## Hirst H.P., Hirst J.L.'s A Primer for Logic and Proof PDF By Hirst H.P., Hirst J.L.

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We can put together a short proof of L A. 1. N EW ∗ 2. N EW ∗ → A 3. 10. MODIFYING L 25 So far, anything provable with the new axiom is provable in the original axiom system. Also, anything provable in L must be provable in the new axiom system. Thus, the new axiom system has exactly the same theorems as L. What does this do for us? The new axiom system satisfies the completeness and soundness theorems. It’s another reasonable axiom system. In reality, it’s just L with a lemma disguised as an axiom.

D) 3 is free for x in A(x) ∨ ∀z(C(z, z) ∧ A(z, y)); A(x) is not in the scope of any quantifiers, and constants can never be captured by quantifiers anyway. 3. B(y) → ∀y(A(x, z) ∧ ∃xC(x, y)) 50 CHAPTER 2. PREDICATE CALCULUS Terms: (a) x is free for x in the formula B(y) → ∀y(A(x, z) ∧ ∃xC(x, y)); the only free occurrence of x is in the A(x, z) predicate, and x is always free for x. (b) y is not free for x in the formula B(y) → ∀y(A(x, z) ∧ ∃xC(x, y)); x occurs free in the A(x, z) predicate, and y will be captured by the ∀y quantifier.

1. Each of the following formulas is logically valid. Mark those that are instances of tautologies. (a) A(x) → (∀yB(y) → A(x)) (b) ∀x(A(x) → (∀yB(y) → A(x))) (c) A(x) → (¬B(y) ∨ B(y)) (d) ∃x(A(x) → (¬B(y) ∨ B(y)) (e) ∃x(¬¬A(x) → (¬B(y) ∨ B(y)) (f) ∃y∃x(¬¬A(x) → (¬B(y) ∨ B(y)) 2. Each of the following formulas is logically valid. Mark those that are instances of tautologies. (a) C(x, y) → C(x, y) (b) ∀x∃y(C(x, y) → C(x, y)) (c) ∀x∃yC(x, y) → ∀x∃y¬¬C(x, y) (d) ∀x∃yC(x, y) → ¬¬∀x∃yC(x, y) 48 CHAPTER 2.