Me-and-my-baby H∀ xLx, MyBaby by Propositional logic.. The following is a deductive analogue in H of the Equivalent replacement theorem 116 and a generalization of the equality axioms, s
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1 Me-and-my-baby H∀ xL(x, MyBaby) by Propositional logic.
2 Me-and-my-baby HL (MyBaby , MyBaby) by 1 and Axiom Ax ∀2.
3 Me-and-my-baby H∀ y(¬ y = Me → ¬ L (MyBaby , y)) by Propositional logic.
4 Me-and-my-baby H¬MyBaby = Me → ¬ L (MyBaby , MyBaby) by 3 and
Axiom Ax ∀2.
5 Me-and-my-baby HL (MyBaby , MyBaby) → MyBaby = Me by 4 and
Propositional logic.
6 Me-and-my-baby HMyBaby = Me QED by 2, 5, and MP.
The following is a deductive analogue in H of the Equivalent replacement theorem
116 and a generalization of the equality axioms, stating that equal terms can provably be replaced for each other in any formula
Theorem 125 (Equivalent replacement) For any formula A(x ) and terms s, t free for
x in A , the following is derivable in H:
Hs=t → A[s/x]↔ A[t/x].
The proof is by structural induction on the formulaA(x), left as an exercise
Theorem 126 (Soundness and completeness of H) Each of the axiomatic systems H
for first-order logic with and without equality is sound and complete, that is, for every first-order formula A1, , A n , C of the respective language (respectively, with or with-out equality):
A1, , A n , H C iff A1, , A n , |= C.
The proof for H with equality is outlined in Section 4.6.
References for further reading
For further discussion and examples on derivations in axiomatic systems for first-order logic, see Tarski (1965), Shoenfield (1967), Hamilton (1988), Fitting (1996), Mendelson (1997), Enderton (2001) For discussion and proofs of Church’s Undecidability Theorem,
see Shoenfield (1967), Jeffrey (1994), Ebbinghaus et al (1996), Enderton (2001), Hedman (2004), and Boolos et al (2007).
Exercises 4.1.1 Show that all instances of the axiom schemes listed in Section 4.1.1 are logically valid, and that the Generalization rule preserves validity in a structure and hence logical validity