What is a non-normal modal logic? Is there a family of semantics that covers both normal and non-normal modal logics?

Question

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?

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What follows is my attempt to answer the question myself and some background on where the question comes from.

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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.

Answer 1

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.

Answer 2

I would point your attention to to Heinrich Wansing's work on impossible worlds semantics for modal logics. In a series of papers Wansing formulates a generalization of standard Kripke semantics for normal modal logics that comprises both normal logics as well as classical modal logics. See his papers “Bemerkungen zur Semantik nicht-normaler möglicher Welten”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 35 (1989): 551–7; “A General Possible Worlds Framework for Reasoning about Knowledge and Belief”, Studia Logica, 49 (1990): 523–39.

I think there has also been much relevant work in the area of neighbourhood semantics for modal logics. One of the main applications of neighbourhood semantics indeed consists in providing an interpretation for non-normal logics (as was originally also the case with the impossible worlds stuff, see some early papers by Kripke and Cresswell). See Pacuit's nice intro text 'Neighbourhood semantics for modal logic'.