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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?

Caleb Stanford
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August
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    if a is false, the implication is true. if a is true, the implication is false. – hamam_Abdallah Oct 20 '16 at 12:31
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    We can translate this into mathematical notation as: $a \implies 0 = a'+0 = a'$. So $a$ is false. Another way to look at it is: true cannot imply false. – Sarvesh Ravichandran Iyer Oct 20 '16 at 12:32
  • Normally, we have $A\implies B$ and $A,B$ have values true or false. So, I guess it is meant that $B$ is false. If $A\implies B$ is true, then $A$ cannot be true, so $A$ must be false. – Peter Oct 20 '16 at 12:32
  • "stumbled upon"? Does that mean you didn't read the chapter before looking at the problems? In constructive logic, "$a \implies \bot$" is equivalent to $\lnot a$. – DanielV Oct 20 '16 at 12:32
  • @DanielV we were only given brief overview. – August Oct 20 '16 at 12:34

5 Answers5

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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.

Bram28
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$$p \to \text{False} \equiv \neg p \lor \text{False} \equiv \neg p$$

Rodrigo de Azevedo
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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.

Supreeth Narasimhaswamy
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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.

Caleb Stanford
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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.

Reese Johnston
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