Double implication in natural language I'm talking about double implication like: (P → Q) → Q I know that this is equivalent to (P ∨ Q), but I don't quite understand why. Let's say I take proposition P to be "having guns", and proposition Q to be "violence", then I would express it in natural language as: "If having guns lead to violence, we would have violence" However it think this implicates some kind of (S ∧ P) → Q, where S is the original (guns → violence), and P is the implicit assumption that we actually have guns. What would be an example without such an implicit assumption, that is easy to hold on to, when intuition fails me?

Intuition works works in your example, but I admit it is not _very_ obvious. The statement "if having guns leads to violence, we would have violence" implies that

* we have guns (because why should we have violence if "guns lead to violence" is true, but we have no guns in the first place), or
* violence is there regardless of whether we have guns or not (an implication is true when the conclusion is true).



So we have guns or violence, or both.