Material implication P -> Q is equivalent to ~P v Q. It is generally agreed that the "if P, then Q" construction in ordinary language is not always the same as material implication. However, when you study mathematics, you're trained to think that, in mathematics, "if P, then Q" really is material implication. Here is an in many ways careful explanation: (1/n)
https://gowers.wordpress.com/2011/09/28/basic-logic-connectives-implies/
I was never sure what to make of this, because I have yet to read a discussion of why material implication is a better model of mathematicians' "if P, then Q" than other alternatives. For example, why not understand "if P, then Q" in mathematics as "necessarily, if P, then Q" and take it to correspond to [](P -> Q), where [] is an operator of modal logic? I'm sure people already thought of this, I just haven't seen the pros and cons of this alternative (and other alternatives) compared to the pros and cons of the material implication. E.g., what about the implication in relevance logic? (2/n)
@imdef the main advantage is simplicity; classical logic gives the only reasonable truth functional definition of connectives, so mathematical statements can be readily understood without reference to elaborate semantics or axiom systems. But e.g. constructive proofs, which are used at least sometimes by all mathematicians, would not allow the inference you discuss (invalid in intuitionistic logic)
@RanaldClouston Fair point, simplicity is a point in favor of material implication. What leaves me unsatisfied with the justifications I've read, though, is that they appear to beg the question by not seriously considering what would be different with a non-truth functional "if-then". E.g. a common argument is that it is convenient to allow vacuously true universal statements, but other non-truth functional implications can also allow these.
@imdef I think there's plenty of literature on this, although I'm not an expert on it. Certainly lots written on intuitionistic vs classical; the relevance logicians have plenty to say although I'm not sure how interested they are in mathematical applications; and I have no idea how Lewis's strict conditional would apply to maths (what would the box mean, precisely?)
@RanaldClouston I'm sure logicians have written a lot, I'm just not so familiar with this literature. I think identifying "if-then" with strict implication would avoid some, but not all, unnaturalness. For example, we could translate from English to formal language, and manipulate the material implication:
1. "If 2|x and 3|x, then 6|x."
2. [](2|x ^ 3|x -> 6|x)
3. []((2|x -> 6|x) v (3|x -> 6|x))
But then we'd be blocked from translating back into English, because for an if-then form of line 3 we'd need the stronger
3'. [](2|x -> 6|x) v [](3|x -> 6|x)
@imdef sure, but what does 'necessarily' mean when applied to mathematics? Most mathematicians would hold that all mathematical truths are necessarily true, which trivialises the modality and hence strict implication collapses to regular implication.
@RanaldClouston At least in simple cases, I think [](P -> Q) says in the object language what P |= Q says in the meta-language. Maybe there's a divergence in complicated cases, I don't know. In the particular case above, I'd take
[](3|x -> 6|x)
to be equivalent to
(for all natural numbers x)(3|x -> 6|x)
@RanaldClouston (With []P meaning "Necessarily, P".)
Vidal also gives the example with a specific value x=4 in the above statement. Here, if I understand the meaning of '2', '4', '6' and the box correctly, the box doesn't really make any difference.