Double implication in natural language

Question

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?

Answer 1

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.

Answer 2

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'