Express universal quantification in terms of existential quantification
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Express universal quantification in terms of existential quantification.
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- (a) Is it possible for an argument to be sound and have a false conclusion? Explainyour answer (in one or two sentences).(b) Is it possible for an argument to be invalid and have a true conclusion? Explainyour answer (in one or two sentences).(c) Explain what a tautology is.(d) Explain what semantic entailment is.(e) State the deduction theorem.(f) If A ⊨ B, then A, C ⊨ B. What is this feature of entailment known as?Prove the following predicate logic statements valid or invalid: A) ∀x(A(x) → B(x)) ∧ ∀x(A(x) ∨ ¬C(x)) ∧ ∃x(¬B(x)) → ¬∃x(¬A(x) ∧ ¬B(x) ∧ ¬C(x)) B) ∀x(A(x) → B(x)) ∧ ∃x(A(x) ∨ B(x)) → ∃x(A(x) ∧ B(x))Consider a proof system P in the style of box-proof which only have one logical connective A. The system only has one rule given by: [X] Y YAX Here, X and Y can represent any formula. Tick each that applies. In the proof system P it is possible to derive the formula ((AAC)AB)AA In the proof system P it is possible to derive AA(AA A) In the proof system P it is possible to derive the formula AAA In the proof system P it is possible to derive (AAA)^ A In the proof system P it is not possible to derive (AA B)A(BAA)