Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 188

164 Logic as a Tool 4.1.2 Prove the logical correctness of the inference rule A1 , , An A ∀xA1 , , ∀xAn ∀xA by showing that if A1 , , An |= A then ∀xA1 , , ∀xAn |= ∀xA 4.1.3 Prove the Deduction Theorem for the extension of H to first-order logic, with respect to the definition of derivations from a set of assumptions given in the text For each of the following exercises on derivations in the axiomatic system H, you may use the Deduction Theorem 4.1.4 Derive the following deductive consequences in H, where P is a unary predicate (a) H ∀xP (x) → ∀yP (y ) (b) H ∃xP (x) → ∃yP (y ) (c) ∀xA(x) H ¬∃x¬A(x) (d) ¬∃x¬A(x) H ∀xA(x) (e) ∀x∀yA(x, y ) H ∀y ∀xA(x, y ) (f) ∃x∃yA(x, y ) H ∃y ∃xA(x, y ) 4.1.5 Suppose x is not free in Q Prove the validity of the following logical consequences by deriving them in H (a) ∀x(P (x) ∨ Q) H ∀xP (x) ∨ Q (b) ∀xP (x) ∨ Q H ∀x(P (x) ∨ Q) (c) ∃x(P (x) ∧ Q) H ∃xP (x) ∧ Q (d) ∃xP (x) ∧ Q H ∃x(P (x) ∧ Q) (e) ∃xP (x) → Q H ∀x(P (x) → Q) (f) ∀x(Q → P (x)), Q H ∀xP (x) (g) ∃x(Q → P (x)), Q H ∃xP (x) (h) ∃x(P (x) → Q), ∀xP (x) H Q (i) ∃x(¬P (x) ∨ Q) H ∀xP (x) → Q (j) ∃x(P (x) ∨ ¬Q) H Q → ∃xP (x) 4.1.6 Determine which of the following logical consequences hold by searching for a derivation in H For those that you find not derivable in H, consider A and B as unary predicates and look for a counter-model, that is, a structure and assignment in which all premises are true while the conclusion is false (a) ∀xA(x) ∧ ∀xB (x) |= ∀x(A(x) ∧ B (x)) (b) ∀xA(x) ∨ ∀xB (x) |= ∀x(A(x) ∨ B (x)) (c) ∀x(A(x) ∧ B (x)) |= ∀xA(x) ∧ ∀xB (x) (d) ∀x(A(x) ∨ B (x)) |= ∀xA(x) ∨ ∀xB (x) (e) ∀xA(x) → ∀xB (x) |= ∀x(A(x) → B (x))