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.
Best Answer
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.