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 |
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]
Formal notation
[edit]This can be written as:
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
[edit]Step | Proposition | Derivation |
---|---|---|
1 | Given | |
2 | Material implication | |
3 | Distributivity | |
4 | Law of noncontradiction | |
5 | Disjunctive syllogism (3,4) |
See also
[edit]References
[edit]- ^ Wansing, Heinrich, ed. (1996). Negation: A Notion in Focus. Berlin: Walter de Gruyter. ISBN 3110147696.
- ^ Haegeman, Lilliane (30 Mar 1995). The Syntax of Negation. Cambridge: Cambridge University Press. p. 70. ISBN 0521464927.