Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.
The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference:
◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT):
p q ~q ~p ◦ Pure Hypothetical Syllogism (HS):
p q q r p r ◦ Disjunctive Syllogism (DS):
p v q ~p q Common strategies for constructing a proof involving the first four rules:
◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived.
◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens.
◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism.
◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference:
◦ Constructive Dilemma (CD):
(p q) • (r s) p v r q v s ◦ Simplification (Simp):
p • q p ◦ Conjunction (Conj):
p q p • q ◦ Addition (Add):
p p v q Common Misapplications
Common strategies involving the additional rules of inference:
◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification.
◦ If the conclusion is a conjunctive statement, consider obtaining it via conjunction by first obtaining the individual conjuncts. ◦ If the conclusion is a disjunctive statement, consider obtaining it via constructive dilemma or addition.
◦ If the conclusion contains a letter not found in the premises, addition must be used to introduce that letter.
◦ Conjunction can be used to set up constructive dilemma. The ten rules of replacement are expressed in terms of pairs of logically equivalent statement forms, either of which can replace each other in a proof sequence. ◦ A double colon (::) is used to designate logical equivalence. ◦ Underlying the use of rules of replacement are Axioms of Replacement, which asserts that within the context of a proof, logically equivalent expressions may replace each other. ◦ By Axioms of Replacement, the rules of replacement may be applied to an entire line or to any part of a line. The First Five Rules of Replacement:
◦ DeMorgan’s Rule (DM) ~(p • q) :: (~p v ~q) ~(p v q) :: (~p • ~q)
◦ Commutativity (Com) (p v q) :: (q v p) (p • q) :: (q • p) ◦ Associativity (Assoc): [p v (q v r)] :: [(p v q) v r)] [p • (q • r)] :: [(p • q) • r)]
◦ Distribution (Dist): [p • (q v r)] :: [(p • q) v (p • r)] [p v (q • r)] :: [(p v q) • (p v r)]
◦ Double Negation (DN): p :: ~~p Common strategies involving the first five rules of replacement: ◦ Conjunction can be used to set up DeMorgan’s rule. ◦ Constructive dilemma can be used to set up DeMorgan’s rule. ◦ Addition can be used to set up DeMorgan’s rule. ◦ Distribution can be used in two ways to set up disjunctive syllogism. ◦ Distribution can be used in two ways to set up simplification. ◦ If inspection of the premises does not reveal how the conclusion should be derived, consider using the rules of replacement to deconstruct the conclusion. The Remaining Five Rules of Replacement:
◦ Transposition (Trans): (p q) :: (~q ~p)
◦ Material Implication (Impl): (p q) :: (~q p)
◦ Material Equivalence (Equiv): (p ≡ q) :: [(p q) • (q p)] (p ≡ q) :: [(p • q) v (~q • ~p) ◦ Exportation (Exp): [(p • q) r] :: [(p (q r)]
◦ Tautology (Taut): p :: (p v p) p :: (p • p) Common strategies involving the remaining five rules of replacement: ◦ Material implication can be used to set up hypothetical syllogism. ◦ Exportation can be used to set up modus ponens. ◦ Exportation can be used to set up modus tollens. ◦ Addition can be used to set up material implication. ◦ Transposition can be used to set up hypothetical syllogism. ◦ Transposition can be used to set up constructive dilemma. ◦ Constructive dilemma can be used to set up tautology.
◦ Material implication can be used to set up tautology.
◦ Material implication can be used to set up distribution. Conditional Proof is a method for deriving a conditional statement (either the conclusion or some intermediate line) that offers the usual advantage of being both shorter and simpler than the direct method. For example:
◦ a (b • c)
◦ ( b v d) e / a e
To Construct a Conditional Proof:
◦ Begin by assuming the antecedent of the desired conditional statement on one line.
◦ Derive the consequent on the subsequent line.
◦ “Discharge” these lines in the desired conditional statement. Every conditional proof must be discharged, otherwise any conclusion can be derived from any premises. Indirect Proof is a technique similar to conditional proof that can be used on any argument to derive either the conclusion or some intermediate line leading to the conclusion. To construct an indirect proof: ◦ Begin by assuming the negation of the statement to be obtained. ◦ Use this assumption to derive a contradiction. ◦ Conclude that the original statement is false. ◦ As in conditional proofs, every indirect proof must be discharged, otherwise any conclusion can be derived from any premises. Indirect and conditional proofs can be combined to derive either a line in a proof sequence or the conclusion of a proof. Both conditional and indirect proof can be used to establish the truth of a logical truth, or tautology.
You can treat tautologies as if they were the conclusions of arguments having no premises. This is suggested by the fact that any argument having a tautology for its conclusion is valid regardless of its premises.