I would like to write the following as a way to make a statement that the equivalence is the same as two implications that work both ways:
$P \Leftrightarrow Q \Leftrightarrow (P \Rightarrow Q \land Q \Rightarrow P)$
But this is circular. I suppose I could write instead:
$P \Leftrightarrow Q = (P \Rightarrow Q \land Q \Rightarrow P)$
But this begs the question:
If $P$ and $Q$ are propositions, what is the difference between $P \Leftrightarrow Q$ and $P = Q$?
Also, what is the difference between:
- $\Leftrightarrow$ and $\leftrightarrow$
- $\Rightarrow$ and $\rightarrow$
I notice that the Wikipedia article on propositional calculus is using the $\leftrightarrow$ and $\rightarrow$ notations exclusively.
What is, then, the proper realm where the $\Leftrightarrow$ and $\Rightarrow$ notations are meant to be used? I remember that we were using $\Leftrightarrow$ and $\Rightarrow$ when solving equations, e.g. writing things like:
$x^2 = y^2 \Leftrightarrow |x|=|y|$
… and:
$x = y \Rightarrow x^2 = y^2$ (since using $\Leftrightarrow$ here would be a mistake)
Typically, $\Rightarrow$ and $\Leftrightarrow$ are used for logical implication and equivalence respectively, while $\rightarrow$ and $\leftrightarrow$ are used for material implication.
The difference is that $\rightarrow$ and $\leftrightarrow$ are logical operators, while $\Rightarrow$ and $\Leftrightarrow$ are meta-logical symbols used to express something about logic statements.
So, you can write $P \leftrightarrow Q \Leftrightarrow (P \rightarrow Q) \land (Q \rightarrow P)$ to express that the two logic statements $P \leftrightarrow Q$ and $(P \rightarrow Q) \land (Q \rightarrow P)$ are logically equivalent.
Unfortunately, not all texts follow this notation, and particularly confusing is the use of $\Rightarrow $ and $\Leftrightarrow $ for material implication.
Also, instead of $\Leftrightarrow$, many texts use $=$ when doing boolean algebra, but others will reserve $=$ to express that two statements are syntactically identical.