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 (${\displaystyle \Box (p\to q)\to (\Box p\to \Box q)}$) and the necessitation principle which states that "a tautology must be necessarily true" (${\displaystyle \vdash A}$ entails ${\displaystyle \vdash \Box A}$).^[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., ${\displaystyle p,q,r}$); logical connectives include negation (${\displaystyle \neg }$), conjunction (${\displaystyle \land }$), disjunction (${\displaystyle \lor }$) and implication (${\displaystyle \to }$). The modalities are most commonly represented with the box (${\displaystyle \Box }$) and the diamond (${\displaystyle \Diamond }$).

A formal grammar for this syntax can minimally be defined using only the negation, disjunction and box symbols. In such a language, ${\displaystyle \varphi ,\psi$ :=p\ |\ \neg \varphi \ |\ \Box \varphi \ |\ \varphi \lor \psi } where ${\displaystyle p}$ is any propositional name.^[3] The conjunction ${\displaystyle \varphi \land \psi }$ may then be defined as equivalent to ${\displaystyle \neg (\neg \varphi \lor \neg \psi )}$. For any modal formula ${\displaystyle \varphi }$, the formula ${\displaystyle \Diamond \varphi }$ is defined by ${\displaystyle \neg \Box \neg \varphi }$. Alternatively, if the language is first defined with the diamond, then the box can be analogously defined by ${\displaystyle \Box \varphi \equiv \neg \Diamond \neg \varphi }$.^[4]

For any propositional name ${\displaystyle p}$, the formulae ${\displaystyle p}$ and ${\displaystyle \neg p}$ are considered propositional literals whilst ${\displaystyle \Box p}$ and ${\displaystyle \neg \Box p}$ 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.

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 ${\displaystyle {\mathcal {M}}}$ is defined with the triple ${\displaystyle \langle {\mathcal {W}},{\mathcal {N}},{\mathcal {V}}\rangle }$ where:^[5][6]

• ${\displaystyle {\mathcal {W}}}$ is a non-empty set of worlds.
• ${\displaystyle {\mathcal {N}}:{\mathcal {W}}\to {\mathcal {PP}}({\mathcal {W}})}$ is the neighbourhood function that maps any world to a set of worlds. The function ${\displaystyle {\mathcal {P}}}$ denotes a power set.
• ${\displaystyle {\mathcal {V}}:Atm\to {\mathcal {P}}({\mathcal {W}})}$ is the valuation function which, given any propositional name ${\displaystyle p}$, outputs a set of worlds where ${\displaystyle p}$ 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

Axiom	Definition	Alternative definition	Corresponding neighbourhood semantics
M	${\displaystyle \Box (A\land B)\to \Box A}$	${\displaystyle \Diamond A\to \Diamond (A\land B)}$	If ${\displaystyle \alpha \in {\mathcal {N}}(w)}$ and ${\displaystyle \alpha \subseteq \beta }$, then ${\displaystyle \beta \in {\mathcal {N}}(w)}$.
C	${\displaystyle \Box A\land \Box B\to \Box (A\land B)}$	${\displaystyle \Diamond (A\lor B)\to \Diamond A\lor \Diamond B}$	If ${\displaystyle \alpha ,\beta \in {\mathcal {N}}(w)}$, then ${\displaystyle \alpha \cap \beta \in {\mathcal {N}}(w)}$.
N	${\displaystyle \Box \top }$	${\displaystyle \neg \Diamond \bot }$	${\displaystyle {\mathcal {W}}\in {\mathcal {N}}(w)}$.

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