Deductive Reasoning in First-order Logic 161 rule to the definition of derivation in H from a set of assumptions Γ, we include the proviso: if the formula A is already in the sequence and x does not occur free in any of the formulae of Γ, then ∀xA can be added to the sequence More generally, we can extend the Generalization rule to work with assumptions on the left, as follows: A1 , , A n H A ∀xA1 , , ∀xAn H ∀xA In the case where x does not occur free in A1 , , An , the vacuous quantification on the left can be achieved by using Axiom (Ax∀3), thus justifying the definition above With the refined definition of derivations from a set of assumptions, the Deduction Theorem still holds for the extension of H to first-order logic The proof of this claim is left as an exercise Here is an example of derivations in H Check that the rules for the quantifiers have been applied correctly Example 122 Derive ∀x(Q → P (x)), ∃x¬P (x) H ¬Q, where x is not free in Q Eliminating ∃, we are to derive ∀x(Q → P (x)), ¬∀x¬¬P (x) H ¬Q derived in the Propositional H H P (x) → ¬¬P (x) H ∀x(P (x) → ¬¬P (x)) from and Generalization by Axiom (Ax∀1) ∀x(P (x) → ¬¬P (x)) H ∀xP (x) → ∀x¬¬P (x) H ∀xP (x) → ∀x¬¬xP (x) by 2, 3, Deduction Theorem, and Modus Ponens ∀x(Q → P (x)) H ∀xQ → ∀xP (x) by Axiom (Ax∀1) by and Deduction Theorem ∀x(Q → P (x)), ∀xQ H ∀xP (x) Q H ∀xQ by Axiom (Ax∀3) by 6,7 and Deduction Theorem ∀x(Q → P (x)), Q H ∀xP (x) H ¬∀x¬¬P (x) → ¬∀xP (x) by and contraposition (derived in the propositional H) by and contraposition 10 ∀x(Q → P (x)), ¬∀xP (x) H ¬Q by 9, 10, and Deduction Theorem 11 ∀x(Q → P (x)), ¬∀x¬¬P (x) H ¬Q 4.1.3 Extension of the axiomatic system H with equality Recall that the equality symbol = is a special binary relational symbol, always to be interpreted as the identity of objects in the domain of discourse For that standard meaning of the equality to be captured in the axiomatic system, the following additional axioms for the equality, mentioned in Section 3.4.4, are needed (though, as noted there, it is not sufficient to express the claim that it is the identity relation), where all variables are implicitly universally quantified Axioms for the equality (Ax= 1) x = x; (Ax= 2) x = y → y = x;