With the concept of negation of strings, we are able to introduce two more desirable properties 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.

| | | | | created: 2018-02-10 00:10:25 | modified: 2018-02-11 09:18:45 | by: *bookofproofs* | references: [656], [7878]

[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