1. A property holding of any deductive system with a modal operator $\square$ that is not normal, that is, a modal logic for which either rule necessitation fails or an instance of the axiom scheme:

• • $\square \left(\phi \to \psi \right)\to \left(\square \phi \to \square \psi \right)$

fails.

2. In Kripke semantics for modal logics, describes a type of world $w$ at which formulae are evaluated in an unusual (and perhaps counterintuitive) fashion. Normal worlds are defined as possible worlds at which formulae of the form $\square \phi$ receive the familiar, Leibnizian truth condition:

• • $\square \phi$ is true at $w$ if for all worlds $w^{\prime }$ accessible from $w$, $\phi$ is true at $w^{\prime }$.

Modal formulae have different truth conditions at non-normal worlds. Thus, in the semantics for the Lewis systems $\text{S}2$ and $\text{S}3$ a non-normal world $w$ is one at which formulae of the form $\square \phi$ are evaluated by the truth condition:

• • $\square \phi$ is not true at $w$.

Non-normal worlds are employed in semantics for non-normal modal logics, explaining their pathological nature. Semantics for many non-normal logics $\text{L}$ must provide models serving as counterexamples to rule necessitation. Hence, the possible worlds framework requires a type of world at which $\square \phi$ can fail for some $\text{L}$-theorem $\phi$. Other deductive systems require further deviations from the evaluation of formulae at normal worlds; hence, other related devices, such as impossible worlds and the absurd world, are often referred to as ‘non-normal’.