I'm not a native English speaker so the distinction (if any) between the colloquial usage and mathematical usage (which is more 'rigid' so to speak') is not obvious to me. It makes more sense that a math text would use if-then statements (and similarly other words in the study of logic) in their logical sense.
If this is the case, can't people just use logic to justify math concepts e.g., "Math statement A can't exist because according to Theorem 1.2: If A then B. and we observe not B therefore by modus tollens, not A"? I've seen a classmate do it but don't know what to make of it since it doesn't address any math concept. It's as if he's using it as someone would use a "black box". But is he justified in using logic since the foundation of math is supposedly built on logic?