Assume that $a \implies \text{false}$. What does that imply about $a$?

Question

The title says most of it. I've recently started learning logical implications, and in my homework I stumbled upon a task that states that,

Assume that $a\implies\text{false}$. What does that imply about $a$?

I tried looking up this problem online, but could not find an explanation about a way solve it, could someone help me?

Answer 1

The statement implies that $a$ is false.

The basic mathematical concept at play here is contradiction. If you assume something, and get a contradiction ("false" is a generic name for a contradiction), then that thing must have been false in the first place. So if $a \implies \text{false}$, then assuming $a$ was true we got a contradiction; therefore, $a$ must not have been true. So we conclude that $a$ is false.

Sometimes, the definition of "not $a$" is taken to be "$a \implies \text{false}$"; the above reasoning shows that this definition makes sense.

Answer 2

For any mathematical statements $A$ and $B$, we have $A\implies B$ if and only if $A$ is false or $B$ is true. For our problem we have $A\equiv a$ and $B\equiv\text{FALSE}$. Since $B$ is $\text{FALSE}$, the only way the implication $A\implies B$ can be true is that $A$ must be false. Thus we can infer that $a$ is false.

Answer 3

$$p \to \text{False} \equiv \neg p \lor \text{False} \equiv \neg p$$

Answer 4

It's like the saying: "if that is true, then pigs fly!", by which we of course mean that we believe the that is not true. So in your logic expression, the $a$ is like the 'that', while the $false$ is the 'pigs fly'. So the logic expression, like this saying, is the same as saying that $a$ is false.

Answer 5

When it comes to propositional logic, students often get caught up in the details and forget the meanings - which is unfortunate, because if you can think clearly about the meanings behind the sentences you're working with, the answer is often obvious. "$a \implies \mathrm{false}$" says, literally, "if $a$ is true, then false is true". "False is true" is nonsense - it's something we know to be false. If I tell you "if it's raining outside then the moon is made of green cheese", what can you conclude about the weather? If it were raining, then the moon would be made of green cheese. Since the moon is not made of any sort of dairy product, this is clearly an impossible state of affairs - so it must not be raining. Likewise, if we know $a \implies \mathrm{false}$, then we can conclude that $a$ is false.