Contingency Logic

Origin. Montgomery and Routley (1966). Contingency as primitive. Δφ: φ is contingent. Non-interdefinability with necessity. Foundation for modal analysis.

Models. Kripke frames. Δφ = ◇φ ∧ ◇¬φ. Alternative: Δ primitive, □ derived. Different expressiveness.

Formalism.

Contingency operator: Δφ: φ is contingent. True at some worlds, false at others.

Standard definition: Δφ ≡ ◇φ ∧ ◇¬φ Contingent = possibly true and possibly false.

Non-contingency: ∇φ ≡ ¬Δφ ≡ □φ ∨ □¬φ Necessarily true or necessarily false.

Δ as primitive: Can we define □ from Δ? No! Δ has less discriminating power. Same Δ-theory, different □-theory possible.

Frame definability: ∇φ → □∇φ: transitive frames. Different axioms than standard modal.

Axioms (over K): Δφ → Δ¬φ (symmetric) ¬Δ⊤, ¬Δ⊥ Δ(φ ∨ ψ) → Δφ ∨ Δψ ∨ Δ(φ ∧ ψ)

Equivalences: □φ ≡ φ ∧ ∇φ (with reflexivity) ◇φ ≡ ¬□¬φ (standard)

Higher contingency: ΔΔφ: contingently contingent. Iterate for complex properties.

Model classes: Different from standard: {reflexive} not definable by Δ alone. Restricted expressiveness.

Symbols.

Metatheory. Weaker than normal modal. Axiomatizable. Decidable. Different frame classes.

Applies to. Philosophy of modality. Free will. Determinism. Modal analysis.

Limitations. Less expressive. Cannot capture reflexivity. Philosophical niche. Limited applications.

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