📂Set Theory

Mathematical Proof of the Counterfactual

Law ¹

$$ p \to q \iff \lnot q \to \lnot p $$

Explanation

If a proposition is true, then its contrapositive is also true; if a proposition is false, then its contrapositive is also false. Of course, if the converse holds, then the inverse of the original proposition also holds through contraposition.

These expressions might be too difficult for those not familiar with mathematics. Let’s understand it through an intuitive example:

• $p$ : The weather is hot
• $q$ : Sweating occurs
• $p \to q$ : If the weather is hot, then sweating occurs

If it is true that sweating occurs when the weather is hot, then if there is no sweating, we can conclude that, at least, it is not because the weather is hot.

Proof

$$ \begin{align*} p \to q \iff & \lnot p \lor q \\ \iff & \lnot p \lor \lnot (\lnot q) \\ \iff & \lnot (\lnot q) \lor \lnot p \\ \iff & \lnot q \to \lnot p \end{align*} $$

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