Fangu
Great Old One
Yes, exactly. They're called Conditional Propositions in Discrete Mathematics (or "computer mathmatics" as I like to call them.) Here's the URL I used for this.I'm not quite sure what you are referring to with 1 - (A | B). I'm not familiar with that notation, sorry!
I think you might have been getting at the fact that a logical equivalence (A is true if and only if B is true) is the same as two separate implications (if A is true then B is true, and if B is true then A is true). In which case, yes, there are multiple ways of proving the equivalence wrong. Only one is needed though
Say
p: VII was your first RPG
q: You like VII
The phrase "if and only if" is represented by the symbol ⇔
p ⇔ q
"You like VII if and only if VII was your first RPG"
p is a necessary and sufficient condition for q.
You can only like VII if VII was your first RPG.
By using Sufficient Conditions in a truth table, it is proved that p ⇔ q only is true when both p and q are true, or both p and q are false.
"You like VII if and only if VII was your first RPG"
"You don't like VII if and only if VII was not you first RPG"
This leaves no room for in betweens. The truth table
p q p ⇔ q
T T T
T F F
F T F
F F T
shows that for statements like
"You don't like VII if and only if VII was your first RPG"
and most importantly
"You like VII if and only if VII wasn't your first RPG"
are untrue.
Which means, like you've found, there can't be people who like VII and who's first RPG wasn't VII. If they exist, the statement is untrue.
(I think I did that right.)