Questions with "of"
In English, when we use "or" in a question, such as "Would you like coffee or milk?", there are two very distinct possible meanings for such a question, and we indicate which it is by the tone of voice and maybe the insertion of a pause. One of the meanings assumes you want one or the other, and is asking which. The other meaning is the same as Latin "vel" or the "v" of the propositional calculus; the answer to it is "yes" or "no", and it is "yes" when either of the branches is true, or both. Do the "of" questions in Nederlands also have these two meanings? Do you indicate which by tone of voice and/or insertion of a pause? If so, are the same tones and/or pause used as in English?
Yes, it can have two possible meanings, but we usually use the meaning of a choice. For example:
"Heb jij een kat of een hond?" - When the emphasize is on 'kat' and 'hond' (and often also a pause after the fist choice) then the asker gives you a choice. Your answer could be "Ik heb een kat." When there is no specific emphasize then it can mean if you have any. You could answer with "Ik heb geen kat of hond." or "Ja, ik heb een kat" or "Ja, ik heb een kat en een hond."
"Wil je suiker of melk in je koffie?" - This question is often asked without any emphasize, because it is also possible that the other wants neither. With no emphasize answers could be "Melk, alstublieft" or "Suiker, alstublieft" or "Melk en suiker alstublieft" or "Nee, dankuwel". When there is emphasize on 'suiker' and 'melk' then the asker leaves you with a choice, sugar or milk, and you can choose only one of them.
So in the first example the question often has an emphasize on the choice. In the second example there isn't often any emphasize. I think these tones are the same as in English. Hope this helped.