The teller of lies

If we call the friend $A$ , the meaningful content of $A$ 's claims can be summarized in two statements.

$1.$ $A$ will not lie today.

$2.$ Statement $1$ is false.

We will evaluate this by cases.

Case 1. $A$ is telling the truth today. Assume $A$ is truthful today. Then $1$ and $2$ should both be true. We note that since $A$ is telling the truth today, $1$ is true. Statements $1$ and $2$ are logically incompatible, in that there is a biconditional relationship such that $1$ if and only if the negation of $2$ . So $2$ must be false. So if $A$ is telling the truth today, then $2$ is true and $2$ is false. This produces a contradiction. Thus, $A$ isn't truthful today.

Case 2. $A$ is lying today. Assume $A$ is lying today. Then $1$ and $2$ should both be false. We note that since $A$ is lying today, $1$ is false. Statements $1$ and $2$ are logically incompatible, in that there is a biconditional relationship such that $1$ if and only if the negation of $2$ . So $2$ must be true. So if $A$ is lying today, then $2$ is false and $2$ is true. This produces a contradiction. Thus, $A$ isn't lying today.

Either assumption leads to a contradiction. Hence, it is an impossible scenario.

It is worth noting that we could have made this determination without having to examine cases by inspecting the problem statement. Since the friend is supposed to make statements that have the same truth value for the entirety of a day, finding any pair of statements such that one is true and the other is false is sufficient for rejection of the hypothesis of the statement, even if it is not known which is true and which is false. We assume, of course, that the statements were made during the same day (i.e., the friend doesn't toggle between truth telling and lying at midnight, with the former question answered at 11:59 PM and the latter answered at 12:00:01 AM).