User alric althoff - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:45:45Z http://mathoverflow.net/feeds/user/5983 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24028/what-is-the-difference-between-the-biconditional-iff-and-equality/24033#24033 Answer by Alric Althoff for What is the difference between the biconditional iff. and equality = ? Alric Althoff 2010-05-09T17:22:56Z 2010-05-18T16:12:51Z <p>In other words, $P(x)\iff Q(x)$ implies that $\forall x \in D, P(x) \leftrightarrow Q(x)$ where $D$ is the relevant domain. Or, that "$\iff$" applies when talking <em>about</em> logic while "$\leftrightarrow$" is used in a formula.</p> <p>But to address the question as to whether $\iff$ can serve as a replacement for $=$, I think that translating it into English is useful. Replace "$\iff$" with "is necessary and sufficient for" and say "$=$" as "equals" (the same goes for "$\equiv$" and "logically equivalent to") and I think that you'll see that they are different, conceptually and syntacticly, only in certain circumstances, or "when doing logic, we speak logic" (I often use them when taking notes on non-math topics, and my brain knows what I mean regardless of symbol choice :) ).</p>