| Negation introduction | |
| Type: | Rule of inference |
| Field: | Propositional calculus |
| Statement: | If a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction. |
| Symbolic Statement: | (P → Q)\land(P → \negQ) → \negP |
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus.
Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.[1] [2]
This can be written as:
(P → Q)\land(P → \negQ) → \negP
An example of its use would be an attempt to prove two contradictory statements from a single fact. For example, if a person were to state "Whenever I hear the phone ringing I am happy" and then state "Whenever I hear the phone ringing I am not happy", one can infer that the person never hears the phone ringing.
Many proofs by contradiction use negation introduction as reasoning scheme: to prove ¬P, assume for contradiction P, then derive from it two contradictory inferences Q and ¬Q. Since the latter contradiction renders P impossible, ¬P must hold.
| Step | Proposition | Derivation | |
|---|---|---|---|
| 1 | (P\toQ)\land(P\to\negQ) | Given | |
| 2 | (\negP\lorQ)\land(\negP\lor\negQ) | Material implication | |
| 3 | \negP\lor(Q\land\negQ) | Distributivity | |
| 4 | \negP\lorF | Law of noncontradiction | |
| 5 | \negP | Disjunctive syllogism (3,4) |