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# Rules of Inference

A rule of inference is a form of argument that is valid.

 Modus Ponens $p \rightarrow q \newline p \newline \therefore q$ $p \rightarrow q \newline p \newline \therefore q$ Modus Tollens $p \rightarrow q \newline \neg q \newline \therefore \neg p$ $p \rightarrow q \newline \neg q \newline \therefore \neg p$ Conjunction $p \newline q \newline \therefore p \land q$ $p \newline q \newline \therefore p \land q$ Contradiction $\neg p \rightarrow c \newline \therefore p$ $\neg p \rightarrow c \newline \therefore p$ Transitivity $p \rightarrow q \newline q \rightarrow r \newline \therefore p \rightarrow r$ $p \rightarrow q \newline q \rightarrow r \newline \therefore p \rightarrow r$ Proof by Division into Cases $p \lor q \newline p \rightarrow r \newline q \rightarrow r \newline \therefore r$ $p \lor q \newline p \rightarrow r \newline q \rightarrow r \newline \therefore r$ Generalization $p \newline \therefore p \lor q$ $p \newline \therefore p \lor q$ $q \newline \therefore q \lor p$ $q \newline \therefore q \lor p$ Specialization $p \land q \newline \therefore p$ $p \land q \newline \therefore p$ $p \land q \newline \therefore q$ $p \land q \newline \therefore q$ Elimination $p \lor q \newline \neg q \newline \therefore p$ $p \lor q \newline \neg q \newline \therefore p$ $p \lor q \newline \neg p \newline \therefore q$ $p \lor q \newline \neg p \newline \therefore q$