What is a non-normal modal logic?
Is there a semantics for modal logic that's broad enough to cover a range of normal and non-normal modal logics and has the normal modal logics as a definable subfamily?
What follows is my attempt to answer the question myself and some background on where the question comes from.
The first section of the Open Logic Project book Boxes and Diamonds covers normal modal logics.
The discussion of validity on pages 10-11 (pdf 25-26) contains the following, which seems like almost a definition of a normal modal logic.
[Validities] represent those modal propositions which are true
regardless of how $□$ and $◇$ are interpreted, as long as the
interpretation is “normal” in the sense that it is generated by some
accessibility relation on possible worlds.
This suggests to me that a normal modal logic is one with a semantics that can be produced by taking the semantics for System K and imposing additional frame conditions. This gives us the familiar family of modal logics including B, S4, and S5.
The problem with this, though, is that I'm having a hard time picturing a semantics for a modal logic that's more general than K's semantics.
This lecture by Graham Priest seems to suggest that a non-normal modal logic is one with a set of impossible worlds $I$ … where every complex formula $\varphi$ is assigned a truth value $\{0, 1\}$ in an arbitrary fashion. I'm not sure whether the complex formula is required to be $\square\lozenge$-free or not. This would make a normal modal logic simply a not-necessarily-normal modal logic where the set of impossible worlds is constrained to be empty.
I think this means we have to expand our Kripke model from $\langle W, R, \Vdash \rangle$ to $\langle I, W, R, \Vdash \rangle$ where $I \subset W$ and the rules restricting how $\Vdash$ behaves for complex formulas are inapplicable to worlds in $I$. This seems kind of unsatisfying though.
Best Answer
Word 'normal' in 'normal modal logic' merely refers to the fact that the the logic satisfies all propositional tautologies, all instances of axiom K $\square(\varphi \to \psi) \to (\square \varphi \to \square \psi)$, and all instances of $\Diamond \varphi \leftrightarrow \lnot \square \lnot \varphi$, plus modus ponens, $\square$-necessitation, and uniform substitution are validity-preserving.
Thus, any logic that fails to satisfy any of the requirements, is non-normal. An example of a non-normal modal logic is alternating-time temporal logic.