Conditional proof

A proof method that temporarily assumes a statement and derives another statement from it, thereby establishing a conditional from the assumption to the conclusion.

Conditional introduction: A natural-deduction rule that infers a conditional by discharging an assumption of its antecedent. This rule formalizes the move that defines conditional proof.
Material conditional: The classical truth-functional connective whose value is false only when its antecedent is true and consequent false. It is one common interpretation of the conditional established by the proof method.
Implication: A relation or connective expressing that one proposition follows from or is entailed by another. Conditional proof constructs a formal implication from an assumed condition to a derived result.
Modus tollens: An inference pattern that derives the negation of an antecedent from a conditional and the negation of its consequent. It reasons from a conditional’s failure case instead of deriving a consequent under an assumption.
Assumption discharge: The removal of a temporary assumption from the undischarged premises of a derivation. Conditional proof ends by discharging its antecedent assumption.
Antecedent: The component proposition that precedes the arrow in a conditional statement. It is the proposition temporarily assumed at the start of the proof.
Proof by cases: A proof method that establishes a conclusion by considering multiple exhaustive alternatives. It can supply the reasoning inside a conditional proof when the assumption admits cases.
Proof by contradiction: A proof method that assumes a claim’s negation and derives a contradiction to establish the claim. It also uses temporary assumptions, but discharges them through contradiction rather than a consequent.
Natural deduction: A proof system organized around introduction and elimination rules for logical connectives. Conditional proof is one of natural deduction’s standard derivation patterns.
Consequent: The component proposition that follows the arrow in a conditional statement. It is what the subproof must derive from the antecedent assumption.

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