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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
Modus Tollens p \rightarrow q \newline \neg q \newline \therefore \neg p
Conjunction p \newline q \newline \therefore p \land q
Contradiction \neg p \rightarrow c \newline \therefore p
Transitivity 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
Generalization p \newline \therefore p \lor q
q \newline \therefore q \lor p
Specialization p \land q \newline \therefore p
p \land q \newline \therefore q
Elimination p \lor q \newline \neg q \newline \therefore p
p \lor q \newline \neg p \newline \therefore q