In symbolic logic, how is a false to true implication equal to true?

if x=3, then 2x=6....true...if x does not = 3, then 2x=6....false!In symbolic logic, how is a false to true implication equal to true?



The best way to understand this is as follows:





When I say to you ';If P then Q'; I am making a statement that says ';the truth of Q is a logical consequence of the truth of P';. That is the statement we are evaluating. So, if it turns out P is false, that just screws up my whole statement. If I was saying ';the truth of Q is a logical consequence of the truth of P';, and then P turned out to be false, then that isn't really a challenge to my statement. That wasn't what I asserted, I was only saying, if P is true, then Q is true. The proper contradiction to my statement would be ';The truth of Q is not a logical consequence of the truth of P'; which is the situation in which P is true and Q is false.





To get back to your example, in order to contradict what is said by ';If x=3 then 2x=6'; you don't want a situation in which x=3 is false, but you would want a situation in which 2x=6 is false while x=3 is true because that just meant ';2x=6 is not a logical consequence of x=3';In symbolic logic, how is a false to true implication equal to true?
The implication


A鈫払


is false when A is true and B is false. It is otherwise true. Specifically, it is true whenever A is false. The thing to remember is that the entire implication A鈫払 is true in that case.





Suppose you're trying to teach a child about fruit, and you say:


';Every apple is a fruit.';


And the child says, ';You mean an onion is a fruit?';


';No, an onion is not a fruit. But an apple is a fruit.';


';But if an onion is not a fruit, how can every apple be a fruit?';


';It really doesn't make any difference, since we're not talking about onions. Every apple is a fruit.';


';So a banana is a fruit?';


';Well, yes, it happens to be that a banana is a fruit, but we were talking about apples ...';


';But if a banana is a fruit and an apple is a fruit, why can't an onion be a fruit?';


';I can see we're going to have to have this conversation another time.';





It's important to remember which implication we're evaluating. In my example, the parent %26amp; child are evaluating the implication:


x is an apple 鈫?x is a fruit.


Suppose x isn't an apple. What does that tell us about whether an apple is a fruit? Nothing at all: something else might be a fruit and it might not, but we're talking about apples, right?





In your example, you are evaluating the implication


x=3 鈫?2x=6.


Now, it's true that you can't find an x other than 3 such that 2x=6. But what does that tell us about x=3 鈫?2x=6? Nothing, really, but we're talking about what happens when x=3, right?