What is the technical difference between $\longleftrightarrow$ and $\iff$?

Question

I understand that they can both be used to mean "if and only if" or similarly link two statements together, but in the general sense, what is the difference between the two symbols ($\longleftrightarrow$ and $\iff$)?

Thank you!

Answer 1

There is no universal standard on this, and different texts may use $\Leftrightarrow$ or $\leftrightarrow$ differently, but many texts do use these two symbols to represent two fundamentally different concepts. So here is what many of those texts do:

The $\leftrightarrow$ is part of the language of formal logic. Like $\land$ and $\lor$, the $\leftrightarrow$ is a truth-functional operator, and it is often referred to as the material biconditional

On the other hand, the $\Leftrightarrow$ is part of the metalanguage we use to talk about logical expressions. In particular, the $\Leftrightarrow$ expresses that two logic expressions are logically equivalent. For example, the formal logic expression $P \land Q$ is logically equivalent to $Q \land P$, and we can express that fact as $P \land Q \Leftrightarrow Q \land P$. This metalogical claim that says something about a relationship between two formal logic claims is different from $(P \land Q) \leftrightarrow (Q \land P)$, which is just a formal logic expression.

Interestingly, it is a metalogical fact that $P \leftrightarrow Q \Leftrightarrow Q \leftrightarrow P$.

Answer 2

Mathematics can be conceived as a series of assemblages of symbols like $$a,b,c, ...,a',b',...,\alpha,\beta,....$$ $$\neg , \vee, \wedge,...$$ $$\in, ...$$

For example, $a\to b$ is the assemblage $\neg a \vee b $ and $a\longleftrightarrow b$ $$a\to b \wedge b\to a$$

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With these assemblages, we form "assertions" to which we give two truth values $$\text{TRUE},\text{FALSE}$$

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$$a\iff b$$means $$a\longleftrightarrow b \text{ is TRUE.}$$