213 Deductive Reasoning in First-order Logic For every n-ary functional symbol f in L and a1 , , an , b1 , , bn ∈ X : if a1 ∼ b1 , , an ∼ bn then f S (a1 , , an ) ∼ f S (b1 , , bn ) For every n-ary predicate symbol r in L and a1 , , an , b1 , , bn ∈ X : if a1 ∼ b1 , , an ∼ bn then r S (a1 , , an ) holds iff rS (b1 , , bn ) holds Definition 163 Let S be any L-structure with domain X and ∼ be a congruence in S The quotient-structure of S with respect to ∼ is the L-structure S∼ with domain X∼ = {[a]∼ | a ∈ X } where [a]∼ is the ∼-equivalence class of a and the non-logical symbols are interpreted in S∼ as follows For every n-ary functional symbol f in L and [a1 ]∼ , , [an ]∼ ∈ X∼ : f S∼ ([a1 ]∼ , , [an ]∼ ) := [f S (a1 , , an )]∼ For every n-ary predicate symbol r in L and [a1 ]∼ , , [an ]∼ ∈ X∼ : r S∼ ([a1 ]∼ , , [an ]∼ ) iff [rS (a1 , , an )] Lemma 164 The definitions above are correct, that is, independent of the representatives of the elements of X∼ Proof This is immediate from the definition of congruence Exercise 4.6.3.1 Canonical structures for first-order theories From this point onward we assume that the language L has at least one constant symbol If not, such can always be added to the language conservatively, that is, without affecting the derivations in the original language Definition 165 (Herbrand structure for a theory) Given a first-order theory Δ, we define the Herbrand structure H(Δ) for Δ as follows For every n-ary predicate symbol r in L, other than the equality, and ground terms t1 , , tn ∈ HL : rH (t1 , , tn ) iff Δ D r(t1 , , tn ) We now define the following binary relation in HL : t1 ∼Δ t2 iff Δ D t1 = t2 Lemma 166 For any first-order theory Δ, the relation ∼Δ is a congruence in H(Δ) Proof This follows from the axioms and rules for the equality in D Exercise