in Mathematical Logic edited by
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In a room there are only two types of people, namely $\text{Type 1}$ and $\text{Type 2}$. $\text{Type 1}$ people always tell the truth and $\text{Type 2}$ people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking the person replies the following

"The result of the toss is head if and only if I am telling the truth"

Which of the following options is correct?

  1. The result is head
  2. The result is tail
  3. If the person is of $\text{Type 2}$, then the result is tail
  4. If the person is of $\text{Type 1}$, then the result is tail
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superb explanation
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Assume Type 1 persom is K(always Telling truth), Type 2 person is N(Always lying).

Statement is $H \leftrightarrow P(K)$ (where H means Head, P(K) means person is K i.e. telling truth )

Case 1 : Assume the person is K.

then $H \leftrightarrow P(K)$ is true, and since P(K) is true in this case, so H is also true. So, Head in this case.

Case 2 : Assume the person is N.

then $H \leftrightarrow P(K)$ is False, and since P(K) is false in this case, so H is true, so, Head in this case also.

Hence, answer A.
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11 Answers

135 votes
135 votes
Best answer

We do not know the type of the person from whom those words are coming from and so can have two cases :

  1. Truth-teller : definitely implies that result of toss is Head.
  2. Liar : the reality will be the negation of the statement.

The negation of $(x\iff y)$ is exactly one of $x$ or $y$ holds. So, we negate the statement : "The result of the toss is head if and only if I am telling the truth". This give rise to two possibilities

  • it is head and lie spoken
  • it is not head and truth spoken

Clearly the second one cannot be true because it cannot be a reality that the liar speaks the truth.

So, this implies that even if we negate the statement to see the reality or don't do that; The reality is that the toss yielded a Head.

Answer = option (A).

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4 Comments

The negation of (x⇔y) is (x⇔¬y) so, we negate the statement : "The result of the toss is head if and only if I am telling the truth" as "The result of the toss is head if and only if I am not telling the truth" or  "The result of the toss is head if and only if I am telling lie". 

Now we can interpret from Type2 as "The result of the toss is head if and only if I am telling lie" and he is lying so the result of toss is head...

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Thanks a lot.Realy nice explanation.

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Very well explained thankyou
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56 votes
56 votes
Person 1 (truth teller), result is head. No doubt here as he is a truth teller. 
Person 2(lie teller). result is head if and only if he is telling truth. He is telling lies. So, the truth is the opposite of his statement. We can analyze his statement as two
  1. If I'm telling the truth result is head 
  2. If result is head I'm telling the truth

Both these are of the form $A \rightarrow B = \neg A \vee B$. Now, the truth will be the negation of these statements which will be $A \wedge \neg B$ which can be expressed as

  1. I'm telling the truth and result is not head
  2. Result is head and I'm telling false

Both of these means, the result is head as the person is lie teller. (Actually even if iff is replaced with if, answer would be A)

So, option A.

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4 Comments

@Arjun Can you please tell me if the below approach is correct?

Let h: result of the toss is head

t: the person is telling truth

h$\Leftrightarrow$t = h't' + ht

Type 1: t is true, so h't' + ht = h

Type 2: t is false, so h't' + ht = h'

Since Type 1 says truth always the answer will be h = head.

Also as Type 2 lies always the answer will not be h' = not not head = head.

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Why truth will be negation of these statements?
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Perfect Sir!!!
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20 votes
20 votes

The question involves understanding the bi-implication completely.

The key statement is

The result of the toss is head if and only if I am telling the truth.

Now let us denote two propositional variables p and q where

p: "The result of the toss is head"

q: "I am telling the truth"

and hence the statement is $p\Leftrightarrow q$

Now we have two cases

Case 1: The person is of Type 1: Here it is very simple case.Since Type 1 person always tells the truth and hence according to him the result of the toss is Head.

Case 2: If the person if Type 2, then he always tells lies. Means, the statement $p \Leftrightarrow q$ given by this person is actually the negation of the original statement which he means 

i.e. $\sim (p \leftrightarrow q) = (p \leftrightarrow \sim q)$

so $p \leftrightarrow \sim q$ is The result of the toss is head if and only if I am not telling the truth.

And, yes this bi-implication is true for this Type 2 person because he never tells truth. So, here in this case also the result is Head.

And hence, the result Head is consistent irrespective of the type of person we choose.

Answer (A)

3 Comments

Perfect :)
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Best answer
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got it
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16 votes
16 votes

P:The result of the toss is head

Q:The person is telling the truth

Statement by the person:P$\Leftrightarrow$Q

Truth table for P$\Leftrightarrow$Q :

P Q P$\Leftrightarrow$Q
T T T
T F F
F T F
F F T

Since the statement is true for either both P and Q are true or both are false,so there are two possibilities:

1.The person tells truth that the result is head.

2.The person tells lie that the result is not head i.e the result is tail,

Both possibilities shows the result is head.

So, the answer is A.

2 Comments

in your table 3rd row also should be true.. becoz in biconditional only P Q = F  rest of all are true
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Awsm explaination
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Answer:

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