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Correlation does not imply causation!
I'm tired of this phrase. It totally does. A statistically significant correlation is very frequently a good candidate for causation, pending other research.
Correlation does not prove causation. But it absolutely implies it, all other things being equal.
"Imply" can mean an indirect suggestion, but it can also mean a logical necessity. Such is language, unfortunately.
Imo mathematical logical notation is a perverse hijacking of actual words. Like in philosophy. Or certain "academic" terms.
Just use the existing words that already have definitions you want. Or if you must use different words, take a random one like "whales" or something. Don't take the thing that has a definition and then use it to mean its opposite. Eventually that usage will trickle into popular usage and just confuse everyone.
For the most part, I believe they Did take existing words that had the definition they needed, then the common definitions changed over time while the sciences definitions stayed the same. The sciences need strict meanings for words, it's a bit like legalese where they use outdated words or meanings, but for good reason - so that they know they are talking about the same thing, when common meanings drift all over.
https://en.m.wikipedia.org/wiki/Material_conditional
Imply is a technical term.
And also a non technical term. Which is how the great majority of people use the phrase.
But the entire phrase is technical. It's discussing science.
Doesn't matter. Most people don't use it that way