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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
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Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.
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If a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.
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456:{\displaystyle (P\rightarrow Q)\land (P\rightarrow \neg Q)\rightarrow \neg P}
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110:{\displaystyle (P\rightarrow Q)\land (P\rightarrow \neg Q)\rightarrow \neg P}
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happy", one can infer that the person never hears the phone ringing.
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use negation introduction as reasoning scheme: to prove ¬
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645:{\displaystyle (\neg P\lor Q)\land (\neg P\lor \neg Q)}
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485:, then derive from it two contradictory inferences
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849:. Cambridge: Cambridge University Press. p.
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577:{\displaystyle (P\to Q)\land (P\to \neg Q)}
700:{\displaystyle \neg P\lor (Q\land \neg Q)}
493:. Since the latter contradiction renders
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841:Haegeman, Lilliane (30 Mar 1995).
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820:. Berlin: Walter de Gruyter.
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738:{\displaystyle \neg P\lor F}
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818:Negation: A Notion in Focus
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481:, assume for contradiction
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362:Existential generalization
167:Biconditional introduction
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16:Logical rule of inference
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401:This can be written as:
353:Universal generalization
193:Disjunction introduction
180:Conjunction introduction
150:Implication introduction
748:Law of noncontradiction
475:proofs by contradiction
881:Propositional calculus
845:The Syntax of Negation
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772:{\displaystyle \neg P}
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388:propositional calculus
212:hypothetical syllogism
133:Propositional calculus
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39:Propositional calculus
782:Disjunctive syllogism
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376:Negation introduction
254:Negation introduction
247:modus ponendo tollens
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20:Negation introduction
797:Reductio ad absurdum
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655:Material implication
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263:Rules of replacement
126:Transformation rules
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886:Rules of inference
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156:elimination (
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219:Constructive
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158:modus ponens
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518:Proposition
501:must hold.
319:Exportation
206:Disjunctive
199:elimination
186:elimination
173:elimination
875:Categories
860:0521464927
827:3110147696
803:References
523:Derivation
232:Absorption
764:¬
730:∨
724:¬
689:¬
686:∧
677:∨
671:¬
634:¬
631:∨
625:¬
619:∧
610:∨
604:¬
566:¬
563:→
554:∧
545:→
448:¬
445:→
436:¬
433:→
424:∧
415:→
326:Tautology
102:¬
99:→
90:¬
87:→
78:∧
69:→
45:Statement
791:See also
857:
824:
784:(3,4)
586:Given
505:Proof
489:and ¬
473:Many
382:, or
378:is a
35:Field
855:ISBN
822:ISBN
513:Step
25:Type
468:not
877::
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63:(
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