A variation of Kripke semantics in which the accessibility relation $R$ between possible worlds is replaced by a relation $R^{\ast }$ between possible worlds and ‘neighbourhoods’ (i.e., nonempty subsets) of possible worlds. The truth conditions for a formula $\square \phi$ are then,

• • $\square \phi$ is true at $w$ if and only if $w$ bears $R^{\ast }$ to the set of possible worlds at which $\phi$ is true.

Neighbourhood semantics provide semantics for many of the systems of logic weaker than the weakest normal system, $\text{K}$.