Full Idea
Any system of proof S is said to be 'negation-consistent' iff there is no formula such that |-(S)φ and |-(S)¬φ.
Clarification
The '(S)' should actually be a subscript
Gist of Idea
For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ
Source
David Bostock (Intermediate Logic [1997], 4.5)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.167
A Reaction
Compare Idea 13542. This version seems to be a 'strong' version, as it demands a higher standard than 'absolute consistency'. Both halves of the condition would have to be established.
Related Idea
Idea 13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]