◀ ▲ ▶Branches / Logic / Definition: Consistency and NegationCompleteness 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 NegationCompleteness 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 $\not\vdash\neg\phi,$
 negationcomplete, 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 negationcompleteness 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 negationcompleteness of logical calculus is the converse  if the drivability of a theorem is necessary for the nonderivability of the negation.
 Consistency and negationcompleteness are desirable properties of logical calculi since such systems avoid contradictions.
Mentioned in:
Axioms: 1
Parts: 2 3
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
 Beierle, C.; KernIsberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000