Negation introduction

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	$(P\rightarrow Q)\land (P\rightarrow \neg Q)\rightarrow \neg P$

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.

Formal notation

This can be written as:

${\Big (}(P\rightarrow Q)\land (P\rightarrow \neg Q){\Big )}\rightarrow \neg P$

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.

Proof

With $\neg P$ identified as $P\to \bot$, the principle is as a special case of Frege's theorem, already in minimal logic.

Another derivation makes use of $A\to \neg B$ as the curried, equivalent form of $\neg (A\land B)$. Using this twice, the principle is seen equivalent to the negation of ${\big (}P\land (P\to Q){\big )}\land \neg (P\land Q)$ which, via modus ponens and rules for conjunctions, is itself equivalent to the valid noncontradiction principle for $P\land Q$.

A classical derivation passing through the introduction of a disjunction may be given as follows:

Step	Proposition	Derivation
1	$(P\to Q)\land (P\to \neg Q)$	Given
2	$(\neg P\lor Q)\land (\neg P\lor \neg Q)$	Classical equivalence of the material implication
3	$\neg P\lor (Q\land \neg Q)$	Distributivity
4	$\neg P\lor \bot$	Law of noncontradiction for $Q$
5	$\neg P$	Disjunctive syllogism (3,4)

See also

• Reductio ad absurdum