Implication Logic Truth Table Explained

The question that has bothered me for a while has been answered and closed here (Implication in mathematics - How can A imply B when A is False?) and probably many other posts. Although all the answers are accurate and correct in their own way of explaining (or went above my head), those still didn't "click" for me so I kept trying to find a more specific example. Here's what I came up with and would like folks to review and comment.

First the question:

In classical logic, why is (p->q i.e. p implies q) True if both p and q are False? I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go. In the truth table, where A->B we obtain this result:

A | B |A->B
------------
T | T | T 
T | F | F 
F | T | T 
F | F | T

Now, my confusion here is regarding why A->B given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.

My conclusion and probably the mistake I've been making to understand here is that this truth table is not about when the result will be true if you use the equivalent (~A V B) logic. What this truth table represents is the fact that if you have a data set (or situations) that results in a false value of (~A V B) then your assumption that A implies B is violated (or is not correct). In simpler words, the true values in the truth table are for the statement A implies B. Conversely, if the result is false that means that the statement A implies B is also false. And now as I read it, I guess, I'm stating the obvious but let's try it with an example.

Let's say we have an assumption that the State of California is the only State with a city named Los Angeles. So we setup two tests; 'A' (City=Los Angeles) and 'B' (State=CA). Now, based on our assumption, whenever we find an address with city of Los Angeles, we can infer that the State must be California (if A is true it will imply that B is also true). However, if we were not correct in making that earlier assumption, and there is another State which has a city of Los Angeles, in that case the test (~A V B) will result in false thus proving that the assumption "City=Los Angeles implies State=CA" is wrong.

My conclusion and probably the mistake I've been making to understand here is that this truth table is not about when the result will be true if you use the equivalent $(\lnot A \lor B)$ logic. What this truth table represents is the fact that if you have a data set (or situations) that results in a false value of $(\lnot A \lor B)$ then your assumption that $A$ implies $B$ is violated (or is not correct). In simpler words, the true values in the truth table are for the statement “$A$ implies $B$”. Conversely, if the result is false that means that the statement “$A$ implies $B$” is also false.

Bear in mind that

1. $(A\to B)$ is just a truth function whose lookup table is defined as $(\lnot A \lor B)$'s truth table.

2. “$A$ implies $B$” means that $(A\to B)$'s truth table's second row has been eliminated.

3. $A$ implying $B$ (e.g., <I won the game> implies that <I scored higher than Jane>) doesn't mean that $A$ causes $B.$

4. The $(F,F)$ and $(F,T)$ cases are perhaps unintuitive to grasp, but the alternative would make even less sense: if we let $F\to T\equiv F,$ then $$\forall n\in\mathbb Z \,\big(n \text{ is a multiple of }4\, \to \,n \text{ is even}\big)$$ would be a false statement (try $n=6$).

Let's say we have an assumption that the State of California is the only State with a city named Los Angeles. So we setup two tests; '$A$' (City=Los Angeles) and '$B$' (State=CA). Now, based on our assumption, whenever we find an address with city of Los Angeles, we can infer that the State must be California (if $A$ is true it will imply that $B$ is also true).

However, if we were not correct in making that earlier assumption, and there is another State which has a city of Los Angeles, in that case the test $(\lnot A \lor B)$ will result in false thus proving that the assumption "City=Los Angeles implies State=CA" is wrong.

✔ ‘Every LA is in CA’ $\implies$ $(A\implies B)$

✔ ‘Not every LA is in CA’ $\implies$ $(A\kern.6em\not\kern -.6em\implies B)$