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AFAIK, normally one would write:

"If c is zero and b nonzero or b2=4 a c, then and only then there exists a unique solution of 0=a+bx+cx2."

Is it possible to interchange the if and the then clauses to emphasize the consequence? Is the following sentence good English?

"Then and only then there exists a unique solution of 0=a+b x+c x2, if c is zero and b nonzero or b2=4 a c."

Looking at the answers to another question I have the impression that even the following should be more correct(?)

"Then and only then exists there a unique solution of 0=a+b x+c x2, if c is zero and b nonzero or b2=4 a c."

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    You could switch the clauses, but you wouldn’t really want to continue using “then and only then”. For example: “A unique solution to ... exists only if c is zero ...” – Jim Apr 14 at 1:57
  • I actually wanted to write "Only then exists there..." to make the reader cautious about the exceptions. Is the word order in that phrase right? I.e., does inversion apply here? – Tobias Apr 14 at 2:04
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    No, that’s boerderline ungrammatical. I highly recommend staying away from “only then” at the beginning of a sentence when the constraint hasn’t been mentioned previously. Burundi you do do it. It needs to be, “only then does there exist a unique solution... – Jim Apr 14 at 2:08
  • @Jim If you copy your comments into an answer and mark my comment as citation there I could accept that answer. That would mark this question as done. Thanks for the fast help. I will follow your recommendations. When you wrote up your answer I will delete this comment. – Tobias Apr 14 at 2:15
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    "Then" must, in any reasonably normal stream of English speech or writing, follow words that establish what logically must be true for the "then" clause to be true. In many other cases you can reverse the order of English clauses, but not in this case. – Hot Licks Apr 14 at 2:38
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Starting at the bottom, your last sentence won't work.

Here are the standard phrases I've seen:

  • There exists a unique solution of 0=a+b x+c x2 if and only if c is zero and b is nonzero, or b2=4 a c.
  • c is zero and b is nonzero, or b2=4 a c, if and only if 0=a+b x+c x2 has a unique solution.

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