160 Logic as a Tool 4.1 Axiomatic system for first-order logic We now extend the propositional axiomatic system presented in Section 2.2 to first-order logic by adding axioms and rules for the quantifiers I denote the resulting axiomatic system again by H In what follows, ∀ is regarded as the only quantifier in the language and ∃ is definable in terms of it An equivalent system can be obtained by regarding both quantifiers present in the language and adding the axiom ∃xA ↔ ¬∀x¬A 4.1.1 Axioms and rules for the quantifiers Additional axiom schemes (Ax∀1) ∀x(A(x) → B (x)) → (∀xA(x) → ∀xB (x)); (Ax∀2) ∀xA(x) → A[t/x] where t is any term free for substitution for x in A; (Ax∀3) A → ∀xA where x is not free in the formula A As an easy exercise, show that all instances of these axiom schemes are valid Additional rule We must also add the following rule of deduction, known as Generalization: A ∀xA where A is any formula and x any variable Note that x may occur free in A, but need not occur free there in order to apply that rule Note also that this rule does not read as “Assuming A is true, conclude that ∀xA is also true” but rather as “If A is valid in the given model (respectively, logically valid) then conclude that ∀xA is also valid in the given model (respectively, logically valid)” that is, the rule preserves not truth but validity (in a model, or logical validity) Respectively, the syntactic/deductive reading of the Generalization rule is: “If A is derived, then derive ∀xA.” 4.1.2 Derivations from a set of assumptions Note that if the Generalization rule is used unrestrictedly in derivations from a set of assumptions, it would derive A H ∀xA for example, which should not be derivable because A ∀xA To avoid such unsound derivations when adding the Generalization