I believe that the formal rules of logic have to be adjusted to account for the real world. There is a reason that we have "the Mathematician's Answer" to questions such as, "Would you like sugar or cream in your coffee?"
For example "A or B" is true if one or both are true, right? So you go to Physics Department and ask the office administrator if Professor Jones is available. The administrator answers, "Either he's in his office, or he's on his way to class." Her answer is true if one is true, right? Shouldn't we expect a third requirement? For example, that she not know which it is? If she knows perfectly well that Professor Jones is in his office, then her statement was deceptive.
Also, ask a question in court. If the witness answers (something) "or 2 + 2 = 4", he's not going to immunize himself from perjury if that (something) is false. In fact, the witness might be penalized for the statement.
Consider this situation in physics, such as a string tied between two walls. One derives the equation, A sin(kL) cos(wt) = 0. That means at least one of A, sin(kL), and cos(wt) is zero. The variable t is time, which progresses forward, so we rule out cos(wt) = 0. This leaves us A=0 or sin(kL)=0. Either could be true, and are perfectly fine solutions. We consider the interesting situation where A doesn't equal zero, so sin(kL) = 0 leading to k=n*pi/L.
Now consider "A and B". True if both statements are true, false otherwise? I put it to you that in the real world, (A and B) could be true, while A without B mentioned somewhere is patently false. That's called, "Omitting critical information". Stating that someone broke into a house at night might get him convicted of burglary. Include the facts that the house was on fire, and a child was trapped inside and the person went in to rescue him, and a conviction of burglary becomes a miscarriage of justice. The accuracy of an account is more than just the accuracy of the individual statements.
We now can see good reason for psychological results showing that people often say that "Sue works as a bank teller and is involved in the feminist movement" is more likely than "Sue works as a bank teller." The issue is an accurate description rather than the literal truth of the statements.
There are situations where specifically avoiding an actual false statement is proof that one is trying to deceive, while making the false statement might be consistent with ignorance but good faith.
"A false statement implies anything." In other words, when A is false, "If A then B" is automatically true. This notion should be discarded in favor of this alternative: "The truth or falsity of A is irrelevant to the truth of `if A then B'." I'm not concerned about statements like, "If 2+2=5 then I am the Pope." I'm thinking of statements like, "If Superman flies at 500 mph, he travels 1000 miles in five minutes." In a science class (or any class, for that matter) we wish to say that that statement is false, regardless that Superman doesn't exist and therefore doesn't fly at 500 mph.
Likewise, a trick question goes, "If a peacock lays 5 eggs on Tuesday and 3 eggs on Wednesday (and no other eggs), how many eggs does he lay?" The correct answer is 8 eggs, even though peacocks don't lay eggs. The reason is the same as above: we disregard the actual falsity of the statement, and instead see what follows from the truth of the statement.
"If A then B" is equivalent to "If B is false, then A is false." This should stand pretty much as it is. It's the heart of being able to figure things out, and the heart of comprehending what is said. The rules of logic are based on the meaning of words, and should be used to understand things, rather than play games and play dumb about what is said.
Communication should be for understanding, not obfuscation.