When should we use "and" and/or "and/or"?

Breaking this down:

• and/or is as official as English gets in the sense that you can use it in extremely formal contexts. There is typically a better way to say whatever is being said but it does convey a specific meaning.

• You should use and/or when both options are applicable in its place. "I would like cake and/or pie" means "I would like one or both of the following: cake; pie."

• The main reason for using and/or is to remove the ambiguity of whether and means "only both" and whether or means "only one." And/or explicitly means "it could be one of these or both of these."

• The confusion is drastically exacerbated by mathematicians, logicians and/or computer scientists who are very familiar with the differences between the logical operators AND, OR, and XOR. Namely, or in English can be either OR or XOR; and/or can only mean OR. As you may have noticed, all of the terms look similar which leads to the confusion in parsing sentences like your title.

EDIT: To strictly answer the question, you should use A and B when you explicitly mean both A and B, and you should use A and/or B when you mean A or B (or both).

In response to a request for pronunciation, I typically treat the / as a hyphen and simply say "and or". This is not always standard for the / symbol, however, and other words or phrases with a / may be different.

There is no official English: English is what its speakers make of it. That said, and/or is terrible English. It should be avoided, and people who use it should be made fun of. It exists because there are three ways to use the words and and or:

1. Eat your peas and carrots.
2. Do you want steak, salad, potatoes, or what?
3. You can either look at your cake, or you can eat it.

In the first, you must eat both your peas and your carrots. In the second, nothing prevents you from choosing steak and potatoes for dinner. In the third, you can't have your cake and eat it too.

Some people, especially lawyers, get the second and third senses confused. The argument is that because and and or are entirely different words, they should have entirely different meanings. Overlap is indicated with a slash, since "you can walk on the red and or or the blue squares" would be unacceptable.

Or means exactly the same thing as and/or. Anyone who tells you otherwise isn't speaking English. The mechanic who says

If your car has a dent or needs an oil change, stop by our shop!

is certainly not excluding those cars that are both dented and need their oil changed. The main difference between or and and is a mild sense of contrast or indifference: "Help yourself to the cakes, the pies, and the tarts" versus "Help yourself to the cakes, the pies, or the tarts."

Still, there are some cases where or is exclusive:

1. Either A or B.
2. A or B, but not both.
3. A or B (where A and B are obviously mutually exclusive)

Context can serve the role of saying "but not both". If your mom says "you can get the jawbreaker or the bubblegum", you know that she (wisely) won't let you have both. But if she intends to let you have both, even when context suggests otherwise, she can say:

1. A or B, or both if you'd like

Or, and I doubt that many will share my taste, you could try omitting the slash, as in the following:

You can walk on the red and or the blue squares.

In summary, avoid and/or and simply use or, they mean the same thing. Context will suggest the correct interpretation of or without the need to be explicit. And if context is misleading and you must be explicit, say "A or B, or both".

And/or is generally used when either one or both of the options may be true. Consider the following three examples:

1. I am going to buy milk and eggs.
2. I am going to buy milk or eggs.
3. I am going to buy milk and/or eggs.

In example #1, I am specifying that I will buy both.
In contrast, example #2 specifies that I will buy only one of them.
Example #3 combines these two and specifies that I may buy one or the other -- or both.

As for whether it is "official English" or not, I would say that it is. It is used within the AP Stylebook, for example.

I have never seen a reference to and/or in any spoken English textbooks, and as such, when answering how it is spoken, I can only speak from personal experience. In my experience, both words are pronounced as normal, i.e. "and or".

In English "or" is usually taken to be exclusive or, if you wish to specifically use inclusive or then use "and/or".