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

If $P$ and $Q$ are statements,

$P \iff Q$

and

The following are equivalent:

$(\text{i}) \ P$

$(\text{ii}) \ Q$

Is there a difference between the two? I ask because formulations of certain theorems (such as Heine-Borel) use the latter, while others use the former. Is it simply out of convention or "etiquette" that one formulation is preferred? Or is there something deeper? Thanks!

Solution 1:

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.

Solution 2:

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.