The Language of Propositional Logic

Building Sentences From Sentences

Propositional logic takes whole declarative sentences as its atoms — written $P$, $Q$, $R$ — and builds compounds from them with connectives. Each connective is truth-functional: the truth value of the compound is fixed entirely by the truth values of its parts, with nothing else relevant.

Two of these need care. Disjunction is inclusive: $P \vee Q$ is true when both disjuncts are true. Ordinary English often means the exclusive "or", "tea or coffee" usually excludes both, but the logical symbol does not.

The conditional is the one that surprises everyone. $P \rightarrow Q$ is defined as false in exactly one case: $P$ true and $Q$ false. So it comes out true whenever $P$ is false, regardless of $Q$. "If the moon is made of cheese then I am the Pope" is true, on this definition. This is the material conditional, and it captures only part of what "if" means in English, but it is the part that makes deduction work, and the surprise is a feature of the definition rather than a mistake.

Brackets matter as much as connectives. $\neg(P \wedge Q)$ and $\neg P \wedge Q$ are different sentences with different truth conditions, and the difference is exactly where ambiguity in ordinary language hides.

Translating from English is where the real work lies. "Unless" is usually $\vee$; "P only if Q" is $P \rightarrow Q$, not $Q \rightarrow P$; "neither P nor Q" is $\neg P \wedge \neg Q$.

Common pitfall: reading "P only if Q" as "if P then Q" reversed. "You may vote only if you are registered" says registration is necessary, not sufficient, so it means voting $\rightarrow$ registered. The "only" marks the consequent, not the antecedent.