- Should I write it like this? Or perhaps like this?
- Should I write it like this? or perhaps like this?
What about after an exclamation mark or semicolon?
Did the sentence end after the question mark?
Yes. Then you need a capital letter to start the new sentence, just as usual.
No. Then the question mark shouldn't have been there, since it ends a sentence.
Right, I should expand on that rather than just be a grumpy old man.
The capitalization rule that we care about here is that the first word of a sentence starts with a capital letter, so the question is really about what ends a sentence. The answer to that is easy: terminal punctuation, i.e. a full stop (or period if you're American), question mark or exclamation mark. There's a visual clue in that '?' and '!' are decorated full stops; you just have to remember that a colon (':') isn't really a decorated full stop, not that you'd ever know by looking at it. Colons, semicolons and commas aren't terminal punctuation, so they don't end a sentence and so don't force the next letter to be a capital. It may be a capital letter for some other reason such as being the start of a proper name, but not because it is starting a sentence.
There are exceptions to this rule, occasions when '?' and '!' become non-terminal punctuation. The most obvious is in quoted speech: if the speaker asks a question or makes an exclamation, the '?' or '!' doesn't have to be terminal if the sentence carries on after the quote.
"Should I write it like this?" he asked. "Or perhaps like this?"
The other class of exception is for what are probably really parenthetical comments. If you have a short phrase that you could have put aside in parentheses or dashes, then a question mark or exclamation mark can be used at the end of that phrase without ending the sentence. Be sparing with this. It looks wrong at a first read, and engenders the sort of argument I had with Jason in the comments below!
Should I write it like this, or abracadabra! like this?
Going back to the examples, and adding a few more for fun:
Should I write it like this? Or perhaps like this?
Should I write it like this? or perhaps like this?
Should I write it like this, or perhaps like that?
"Should I write it like this?" he asked, "or perhaps like that?"
Example 2 with the lowercase "or" is just plain wrong.
Crusty old grammarians who disapprove of starting sentences with conjunctions may frown at example 1 all they like, but it's a perfectly acceptable fragmentary sentence. Whether it's the right answer or not is another question entirely. Example 1 makes the point that the questions are distinct, though they are strongly linked otherwise the whole structure wouldn't work.
Example 3 on the other hand emphasizes that the two questions are options in a common situation, as well as reflecting a different way of saying them. That is obvious in this case because the two questions really are tightly coupled alternatives. However, consider the following:
Are the lights green? Or is the switch up?
Are the lights green, or is the switch up?
Both of these examples imply that the state of the lights and the state of the switch are related somehow. Version 2 couples them more tightly; I would usually assume (without more context) that either this is the same question being asked in two different ways (i.e. that the switch being up should cause the lights to be green), or that they are an exhaustive list of possibilities (either the switch is up or the lights are green, but not both or neither). This isn't an absolute rule, but it is quite strongly implied.
Example 4 is also wrong, though it has a better disguise. If you unwrap the quotes, what you get is:
Should I write it like this? or perhaps like this?
which is our duff example 2 back again. What you actually want is one of:
"Should I write it like this?" he asked. "Or perhaps like this?" (i.e. example 1)
"Should I write it like this," he asked, "or perhaps like that? (i.e. example 3)
depending this time on how much of a break there is supposed to be in the middle of what was said. But speech-marking is a topic for another question :-)
Exclamation marks work just like question marks for this purpose! Semicolons don't; they end a clause, not a sentence.
The word after a question mark (or exclamation point) is capitalized if it starts a new sentence.
If the next word does not start a new sentence, because the question mark is part of a quotation, a title, an interruption, etc., then it is not capitalized. Usually such things are set off with quotes, dashes, parentheses, or italics; but not always.
Your option #2 is a single interrogative sentence with two parts and a question mark at the end of each part. This is rare today, but it was apparently a lot more common historically, when sentences were longer. It is not at all hard to find examples with the word or:
Do you not know what beauty is? or do you not know whether you think him handsome? (1814)
Who lives in a more elegant style? or receives more illustrious visitors? (1815)
Should he anticipate the savages? or should he lie still, and abide their assault? (1822)
Could you have done as much? you — or any other man? or any other woman that ever breathed? (1833)
Does the science of medicine prevent death? or the institution of government prevent crime? (1847)
“Where are we scheduled to walk? or are we to have something to say about it ourselves?” (1896)
These examples are all from the Corpus of Historical American English, which only goes back to 1810. There are no doubt plenty of examples in later decades – I stopped looking – but this way of writing questions has all but disappeared. I rather like it, but to some people it’s just wrong, and to others it might seem literary or pretentious.
Update: Consecutive sentences in the prologue of Shakespeare’s Henry V contain a non-final question mark and a non-final exclamation point.
[...] But pardon, and gentles all,
The flat unraised spirits that have dared
On this unworthy scaffold to bring forth
So great an object: can this cockpit hold
The vasty fields of France? or may we cram
Within this wooden O the very casques
That did affright the air at Agincourt?
O, pardon! since a crooked figure may
Attest in little place a million;
And let us, ciphers to this great accompt,
On your imaginary forces work.
It depends on whether the two questions are related or not:
Should I write it like this? or like this?
If different questions:
Should I write it like this? What if there is an exclamation point?
If it's not a question mark, but rather a semicolon:
You shouldn't capitalize after a semicolon; just leave it.
Or if it's an exclamation point:
Just capitalize it! Like this!
In modern English it is the norm to capitalize the letter following a question mark. But archaically, question clauses embedded in a sentence were used without a capital after. Have a look at the Pilgrim's Progress for good examples.