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My writing a proof of a complex proposition of the type "$A \iff B$" for a paper. I want to close a block of argumentation with a sentence like "Which concludes the first implication". Here I am calling the "$\implies$" as the first implication, but I don't know if this is the correct term to use.

What is the correct term to refer to the "$\implies$" and the "$\impliedby$" implication in the mentioned context?

Cotton Headed Ninnymuggins
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Renato Fernandes
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    You can also refer to them as forward and reverse implications. – Giorgos Giapitzakis Nov 11 '22 at 00:14
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    "with the forward direction shown, we now prove the other direction" – Brevan Ellefsen Nov 11 '22 at 00:15
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    Another way I've seen people do this in papers is to put $(\Rightarrow)$ and $(\Leftarrow)$ at the beginning of paragraphs, almost as bullet points. You would see an indent, then $(\Rightarrow)$:, followed by a space, then the proof that $A \implies B$ would begin from there. – Theo Bendit Nov 11 '22 at 00:24
  • @TheoBendit, It is a good suggestion, but I was trying to avoid "splitting" the demonstration into two parts for aesthetic reasons. – Renato Fernandes Nov 11 '22 at 00:49

3 Answers3

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The $\implies$ direction is called sufficiency, and the $\impliedby$ direction is called necessity.

RobPratt
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I would say something as: ($\dots$) which concludes the previous "if A, then B" statement.

Remark 1: From here on, we indicate the forward/direct implication as "$A \Rightarrow$ B", whereas we use "$A \Leftarrow$ B" for its opposite relation (i.e., a reverse relation between proposition A and proposition B).

Marco Ripà
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@BrevanEllefsen answered in comments, $\Rightarrow$ is the forward implication while $\Leftarrow$ is the reverse implication.

In practice, I like @TheoBendit's suggestion of preceding the paragraph where you start proving one of the implications by $(\Rightarrow)$ or $(\Leftarrow)$ (with appropriate formatting like indent,...) very much, especially on my personal notes or drafts.

On the other hand, if your proof is well-constructed, this could be superfluous. Frame the proof of each implication by the necessary assumptions and conclusion ("Assume A. Then [...]. This implies that B") and separate the two proofs by "Conversely", or similar expressions.

Taladris
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