False implies anything [duplicate]

Solution 1:

1. The fact that $$P\to Q$$ is true whenever its antecedent $P$ is false (principle of explosion; vacuous truth) is actually so by definition:

         $P\to Q\,$ is the truth function that is tautologically equivalent to $\,\lnot P\lor Q.$

   So, $P\to Q$ is false precisely when $P$ is true but $Q$ false.

2. To be clear: when $P$ is false, $\,P\to Q\,$ (is always true so) never gives any information about $Q.$

3. Summarising these two explanations of the motivation for the above definition:

   if we insist, to the contrary, that  False$\to$True  be false, then, unfortunately, these violations of natural deduction arise: $$\text{$A$ is true and $B$ is false $\implies\bigg(\big[(A\land B)\to A\big]\;$is false}\bigg)$$ and $$\big[\forall n\in\mathbb Z \,\big(n \text{ is a multiple of }4\, \to \,n \text{ is even}\big)\big]\;\text{is false}.$$

4. It is worth noting that in logic/mathematics, $P$ need not cause $Q$ for $P$ to imply $Q$ (i.e., for the material conditional $P\to Q$ to actually be true); after all, $\to$ is defined truth-functionally rather than based on interpretation (of the meanings of $P$ and $Q$).