And, or, not

On a menu, “soup or salad” usually means one of the two. In logic, “p or q” stays true when both are true, and that single rule settles most arguments about the word or.

And: both parts must hold

Take two statements, p: “The shop sells bread” and q: “The shop sells milk.” The conjunction p ∧ q, read “p and q”, says both at once. It is true in one situation only, when p and q are both true. If the shop has bread but no milk, the sentence “The shop sells bread and milk” is false, because half of what it promised is missing. With two statements there are four possible situations (TT, TF, FT, FF), and a conjunction is true in exactly one of them.

English and Korean have many words that do the same logical job as and. “The flat is small but bright” claims that the flat is small and that it is bright; the word but adds a sense of contrast, and forall x treats it as a conjunction for that reason. Although, yet, also and while (in the sense of whereas) usually work the same way. Logic keeps the truth conditions and drops the tone.

Or: at least one part holds

The disjunction p ∨ q, read “p or q”, is true when at least one of p and q is true. It is false only when both are false. This is called the inclusive or, and the symbol ∨ always means it. Suppose a job advert says “Applicants must speak Japanese or Chinese.” Someone who speaks both is plainly not turned away. That is the inclusive reading at work.

Sometimes we do mean “one or the other, but not both.” A set lunch that offers “coffee or tea” rarely lets you take both. This is the exclusive or. Logic has no separate basic symbol for it in forall x; you build it from the others as (p ∨ q) ∧ ¬(p ∧ q), meaning at least one, and not both. When a sentence is ambiguous, the safest habit is to ask which reading the speaker intends, and write the exclusive version out in full if that is the one.

Not, and De Morgan’s laws

Negation flips one truth value. The interesting part is what happens when you negate a whole conjunction or disjunction. “It is not the case that the flat is cheap and near the station” is true whenever at least one part fails: the flat is not cheap, or it is not near the station, or both. In symbols, ¬(p ∧ q) has the same truth value as ¬p ∨ ¬q in every row. In the same way, “It is not the case that you can pay by cash or by card” means you can pay by neither: ¬(p ∨ q) matches ¬p ∧ ¬q. These two equivalences are called De Morgan’s laws, after the nineteenth-century logician Augustus De Morgan.

The common mistake is to push the not inside and forget to swap the connective. “Not (cheap and near)” does not mean “not cheap and not near.” A flat that is cheap but far from the station makes the first sentence true and the second false. Whenever a not moves inside brackets, and turns into or and or turns into and.

Brackets matter for the same reason they matter in arithmetic. ¬p ∧ q means “not p, and q”, while ¬(p ∧ q) means “not both.” In the first, the not touches only p. When you translate an ordinary sentence, decide first which words the not is meant to cover, then place the brackets.