Question 1

Determine whether the following argument is valid or invalid. If the argument is invalid, upload a counterexample world. if it is valid, please upload an informal proof of the conclusion from the premises.

1. Larger (a, b)
2. Small(a) v Medium(b)

/:.larger(a)

Proof

Valid

(Small (a) ∧ Smaller (a, b)) v (Large (b) ∧ Smaller (a, b)), c = b} |= Smaller (a, c) ∧ c = b

1. (Small (a) ∧ Smaller (a, b)) v (Large (b) ∧ Smaller (a, b))

2. c = b

3. Small (a) ∧ Smaller (a, b)

4.Smaller(a, b)                           ∧Elim: 3

5. Large (b)       ∧ Smaller (a, b)

6. Smaller(a, b)                            ∧Elim: 5

 7. Smaller(a, b)                          ∨Elim: 1, 3-4. 5-6

 8. c = c Refl=

 9. b = c Ind. Id: 8, 2

 10. Smaller(a, c) Ind. Id.: 7, 9

11. Smaller(a, c) ∧ c= b ∧Intro: 10, 2

Question 2

Determine whether the following argument is valid or invalid. If the argument is invalid, upload a counterexample world. if it is valid, please upload an informal proof of the conclusion from the premises.

1. Between(a, b, c)
2. LeftOf(c, a)

/:.LeftOf(a,b)BackOf(b,a)

Proof

Valid

Between (a, b, c) → {a → b, a → (b → c), b → (c → d) } |= a → d

1. a → b

2. a → (b → c)

3. a → (b → c)

 4. a [(LeftOf (c, a)]}

5. b                  →Elim: 4, 1

6. b → c                      →Elim: 4, 2

7. a → c                       →Elim: 5, 3

8. c (LeftOf(b, d))                    →Elim: 5, 6

9. a                  →Elim: 8, 7

10. a → c                     →Intro: 4-9