On the mathematical convention used to talk about biconditional proofs

Question

Given $P \Longleftrightarrow Q$, the following does apply:

• $P \Rightarrow Q$ is equivalent to:
  $P$ is a sufficient condition for $Q$,
  $Q$ is a necessary condition for $P$.

• $Q \Rightarrow P$ is equivalent to:
  $Q$ is a sufficient condition for $P$,
  $P$ is a necessary condition for $Q$.

Still, when we read a proof of a statment of the form of a biconditional, we find the authors referring to one direction as THE sufficient condition, and the other as THE necessary condition, when – after all – as arises naturally from what written above, it is basically relative.

Still, I found that in general (I do not remember an instance where I noticed a different pattern) we have that:

• ($\Longrightarrow$) is associated to the necessary condition (only if),
• ($\Longleftarrow$) is associated to the sufficient condition (if).

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Here there is an example from Dudley's "Real analysis and probability" that should make more perspicuous what my point is all about:

2.1.3. Theorem: [Let $(S, \mathcal{T})$ be any topological space] A set $U \subset S$ is open iff for every $x \in U$ and net $x_i \to x$ there is some $j$ with $x_i \in U$ for all $i \geq j$.

Proof:
“Only if” follows from the definition of convergence of nets.
“If”: suppose a set $B$ is not open. Then for some $x \in B$, by (b) there is a net $x_i \to x$ with $x_i \notin B$ for all $i$.

(Here this is part (c) of the theorem, and the (b) in the proof that is mentioned is another result previously obtained that still is irrelevant for my point)

As it can be noticed, indeed Dudley uses "only if" to refer to a certain direction, while "if" to refer (obviously) to the opposite direction. Still, this looks to me as a convention, namely the one described above.

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Hence, is my induction concerning this convention correct?
Is true that this is the standard convention in mathematics when writing about biconditionals?

Any feedback is most welcome.
Thank you for your time.

Answer 1

The following are all exactly the same:

• If $P$, then $Q$
• $P \Rightarrow Q$
• $P$ is sufficient for $Q$
• $Q$ is necessary for $P$
• $Q$ if $P$
• $P$ only if $Q$.

The following are all exactly the same:

• If $Q$, then $P$
• $Q \Rightarrow P$
• $Q$ is sufficient for $P$
• $P$ is necessary for $Q$
• $P$ if $Q$
• $Q$ only if $P$.

To show $$A \text{ iff } B,$$ it doesn't matter whether we phrase it as "A iff B" or "B iff A". However, once the theorem is stated as "A iff B", then we may say "We prove the 'if' direction" and we mean we are showing "A if B": that is, "B implies A". Similarly, if we prove the "only if" direction in that case, that means we are showing "A only if B", or "A implies B".