Negation introduction

Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus.

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

Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.^[1][2]

Formal notation

This can be written as:

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

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

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

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

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

See also

• Reductio ad absurdum