formal logic
Question Description
Question 1
Use truth tables to prove that the following propositional formulas have the corresponding properties. In each case, also explain (in words) why the formula has the relevant property.
(i) (p ∧ (q ∨ r)) → ((p ∧ q) ∨ r) is valid (ii) (p ∧ ¬q) ↔ (¬p ∨ q) is unsatisfiable
(iii) (p → p) → (q ∧ (q → p)) is satisfiable and is falsifiable (20)
Question 2
Prove that the formulas given in Question 1 (i) and (ii) above have the corresponding properties, by means of semantic arguments in terms of the truth values of formulas. (20)
Question 3
Prove the following theorem: Given a set of formulas U and a single formula A, U |= A iff U ∪ {¬A} is unsatisfiable. (20)
Question 4
Prove that the propositional formulas given in Question 1 (i), (ii) and (iii) above have the corresponding properties, by means of semantic tableaux. (30)
Normally, you shouldn’t use the optimized tableau algorithm described in Section 2.6.4, but you may do so for part (ii) of this question. It will make the tableau shorter and simpler.
Question 5
Prove that the formula given in Question 1 (i) above is a theorem of the Gentzen system
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