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?
 A: 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.
A: The trouble is that the material implication $\rightarrow$ does not always perfectly match the English 'if ... then...'.
This mismatch is called the Paradox of Material Implication.
So, while given the mathematical definitions of the truth-functional operators $\rightarrow$ and $\lor$ it is true that $(P \rightarrow Q) \rightarrow Q \Leftrightarrow P \lor Q$, this does not readily make sense when interpreting this in terms of English conditionals.
Here is another example:
According to the way we mathematically defined the truth-functional operator $\rightarrow$, we have that:
$$(P \land Q) \rightarrow R \Leftrightarrow (P \rightarrow R) \lor (Q \rightarrow R)$$
Now, does that make any intuitive sense? No. For example, we believe that 'If one is a male and unmarried, then one is a bachelor', but we don't believe that either 'If one is a male then one is a bachelor' or that 'If one is unmarried then one is a bachelor'
