The proof is by structural induction on the formulaAx, and is left as an exercise for the reader.. 3.4.5 Logical equivalence in first-order logic Logical equivalence in first-order logic
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Theorem 116 (Equivalent replacement) For any formula A(x ) and terms s, t free for
x in A , the following holds:
|= s=t → A[s/x]↔ A[t/x].
The proof is by structural induction on the formulaA(x), and is left as an exercise for the reader
3.4.5 Logical equivalence in first-order logic
Logical equivalence in first-order logic is based on the same idea as in propositional logic:
the first-order formulaeAandBare logically equivalent if always one of them is true if and only if the other is true This is defined formally as follows
Definition 117 The first-order formulae A and B are logically equivalent, denoted A ≡
B , if for every structure S and variable assignment v in S:
S , v |= A if and only if S , v |= B.
In particular, if A and B are sentences, A ≡ B means that every model of A is a model
of B , and every model of B is a model of A
The following theorem summarizes some basic properties of logical equivalence
Theorem 118
1 A ≡ A
2 If A ≡ B then B ≡ A
3 If A ≡ B and B ≡ C then A ≡ C
4 If A ≡ B then ¬ A ≡ ¬ B , ∀ xA ≡ ∀ xB , and ∃ xA ≡ ∃ xB
5 If A1 ≡ B1and A2 ≡ B2then A1◦ A2≡ B1◦ B2where ◦ is any of ∧ , ∨ , → , ↔.
6 The following are equivalent:
(a) A ≡ B ;
(b) |= A ↔ B ; and
(c) A |= B and B |= A
Theorem 119 The result of renaming of any variable in any formula A is logically equiv-alent to A Consequently, every formula can be transformed into a logically equivalent clean formula.
Example 120 Some examples of logical equivalences between first-order formulae are
as follows.
• Any first-order instance of a pair of equivalent propositional formulae is a pair of log-ically equivalent first-order formulae For example:
¬¬∃ xQ(x, y)≡ ∃ xQ(x, y ) (being an instance of ¬¬ p ≡ p );
∃ xP(x)→ Q(x, y)≡ ¬∃ xP(x)∨ Q(x, y ) (being an instance of p → q ≡ ¬ p ∨ q ).