Full Idea
Any system of proof S is said to be 'absolutely consistent' iff it is not the case that for every formula we have |-(S)φ.
Gist of Idea
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
Source
David Bostock (Intermediate Logic [1997], 4.5)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.167
A Reaction
Bostock notes that a sound system will be both 'negation-consistent' (Idea 13541) and absolutely consistent. 'Tonk' systems can be shown to be unsound because the two come apart.
Related Idea
Idea 13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]