Full Idea
If we abandon classical logic in favour of truth-value gaps and try to strengthen the theory, this leads to the Kripke-Feferman theory of truth, and variants of it.
Gist of Idea
Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
Source
Leon Horsten (The Tarskian Turn [2011], 01.4)
Book Reference
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.8
Related Idea
Idea 15330 Friedman-Sheard theory keeps classical logic and aims for maximum strength [Horsten]