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Mathematics

Mathematical Logic

Classification of Statements

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Logical Equivalence

Two statements are logically equivalent if they have the same truth value in every possible interpretation. An example is $p \rightarrow q$ is logically equivalent to $\neg p \lor q$ .

Tautology

A tautology is a formula that is true in every possible interpretation. An example of a tautology is $p \lor \neg p$ (for any proposition $p$ , either $p$ is true or $p$ is false).

Contingency

A contingency is a formula that can be either true or false depending on the interpretation. An example is the proposition $p \lor q$ (which is true if either $p$ or $q$ are true, false otherwise).

Contradiction

A contradiction is a formula that is false under every possible interpretation. An example of a contradiction is $p \land \neg p$ (a proposition $p$ cannot be both true and false at the same time).

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