One can use it, as one might expect, to translate "j or y", where the "or" is being used inclusively; that is to say, one intends the whole sentence to be true when either or both constituents are true, and false only when they are both false.
So one can translate "John is either a fool or a knave" as "[John is a fool Ú John is a knave]".
"Or" is sometimes used exclusively. In spoken English the effect can sometimes be achieved by giving the word special emphasis. In written English one can sometimes achieve the effect by using bold type, or capitals, or underlining. But the easiest way of conveying that one intends one's sentence to be false when both the constituents are true is by saying, for example, "John is a fool or a knave, but not both" - where it doesn't matter whether the "or" is inclusive or exclusive. Similarly, if one wants to avoid any doubt that one intends one's sentence to be true if both constituents are, one can say, "John is a fool or a knave, or both" - where again it doesn't matter whether the occurrences of "or" are inclusive or exclusive.
How can one tell if "or" is being used inclusively? Well, if, after a sentence of the form "j or y" one can add, with no hint of contradiction, "and possibly both", that is a pretty sure sign that the "or" is inclusive. (If it were exclusive, there would be a hint of contradiction, because one's use of "or" would have ruled out the possibility of both.)
Again, if after a sentence of the form "j or y" one can add "but not both" without any hint of repeating oneself, that again would suggest that the "or" is inclusive. (If it were exclusive, there would be a hint of repetition, because one's use of "or" would already have implied "not both".)
In fact "or" is most commonly used inclusively, and is hardly ever used unambiguously exclusively. You should certainly beware of thinking that it must be used exclusively in "Either is raining or it is not raining", on the grounds that it cannot be both raining and not raining. If that were a valid reason, one could equally well argue that the expression "[It is raining Ú it is not raining]" involved some sort of contradiction, because "[jÚy]" is inclusive. But it involves no contradiction. On the contrary, it is a necessary truth.
The fact is that, although it is perfectly obviously true that it can't be both raining and not raining, that does not mean that anyone who says, "Either it is raining or it is not raining" must be saying that it isn't both raining and not raining. If anything it is unnecessary to say it just because it is obvious. In any case, one could perfectly well say, "Either it is raining, or it is not raining, but not, of course, both" without any hint of repeating oneself.
Another example of something which can be translated with "[jÚy]" is "John will not come unless Mary comes", which can be translated as "[John will not come Ú Mary will come]". Sometimes people protest that "unless" ought to be treated as an exclusive "or". If that were the case with this example, it would be equivalent to "John will not come unless Mary comes, in which case he will." But certainly "unless" does not have to be used exclusively.
For instance "You will not win the lottery jackpot unless you buy a ticket" is obviously true. But if the "unless" were used exclusively, it would be equivalent to, "You will not win the lottery unless you buy a ticket, in which case you will", which is obviously false. Again, a sure indication of an inclusive "unless" is if one can add "and maybe not even then" without a hint of contradiction.
Are there any cases where "unless" is being used exclusively? Certainly there are cases where in saying something of the form "j unless y" one implies "in which case not j". Perhaps, if someone says to his son, "I shall be angry with you, unless you tidy your room", he implies that he won't be angry if he does. But it is not so clear that this is logical implication, as opposed to implicature.
Can one translate "I will wake up unless my alarm clock breaks" as "[I will wake up Ú my alarm clock will break]"? One might argue that this would be wrong on the grounds that, if it were correct, it would also be correct to translate "My alarm clock will break unless I wake up" as "[My alarm clock will break Ú I will wake up]". But the problem is that the two translations are equivalent; whereas the two original sentences seem not to be. Well, they certainly do not mean the same thing. But one might argue the difference lies in the fact that what they implicate differs (perhaps something about time order, or dependence) and not in what they logically imply.
Somewhat similarly one might say that it is perfectly all right to translate "John caught a train and went to London" as "[John caught a train Ù John went to London]", in spite of the fact that it differs in meaning from "John went to London and caught a train".