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Suppose you are proving $p \leftrightarrow q$. In your first paragraph you prove $p \rightarrow q$. Your second paragraph begins, “For the converse, assume $q$ holds.”

In this situation, we have a very precise way of referring to the statement $q \rightarrow p$: “the converse.” But we have no good way to refer to $p\rightarrow q$. Sometimes we say “the forward implication,” but I am not a big fan of this phrase and am wondering if there is a single latinate word which means the same thing. (Obverse? Inverse? Contrapositive? No, none of those mean “$p \rightarrow q$.”)

Granted, the word “converse” only works in context, since you are implicitly saying “the converse to $p \rightarrow q$,” and that context can only be inferred if you are using the word “converse” right after you have proven “$p \rightarrow q$.” But, anyway, I am simply asking if there is a better way to say “the forward implication.”

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    $\begingroup$ You can just directly say "the statement" $\endgroup$
    – Henry
    May 15, 2016 at 1:03
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    $\begingroup$ If you’re starting with $P\Longrightarrow Q$, then it’s an (or the) implication. Its converse is the implication in the opposite direction. But something is a converse only in relation to a previously stated implication. So if you’re starting with $P\Longleftarrow Q$, that’s not a converse, it’s an (or the) implication. $\endgroup$
    – Lubin
    May 15, 2016 at 1:12
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    $\begingroup$ In direct reference to your response to @HenningMakholm, I would start the whole proof with the words, “Suppose $p$ holds.” $\endgroup$
    – Lubin
    May 15, 2016 at 1:14
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    $\begingroup$ @Lubin I totally agree that “converse” is a purely relative term. There is no such thing as THE converse, and when you write “the converse” in a proof, what you’re referring to is the converse of whatever implication you were talking about previously. But see my comment above for an explanation of what I’m looking for. $\endgroup$
    – Zach Blumenstein
    May 15, 2016 at 1:15
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    $\begingroup$ Well, yes, I guess I see your point. Then appropriate words might be “the direct implication” or “the original”. Once upon a time one used the words “necessary” and “sufficient”, but nobody seems ever to have known which was which. $\endgroup$
    – Lubin
    May 15, 2016 at 1:29

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Why do you need a special word? If you are proving $P\iff Q$ you can just talk about the statement $P\implies Q$ and its converse.

If you think about it, the converse (and inverse, and contrapositive, and negation) all implicitly refer to the original statement. It is not just "the converse" it is "the converse of the statement."

You can't get around establishing what the statement is before you start saying "converse", so there is no sense avoiding it, and there does not really seem to be any evidence more terminology is needed for it, either.

Even if you don't buy that argument for some reason, you should still consider the possibility you are not using the words as they were intended. "Converse" is not really a label for $Q\implies P$, it is a statement about changing an existing statement, just like negation, inversion, and contraposition are. Would one similarly ask "what is the antonym of the contrapositive?" Probably not.

This is supported by the presentation on the wiki article on contraposition, where contraposition, inversion and conversion and negation are all thought of as things you can do to a given statement.

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    $\begingroup$ See the comment thread above. But also “the statement” couldn’t work because it’s ambiguous. If you’re proving $p \leftrightarrow q$, then does “the statement” refer to the statement you’re trying to prove, or the forward implication ($p \rightarrow q$)? $\endgroup$
    – Zach Blumenstein
    May 15, 2016 at 1:21
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    $\begingroup$ Sometimes I use "the conditional $p \rightarrow q$ and its converse $q \rightarrow p$". $\endgroup$
    – J. M. ain't a mathematician
    May 15, 2016 at 2:52
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    $\begingroup$ @ZachBlumenstein My response is: I don't see the need for a word ( which we are probably showing here does not even exist) to complete a contrived sample sentence. It is all perfectly avoidable by rewriting in one of many simple ways people have already described. $\endgroup$
    – rschwieb
    May 16, 2016 at 5:31
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It's not as nice as a simple statement but in each paragraph you can refer to the other p->q or q->p as the converse. I.e. the converse of the converse is the original.

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I've seen in some proofs of $P\iff Q$:

"First, we prove the direct…

Where $P\implies Q$ is the direct.

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(Prove) (the) "implication", "direction" or "direct implication" $p \to q$

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