# is \iff the same as \equiv? When to use which?

Is there a difference between using $\iff$(\iff) and $\equiv$ (\equiv)?

When should I use one or the other?

• What about $\leftrightarrow$? – Hagen von Eitzen Nov 5 '17 at 20:45

Unfortunately there does not seem to be a strict rule (or even all that great of a consensus) as to what symbol to use where and when. The most important thing is that you understand the difference between a logic symbol and a meta-logic symbol:

The material biconditional is a truth-function that takes two logic expressions and combines them into a new one. As such, it is part of a logic statement. You often see $P \leftrightarrow Q$, but some books use $P \equiv Q$, and some use $P \Leftrightarrow Q$

On the other hand, logical equivalence is a statement about two logic statements, namely that they have the same truth-conditions. We know, for example, that the logic statement $\neg (P \lor Q)$ is logically equivalent to $\neg P \land \neg Q$. We can write a symbol between them to say this about them, but as such we don't get a new logic statement, but rather a meta-logical statement. For this, I can't recall ever having seen the $\leftrightarrow$ used, but you'll see both $\neg (P \lor Q) \equiv \neg P \land \neg Q$ and $\neg (P \lor Q) \Leftrightarrow \neg P \land \neg Q$. I've also seen $\neg (P \lor Q) :: \neg P \land \neg Q$

Anyway, context should make it clear what is meant by which symbol, and which symbol you are therefore supposed to use.

Let's take a look at the definition from Rosen's Discrete Mathematics and its Applications.

Definition:

The compound propositions $$p$$ and $$q$$ are called logically equivalent if $$p \iff q$$ is a tautology. The notation $$p \equiv q$$ denotes that $$p$$ and $$q$$ are logically equivalent.

That is, the symbol $$\equiv$$ is not a logical connective, and $$p \equiv q$$ is not a compound proposition but rather is the statement that $$p \iff q$$ is a tautology.

• Please do not recommend Rosen's textbook, as it is full of conceptual errors and imprecision. This is well known to many mathematicians but not to most students. – user21820 Mar 15 '20 at 15:48