false implication imples truth of any premise

Question

I have dilemma with false implication.

Say implication $A\to B$ is false, then this means that $A$ is true right? (based on truth table)

So I am confused how in practice from false conclusion you can prove that any premise is true?

Examples.

If Jon is guilty, he had accomplice.

Now imagine that this implication is false because someone who said it lied for example.

Then it means that Jon is guilty right? But clearly in practice you can't prove that Jon is guilty?

So my question is more on implication and its relation to truths in the real world.

Or another example.

If $X$ is an even number, then $X$ is prime.

This is false implication right? Which means that $X$ is even. But it is nonsense because $X$ is ANY number.

Or feel free to come up with other examples, I hope I made clear what confuses me?

Answer 1

I use here your second example: You should not try do translate an implication by "if...then"

When you have $A\rightarrow B$, your A is a statement which can have either the value true or the value false (at least in predicat logic). You understand now that you can then not say "X is an even number" for the predicat A, because you can not attribute a value true or false to it.

You should better say: let X be any number but a fixed one. Then if X is even, then X is prime. And if this statement is false, it is then for a certain X, and this one should be even.

Let us now consider the first example you did. The difficulty is here to understand what do we mean in mathematical logic and what do we mean in reality. When one say in the reality "if John is guilty then he had a complice" means that he has a reasonnig proving that for any people in the same situation with the same clues and so on, we can do this implication. The trick here, is that when we say in the REALITY that the implication is false, we do not mean really that the implication is false, but that the rasonning used to conclude that is was right has a mistake. I beleave that your problem comes from here.

Answer 2

We say person X lies when uttering statement Y at time T exactly when X believes that Y is false at time T.

This means that if X lies by saying "If Jon is guilty, he had an accomplice", then at the time that person said this statement, X would have to believe that "Not (If Jon is guilty, he had an accomplice)$, which is logically equivalent to "Jon is guilty and did not have an accomplice."

But clearly, X believing Jon is guilty doesn't necessarily imply Jon actually is guilty. And even if Jon were guilty, the authorities may never be able to actually prove that he is (or even come to the conclusion that he is).

Also, the implication "if X is even, then X is prime" is not necessarily false. Aside from the example of X = 2, the statement is true for any odd X. The statement "if 3 is even, then 3 is prime" is perfectly true since 1 is not, in fact even.

I hope this analysis proves useful to you.

Answer 3

Logical statements can be open or closed. Which is to say, they have a variable / unknown (or several) in them or they don't. For instance, your

If X is an even number, then X is prime

is open because it has a variable X that could be anything. On the other hand,

If 5 is an even number, then 5 is prime

is a closed statement, as there is no variable / unknown in it.

Only closed statements have truth values. This is easier to see if we simplify our example a little:

X is an even number

is neither true nor false. We don't know, because X could be anything. On the other hand,

5 is an even number

does have a truth value (it's false).

So how can one say whether an open implication, like yours, is true or false? Technically, one can't. But very often, there is an implied "For any relevant X" in front of those statements. It's a bad habit to elide it, but unfortunately, many do. So, if you hear someone say that

"If X is an even number, then X is prime" is false

they actually mean

"For any relevant X, if X is an even number, then X is prime" is false

where "relevant X" here probably means "integer X" or "natural number X". Note that this is a closed statement. There is no free variable here, as this is a logical statement about the set of "relevant X". And they are right that this statement is false, because there are counterexamples, like X=4:

If 4 is an even number, then 4 is prime

is a closed statement, and it is false. And in fact, as your truth table analysis shows, any counterexample like this must necessarily have an even X. Because that is the only way you can make the implication false.