Practice Set 5.1

I. Prove the following theorems:
1. ∴ ~A ⊃ ~(A · B)
2. ∴ A ≡ ~ ~ A [The Law of Double Negation]
3. ∴ (A ⊃ B) ≡ (~B ⊃ ~A) [The Law of Transposition]
4. ∴ (A ∨ B) ∨ ~A

II. Find the following proofs for the following arguments.
1. ~D ⊃ C / C ⊃ (~F ⊃ D) / ~F ∴ D
2. ~(A · ~B) ∴ A ⊃ B
3. ~L ⊃ (M · N) / M ⊃ (~L ⊃ ~O) / O ∴ L
4. (R ⊃ S) · (F ⊃ W) / ~(S ∨ W) ∴ ~(F ∨ R)
5. A ∴ ~A ⊃ G
6. A ∨ B ∴ ~(~A · ~B) [Hint: you will use CD and RAA, and the order in which you make your subproofs will determine whether the proof is manageable or really long]
7. ~(~A · ~B) ∴ A ∨ B [Hint: the basic strategy is RAA, but you will need to use RAA twice more, inside the main RAA strategy]

III. If p and q are both theorems in our propositional logic, is the sentence p · q also a theorem? Explain.

IV. Show that the following argument is SL valid (hint: be very aware of what your atomic sentences are!):
Russell’s Paradox:
If the usual assumptions of intuitive set theory are all true, then the set of all sets that are not members of themselves is a member of the set of all sets that are not members of themselves if and only if the set of all sets that are not members of themselves is not a member of the set of all sets that are not members of themselves. Therefore, the usual assumptions of intuitive set theory are not all true.

Practice Set 4.3

Proofs for Practice Set 4.3

(1) 1. F ∨ F A Deduce: F


(2) 1. ~A ⊃ B A
2. C ⊃ ~D A
3. ~A ∨ C A Deduce: B ∨ ~D


(3) 1. (H ∨ ~B) ⊃ R A
2. (H ∨ ~M) ⊃ P A
3. H A Deduce: P · R


(4) 1. (D ⊃ B) · (C ⊃ D) A
2. (B ⊃ D) · (E ⊃ C) A
3. B ∨ E A Deduce: D ∨ B


(5) 1. D ∨ (E ∨ G) A Deduce: E ∨ (D ∨ G)


(6) 1. T ∨ (P · ~S) A
2. ~S ⊃ T A
3. ~P ⊃ ~T A Deduce: P ≡ T


(7) 1. D ⊃ (B ≡ A) A
2. D ∨ C A
3. ~A ⊃ ~C A Deduce: B ⊃ A

Practice Set 4.2

Proofs section of 4.2 (and all that is due for Exam 2)

Construct proofs for the following arguments.
(1) 1. A ⊃ B A Deduce: ~B ⊃ ~A

(2) 1. D ⊃ (E ⊃ F) A Deduce: E ⊃ (D ⊃ F)

(3) 1. A ⊃ B A Deduce: (~A ⊃ ~C) ⊃ (C ⊃ B)

(4) 1. (D ∨ A) ⊃ B A
2. A A Deduce: B

(5) 1. C ≡ D A
2. D A Deduce: C

Practice 4.1

Shortened Practice set 4.1 (this is the proof section for set 4.1 and all that is due for exam 2]

III. Construct proofs for the following arguments.
(1) 1. A · B A
2. C · D A Deduce: A · D

(2) 1. D ⊃ E A
2. E ⊃ F A
3. ~F A Deduce: ~D

(3) 1. A ⊃ B A
2. C · A A Deduce: B

(4) 1. D ⊃ (A · B) A
2. E ⊃ ~A A
3. D A Deduce: ~E

(5) 1. (B · A) ⊃ E A
2. ~ ~A · B A Deduce: E

(6) 1. D ⊃ (B ⊃ A) A
2. D · ~A A Deduce: ~B

(7) 1. E · (D ∨ B) A
2. (D ∨ B) ⊃ ~ ~A A Deduce: A

(8) 1. D ⊃ (B · A) A
2. ~E · D A
3. ~C ⊃ ~B A Deduce: C