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I have a sentence like:

Second, we propose a new mechanism that we call XYZ, which aims to solve problem X. We implement and evaluate a proof-of-concept for it.

However, I’m not sure if it should be:

a proof of concept for it.

or:

a proof of concept of it.

Can you clarify when to use “for” versus “of” in this context?

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    We usually do something for something. But your examples would be better with some more context and without pronouns like "it" where we don't know what "it" is. – user3169 Nov 10 '18 at 3:36
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    Then think of it this way: "evaluate (do) a proof-of-concept (something) for a new mechanism (something)". – user3169 Nov 10 '18 at 16:41
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If I had to answer in the terms in which you ask, then I’d say the preferred form in this case is, “of it”.

But I really don’t like either of the two. It’s obvious the PoC applies to XYZ, so you don’t need to specify using “for it”, “of it”, or “[anything else] it”. You could just use, “We implement and evaluate a proof of concept.” Or take the opportunity presented by chopping off those two words, to further simplify a little with a conjunction:

Second, we propose a new mechanism that we call XYZ, which aims to solve problem X, and we implement and evaluate a proof of concept.

But to be honest, even that is jarring to my ears. So if I was the author, I would step back and re-consider the whole thing. In particular I’d try to leave out that “proof of concept” term. It is one of those lazy (albeit common) new-fangled idioms that can often end up making technical English so dry and hard to read. And I suspect it is part of the reason you are struggling to choose between “of” and “for” in the first place. You are struggling not because you don’t understand English, but precisely because you do, and your ear is objecting to having to use a piece of gnarly neologism in the form of “proof of concept”! 🤓

So, I’d need to know the subject matter a little more to be sure of the meaning, but instead of:

Second, we propose a new mechanism that we call XYZ, which aims to solve problem X. We implement and evaluate a proof-of-concept for it.

Perhaps something as simple as the following might work?

Second, we evaluate a prototype of XYZ, a new potential solution to X.

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