Logic of Actuality

Axioms

Non-modal Axioms for the Actuality Operator

A Modally Fragile Axiom

We take only the instances of the universal closures of the following as axioms:

• 𝒜φ → φ

This axiom can't be necessitated and so its modal closures are not axioms and should not be derivable. We mark this axiom and any theorem derived from it with a ★. Though we take the universal closures of the above as axioms, we don't need to take the actuality closures (i.e., instances prefaced with any string of actuality operatores) because these can be derived.

Non-modal Axioms for Actuality that Are Necessitatable

We take the all the closures of all the instances of the following axioms:

• 𝒜¬φ ≡ ¬𝒜φ
• 𝒜(φ → ψ) ≡ (𝒜φ → 𝒜ψ)
• 𝒜∀αφ ≡ ∀α𝒜φ
• 𝒜φ ≡ 𝒜𝒜φ

Modal Axioms for the Actuality Operator

We take the all the closures of all the instances of the following axioms:

• 𝒜φ → □𝒜φ
• □φ ≡ 𝒜□φ

Metarule: The Rule of Actualization

Rule of Actualization (RA):

• If Γ ⊢ φ, then 𝒜Γ ⊢ 𝒜φ

When Γ is empty, the rule state:

• If ⊢ φ, then ⊢ 𝒜φ