Section # 2: Use the eight rules of inference to prove the validity of the following arguments 7. 1. (R v S ) É ( E É Z ) 2. ( R v B ) É ( G É ~ H ) 3. ( R v U ) É (E v G ) 4. R // Z v ~ H 7. 1. (R v S ) É ( E É Z ) 2. ( R v B ) É ( G É ~ H ) 3. ( R v U ) É (E v G ) 4. R // Z v ~ H
Added by Emily W.
Step 1
The goal is to derive \( Z \lor \sim H \) from the premises provided. Given Premises: 1. \( (R \lor S) \rightarrow (E \rightarrow Z) \) 2. \( (R \lor B) \rightarrow (G \rightarrow \sim H) \) 3. \( (R \lor U) \rightarrow (E \lor G) \) 4. \( R \) Show more…
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Use propositional logic to prove the validity of the arguments in Exercises 25–33. These will become additional derivation rules for propositional logic, summarized in Table 1.14. 25. (P ∨ Q) ∧ P' → Q 26. (P → Q) → (Q' → P') 27. (Q' → P') → (P → Q) 28. P → P ∧ P 29. P ∨ P → P (Hint: Instead of assuming the hypothesis, begin with a version of Exercise 28; also make use of Exercise 27.) 30. [(P ∧ Q) → R] → [P → (Q → R)] 31. P ∧ P' → Q 32. P ∧ (Q ∨ R) → (P ∧ Q) ∨ (P ∧ R) (Hint: First rewrite the conclusion.) 33. P ∨ (Q ∧ R) → (P ∨ Q) ∧ (P ∨ R) (Hint: Prove both P ∨ (Q ∧ R) → (P ∨ Q) and P ∨ (Q ∧ R) → (P ∨ R); for each proof, first rewrite the conclusion.) TABLE 1.14 More Inference Rules From Can Derive Name/Abbreviation for Rule P → Q, Q → R P → R [Example 16] Hypothetical syllogism — hs P ∨ Q, P' Q [Exercise 25] Disjunctive syllogism — ds P → Q Q' → P' [Exercise 26] Contraposition — cont Q' → P' P → Q [Exercise 27] Contraposition — cont P P ∧ P [Exercise 28] Self-reference — self P ∨ P P [Exercise 29] Self-reference — self (P ∧ Q) → R P → (Q → R) [Exercise 30] Exportation — exp P, P' Q [Exercise 31] Inconsistency — inc P ∧ (Q ∨ R) (P ∧ Q) ∨ (P ∧ R) [Exercise 32] Distributive — dist P ∨ (Q ∧ R) (P ∨ Q) ∧ (P ∨ R) [Exercise 33] Distributive — dist
Supreeta N.
Instructions Choose two of the arguments below and write a direct proof using the eight rules of inference introduced in section 8.1 of the textbook. You can do argument 1 or argument 2, but not both, then any of arguments 3-6. Note that commas are used to separate the premises from each other. 1. ~M, (~M • ~N) → (Q → P), P → R, ~N, therefore, Q → R 2. ~F → ~G, P → ~Q, ~F v P, (~G v ~Q) → (L • M), therefore, L 3. ~(Z v Y) → ~W, ~U → ~(Z v Y), (~U → ~W) → (T → S), S → (R v P), [T → (RvP)] → [(~R v K) • ~K], therefore, ~K 4. (S v U) • ~U, S → [T • (F v G)], [T v (J • P)] → (~B • E), therefore, S • ~B 5. ~X → (~Y → ~Z), X v (W → U), ~Y v W, ~X • T, (~Z v U) → ~S, therefore, (R v ~S) • T 6. (C → Q) • (~L → ~R), (S → C) • (~N → ~L), ~Q • J, ~Q → (S v ~N), therefore, ~R Natural deduction is so called because it is a model for how we naturally reason. This often comes as a surprise to students because all of the symbols seem anything but natural. The symbols, however, allow us to focus on the form of the argument without getting bogged down by content. Recall that each sentence letter represents a simple sentence in English. After writing your direct proofs, construct a translation key for your argument by assigning each letter a simple sentence, and use that key to fill in the content of the argument.
Sri K.
Use the rules of inference and the laws of propositional logic to prove that each argument is valid. Number each line of your argument and label each line of your proof "Hypothesis" or with the name of the rule of inference used at that line. If a rule of inference is used, then include the numbers of the previous lines to which the rule is applied.
Vincenzo Z.
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