Abstract
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.
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