  We study propositional logical systems arising from the language of
Johansson's minimal logic and obtained by weakening the requirements for the
negation operator. We present their semantics as a variant of neighbourhood
semantics. We use duality and completeness results to show that there are
uncountably many subminimal logics. We also give model-theoretic and algebraic
definitions of filtration for minimal logic and show that they are dual to each
other. These constructions ensure that the propositional minimal logic has the
finite model property. Finally, we define and investigate bi-modal companions
with non-normal modal operators for some relevant subminimal systems, and give
infinite axiomatizations for these bi-modal companions.
