Interpreting A⇔B⇔C as (A⇔B)⇔C is utterly non-standard, but admittedly interesting.
A⇔B⇔C is usually not a valid expression in any mathematical formalism. But if used informally, A⇔B⇔C is usually meant to mean (A⇔B)∧(B⇔C), just as when you say A=B=C when you actually mean (A=B)∧(B=C).
The discussion is the meaninglessness of the parentheses. ⇔ is indeed commutative and associative which means that we can define removing the parentheses as equivalent. However it is equally valid to not do this and define an n-ary '⇔' operator that evaluates ⇔(a1..an) as all ai have the same value. The result is different semantics for A⇔B⇔C.
I've taught and always assume ((A⇔B)⇔C). I've apparently used (A⇔B)&(B⇔C) once.
Why not? The sky if green if and only if pigs can fly if and only if water is wet.
The sky if green if and only if pigs can fly is true.
True if and only if water is wet is true.
Ok, you've got a point, I've just never seen it used that way (though too be fair, I haven't often seen it used, probably for the reasons we're discussing).
