How would you construct a proof to show that the following arguments are valid?
1. TvH
2. T→G
3. ~H ֶG
1. ~~A
2. ~A v C ֶC
1. S • T
2. (S • T) → G
3. G →(H v B) ֶ H v B
1. A →B
2. C →D
3. A v C
4. ~D ֶ B
To construct a proof to show the validity of the given arguments, we can use the rules of logic, specifically, the rules of inference and the laws of propositional logic. Here are the steps to construct proofs for each argument:
Argument 1:
1. TvH (Given)
2. T→G (Given)
3. ~H (Given)
ֶG (To be proved)
To prove that "G" is valid, we can use the law of the hypothetical syllogism (HS), which states that if we have two conditional statements, and the consequent of the first statement matches the antecedent of the second statement, then we can infer the consequent of the second statement.
Proof:
4. T (Assumption, using the law of simplification on premise 1)
5. G (Modus Ponens on T→G (premise 2) and T (step 4))
Therefore, we have proved G from the given premises.
Argument 2:
1. ~~A (Given)
2. ~A v C (Given)
ֶC (To be proved)
To prove "C" is valid, we can use the rule of disjunctive syllogism (DS), which states that if we have a disjunction (A v B), and we know one of the disjuncts is false (~A), then we can infer the truth of the other disjunct (B).
Proof:
3. ~A (Double negation on ~~(A) (premise 1))
4. C (Disjunctive syllogism on ~A v C (premise 2) and ~A (step 3))
Therefore, we have proved C from the given premises.
Argument 3:
1. S • T (Given)
2. (S • T) → G (Given)
3. G → (H v B) (Given)
ֶH v B (To be proved)
To prove "H v B" is valid, we can use the rule of hypothetical syllogism (HS) again.
Proof:
4. G (Modus Ponens on (S • T) → G (premise 2) and S • T (premise 1))
5. H v B (Modus Ponens on G → (H v B) (premise 3) and G (step 4))
Therefore, we have proved H v B from the given premises.
Argument 4:
1. A → B (Given)
2. C → D (Given)
3. A v C (Given)
4. ~D (Given)
ֶB (To be proved)
To prove "B" is valid, we can use the law of disjunctive syllogism (DS) again.
Proof:
5. ~C (Assumption, using the law of disjunction elimination (DE) on premise 3)
6. ~A (Assumption, using the law of disjunction elimination (DE) on premise 3)
7. ~A • ~C (Conjunction of steps 5 and 6)
8. ~D • (~A • ~C) (Conjunction of ~D (premise 4) and ~A • ~C (step 7))
9. ~D • (~C • ~A) (Commutation of step 8)
10. ~(D • (C • A)) (De Morgan's law on step 9)
11. ~(~(D • (C • A))) (Double negation on step 10)
12. D • (C • A) (Double negation elimination on step 11)
13. (C • A) • D (Associativity on step 12)
14. C • (A • D) (Associativity on step 13)
15. A • D (Simplification on step 14)
16. B (Modus Ponens on A → B (premise 1) and A • D (step 15))
Therefore, we have proved B from the given premises.
These proofs demonstrate the validity of the provided arguments using logical rules and the laws of propositional logic.