Mathematical Statements and Logic

The Precision of Mathematical Language

Mathematics is written in a language of unusual precision. Its sentences are statements (propositions): declarative assertions that are unambiguously either true or false — never both, never neither. "$7$ is prime" is a true statement; "$4 > 9$" is a false one; "$x + 1$" and "is this prime?" are not statements, because they assert nothing with a definite truth value.

Simple statements combine into compound ones using logical connectives, each with a fixed meaning:

• Negation ($\neg p$, "not $p$") — reverses the truth value.
• Conjunction ($p \land q$, "$p$ and $q$") — true exactly when both are true.
• Disjunction ($p \lor q$, "$p$ or $q$") — true when at least one is true. Note this is the inclusive or: $p \lor q$ is true even when both hold.

The meaning of a connective is captured completely by its truth table, which lists the result for every combination of inputs. Because each component is true or false, a compound of $n$ simple statements has $2^n$ rows.

Two statements are logically equivalent when they have identical truth tables — they say the same thing, however differently phrased. This is the bedrock of proof: replacing a statement by an equivalent one preserves truth. Mathematical logic gives us the rules that guarantee our reasoning transports truth from premises to conclusions without leaks.

Common pitfall: reading mathematical "or" as the everyday exclusive or ("one or the other, but not both"). In mathematics $p \lor q$ is inclusive: it is true when $p$ is true, when $q$ is true, and when both are true. "$x > 0$ or $x < 5$" holds for $x = 3$ even though both parts are true. Only explicitly stated exclusive-or means "not both."

A truth table for p OR q with all four rows shown, the both-true row (p=1, q=1) highlighted in accent as still TRUE — the inclusive nature of mathematical disjunction.

$$p \lor q \text{ is true} \iff p \text{ or } q \text{ (or both)}$$