A system $${\sf WF}$$ of subintuitionistic logic is introduced, weaker than Corsi’s basic subintuitionistic system $${\sf F}$$. A derivation system with and without hypotheses is given in line with the authors’ derivation system for $${\sf F}$$. A neighbourhood semantics is introduced with a somewhat more complex definition than the neighbourhood semantics for non-normal modal logics. Completeness is proved for $${\sf WF}$$ with respect to this neighbourhood semantics, and similarly for some logics between $${\sf WF}$$ and $${\sf F}$$ which characterize nice frame classes. The study by the authors of the conservativity of $${\sf IPC}$$ over $${\sf F}$$ with respect to some classes of implications is extended to $${\sf WF}$$, 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.