Logic : One person speaks Truth only but the other one only Lies.

In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and 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.”

How do I understand that the result is Head. No matter who speaks those words.

Not able to grasp it. Although, it is clear that if these words are from truth-teller then definitely the result is Head. But what if it was the Liar who said them?

$X={}$“The result of the toss is head if and only if I am telling the truth.” $$\begin{array}{|c|c|c|} \hline \text{type of person} & \text{coin outcome} & \text{Could say X?} \\ \hline \text{truth-teller} & \text{head} & \text{yes} \\ \text{truth-teller} & \text{tail} & \text{no} \\ \text{liar} & \text{head} & \text{yes} \\ \text{liar} & \text{tail} & \text{no} \\ \hline \end{array}$$ The third column is "yes" when the outcome is a "head"; otherwise it's "no".

If a liar says this, then the truth is the negation of the statement.

In general, $$p\iff q\equiv (p\Leftarrow q)\wedge(q\Leftarrow p)\equiv(q\vee\neg p)\wedge(p\vee\neg q),$$ and so $$\neg(p\iff q)\equiv\neg(q\vee\neg p)\vee(p\vee\neg q)\equiv(p\wedge\neg q)\vee(q\wedge\neg p)$$ by DeMorgan's laws.

In this particular situation, then, we let $p$ be the statement "the result of the toss is heads"; $q,$ "I am telling the truth." Since $q$ is false, then so is $q\wedge\neg p,$ but $(p\wedge\neg q)\vee(q\wedge\neg p)$ is true, so what can we conclude?

What I believe is the following: In case the person is telling a lie, either of the following statements must be true (i.e. the statement of the person is to be "negated"):

$$\mathbb{The \ result \ of \ the \ toss \ is \ a \ head \ but \ the \ person \ is \ not \ telling \ the \ truth}$$ or $$\mathbb{The \ person \ is \ telling \ the \ truth \ but \ the \ result \ of \ the \ toss \ is \ a \ tail}$$ The second statement cannot be true since the person cannot be telling the truth (Note this is the case when the person is a liar) and thus the statement that the result of the toss is a head but the person is lying must be true.

We do not know what is the person from whom those words are coming from, we can have two cases :

Truth-teller : definitely implies that result of toss is Head.

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 imply that even if we negate the statement to see the reality or don't negate that;

The reality in any case is that the toss yielded a Head.