Ironing out my intuitive understanding of $\implies$ I'm at the end of my doctoral degree in mathematics. As far as I'm aware, I've been using the symbol $\implies$ effectively for 10-years of higher education (for example, effectively enough to get research papers in mathematical physics through peer review). I thought I would brush up on my formal logic, and it turns out that my intuitive understanding of what $\implies$ means is not enough to understand the logic table of $\implies$;
$$
    \begin{matrix}
    A & B & A \implies B\\
    T & T & T \\
    T & F & F \\
    F & T & T \\
    F & F & T
    \end{matrix}
$$
Suppose I have the following three propositions:
$$ A: 2^2 = 4
$$
$$B: \sin(n\pi) = 0 \;\forall \; n \in \mathbb{Z}$$
$$C: \sin(0) = 0$$
All of these statements are true. B is a more general version of C, and A is unrelated to B and C.
I see that B is true, and C is true, and if I know B then I will know C. So I agree with the truth table if I apply it to B and C.
However, A and B are both true, but if I know A it doesn't tell me anything about B. So I don't see why I would want to say that A $\implies$ B is true in this case. My intuitive understanding of $\implies$ in relation to A and B is then that A doesn't imply B, and I'd say that  A $\implies$ B is false.
Can anybody give me some intuitive explanation about why I would want to assign this truth table to $\implies$, preferably with some examples?
The lecture notes I've been reading seem to think that "snow is black implies grass is red" is a good example for understanding this truth table. Apparently from the fact that "snow is not black" and "grass is not red", it should be intuitively obvious to me that "snow is black implies grass is red" is true. I run into the same problem here; as far as I can see the colour of snow doesn't tell me anything about the colour of grass.
I think my confusion is coming from the fact that I'm assigning some kind of causality to $\implies$, when I'm not supposed to do that? That's what my reading told me anyway, but I don't understand it.
Any explanation about what operation what my intuitive understanding may be referring to would also be helpful for me. I think this is probably some concept from probability or information theory (if you guessed that my research so far hasn't been too closely related to these fields, you'd be correct).
Edit;
I'm looking for some explanation about what concept from probability or information theory captures the intuitive understanding of the word implies that I've tried to express here, additionally to the explanation of the formal logic
A: The $\implies$ symbol’s meaning is if A then B.
I.e. if $sin(n\pi =0),\forall n \in \mathbb Z$ then $sin(0)=0$.
Your problem is that you are assuming, what you intended to say is that if non-else is known, does there exist a possibility that A is true while B false? In which case, as long as neither A or B are assumed your intuition holds.
The more formal(and less helpful) way to phrase it is:
For some base axiom system $P$, does adjoining the axiom $A \land \lnot B$ Result in a contradiction?
