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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 Ref
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]
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| 13341 | Using the definition of truth, we can prove theories consistent within sound logics [Tarski] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 12656 | P-and-Q gets its truth from the truth of P and truth of Q, but consistency isn't like that [Fodor] |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| 18785 | Consistency is semantic, but non-contradiction is syntactic [Mares] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |