B
2: C or D
3: C
4: D
5: A

Wildcard
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### Re: Six statements. How many are true?

I get
Spoiler:
A, B, C are true and D, E, F are false.
However, we can rephrase the statements to be simpler:

A. Statement B is true.
B. Less than two of the statements below this one are true.
C. If either A or B is false (or both), then at least one of {D, E, F} is true.
D. Exactly four of these statements are true.
E. Exactly one of statements D and F are true.
F. Statement A is false.

C is hard to rephrase. Anyone have a simpler rephrasing for that one?
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ingdas
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### Re: Six statements. How many are true?

I formulated this problem in first order logic (substituting 1-6 for A-F) and IDP confirms there is only one model

dtai.cs.kuleuven.be/krr/idp-ide/?src=b487d0bea2a0c3912b8e

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### Re: Six statements. How many are true?

Wildcard wrote:However, we can rephrase the statements to be simpler:

A. Statement B is true.

It's not quite the same... the system:
A: Statement B is true
B: <something>
can only be consistent if A and B have the same truth value. However, the system:
A: Statement A is true iff statement B is true
B: <something>
can only be consistent if B is true (and A can be either).

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