I truly have no idea and wish I did, haha. It looks like a shorthand for which operation is being followed, maybe like a group theory thing, but I really don't know.

Samesies

I think 'implies' asks whether it's possible that A causes B to be true. In other words, it is false if there is evidence that A does not cause B.

So:

If A is true and B is false, then the result is false, since A could not cause B to be true.

If A and B are both true, then the result is true, since A could cause B.

If A is false and B is true, then the result is true since A could or could not make B true (but another factor could also be making B true)

If A and B are both false we don't have any evidence about the relationship between A and B, so the result is true.

I don't know for sure, though. I'm not a mathematician.

“A implies B” means if A is true then B must be true; if A is false, then B can be anything. In other words, the only state not allowed is A being true and B being false. Therefore, the only “hole” is the part of A that doesn’t include B.

Yeah, that one's always bothered me too. I think the difference in meaning from colloquial "implication" and logical "material implication" are also involved in the raven paradox.

So the statement that "all ravens are black" can be taken as "if RAVEN then BLACK". Is this statement true? If you see a black raven then trivially yes, if you see a white raven then trivially false (via counter example).

However if you see a non-raven, it is evidence for the truth of the statement because it doesn't go against it: not-ravens being black-or-not-black both reduce the universe of possible objects without proving not-black-ravens exist.

Or something like that, I think it's stupid too. Trinary logic can adopt a more sensible (IMHO) definition of implication that makes A being false always lead to the third value (usually defined as indeterminate or neither-true-nor-false).

It comes from boolean logic, not from set theory (which is what it's used for here).

A implies B means that "If A is true, B is also true", so there's the implication. "If the thing we are talking about is a desk, it is also a piece of furniture".

The implication has no effect if A is not true: "If the thing we are talking about is not a desk, that doesn't say anything about whether it's furniture or not".

So the implication is only false if A is true and B is false. In any other case the implication is satisfied and the result is true.

Set theory just took over all of the boolean logic functions, since they work exactly the same in set theory, so they also took over the naming even if the name "implies" doesn't really make much sense in set theory.

"(!Trick) && Treat" is probably the binding they meant