Lukasiewicz Logic Tautology

Question

Suppose a statement form $\varphi$ always has value T or U. Show $\varphi$ is a classical tautology.

Answer 1

Suppose that a statement $\varphi$ of propositional logic (using only $\wedge$, $\vee$ and $\to$) is not a propositional tautology in classical logic. It follows that some row of the truth table for $\varphi$ has value $F$.

Now, consider the truth table of $\varphi$ in Łukasiewicz three-valued logic, which is called Kleene logic on the Wikipedia page. The key observation is that Łukasiewicz three-valued logic agrees with classical logic on classical logic input. In other words, if all the propositional variables of $\varphi$ are given classical T/F values, then the truth value of $\varphi$ will agree with the classical logic value. Thus, since $\varphi$ had a classical row with value $F$, the very same row will have value F in the Łukasiewicz truth table of $\varphi$, contrary to the assumption that $\varphi$ had only values T or U.