Negation introduction: Difference between revisions
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| 4 || <math>\neg P\lor ((\neg P \lor Q)\land \neg Q)</math> || [[Conjunction elimination]] (3) |
| 4 || <math>\neg P\lor ((\neg P \lor Q)\land \neg Q)</math> || [[Conjunction elimination]] (3) |
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| 5 || <math>\neg P \lor ((\neg P\land \neg Q) \lor (Q \land \neg Q))</math> || |
| 5 || <math>\neg P \lor ((\neg P\land \neg Q) \lor (Q \land \neg Q))</math> || Distributivity |
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| 6 || <math>\neg(Q \land \neg Q)</math> || [[Law of noncontradiction]] |
| 6 || <math>\neg(Q \land \neg Q)</math> || [[Law of noncontradiction]] |
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| 7 || <math>\neg P \lor (\neg P \land \neg Q)</math> || [[Disjunctive syllogism]] (5,6) |
| 7 || <math>\neg P \lor (\neg P \land \neg Q)</math> || [[Disjunctive syllogism]] (5,6) |
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| 8 || <math>\neg P \lor \neg P</math> || |
| 8 || <math>\neg P \lor \neg P</math> || Conjunction elimination (7) |
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| 9 || <math>\neg P</math> || [[Idempotence|Idempotent laws]] |
| 9 || <math>\neg P</math> || [[Idempotence|Idempotent laws]] |
Revision as of 10:55, 24 February 2019
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
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 "When the phone rings I get happy" and then later state "When the phone rings I get annoyed", the logical inference which is made from this contradictory information is that the person is making a false statement about the phone ringing.
Proof
Step | Proposition | Derivation |
---|---|---|
1 | Given | |
2 | Material implication | |
3 | Distributivity | |
4 | Conjunction elimination (3) | |
5 | Distributivity | |
6 | Law of noncontradiction | |
7 | Disjunctive syllogism (5,6) | |
8 | Conjunction elimination (7) | |
9 | Idempotent laws |
References
- ^ 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.