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With the concept of negation of strings, we are able to introduce two more desirable properties of a logical calculus:
Definition: Consistency and Negation-Completeness of a Logical Calculus
A logical calculus $L$ is called
- consistent, if for every derivable statement its negation is not derivable, formally $\vdash \phi$ implies $\not\vdash\neg\phi,$
- negation-complete, if the negation of a statement is not derivable, then the statetment is derivable, formally $\not\vdash\neg\phi,$ implies $\vdash \phi.$
- Consistency and negation-completeness are syntactical properties of a logical calculus.
- A logical calculus is consistent if it is not possible in it to derive both - a theorem and its negation.
- The negation-completeness of logical calculus is the converse - if the drivability of a theorem is necessary for the non-derivability of the negation.
- Consistency and negation-completeness are desirable properties of logical calculi since such systems avoid contradictions.
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Thank you to the contributors under CC BY-SA 4.0!
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
- Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000