If Las Vegas is the capital of Fiji, then $x^2=4$.

Question

If Las Vegas is the capital of Fiji, then $x^2=4$.

I was asked to state either the above claim is true or false. I must give a proof if it is true and counter example if it is false.

I prove its contrapositive: if not $q$ then not $p$ in the truth value table which is true always and is false only when $q$ or conclusion is false.

So since $x^2=4$ is false because the value of $x$ is unknown.
I say the claim is false.
Am I correct?

Answer 1

This is a vacuous truth, since the condition "Las vegas is capital of Fiji" is never satisfied, so for some $x$, this is equivalent to $$ \text{False} \implies \text{True} $$ and for some $x$, this is $$ \text{False} \implies \text{False} $$ Both statements are true, so this is true.

Answer 2

This is vacuously true by definition of material implication: https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference)

It is true because the law of the excluded middle demands implication to have a truthy or falsy value. The truth table of implication is such that if P is false, then regardless of the validity of Q, the whole statement is true. Vacuous truths, however, are not of much interest beyond this.

Answer 3

Differing opinion here. What has been presented is not a proposition at all, so it can not be either true or false. It is not a proposition because $x^2=4$ is not a proposition. It includes a "variable" but no quantifier. $x^2 = 4$ for x=3 is a proposition. So is $x^2 = 4$ for all $x \in \mathbb R$. (As it happens, both these propositions are false). $x^2 = 4$ by itself, with no binding of $x$ or universal or existential quantifier to $x$, is simply not a proposition, so the implication is not a proposition either. So to ask whether it is true or false doesn't make sense (is undefined).