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 that the given arguments are valid, we can use the rules of logical inference. The goal is to derive the conclusion from the premises using valid reasoning steps. I will explain how to approach each argument individually.
1. To prove the argument:
1. TvH
2. T→G
3. ~H ֶ G
We can use a proof by cases and conditional proof to show the validity.
Step-by-step proof:
1. Assume T (for proof by cases).
2. Using premises 1 and 2, we can infer G by modus ponens.
3. Discharge the assumption made in Step 1.
4. Assume ~T (for proof by cases).
5. Using premises 1 and 3, we can infer H by disjunctive syllogism.
6. Using premises 2 and H (derived in Step 5), we can infer G by modus ponens.
7. Discharge the assumption made in Step 4.
8. Using proof by cases, we have shown that in both cases, G is true, satisfying the conclusion.
Therefore, the argument is valid.
2. To prove the argument:
1. ~~A
2. ~A v C ֶ C
We can use proof by contradiction and disjunctive syllogism to prove the validity.
Step-by-step proof:
1. Assume ~C (for proof by contradiction).
2. Using premise 2 and ~C, we can infer ~A by disjunctive syllogism.
3. Using premise 1 and ~~A, we can infer A by double negation.
4. Contradiction: We derived both A and ~A.
5. Conclude that ~C is false (negation of the assumption made in Step 1).
6. Infer C by disjunctive syllogism using premise 2.
Therefore, the argument is valid.
3. To prove the argument:
1. S • T
2. (S • T) → G
3. G → (H v B) ֶ H v B
We can use modus ponens to prove the validity.
Step-by-step proof:
1. Using premises 1 and 2, we can infer G by modus ponens.
2. Using premises 3 and G (derived in Step 1), we can infer H v B by modus ponens.
Therefore, the argument is valid.
4. To prove the argument:
1. A→B
2. C→D
3. A v C
4. ~D ֶ B
We can use disjunctive syllogism and modus tollens to prove the validity.
Step-by-step proof:
1. Using premises 3 and 1, we can infer B by disjunctive syllogism.
2. Using premises 3 and 2, we can infer D by disjunctive syllogism.
3. Contradiction: We derived both ~D and D.
4. Conclude that ~C, negating the assumption made in Step 2.
5. Infer B by modus tollens using ~D (premise 4) and ~C (derived in Step 4).
Therefore, the argument is valid.