Consider the following truth–tellers and liars puzzle:
Three students are talking over a beer, discussing mathematics.
Ralph rounds a cigarette with his forefingers and thumbs, and announces with satisfaction: “If interior angles of a triangle add up to $2\pi$, then interior angles of a quadrilateral add up to $\pi$.”
“That statement is true,” says Morgan and smiles admiringly. “It is indeed.”
“It is pure nonsense!” The resistance in Raymond’s voice is almost frantic.
Who is a truth–teller and who is a liar?
Obviously, as long as standard units of angular measures are applied, all interior angles of a triangle add up to $\pi$ whereas all interior angles of a quadrilateral add up to $2\pi$.
As far as Euclidean geometry is concerned, Raymond is the only truth–teller regardless of whom he opposes. Ralph is a liar, and so is Morgan as a supporter of Ralph’s statement. A diagonal divides a quadrilateral into two triangles for which it is the common edge. This implies that the sum of all interior angles of the quadrilateral is the sum of interior angles of both triangles. This in turn implies, that the sum of all interior angles of a quadrilateral is double the sum of all interior angles of a single triangle, not a half of it, as Ralph’s statement suggests. This answer seems valid whether we take (1) into account or not.
As far as Propositional calculus is concerned, the answer is quite different:
- $p$ is the antecedent of Ralph’s statement: interior angles of a triangle add up to $2\pi$,
- $q$ is the consequent of the statement: interior angles of a quadrilateral add up to $\pi$,
- taking (1) into account, $p$ is always false, therefore $p \rightarrow q$ is always true regardless of the value of $q$ which makes Ralph a truth–teller,
- Morgan is a truth–teller as his statement ($r$) about $p \rightarrow q$ is true,
- Raymond is a liar as his statement ($s$) is false regardless whether it refers to $p \rightarrow q$ or to $r$.
Is my reasoning correct?
Based on the remarks provided in the comments section, points (1) and (3) of the original description present a correct solution of the puzzle.