Constructive dilemma

Constructive dilemma is an inference rule that combines a disjunction of antecedents with conditional premises to derive a disjunction of their consequents. Its standard form is: if P or Q, and P implies R while Q implies S, then R or S.

Disjunction: A logical connective that is true when at least one of its component propositions is true. The rule begins with a disjunction of antecedents and concludes with a disjunction of consequents.
Modus ponens: An inference rule deriving Q from the premises P and if P, then Q. Each conditional branch of constructive dilemma applies this basic pattern to a possible antecedent.
Destructive dilemma: An inference rule deriving a disjunction of negated antecedents from conditional premises and a disjunction of negated consequents. It reverses the direction of constructive dilemma by reasoning from threatened outcomes to denied starting points.
Natural deduction: A proof system that derives conclusions through rules governing the introduction and elimination of logical connectives. Constructive dilemma can be represented through ordinary disjunction and conditional reasoning in this proof framework.
Conditional: A connective expressing that one proposition is sufficient for another, commonly written as if P, then Q. Each branch of the rule links an antecedent to its corresponding consequent.
Modus tollens: An inference rule deriving not P from if P, then Q and not Q. It is a foundational conditional rule that helps distinguish constructive from destructive forms.
Hypothetical syllogism: An inference rule combining two conditionals, if P then Q and if Q then R, to derive if P then R. It chains conditionals in sequence, whereas constructive dilemma branches from alternative antecedents.
Sequent calculus: A formal proof system whose judgments express entailment between collections of premises and conclusions. Its inference rules provide another precise setting for expressing the same entailment.
Propositional calculus: A formal system that studies propositions and their logical connectives. Constructive dilemma is a rule within this broader system of inference.
Logical consequence: A relation in which a conclusion must be true whenever all its premises are true. The rule is valid because its conclusion is a logical consequence of its premises.

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