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.