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?

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'
