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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?

Em.
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Surdz
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    "If $P$ then $Q$" is always true when $P$ is false. So regardless of what $x$ is, the statement is true. – User8128 Mar 28 '16 at 02:05
  • Small notes: Proof is a noun, prove is a verb, and you can find formatting tips here. – Em. Mar 28 '16 at 02:20
  • I would say it is neither true nor false, since our expression is a formula, but not a sentence. Some interpret free occurrences of variables as implicitly universally quantified. For those people, the formula is true. – André Nicolas Mar 28 '16 at 02:34

3 Answers3

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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.

Henricus V.
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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).

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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.

q.Then
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