3

How is the $\Rightarrow \Leftarrow$ symbol actually used in practice? I think my issue here is that I don't know what the symbol is meant to mean. For example, I know that $\implies$ means "which implies", and $\iff$ means "if and only if" or "iff". What does $\Rightarrow \Leftarrow$ mean? I'm thinking either "a contradiction" or "which implies a contradiction".

Is Ne
  • 2,758
  • It's (as far as I know) a symbol only used to mean "contradiction." – TravisJ Mar 17 '15 at 12:58
  • 1
    Oh. In that case, I personally just use it in place of the $\square$ at the end of a proof. You might have something like: $A$ implies $B$. $B$ implies $C$. $C$ implies not $A$. $\Rightarrow\Leftarrow$. – TravisJ Mar 17 '15 at 13:01
  • 1
    I hesitate to assert that it is completely standard, but I certainly would not use it to replace the word contradiction in a sentence (using symbol to replace a word as in your previous example). – TravisJ Mar 17 '15 at 13:04
  • 1
    @Ark The use of arrows is an abuse of notation $\Rightarrow \Leftarrow$ in the sense that it has nothing to do with implication, reverse implication. I think you're trying to read more into the expression, which is not standard, but rather, shorthand. – amWhy Mar 17 '15 at 13:05
  • Are you asking how to pronounce the symbol when you see it in somebody else's writing, or how to use it yourself? In the latter case, the answer is: Don't. – hmakholm left over Monica Mar 17 '15 at 13:07

1 Answers1

4

The symbol means nothing more and nothing less than a casual declaration that a contradiction has been reached in an argument/proof. It has no inherent meaning, logically. I've see some use it in classrooms, writing it precisely following the point at which a contradiction has been revealed, as a sort of shorthand.

To be honest, the use of arrows, as in$\Rightarrow \Leftarrow$, is an abuse of notation, in the sense that it has nothing to do with implication, nor reverse implication. I think you're trying to read more into the expression, which is not standard, but rather, shorthand, as suggested in my first paragraph.

amWhy
  • 210,739
  • 1
    Casually, you may implement the shorthand immediately following the point at which a contradiction has been reached. It is not a logical symbol in the same way that $\rightarrow$ and $\leftrightarrow$ are. Indeed, it is never okay in propositional logic, for example, to write $p \Rightarrow\Leftarrow \lnot p$ – amWhy Mar 17 '15 at 13:16
  • 1
    @Ark: Take a look at my question here http://math.stackexchange.com/q/160039/8581 – Mikasa Jun 03 '16 at 10:21