In mathematics, distinguishing material implication ('$\to$') from logical implication ('$\Rightarrow$') Can anyone exhibit a mathematical sentence in which a conditional (not necessarily the main connective) has to be STRICTLY understood as a MATERIAL one, and would become false if the material conditional was understood as logical implication instead?
Some context:
In logic, strictly speaking, material implication (' → ') has to be carefully distinguished from logical implication (' ⇒ '). However, I have noticed that in mathematics books, the distinction is not emphasized, as if, in that field, all implications are logical implications. Is it actually the case? (Reference at Archive.org: On this distinction and on the symbols I use , Seymour Lipschutz, Schaum's Outline of Set Theory , ch. 14 " Algebra Of Propositions".)
To illustrate the difference between material and logical implication, consider the sets A={ x | x is a mathematician → x is a musician } and B={ x| x is a mathematician ⇒ x is a musician }. A is simply the set of people who (contingently) happen not to be both mathematician and non-musician, since its conditional is a material one. However, B is the set of people such that for each member, it is or would have been logically impossible for them to be mathematician without being musician; depending on one's opinion  concerning the relationship between mathematics and fine arts, one will probably tend to answer either that B is either the universal set (a mathematician is necessarily a musician) or the empty set.
I think that substituting ' → ' for ' ⇒ ' cannot lead to important problems, since, if A logically implies B, then A should also materially imply B ("A ⇒ B" meaning that (A → B) is true in all possible cases, all possible "interpretations"). Here I'm asking the reverse question: is it always correct to substitue ' ⇒ ' for ' → ' in mathematics, in other words, is it correct to use always " ⇒ " in mathematics?
My question is not on symbols.
 A: Consider the statement 

"If x is even, then x is divisible by $2$" 

In 'math world' we regard this statement as true.
But this is not a logical truth. That is, logically one is allowed to interpret 'even', 'divisible by' and '$2$'  in a way that would make the statement false.
So, the statement is a mathematical truth, but not a logical truth. More to the point: the 'if' part does not logically imply the 'then part. Indeed, if we were to symbolize it, we should be using the material implication, and not the logical implication.
Of course, if we are given the (normal!) definitions of 'even', 'divisible by' and '$2$', then we can logically infer the truth of the statement above as a whole. That is, the statement as a whole is logically implied by the relevant definitions.
Also, if we fill in a specific value for $x$, say $4$, then the statement becomes:

"If $4$ is even, then $4$ is divisible by $2$" 

And now, given the standard definitions/axioms (let's refer to that as a set of statements $A$), we have that $A$ together with "$4$ is even" logically implies that "$4$ is divisible by $2$" ... but we still don't have that "$4$ is even" by itself logically implies that "$4$ is divisible by $2$"
A: The symbols you want are $\to$ (\to) for material implication and $\implies$ (\implies) for logical implication. Insofar as mainstream mathematics distinguishes them, $p\implies q$ means that $p\to q$ is (a) true in all models of a theory of interest (however, in that context we'd usually write $\models$ (\models) instead of $\implies$ to make it clear) or (b) a tautology. And in modal logic, we can rewrite $p\implies q$ as $\Box(p\to q)$ (note the use of \Box). But in practice, $\implies$ is often used in proofs to indicate an inference from what was already known.
A: In mathematics based on classical logic, there appears to be no difference between material and logical implication.
Consider, for example, the implication: If it is raining ($R$), then it is cloudy ($C$). 
$$R \implies C$$
This implication does not mean that rain causes cloudiness, or that cloudiness causes rain. It means only that, at the moment, it is not both raining and not cloudy.
$$\neg [R \land \neg C]$$
This is often used in one form or another as The Definition of $\implies$ in introductory textbooks, but it can also be derived from other widely accepted properties of implication, conjunction and negation in classical logic:


*

*Introducing and Eliminating $\land$

*Eliminating $\neg\neg$

*Conditional proof

*Proof by contradiction

*Detachment (Modus Ponens)


See my formal proof.
So, in classical logic anyway, the above "definition" would seem to apply to every implication. 
