Logic

A Primer for Logic and Proof by Hirst H.P., Hirst J.L.

By Hirst H.P., Hirst J.L.

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1. Use the Completeness Theorem for L to show that L can prove Theorem L7. 2. Use the Completeness Theorem for L to show that L can prove Theorem L13. 3. Use the Completeness Theorem for L to show that L can prove Theorem L15. 4. Is there a proof in L of (A ∨ B) → B? 5. Is there a proof of B L A ∨ B? 6. Is there a proof of (A ∧ ¬A) L B? 7. Is there a proof in L of A ∧ ¬A? 10 Modifying L As noted in the previous section, L has some nice properties. How can we modify L, retaining soundness and completeness?

How could we adapt the first version to handle this? ” Let N (x) stand for this: ∀x(N (x) → P (x, s(x))). Example. Now let’s try a more complex example: Socrates is a man. All men have ugly feet. Socrates has ugly feet. Again, we can proceed in several ways. Let the universe be the set of all people. Let U (x) be the predicate x has ugly feet. Let M (x) be the predicate x is a man. Let s be the constant in the universe representing the man Socrates. Then the three statements above translate as follows.

Not every logically valid formula is an instance of a tautology. Indeed there are logically valid formulas that simply cannot be built using the techniques of this section. Exercises. 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.

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