Chapter 8: Identity and Definite Descriptions

Atomic formulae of 𝐿₌

All atomic formulae of 𝐿₂ are atomic formulae of 𝐿₌. Furthermore, if s and t are variables or constants, then s = t is an atomic formula of 𝐿₌.

Formulae of 𝐿₌

• All atomic formulae of 𝐿₌ are formulae of 𝐿₌.

• If Φ and Ψ are formulae of 𝐿₌, then ¬Φ, (Φ ∧ Ψ), (Φ ∨ Ψ), (Φ → Ψ), and (Φ ↔ Ψ) are formulae of 𝐿₌.

• If v is a variable and is a formula of 𝐿₌, then ∀vφ and ∃vφ are formulae of 𝐿₌

Satisfaction in 𝐿₌

As in the definition of satisfaction in 𝐿₂ with the additional clause:

If provable by natural deduction, they are valid.

Assume that φ and all elements of Γ are 𝐿₌-sentences. then Γ⊢φ if and only if Γ ⊨ φ.