Consistency and Negation-Completeness of a Logical Calculus

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

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.

[7878] Beierle, C.; Kern-Isberner, G.: “Methoden wissensbasierter Systeme”, Vieweg, 2000

[656] Hoffmann, Dirk W.: “Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011