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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 $\vdash \phi.$

Notes:

• 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.

Mentioned in:

Axioms: 1
Parts: 2 3

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References

Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
2. Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000