## Ambiguous Teller Riddle

• 5.4k
I was reading various riddles and puzzles on the internet. One of the formulations of the riddles state: formulate it [the riddle] with first-order logic. Is it possible to formulate it using first-order logic?

The riddle goes as follows:

Among persons A, B and C one person always lies, one person always tells the truth and one person sometimes speaks the truth, hence being ambiguous.

We get the following statements:

• Person A claims person B always tells the truth.
• Person B claims person B (himself) sometimes tells the truth.
• Person C claims person B always lies.

I made an attempt to formulate the 3 statements:

A ∧ B. A is true because B is true.
B ∨ = ¬B. B says the truth or not but not both.
¬(C ∨ B). C says the truth as a negation of B.

My first inquiry is if I accurately formulated the statements. If you see any errors or hiccups, please correct me.

Secondly, I want to solve this riddle. Who is the liar?

Statement like "A claims P" for some proposition P becomes a pair of statements:

P ∧ ¬A. if P is true, A cannot be the liar.
¬P ∧ ¬A. if P is false, A cannot be the truth teller.

But here we have some contradictions. Assume A is true always, but he says b is true always.
On the other hand, assume B is true always, but he himself admits that he is ambiguous. This is all contradictory.

How would you formulate this riddle?

I think the statements of A and B are contradictory. Do you agree?

• 1.8k
Person A claims person B always tells the truth.
Person B claims person B (himself) sometimes tells the truth.
Person C claims person B always lies.

Secondly, I want to solve this riddle. Who is the liar?

When person B says he "sometimes tells the truth", is that consistent with the statement "person B always tells the truth"? I just want to get that question clarified first, because it could be a weird technicality - when he says he sometimes tells the truth, is he saying he's definitely not the guy who always tells the truth?

Because there's an interpretation of "sometimes" that's consistent with "always". If "some" just means more than one, then "always" counts as a possible validation of "sometimes".
• 1.8k
But if "b always tells the truth" and "b sometimes tells the truth" are interpreted to be mutually exclusive statements, the riddle has an immediate solution.

C tells the truth
B is the liar
A sometimes tells the truth

If they aren't mutually exclusive there's another possible answer (maybe more than one)
• 5.4k
when he says he sometimes tells the truth, is he saying he's definitely not the guy who always tells the truth?

B says the truth or not but not both. B says the truth often. B also lies often. B doesn’t say both. The ambiguity of B is the core of this riddle. I copied and pasted it as it is written on the internet. I haven’t altered anything. We can conclude he is not definitely the person who always tell the truth, yes. Nonetheless, A claims B always tells the truth.

This is a clear contradiction to the riddle. Right?

B is the liar

Exactly.
• 1.8k
I'm not sure you answered my question. Can you give me a yes or no?
• 5.4k
Okay. I will be more clear.

is he saying he's definitely not the guy who always tells the truth?

Yes.
• 1.8k
okay, then if they're mutually exclusive, we can use the following logic:

B definitely CANNOT be the guy who always tells the truth, since that would make Bs statement about himself a lie

A definitely CANNOT be the guy who always tells the truth, since that would mean B always tells the truth, which we know is false

That only leaves C as the guy who always tells the truth

The rest naturally follows.
• 5.4k
That only leaves C as the guy who always tells the truth

I had the same thoughts about C. A is a liar, and B is ambiguous , so I believe C is the lone truth-teller. But I was wondering if I had properly written those in logic language because the riddle statement requested if it could be formulated in first-order logic or not. Since you did not criticise my initial question, I assume I formulated it correctly.

The rest naturally follows

The liar is B, but also A. Is it contradictory or ambiguous?
• 1.8k
if we know c is the truth teller, and c says b is the liar, then b is the liar. Easy as that.

A sometimes tells the truth, and his statement in this riddle just happens to be a lie. Presumably one can imagine a has told the truth at some other occasion.
• 5.4k
A sometimes tells the truth, and his statement in this riddle just happens to be a lie. Presumably one can imagine a has told the truth at some other occasion.

Exactly. That is why I questioned whether A contradicts his own assertion or if A, like B, is simply ambiguous. At least, we both believe that C is the lone truth-teller.
Then B is always false and ambiguous. But what happens to A?
• 1.8k
Then B is always false and ambiguous

No, just false

But what happens to A?

He sometimes lies, and sometimes tells the truth.
• 5.4k
Are you suggesting that A is actually ambiguous and not B?
• 1.8k
I've been saying that consistently since my second post in this thread, yeah
• 1.8k
do you think it could be some other way?
• 5.4k
I back up your argument, and I also see A as ambiguous. A claims person B always tells the truth. B claims that person B (himself) sometimes tells the truth. Then, A tells the truth (ambiguous). But consider for a second this:

Assume A is true always, but he says B is true always. Is A still ambiguous or contradictory?

Assume B is always true, but he himself admits that he is ambiguous (because B stated that he sometimes tells the truth). Then B is the liar, though not always. He is ambiguous in this context.
• 15k
Among persons A, B and C one person always lies, one person always tells the truth and one person sometimes speaks the truth, hence being ambiguous.

We get the following statements:

Person A claims person B always tells the truth.
Person B claims person B (himself) sometimes tells the truth.
Person C claims person B always lies.

If "sometimes tells the truth" entails "sometimes lies" then:

If Person A is the person who always tells the truth then Person B is the person who always tells the truth. This is a contradiction. Therefore, Person A is not the person who always tells the truth.

If Person A is the person who always lies then Person C is the person who always tells the truth. If Person C is the person who always tells the truth then Person B is the person who always lies. This is a contradiction. Therefore, Person A is not the person who always lies.

Therefore, Person A is the person who sometimes tells the truth, Person B is the person who always lies, and Person C is the person who always tells the truth.
• 1.8k
you've just laid out two scenarios we already know aren't the case. I'm not really sure what the point of assuming they are the truth tellers, when we already know they can't be the truth teller.

Assuming either one of them is the truth teller leads to contradiction, so we don't.
• 486
(1) Person A claims person B always tells the truth.
(2) Person B claims person B (himself) sometimes tells the truth.
(3) Person C claims person B always lies.

In first-order logic, you may want a predicate is(person,identity). Example, is(A,liar) evaluating to true if A is liar.

In fact, the solution space is very small, just 3! = 6 possibilities. Therefore, brute force is a perfectly viable solution strategy:

(truth,random,liar)

A B C
0 truth random liar
1 random truth liar
2 liar random truth
3 random liar truth
4 truth liar random
5 lair truth random

Brute force evaluation:

0 truth random liar
(1) truth says about random that he is truth -> false, abort

1 random truth liar
(1) random says about truth that he is truth -> consistent
(2) truth says about truth that he is random -> false, abort

2 liar random truth
(1) liar says about random that he is truth -> consistent
(2) random says about random that he is random -> consistent
(3) truth says about random that he is liar -> false, abort

3 random liar truth
(1) random says about liar that he is truth -> consistent
(2) liar says about liar that he is random -> consistent
(3) truth says about liar that he is liar -> consistent
-- this is a legitimate solution --

4 truth liar random
(1) truth says about liar that he is truth -> false, abort

5 liar truth random
(1) liar says about liar that he is truth -> consistent
(2) truth says about truth that he is random -> false, abort

solution predicate:

is(person,identity) ≡ (person=A ∧ identity=random) ∨ (person=B ∧ identity=liar) ∨ (person=C ∧ identity=truth)
• 5.4k
Yes. I agree and I see the point. But what happens to A then? B is the only one who says he tells the truth sometimes. Nonetheless, it seems the riddle turned out with A being the ambiguous person, and this is very tricky to me. B cannot be the truth-teller. Therefore, A did not speak the truth  and therefore, A is not the truth-teller either. We all agree that C is the truth-teller. Then, is A the contradictory, ambiguous, or unreliable person here?

For me, it is like pulling a rabbit out of the hat. It is surprising that A is actually the problem and not B. Or did you see the point from the beginning?
• 5.4k
Assuming either one of them is the truth teller leads to contradiction, so we don't.

That’s what I tried to argue! If A is contradictory, then C is the truthteller, and B is ambiguous. I mean, according to this context, B could be the one who sometimes tells the truth.
• 15k
But what happens to A then

A is the person who sometimes tells the truth.

B cannot be the truth-teller.

That depends on whether or not "I sometimes tell the truth" entails "I sometimes lie". If it does then B is not the truth-teller. If it doesn't then the answer is undecidable.

Nonetheless, it seems the riddle turned out with A being the ambiguous person

No, B is the ambiguous person given the ambiguity of the phrase "I sometimes tell the truth".
• 1.8k
according what context? No b couldn't
• 1.8k
No, B is the ambiguous person given the ambiguity of the phrase "I sometimes tell the truth".

He's using unclear wording, but when he says "ambiguous person" he means "the person who sometimes tells the truth". He doesn't mean "the person whose role is ambiguous".
• 15k
He's using unclear wording, but when he says "ambiguous person" he means "the person who sometimes tells the truth". He doesn't mean "the person whose role is ambiguous".

Gotcha, thanks.
• 5.4k
I thought B was ambiguous at first glance, but consider this, Michael:

Person A claims person B always tells the truth. Person B claims person B (himself) sometimes tells the truth. Okay, then, A tells the truth and B always lies. A is the ambiguous person here. I tried to explain that he is just contradictory and B is ambiguous.
But I am starting to realise that A is dragged down by the ambiguity of B. And then I asked myself: does this make A ambiguous or just contradictory with his statement?
• 9k
Interested puzzlers are recommended to search out the books of Raymond Smullyan.
• 15k

I've made a few edits to my first comment since I first posted it. If you haven't already, refresh the page and check it out. It should answer everything very clearly.
• 5.4k
The context where A is always true and not sometimes.
• 1.8k
But that's not the context. A doesn't always tell the truth. So... why are you saying that's the context? We know he can't be the one who always tells the truth, because that would lead to a contradiction.
• 5.4k
Excellent! :up: Thanks to your contribution to this thread. Also, to @flannel jesus for keeping the discussion alive. I am enjoying this.
• 5.4k
I am trying to make an approach. I claim that A is just contradictory in his statement, but you are defending that A is precisely the ambiguous here because he ‘sometimes tells the truth’ and B is always the liar. I defend the opposite: B is ambiguous and A is contradictory for always telling the truth. It is B who often says the truth and others not, but not both. Then, when B tells the truth, A tells the truth as well. B is the ambiguous.
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