and:

- between two objects, P and Q means the set \(\{P, Q\}\).
- between two sets, P and Q means the set \(P\cup Q\).
- between two propositions, P and Q means that both P is true and Q is true. (It is difficult to avoid a self-referencing definition.)

A possible objection to use-case 2 for "and" is that the conjunction "and" is used in the definition of intersection and so P and Q on sets should refer to set intersection, not union. However, a set or a list or the designation of a category refer to objects (conceptual ones or otherwise). Our best analogy is with with use-case 1, where we have both items included in the set. This conforms to customary usage. For example, in the designation The Department of Math and Computer Science, customary usage indicates this means "the department which contains math courses and computer science courses", or, more expansively, "the department which contains math courses and contains Computer Science courses". We recognize this rendering as use-case 3, which serves as the argument for use-case 2. If we called it The Department of Math or Computer Science or The Department of Math and/or Computer Science, this would imply that the course name would be valid if it contained math courses only, computer science courses only, or both types of courses. The name of such a department would not inspire confidence in the applicant interested in one of these areas specifically, that the department offered the kind of courses he/she was interested in.

or:

- between two objects, exclusively, either P or Q means exactly one element of the set \(\{P, Q\}\).
- between two object, inclusively, P or Q means any element of the set \(\{P, Q\}\).
- between two sets, exclusively, either P or Q normally means either an element of P or an element of Q, according to the sense of use-case 1.
- between two sets, inclusively, P or Q normally means an element of \(P\cup Q\).
- between propositions, exclusively, either P or Q means exactly one of P or Q is true.
- Yes, probably self-referencing again.
- between propositions, inclusively, P or Q means any of P is true, Q is true, or both P and Q are true.
- Still self-referencing, I guess.

At least some of the above inclusive cases are sometimes expressed in English using "and/or" in place of "or" as a means of indicating the inclusive nature of the conjunction intended. The exclusive uses are sometimes emphasized by using the word "either", as above.

At this point, we start to ask ourselves, "where is set intersection in all this?" Given that union occurs in our definitions, it seems reasonable that we should find set intersection somewhere in the mix. After all, we use "and" and "or" for defining intersection and union, respectively. We can see intersection in restating some of the use-cases given.

We can take our example of The Department of Math and Computer Science and express it in a way that uses set intersection. Let M be the set of departments in a university offering the majority of mathematics courses and let C be the set of departments offering the majority of Computer Science courses. Then \(M\cap C\) is The Department of Math and Computer Science. If we get the empty set, then there is no such department at the given university. This is use-case 3 for "and", expressed differently.

We can also see set intersection in the exclusive use-cases of "or". Use-case 3, in particular, can be expressed as \(P\cup Q - P\cap Q\). Depending on the context, this might appear redundant. For example, we might say that some bathrooms are made for either males or females. We take M as bathrooms intended for male occupancy and F as bathrooms intended for female occupancy. This can be seen as use-case 3, but the intersection is empty. In fact, the nature of the statement may be to emphasize the fact that the intersection is empty and the speaker is really saying \(M\cup F = M\cup F - M\cap F\), that is, "all of the bathrooms I have in mind are for single sex occupancy, none of the bathrooms I have in mind are accepting of both male and female". There may be other bathrooms where the intersection is non-empty (e.g., single occupancy), but not the bathrooms the speaker has in mind.

At this point, we start to ask ourselves, "where is set intersection in all this?" Given that union occurs in our definitions, it seems reasonable that we should find set intersection somewhere in the mix. After all, we use "and" and "or" for defining intersection and union, respectively. We can see intersection in restating some of the use-cases given.

We can take our example of The Department of Math and Computer Science and express it in a way that uses set intersection. Let M be the set of departments in a university offering the majority of mathematics courses and let C be the set of departments offering the majority of Computer Science courses. Then \(M\cap C\) is The Department of Math and Computer Science. If we get the empty set, then there is no such department at the given university. This is use-case 3 for "and", expressed differently.

We can also see set intersection in the exclusive use-cases of "or". Use-case 3, in particular, can be expressed as \(P\cup Q - P\cap Q\). Depending on the context, this might appear redundant. For example, we might say that some bathrooms are made for either males or females. We take M as bathrooms intended for male occupancy and F as bathrooms intended for female occupancy. This can be seen as use-case 3, but the intersection is empty. In fact, the nature of the statement may be to emphasize the fact that the intersection is empty and the speaker is really saying \(M\cup F = M\cup F - M\cap F\), that is, "all of the bathrooms I have in mind are for single sex occupancy, none of the bathrooms I have in mind are accepting of both male and female". There may be other bathrooms where the intersection is non-empty (e.g., single occupancy), but not the bathrooms the speaker has in mind.