"IFF" (if and only if) vs. "TFAE" (the following are equivalent)

As Brian M. Scott explains, they are logically equivalent.

However, in principle, the expression $$(*) \qquad A \Leftrightarrow B \Leftrightarrow C$$ is ambiguous. It could mean either of the following.

  1. $(A \Leftrightarrow B) \wedge (B \Leftrightarrow C)$

  2. $(A \Leftrightarrow B) \Leftrightarrow C$

These are not equivalent; in particular, (1) means that each of $A,B$ and $C$ have the same truthvalue, whereas (2) means that either precisely $1$ of them is true, or else all $3$ of them are true. Also, you can check for yourself that, perhaps surprisingly, the $\Leftrightarrow$ operation actually associative! That is, the following are equivalent:

  • $(A \Leftrightarrow B) \Leftrightarrow C$
  • $A \Leftrightarrow (B \Leftrightarrow C)$.

In practice, however, (1) is almost always the intended meaning.


They are exactly equivalent. There may be a pragmatic difference in their use: when $P$ and $Q$ are relatively long or complex statements, the second formulation is probably easier to read.