TITLE: Associativity of $\iff$
QUESTION [28 upvotes]: In this answer, user18921 wrote that the $\iff$ operation is associative, in the sense that

$(A\iff B)\iff C$
$A\iff (B\iff C)$ 

are equivalent statements. 
One can brute-force a proof fairly easily (just write down the truth table). But is there a more intuitive reason why this associativity must hold?

REPLY [2 votes]: The reason it's hard to intuit this is that we're used to thinking of "$(A\Leftrightarrow B)\Leftrightarrow C$" as stating something about the sentences "$A\Leftrightarrow B$" and "$C$", but nothing about "$A$" (or "$B$") directly: converting it to a sentence, "$A\Leftrightarrow(B\Leftrightarrow C)$", that speaks about "$A$" directly, is counterintuitive.
So to understand the equivalence between "$(A\Leftrightarrow B)\Leftrightarrow C$" and "$A\Leftrightarrow(B\Leftrightarrow C)$", we need to think of them not as sentences about "$A\Leftrightarrow B$" and "$C$", respectively "$A$" and "$B\Leftrightarrow C$", but as sentences about "$A$", "$B$", and "$C$" directly.
What does "$(A\Leftrightarrow B)\Leftrightarrow C$" say about "$A$", "$B$", and "$C$"?

If "$C$" and "$A$" hold, so does "$B$".
If "$B$" and "$C$" hold, so does "$A$".
If "$A$" and "$B$" hold, so does "$C$".
If "$A$" and "$B$" both are false, "$C$" holds.

So the crucial point in understanding the equivalence between "$(A\Leftrightarrow B)\Leftrightarrow C$" and "$A\Leftrightarrow(B\Leftrightarrow C)$" is understanding the equivalence between
4. If "$A$" and "$B$" both are false, "$C$" holds.
and
4$'$. If "$B$" and "$C$" both are false, "$A$" holds.
(which is clear).
The trick here was looking only at the direct statements (e.g. if "$C$" and "$A$" hold, so does "$B$"): their contrapositives (e.g. if "$C$" holds but "$B$" does not, then "$A$" is false) are less intuitive.