TITLE: OR in real life vs OR in Mathematical Logic
QUESTION [13 upvotes]: I stumbled upon this issue.
Imagine sentence:

David take cake OR gift

In my understanding of English language David can now take either cake or gift but not both.
Am I right?
If I am right does this mean that OR as used in English language is different from OR in mathematical logic? Because in mathematical logic David could take both cake and gift.
But I came to another situation also.
Imagine:

Either David or Nick will be at home.

In this case, I think if both of them will be at home, above sentence is still true.
So why is in this case OR similar to OR from mathematical logic?
Did I use different ORs in above two sentences? What am I missing?
Do operations like OR, AND in mathematical logic need to resemble OR and AND from English language? and what are consequences if they don't?

REPLY [5 votes]: The English word "or" has lots of different meanings, and the logical OR operation is only one of those meanings. 
Here are some sentences that seem to use the word "or" to indicate the logical OR operation:

Either David or Nick will be at home.
I hear a hissing noise. Either the air is leaking out, or there are snakes in here. 

Sometimes the word "or" indicates the logical XOR operation:

I see that each person is only taking a slice of cake or a slice of pie. Why not both? 

The word "or" can be used to indicate a list of options, in which case the meaning is neither the logical OR nor the logical XOR. In fact, the meaning is more similar to a logical AND.

You can have the soup or a salad. (You are allowed to choose the soup, and you are allowed to choose the salad, but you are not allowed to choose both.)
This car has two color options: blue, or red. (Blue is one of the color options, and red is one of the color options.)
The blue car costs \$20,000 and the red car costs \$30,000. I have \$40,000 to spend, so I can afford either the blue car or the red car. (I can afford the blue car, and I can afford the red car, but I can't afford both.)

The word "or" is used to ask "which one?" In this usage, the word "or" definitely doesn't indicate any logical operator, because it doesn't form a yes-or-no question:

Would you like cream or lemon, Mr. Feynman? 

The word "or" can be used to make a threat:

Give me some ice cream, or I'll scream! 

All of these are different meanings of the word "or", and only the first meaning corresponds to the mathematical "or". 


Do operations like OR, AND in mathematical logic need to resemble OR and AND from English language? and what are consequences if they don't?

Well, the logical OR and AND operators are named after the corresponding English words, because the meanings of the logical OR and AND operators resembles the meanings of "or" and "and" in English. 
There doesn't have to be a correspondence. For example, the logical IF-THEN (material implication) operator doesn't correspond to any common meaning of the word "if" in English. The consequence of this is that millions of students are confused by this operator. 
Here's an example of a confusing IF-THEN sentence. Using the logical IF-THEN operator, the sentence "IF there are people on Mars, THEN there are no people on Mars" is true. But using the ordinary English meaning of the word "if", the sentence "if there are people on Mars, then there are no people on Mars" is false, or at least disagreeable.