140 Logic as a Tool As noted earlier, logical consequence in first-order logic satisfies all basic properties of propositional logical consequence Further, some important additional properties related to the quantifiers hold They will be used as rules of inference in deductive systems for first-order logic, so I give them more prominence here Theorem 112 For any first-order formulae A1 , , An , A, B , the following hold If A1 , , An |= B then ∀xA1 , , ∀xAn |= ∀xB If A1 , , An |= B and A1 , , An are sentences, then A1 , , An |= ∀xB , and ¯ , where B ¯ is any universal closure of B hence A1 , , An |= B If A1 , , An |= B [c/x] where c is a constant symbol not occurring in A1 , , An , then A1 , , An |= ∀xB (x) If A1 , , An , A[c/x] |= B where c is a constant symbol not occurring in A1 , , An , A, or B , then A1 , , An , ∃xA |= B For any term t free for substitution for x in A: (a) ∀xA |= A[t/x] (b) A[t/x] |= ∃xA The proofs of these are easy exercises using the truth definition 3.4.4 Using equality in first-order logic First-order languages (especially those used in mathematics) usually contain the equality symbol =, sometimes also called identity This is regarded as a special binary relational symbol, and is always meant to be interpreted as the identity of objects in the domain of discourse Example 113 The equality is a very useful relation to specify constraints on the size of the model, as the following examples in the first-order language with = show The sentence λn = ∃x1 · · · ∃xn ¬xi = xj 1≤i=j ≤n states that the domain has at least n elements The sentence μn = ¬λn+1 or, equivalently, μn = ∀x1 · · · ∀xn+1 xi = xj 1≤i=j ≤n+1 states that the domain has at most n elements The sentence σn = λn ∧ μn states that the domain has exactly n elements The proofs of these claims are left as an exercise (Exercise in Section 3.4.8)