Polyadic Modal Logic

Origin. Venema (1991), expanding modal operators to multiple arguments. Generalized modalities. n-ary accessibility. Foundation for complex modal interactions.

Models. Kripke frames with n+1-ary relations. R(w, v₁, ..., vₙ) relates world to n-tuple. Operators take n formulas. Covers products and fusions.

Formalism.

n-ary modal operator: ◇ₙ(φ₁, ..., φₙ): n-ary diamond □ₙ(φ₁, ..., φₙ): n-ary box

Semantics: w ⊨ ◇ₙ(φ₁,...,φₙ) iff ∃v₁...vₙ. R(w,v₁,...,vₙ) ∧ v₁⊨φ₁ ∧ ... ∧ vₙ⊨φₙ

Duality: □ₙ(φ₁,...,φₙ) = ¬◇ₙ(¬φ₁,...,¬φₙ) Generalized De Morgan.

Binary modality examples: ◇₂(φ,ψ): both accessible Conditional: ψ ⇒ φ as □₂(¬ψ,φ) Until: φ U ψ as binary temporal.

Correspondence: n-ary frame conditions. More expressive than unary. Captures complex dependencies.

Composition: Product: ◇₁◇₂ as ◇₂ Fusion: ◇₁ + ◇₂ Polyadic unifies.

Arrow logic: Binary modality. Composition of transitions. Applications to processes.

Symbols.

Symbol	Unicode	Name	Meaning
◇ₙ	U+25C7	n-Diamond	n-ary possibility
□ₙ	U+25A1	n-Box	n-ary necessity
R	—	Relation	n+1-ary accessibility

Metatheory. Decidable for many cases. Bisimulation extends. Correspondence theory. Algebraic semantics.

Applies to. Arrow logic. Process composition. Complex accessibility. Unified modal framework.

Limitations. Complexity increases with arity. Axiomatization harder. Less intuitive. Specialized applications.