Logic and Statements

Precise definition

A proposition is a declarative sentence with a definite truth value, true or false. From propositions $p$ and $q$, negation $\neg p$, conjunction $p\land q$, disjunction $p\lor q$, implication $p\to q$, and biconditional $p\leftrightarrow q$ form compound propositions. In standard inclusive logic, $p\lor q$ allows both to be true.

Notation and mathematical language

The implication $p\to q$ is false only when $p$ is true and $q$ is false. Its converse is $q\to p$, inverse is $\neg p\to\neg q$, and contrapositive is $\neg q\to\neg p$. Only the contrapositive is logically equivalent to the original implication in general.

Conceptual picture

Logical form separates a pattern of reasoning from a sentence's subject matter. Vacuous truth may feel unfamiliar: if $p$ is false, no counterexample of the form 'true premise, false conclusion' occurs, so $p\to q$ is true. This convention makes implication compatible with proof and set inclusion.

Fully worked example

Interpretation and application

Formal logic supports proof, database queries, digital circuits, and software conditions. In a program, confusing 'necessary' with 'sufficient' can admit invalid input or reject valid input; writing the intended implication before coding reduces that risk.

Common mistakes