Weak subintuitionistic logics

A system WF of subintuitionistic logic is introduced, weaker than Corsi’s basic subintuitionistic system F. A derivation system with and without hypotheses is given in line with the authors’ derivation system for F. A neighbourhood semantics is introduced with a somewhat more complex definition than the neighbourhood semantics for non-normal modal logics. Completeness is proved for WF with respect to this neighbourhood semantics, and similarly for some logics between WF and F which characterize nice frame classes. The study by the authors of the conservativity of IPC over F with respect to some classes of implications is extended to WF, and shows clearly the difference in strength between the two logics. Study of translations of these weak subintuitionistic logics into non-normal modal logics turned out to be hard because of the difference between their respective neighbourhood structures and leaves us with some open problems.