Possible Worlds: An introduction to Logic and Its Philosophy

Answers to exercises on page 306.

EXERCISE #1.

 All forms of  "A ∨ (B ● C)"  arranged in order of length. All forms of  "(B ∨ C) ∨ (B ● C)"  arranged in order of length. P length 1 P P ∨ Q length 3 P ∨ Q P ∨ (Q ● R) length 7 P ∨ (Q ● R) (P ∨ Q)  ∨ R — length 11 (P ∨ Q) ∨ (P ● Q) (P ∨ Q) ∨ (P ● R) (P ∨ Q) ∨ (R ● Q) (P ∨ Q) ∨ (R ● S)

The first sentence shares all of its forms with the second sentence (see the first three lines in the table above.) The second sentence has five forms (see the last five lines of the table above) that are not shared with the first sentence. The specific form of the first sentence is "P  (Q ● R)"; the specific form of the second sentence is "(P  Q)  (P ● Q)".

EXERCISE #2.

 All forms of  "(A ∨ ~B) ⊃ (A  ≡  ~B)"  arranged in order of length. P 1 @ length 1 P ⊃ Q 1 @ length 3 P ⊃ (Q  ≡  R) (P ∨ Q) ⊃ R 2 @ length 7 P ⊃ (Q  ≡  ~R) (P ∨ ~Q) ⊃ R 2 @ length 8 (P ∨ Q) ⊃ (P  ≡  Q) (P ∨ Q) ⊃ (P  ≡  R) (P ∨ Q) ⊃ (R  ≡  Q) (P ∨ Q) ⊃ (R  ≡  S) 4 @ length 11 (P ∨ Q) ⊃ (P  ≡  ~R) (P ∨ Q) ⊃ (R  ≡  ~S) (P ∨ ~Q) ⊃ (P  ≡  R) (P ∨ ~Q) ⊃ (R  ≡  S) 4 @ length 12 (P ∨ ~Q) ⊃ (P  ≡  ~Q) (P ∨ ~Q) ⊃ (P  ≡  ~R) (P ∨ ~Q) ⊃ (R  ≡  ~Q) (P ∨ ~Q) ⊃ (R  ≡  ~S) 4 @ length 13