# Why is implication $A\Rightarrow B$ defined as $(\neg A) \lor B$? [duplicate]

I don't get the idea behind it. There are some things in the truth table which disturbs me. As if $$A$$ is false, the $$A\Rightarrow B$$ is always true. I read that implication was the negation of ($$A \land (\neg B)$$). Thus indicating that if $$A$$ is true and $$B$$ false, then the implication is false and that is all we care about, the other cases are just a consequence of the definition. Is this right?

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• Yes, the negation of (a ^ not b) is not a or b. For the implication to hold, either a is false (and not a true) or a is true, requiring b to be true. – RJM Feb 10 at 22:53
• Think about it this way. When is $A\to B$ false? To put it in natural language and exemplifying, when will I have lied if I say that "If it rains, then I take an umbrella"? Do you really want to say that I lied when it doesn't rain? – Git Gud Feb 10 at 23:13

It is a matter of convenience. The idea is that $$a\Rightarrow b$$ is false only when $$a$$ is true and $$b$$ is false. A false premise corresponds (in terms of sets) to the empty set. It is convenient to consider the empty set as a subset of any set, and it is convenient to identify properties (like $$a$$) with the subset of elements having the property. For example, it is true that "if $$T$$ is a right equilateral triangle, then $$T$$ is red" just because there is no triangle satisfying your hypothesis.

You could define the implication otherwise, but then this correspondence is lost and many simple propositions would have exceptions, etc.

The teach us think "if conditions hold" then "Are these statements true" but I find it better and more into intuitive to think if these statements are true, what is possible.

If $$A \implies B$$ what is possible?

Is $$A$$ possible? yes but only if $$B$$ is; If $$A$$ then $$B$$.

Is $$A$$ being false possible? Sure, $$A\implies B$$ doesn't say anything about what happens if $$A$$ is false.

Is $$B$$ possible? Sure, if $$A$$ then $$B$$ must be true. And if not $$A$$ then anything is possible.

Is $$B$$ false possible? Well, not if $$A$$ is true it's not but if $$A$$ is false that's just fine.

What about $$A$$ and $$B$$ etc. Well, $$A$$ and $$B$$ and $$A$$ and not $$B$$ and not $$A$$ and not $$B$$ are all possible but only $$A$$ and not $$B$$ are impossible. That can't ever happen (if we accept $$A \implies B$$) so one way or another we are either going to not have $$A$$ or we will have $$B$$.

Now do the same for $$\lnot A \lor B$$. You get the same results. That two statements can only be true under the same conditions.