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 ( ◻ ( p → q ) → ( ◻ p → ◻ q ) {\displaystyle \Box (p\to q)\to (\Box p\to \Box q)} ) and the necessitation principle which states that "a tautology must be necessarily true" ( ⊢ A {\displaystyle \vdash A} entails ⊢ ◻ A {\displaystyle \vdash \Box A} ). 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. 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., p , q , r {\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,
φ , ψ := p | ¬ φ | ◻ φ | φ ∨ ψ {\displaystyle \varphi ,\psi :=p\ |\ \neg \varphi \ |\ \Box \varphi \ |\ \varphi \lor \psi } where p {\displaystyle p} is any propositional name. The conjunction φ ∧ ψ {\displaystyle \varphi \land \psi } may then be defined as equivalent to ¬ ( ¬ φ ∨ ¬ ψ ) {\displaystyle \neg (\neg \varphi \lor \neg \psi )} . For any modal formula φ {\displaystyle \varphi } , the formula ◊ φ {\displaystyle \Diamond \varphi } is defined by ¬ ◻ ¬ φ {\displaystyle \neg \Box \neg \varphi } . Alternatively, if the language is first defined with the diamond, then the box can be analogously defined by ◻ φ ≡ ¬ ◊ ¬ φ {\displaystyle \Box \varphi \equiv \neg \Diamond \neg \varphi } . For any propositional name p {\displaystyle p} , the formulae p {\displaystyle p} and ¬ p {\displaystyle \neg p} are considered propositional literals whilst ◻ p {\displaystyle \Box p} and ¬ ◻ p {\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.
Hilbert calculus The Hilbert calculus for logic E is built upon the one for classical propositional logic with the congruence rule (RE): A ↔ B ◻ A ↔ ◻ B {\displaystyle {\frac {A\leftrightarrow B}{\Box A\leftrightarrow \Box B}}} . Alternatively, the rule can be defined by A ↔ B ◊ A ↔ ◊ B {\displaystyle {\frac {A\leftrightarrow B}{\Diamond A\leftrightarrow \Diamond B}}} . Logics containing this rule are called congruential.
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