A non-normal modal logic is a variant of modal logic that deviates from the basic principles of normal modal logics.

Normal modal logics adhere to the distributivity axiom () and the necessitation principle which states that "a tautology must be necessarily true" ( entails ).[1] On the other hand, non-normal modal logics do not always have such requirements. The minimal variant of non-normal modal logics is logic E, which contains the congruence rule in its Hilbert calculus or the E rule in its sequent calculus upon the corresponding proof systems for classical propositional logic. Additional axioms, namely axioms M, C and N, can be added to form stronger logic systems. With all three axioms added to logic E, a logic system equivalent to normal modal logic K is obtained.[2]

Whilst Kripke semantics is the most common formal semantics for normal modal logics (e.g., logic K), non-normal modal logics are often interpreted with neighbourhood semantics.

Syntax

The syntax of non-normal modal logic systems resembles that of normal modal logics, which is founded upon propositional logic. An atomic statement is represented with propositional variables (e.g., ); logical connectives include negation (), conjunction (), disjunction () and implication (). The modalities are most commonly represented with the box () and the diamond ().

A formal grammar for this syntax can minimally be defined using only the negation, disjunction and box symbols. In such a language, where is any propositional name.[3] The conjunction may then be defined as equivalent to . For any modal formula , the formula is defined by . Alternatively, if the language is first defined with the diamond, then the box can be analogously defined by .[4]

For any propositional name , the formulae and are considered propositional literals whilst and are considered modal literals.

Proof Systems

Logic E, the minimal variant of non-normal modal logics, includes the RE congruence rule in its Hilbert calculus or the E rule in its sequent calculus.

Hilbert Calculus

The Hilbert calculus for logic E is built upon the one for classical propositional logic with the congruence rule (RE): . Alternatively, the rule can be defined by . Logics containing this rule are called congruential.

Sequent Calculus

The sequent calculus for logic E, another proof system that operates on sequents, consists of the inference rules for propositional logic and the E rule of inference: .

The sequent means entails , with being the antecedent (a conjunction of formulae as premises) and being the precedent (a disjunction of formulae as the conclusion).

Resolution Calculus

The resolution calculus for non-normal modal logics introduces the concept of global and local modalities. The formula denotes the global modality of the modal formula , which means that holds true in all worlds in a neighbourhood model. For logic E, the resolution calculus consists of LRES, GRES, G2L, LERES and GERES rules.[3]

The LRES rule resembles the resolution rule for classical propositional logic, where any propositional literals and are eliminated: .

The LERES rule states that if two propositional names and are equivalent, then and can be eliminated. The G2L rule states that any globally true formula is also locally true. The GRES and GERES inference rules, whilst variants of LRES and LERES, apply to formulae featuring the global modality.

Given any modal formula, the proving process with this resolution calculus is done by recursively renaming a complex modal formula as a propositional name and using the global modality to assert their equivalence.

Semantics

Whilst Kripke semantics is often applied as the semantics of normal modal logics, the semantics of non-normal modal logics are commonly defined with neighbourhood models. A standard neighbourhood model is defined with the triple where:[5][6]

  • is a non-empty set of worlds.
  • is the neighbourhood function that maps any world to a set of worlds. The function denotes a power set.
  • is the valuation function which, given any propositional name , outputs a set of worlds where is true.

The semantics can be further generalised as bi-neighbourhood semantics.[7]

Additional Axioms

The classical cube of non-normal modal logic considers axioms M, C and N that can be added to logic E defined as follows.[6]

Additional axioms for non-normal modal logics
AxiomDefinitionAlternative definitionCorresponding neighbourhood semantics
MIf and , then .
CIf , then .
N.

A logic system containing axiom M is monotonic. With axioms M and C, the logic system is regular. Including all three axioms, the logic system is normal.

With these axioms, additional rules are included in their proof systems accordingly.

References

  1. Garson, James (2023). Zalta, Edward Nouri; Nodelman, Uri (eds.). "Modal Logic". The Stanford Encyclopedia of Philosophy. Retrieved 24 December 2023.
  2. Dalmonte, Tiziano; Negri, Sara; Olivetti, Nicola; Pozzato, Gian Luca (September 2021). Theorem Proving for Non-normal Modal Logics. OVERLAY 2020. Udine, Italy. Retrieved 24 December 2023.
  3. 1 2 Pattinson, Dirk; Olivetti, Nicola; Nalon, Cláudia (2023). Resolution Calculi for Non-normal Modal Logics. TABLEAUX 2023. Lecture Notes in Computer Science. Vol. 14278. pp. 322–341. doi:10.1007/978-3-031-43513-3_18. Retrieved 24 December 2023.
  4. Carnap, Rudolf (June 1946). "Modalities and Quantification". The Journal of Symbolic Logic. Association for Symbolic Logic. 11 (2): 33–64. doi:10.2307/2268610. Retrieved 27 December 2023.
  5. Pacuit, Eric (November 2017). Neighborhood Semantics for Modal Logic. Springer. doi:10.1007/978-3-319-67149-9. ISBN 978-3-319-67149-9. Retrieved 24 December 2023.
  6. 1 2 Dalmonte, Tiziano (2020). Non-Normal Modal Logics: Neighbourhood Semantics and their Calculi (PDF) (Thesis). Aix-Marseille Université. Retrieved 24 December 2023.
  7. Dalmonte, Tiziano; Olivetti, Nicola; Negri, Sara (August 2018). "Non-normal modal logics: bi-neighbourhood semantics and its labelled calculi". Advances in Modal Logic 2018. Berne, Switzerland. ISBN 978-1-84890-255-8.
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