Modal verbs express possibility/probability, necessity/obligation or ability/willingness. They show up as a base verb with a modal auxiliary, such as may, might, should, ought to, must, can, could, would, wish to. Shall and will function sometimes as modals and sometimes as simple future tense markers, depending on context.
There's a lot of nuance in modals, and here we'll take a look at two that are close but not exactly the same: may and might.
If you say "I may order that pasta," you're likely — but not certain — to order it. Perhaps you'll end up not ordering it, but you're indicating a strong-ish possibility.
If you say "I might order that pasta," there's a little more hesitation, you'll order it if it doesn't have mushrooms in it, if it comes with a tasty sauce, if the special isn't something you'd rather have, etc.
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Some sentences are too close to call, with the may/might choice not yielding much difference: "We may/might go swimming tomorrow, depending on the weather."
And many people don't distinguish these two in their speech or writing, so it's hard to put a lot of stock in the difference if you don't know how closely the source adheres to the distinction. Use context as your guide.
Problems with may and might arise in two areas: past tense and negatives.
In the past tense, the difference is amplified, with "may" implying possibility and "might" implying something that didn't happen but could have.
Here's an example: "Tornado sirens may have given the family time to get to the basement before the storm hit." From this, we know they survived the storm and could credit sirens for that. But if the sentence is: "Tornado sirens might have given the family time to get to the basement before the storm hit," we know there weren't any sirens and the family didn't make it to the basement in time.
Also, may has might as its simple past-tense form, which shows up when there's another past-tense verb in the sentence: "We thought we might have time to see the Wax Museum." "Luella said she might be able to bring a mince pie."
By the same token, in hypotheticals and contrary-to-fact situations, which call for the subjunctive, might is the only choice. "One might think someone with that attitude would have been fired long ago." "I might be going to the picnic, if liked blistering heat, sticky humidity and ants."
With negatives, the permission sense of "may" is often stronger than the possibility sense: "You may not eat that squid; it's for the party." Meaning you are not allowed to do so, though it is certainly physically possible.