B = (A or B) and (if A then B)
In the following table we have the other three operations which give different truth values to what you call the vacuous case. If "if" is replaced by "x" or "y" in the following table than that factorization does not work, "z" still does work but I'd be hard pressed to find anyone who thinks it represents causal implication better than "if" or "x". | A | B | x A B | y A B | z A B | If A then B |
|---+---+-------+-------+-------+-------------|
| T | T | T | T | T | T |
| T | F | F | F | F | F |
| F | T | F | F | T | T |
| F | F | F | T | F | T |
I'm not using the common names for the operations to not bias people.Or to put it another way: No one outside a logic course seriously thinks that A implies B means the same thing as not A or B.