# Non-normal modal logic
A non-normal modal logic is a modal logic that is weaker than the smallest normal modal logic K: it lacks the K axiom, the rule of necessitation, or both, and is interpreted not over Kripke frames but over neighborhood (Scott–Montague) models. The family descends from [C. I. Lewis](https://www.edgepedia.org/c-i-lewis)'s systems S1, S2, and S3, which are non-normal because they do not contain the rule of Necessitation[1](https://plato.stanford.edu/entries/logic-modal-origins/). Today the central systems are the classical logics E, M, and R, and their axiom extensions, which trade the strong closure properties of normal logics for a better fit with deontic reasoning, non-omniscient agents, and notions of ability.[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18)
| Key fact | Detail |
|---|---|
| Minimal system | Logic E extends classical propositional logic with the congruence rule RE alone[3](https://theses.fr/2020AIXM0314.pdf) |
| The classical cube | Extending E with any combination of axioms M, C, and N yields eight non-equivalent systems; the strongest, EMCN, is exactly normal modal logic K[3](https://theses.fr/2020AIXM0314.pdf) • [2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18) |
| Regularity | The system with both M and C is equivalent to the regular system R[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18) |
| Semantics | Neighborhood models assign each world a set of neighborhoods, each neighborhood being a set of worlds[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18) |
| Complexity | Satisfiability is in NP for E with any subset of {M, N}, and in PSPACE for EC with any subset of {M, N}[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18) |
| Motivating paradoxes | Normal deontic logic yields Ross's paradox and Forrester's gentle murder; non-normal systems avoid them[4](http://www.aiml.net/volumes/volume12/Dalmonte-Olivetti-Negri.pdf) • [5](https://www.mv.helsinki.fi/home/negri/negri_logica_16_final.pdf) |
| Relational alternative | Kripke's non-normal worlds characterize Lewis's and Lemmon's systems where necessitation fails or holds only in restricted form[3](https://theses.fr/2020AIXM0314.pdf) |
## The K axiom and necessitation under scrutiny
A normal modal logic contains classical propositional logic, the axiom K (□(A → B) → (□A → □B)), and the rule of necessitation, which infers □A from A; K is the smallest such system[1](https://plato.stanford.edu/entries/logic-modal-origins/). Under an epistemic reading, the closure built into these principles is too strong: a non-omniscient agent should reject the monotonicity rule, that A → B implies □A → □B, and possibly necessitation as well, since the latter would make the agent know every validity[4](http://www.aiml.net/volumes/volume12/Dalmonte-Olivetti-Negri.pdf). This is the problem of logical omniscience, which traces back to Hintikka's 1962 standard approach to modeling knowledge; [Ronald Fagin](https://www.edgepedia.org/ronald-fagin), in his 1990 TARK paper, showed that agents in the structures of Levesque and Lakemeyer are, in a precise sense, perfect reasoners in a nonstandard logic NPL, one of several responses to the problem[6](http://www.tark.org/proceedings/tark_mar4_90/p41-fagin.pdf).
Deontic readings fail for a different reason. Necessitation conflicts with the principle of deontic contingency, stated as "A tautologous act is not necessarily obligatory", and it excludes empty normative systems by making all tautologies obligatory[3](https://theses.fr/2020AIXM0314.pdf). The normal deontic base, K plus the axiom ¬O⊥, generates well-known paradoxes such as Forrester's gentle murder (1984); the non-normal deontic systems ED, MD, RD, and KD, extensions of E, M, N, and C with ¬O⊥, were proposed as a way out[5](https://www.mv.helsinki.fi/home/negri/negri_logica_16_final.pdf). Non-normal systems also allow conflicting obligations □A and □¬A without trivializing, and avoid Ross's paradox[4](http://www.aiml.net/volumes/volume12/Dalmonte-Olivetti-Negri.pdf).
## Neighborhood (Scott–Montague) semantics
Neighborhood semantics was introduced by [Dana Scott](https://www.edgepedia.org/dana-scott) and Richard Montague and generalises relational (Kripke) semantics; a neighborhood frame assigns each world a set of subsets of worlds, interpretable as the propositions that are brought about, known, or obligatory in that world[7](https://ar5iv.labs.arxiv.org/html/2307.12265). Formally, each world w is associated with a set of neighborhoods N(w), where each neighborhood is itself a set of worlds[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18). The modal clause is: □A is true at w if the set of worlds which make A true belongs to the neighborhoods of w[8](http://hdl.handle.net/10138/327396).
Because nothing forces N(w) to be closed upward or to contain the set of all □A-worlds as a unit tied to a binary accessibility relation, the semantics validates fewer principles than [Kripke semantics](https://www.edgepedia.org/kripke-semantics), and different closure conditions on the neighborhoods yield the different systems E, M, R, and their extensions[3](https://theses.fr/2020AIXM0314.pdf). A bi-neighborhood variant associates each world with pairs of neighborhoods, the two components providing independently positive and negative support for a modal formula; this semantics is equivalent to the standard one but makes countermodels easier to generate[4](http://www.aiml.net/volumes/volume12/Dalmonte-Olivetti-Negri.pdf). Dalmonte, Charles, Olivetti, and Negri's theorem-prover PRONOM builds on these models for proof search and countermodel generation[9](https://doi.org/10.3233/ia-200052).
## The systems E, M, R and the classical cube
The minimal classical non-normal modal logic E extends classical propositional logic with the congruence rule RE (from A ↔ B infer □A ↔ □B) and nothing else[3](https://theses.fr/2020AIXM0314.pdf). The standard non-normal logics extend E with axioms from a small set: M (monotonicity), C (aggregation), and N (necessitation); logic EM is called monotone logic M, and MCN is modal logic K[10](http://tableaux2017.cic.unb.br/T2_L4-Lellmann). The result is a lattice of eight non-equivalent systems, the classical cube; the strongest, defined by all three axioms M, N, and C, is just the normal modal logic K[3](https://theses.fr/2020AIXM0314.pdf) • [2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18). Monotonicity and aggregation together correspond to regularity: the system with both M and C is equivalent to the regular system R[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18).
Every non-normal modal logic in this family is sound and complete with respect to its corresponding class of bi-neighborhood models, and the E* logics are sound and complete with respect to standard neighborhood models[11](https://ar5iv.labs.arxiv.org/html/2006.05436) • [8](http://hdl.handle.net/10138/327396). Specifically, a formula A is a theorem of E if and only if it is valid in all bi-neighborhood models, and the correspondence carries over to extensions with M, C, and N[8](http://hdl.handle.net/10138/327396).
## By the numbers: complexity and the lattice of logics
The derivability problem for E and its extensions with axioms M, C, N, P, T, and 4 is decidable: it is coNP-complete for the systems lacking the axiom C, and in PSPACE for the systems containing C (a result of Vardi)[3](https://theses.fr/2020AIXM0314.pdf). For systems with axiom D and the rule RD+n but without axiom 4, derivability is in PSPACE by a general result of Schröder and Pattinson for non-iterative axioms[3](https://theses.fr/2020AIXM0314.pdf). On the satisfiability side, the problem for E with any subset of {M, N} is in NP, while for EC with any subset of {M, N} it is in PSPACE[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18). The hypersequent calculi of the 2006.05436 work give an optimal coNP decision procedure for logics without C and an optimal PSPACE procedure for logics with C[11](https://ar5iv.labs.arxiv.org/html/2006.05436).
The complexity split tracks the lattice: many classical systems are NP-complete, as opposed to the "usual" PSPACE-completeness for normal modal logics[12](https://www-cs.stanford.edu/~epacuit/classes/esslli/nbhdesslli.pdf). On the model theory of monotonic neighborhood models the situation is well understood, while the more general setting still holds open questions[12](https://www-cs.stanford.edu/~epacuit/classes/esslli/nbhdesslli.pdf).
## How it compares with normal modal logics
The logic K is the smallest normal, or Kripkean, modal logic, and the logics E, EM, and their siblings are strictly weaker than K[12](https://www-cs.stanford.edu/~epacuit/classes/esslli/nbhdesslli.pdf). On the normal side, S4 adds axiom 4 to system T, and S5 adds axioms B and 4, or alternatively axiom E[1](https://plato.stanford.edu/entries/logic-modal-origins/); these systems include the rule of necessitation, which is exactly what the non-normal systems such as Lewis's S1, S2, and S3 give up. Non-normal modalities nonetheless arise naturally in the modal logic of derivability, logics of strategic ability, and weak epistemic logics without logical omniscience[12](https://www-cs.stanford.edu/~epacuit/classes/esslli/nbhdesslli.pdf).
Non-normal logics can also be given relational semantics. Kripke introduced relational models with so-called non-normal worlds to characterize Lewis's and Lemmon's systems in which necessitation fails or is validated only in a restricted form; these models were later reformulated by Fitting and Priest[3](https://theses.fr/2020AIXM0314.pdf). At a non-normal world, all modal formulas □A are stipulated false, while propositional formulas continue to be evaluated by the valuation and the classical truth conditions; this is what makes the necessitation of a theorem fail at such worlds, so necessitation cannot be applied universally[13](https://sites.ualberta.ca/~francisp/Phil428.526/NonNormalLogics.pdf).[17](https://grahampriest.net/?ddownload=2101)
## Applications: deontic, epistemic, and beyond
Non-normal modal logics appear in epistemic reasoning, where they offer a simple preliminary solution to the problem of logical omniscience, and in deontic logic, where they avoid some well-known paradoxes of classical deontic logic and enable the representation of conflicting obligations[2](https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18). They are also used in reasoning about games and about "truth in most of the cases"[8](http://hdl.handle.net/10138/327396).
Two readings illustrate why the axioms fail. Read □A as "the agent can bring about A": Ann can draw a card that is red or black, but cannot ensure drawing a red card or a black one, so the C axiom fails for ability[9](https://doi.org/10.3233/ia-200052). Read □A as "A is true in almost all cases" or "true with high probability": A and B can each hold in almost all cases without A∧B doing so, so C fails again[9](https://doi.org/10.3233/ia-200052). Monotonic logic extended with a coalition axiom is a particular case of coalition logic, connecting the family to game theory[9](https://doi.org/10.3233/ia-200052).
## References
1. Modern Origins of Modal Logic, Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logic-modal-origins/
2. Resolution Calculi for Non-normal Modal Logics, TABLEAUX 2023. https://link.springer.com/chapter/10.1007/978-3-031-43513-3_18
3. T. Dalmonte, Non-Normal Modal Logics: Neighbourhood Semantics and Its Calculi, PhD thesis, Aix-Marseille. https://theses.fr/2020AIXM0314.pdf
4. Dalmonte, Olivetti, Negri, Non-Normal Modal Logics: Bi-Neighbourhood Semantics and Its Labelled Calculi, AiML volume 12. http://www.aiml.net/volumes/volume12/Dalmonte-Olivetti-Negri.pdf
5. Negri, Orlandelli, Non-normal modal logics: a challenge to proof theory. https://www.mv.helsinki.fi/home/negri/negri_logica_16_final.pdf
6. R. Fagin, A Nonstandard Approach to the Logical Omniscience Problem, TARK 1990. http://www.tark.org/proceedings/tark_mar4_90/p41-fagin.pdf
7. Non-Normal Modal Description Logics (extended version). https://ar5iv.labs.arxiv.org/html/2307.12265
8. Dalmonte, Charles, Olivetti, Negri, Theorem proving for non-normal modal logics. http://hdl.handle.net/10138/327396
9. Proof-search and countermodel generation for non-normal modal logics: The theorem prover PRONOM. https://doi.org/10.3233/ia-200052
10. B. Lellmann, General Methods in Proof Theory for Modal Logic, Lecture 4, TABLEAUX 2017. http://tableaux2017.cic.unb.br/T2_L4-Lellmann
11. Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity. https://ar5iv.labs.arxiv.org/html/2006.05436
12. E. Pacuit, Neighbourhood semantics, ESSLLI course notes. https://www-cs.stanford.edu/~epacuit/classes/esslli/nbhdesslli.pdf
13. G. Priest, Non-Normal Propositional Modal Logics and their Tableaux Rules, lecture notes. https://sites.ualberta.ca/~francisp/Phil428.526/NonNormalLogics.pdf
14. Inquisitive Neighborhood Logic, Journal of Logic, Language and Information, 2025. https://link.springer.com/article/10.1007/s10849-025-09440-0
15. Uncountably many maximally consistent neighborhood complete congruential modal logics. https://arxiv.org/abs/2609.04508
16. On Modal Logics of Full Products of Neighborhood Frames. https://arxiv.org/html/2606.31852
17. grahampriest.net. https://grahampriest.net/?ddownload=2101
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Logical calculi and logical syntax › Modal and temporal logic › Non-normal and multimodal logics*
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