The word “Or”

English uses the word "Or" in two genuinely different senses, and mathematics needs to distinguish them before any compound statement built from "or" can be judged true or false.

• Exclusive Or allows exactly one of the two alternatives, never both. "An ice cream or a pepsi is available with a Thali" means you may choose one of the two — not both together.
• Inclusive Or allows one alternative, the other, or both together. "A student who has taken biology or chemistry can apply for the M.Sc. microbiology programme" clearly admits a student who has taken both subjects as well as one who has taken only one — so this is inclusive.

Whichever sense is meant, the compound statement built with "Or" follows one rule for truth:

1. A compound statement joined by "Or" is true when at least one component statement is true (one being true, or both being true, both count).
2. It is false only when both component statements are false. …