For all $x$, if $x^2$ is even, then $x$ is even.
The contrapositive to this statement is:
For all $x$, if $x$ is odd, then $x^2$ is odd.
Why do we ignore the "For all $x$" and not say "For some $x$..."?
$\endgroup$ 32 Answers
$\begingroup$Because the contrapositive refers to an equivalent form of the implication, it is thus a tautological equivalence. It refers to the "realm" of propositional logic and not first order logic.
For example, I'm sure you know that an equivalent form of $\phi \to \psi$ is $\lnot \phi \lor \psi$. Would you not then substitute in $\forall x (Fx \to Qx)$ for $\forall x (\lnot Fx \lor Qx)$?
$\endgroup$ $\begingroup$Symbolically
$~~~~\forall x \in N: [x^2 \in E \implies x \in E]$$
$~~~\equiv ~~~\forall x \in N: [x \notin E \implies x^2 \notin E]$
$~~~\equiv ~~~\forall x \in N: [x \in O \implies x^2 \in O]$
where $N$ is the set of natural numbers, $E$ is the set of even numbers, and $O$ is the set of odd numbers.
We use the contrapositive of the original implication. The quantifier is unaffected.
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