"and" and "or" in Set Theory and English

Relating the English words "and" and "or" to concepts in set theory and logic can be done by starting with basic examples and reasoning by analogy to more complex cases. It must also be remembered that human language is generally less precise than formal logic. In natural language, we recognize a range of meaning for a word and so we must look not for one definition, but a few definitions. Here, below, I attempt to give a listing of the use-cases of the words "and" and "or".

and:

1. between two objects, P and Q means the set \(\{P, Q\}\).
2. between two sets, P and Q means the set \(P\cup Q\).
3. between two propositions, P and Q means that \(\{P, Q\}\) is contained in the set of all true propositions.

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:

1. between two objects, exclusively, either P or Q means exactly one element of the set \(\{P, Q\}\).
2. between two object, inclusively, P or Q means any element of the set \(\{P, Q\}\).
3. between two sets, exclusively, x is in either P or Q means \(x\in P\cup Q - P\cap Q\).
4. between two sets, inclusively, x is in P or Q means \(x\in P\cup Q\).
5. between propositions, exclusively, either P or Q means \(|\{P, Q\}\cap T| = 1\), where \(T\) is all true statements.
6. between propositions, inclusively, P or Q means \(\{P, Q\}\cap T\ne \emptyset\).

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.

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 \(M\cap C\) will either be a one element set containing The Department(s) of Math and Computer Science or the empty set, meaning there is no such department at the given university. However, this is forcing the issue and use case 2 for "and" is the more natural expression.

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 a washroom is made for either males or females. We take M as washrooms intended for male occupancy and F as washrooms intended for female occupancy. This can be seen as use-case 3, but if the context of the conversation is public, multiple-occupancy washrooms 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 washrooms I have in mind are for single sex occupancy, none of the washrooms I have in mind are accepting of both male and female". There may be other washrooms where the intersection is non-empty (e.g., single occupancy), but not the washrooms the speaker has in mind.