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She may/might be honest.

This sentence already has the meaning that

She may/might not be honest.

1. Why do we sometimes say:

She might and/or might not be honest.

2. Which conjunction is (more) correct: "and" or "or"?

So, the question is not on when to use "might" and when to use "may", it is about why each comes along with its negation, and what difference that usage makes to the meaning. Is it just about emphasis?

The second question is about the conjunction that should be used between them: "and" or "or"? Both versions are used frequently on the web.

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    When you emphasize the "might not" as much as the "might" you make clear that either outcome is a possibility instead of merely casting some uncertainty on a given statement. This is actually important when either outcome is acceptable and the structure is frequently used when conveying something close to "regardless" or mentioning something is irrelevant. Even if not strictly necessary in a logical sense it adds emphasis. – Tom22 Jun 24 '17 at 23:57
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    What @Tom22 said. If something might be true (possibility A), this obviously implies the alternate that it might not be true (possibility B). Until such times as either of A or B is established to be true or false, A and B are both possible. But although by strict pedantic/anal logic one could/should therefore use and, we rarely do. – FumbleFingers Jun 25 '17 at 0:33
  • @FumbleFingers So you really think the question is a duplicate, even after the last edition? – Sasan Jun 25 '17 at 0:57
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    about 12,200 results in Google books for might and might not and 400,000 hits for might or might not – Mari-Lou A Jun 25 '17 at 5:12
  • @Sasan: That should be after the last edit, not edition. Which in itself is further support for my closevote - I voted to migrate this question to English Language Learners, not to close it as a duplicate. It seemed to me the "side question" about may/might was trivial, so I ignored it, but the primary and/or question is almost meaningless to native speakers - which is presumably why other closevoters didn't really address it at all. – FumbleFingers Jun 25 '17 at 11:52