Modus ponens

Modus ponens is a rule of inference that derives Q from the premises “if P then Q” and P. It is valid in classical logic and many other logical systems.

Material implication: A truth-functional connective that makes a conditional false only when its antecedent is true and its consequent false. This interpretation of “if P then Q” makes the rule’s truth-preserving character explicit.
Proposition: A declarative statement that can be evaluated as true or false. P and Q are propositions in the standard statement of the rule.
Modus tollens: A rule of inference that derives not-P from “if P then Q” and not-Q. It reasons backward from a denied consequent, unlike modus ponens.
Peirce's law: The classical logical formula (((P → Q) → P) → P), which is not generally valid in intuitionistic logic. Classical derivations involving this principle can use modus ponens to apply nested conditionals.
Conditional proof: A proof method that establishes a conditional by assuming its antecedent and deriving its consequent. It can establish the conditional premise that a later modus ponens step uses.
Logical connective: An expression that combines or modifies propositions to form more complex propositions. The conditional connective joins P to Q in the rule’s first premise.
Affirming the consequent: The invalid inference from “if P then Q” and Q to P. It resembles modus ponens but wrongly treats the consequent as sufficient for the antecedent.
Intuitionistic logic: A logic that rejects unrestricted use of classical principles such as the law of excluded middle. Modus ponens remains valid even though some classical inference principles do not.
Rule of inference: A formal pattern that licenses deriving a conclusion from specified premises. Modus ponens is one member of this broader family of proof rules.
Premise: A statement supplied as a starting point for an argument or proof. The conditional and its antecedent are the two premises from which the conclusion follows.

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