### Possible Worlds: An introduction to Logic and Its Philosophy

Answers to exercises on pages 48 and 55-57

Exercise on page 48
1. Modal relations between proposition-sets:

1. is inconsistent with v.
2. is inconsistent with none (i.e. is inconsistent with none of i-v).
3. is inconsistent with i, ii, iii, iv, v.
4. is inconsistent with none.
5. is inconsistent with none.

1. is consistent with i, ii, iii, iv.
2. is consistent with i, ii, iii, iv, v.
3. is consistent with none.
4. is consistent with i, ii, iii, iv, v.
5. is consistent with i, ii, iii, iv, v.

1. implies i.
2. implies none.
3. implies i, ii, iii, iv, v.
4. implies none.
5. implies iv.

1. is equivalent to none.
2. is equivalent to none.
3. is equivalent to none.
4. is equivalent to none.
5. is equivalent to iv.

2. iii.

3. {Sylvia is Diane's mother, Sylvia is female and is a parent of Diane's, Diane is a child of Sylvia who is female.}

4. {Two plus two equals four, It is raining or it is not raining, Whatever is blue is colored}

Exercise on pages 55-57
Part A

 1 P 1, 2, 5 2 Q 1, 3, 6 3 P 6, 7, 9, 10, 11, 12, 13, 14, 15 4 Q 5, 8, 9, 10, 11, 12, 13, 14, 15 5 P is actually true 1, 2, 5 6 P Q 1, 3, 4, 6, 8, 9, 11 7 Q P 1, 2, 4, 5, 7, 9, 13 8 P O Q 1, 5, 6, 9, 11, 13, 14, 15 9 P Ø Q 2, 3, 4, 7, 8, 10, 12 10 Contradictories 2, 3, 10 11 Contraries 4, 7, 8, 12 12 Subcontraries 1, 5, 6, 14 13 P Q 1, 4, 9 14 P and Q 1, 5, 6, 9, 10, 11, 12, 13, 14, 15 15 P, Q, and P Ø Q 10, 12 16 ~(P Q) and ~(Q P) 10, 12, 14, 15 17 (P Q) and (P O Q) 1, 6, 9, 11 18 (P Q) and (P Ø Q) 3, 4, 8 19 (P Q) and ~(Q P) 3, 6, 8, 11 20 (P O Q) and ~(P Q) 5, 13, 14, 15 21 (P Ø Q) and ~(P Q) 2, 7, 10, 12 22 P P 1 - 15 (i.e. every case) 23 P, Q, and (P Q) 6 24 P, Q, and (P Q) none (this is impossible) 25 (P Q) and (P Ø Q) 4 26 P ~P 3, 4, 8 27 ~P P 1, 2, 5 28 P O ~P none (an impossible case) 29 (P ~Q) and (P O Q) none (an impossible case) 30 P and ~P none (an impossible case)
 31 WD1 P: All squares have four sides. Q: Whatever is green, is red or green. 32 WD2 P: 2 + 2 = 4 Q: 5 x 7 = 57 33 WD3 P: Some triangles have five sides. Q: All aunts are females. 34 WD4 P: It is raining and it is not raining. Q: The Prime Minister of Canada knows that 2 + 2 = 7. 35 WD5 P: If anyone knows that it is raining, then it is raining. Q: It is raining. 36 WD6 P: Two liters of water mixed with two liters of alcohol yields four liters of liquid. Q: 2 + 2 = 4 37 WD7 P: Today is Tuesday. Q: Today is Tuesday and the day before Saturday. 38 WD8 P: There are exactly five women here and the women outnumber the men (exactly) by a factor of two. Q: Someone knows all the words to the national anthem. 39 WD9 P: John is the same age as Judy. Q: Judy is the same age as John. 40 WD10 P: Canada is a member of the British Commonwealth. Q: It is false that Canada is a member of the British Commonwealth. 41 WD11 P: Judy loves strawberry ice cream. Q: Judy loves at least one flavor of ice cream. 42 WD12 P: Aziz is exactly 15 years old. Q: Aziz is exactly 19 years old. 42 WD13 P: Judy loves at least one flavor of ice cream. Q: Judy loves strawberry ice cream. 44 WD14 P: It is false that Aziz is exactly 15 years old. Q: It is false that Aziz is exactly 19 years old. 45 WD15 P: He is stubborn. Q: You are dogmatic.

Part B

 46 ~(P Q) 2, 5, 7, 10, 12, 13, 14, 15 47 (~P Q) 1, 2, 3, 5, 6, 10, 14 48 (P Q) and (~P Q) 1, 3, 6

Part C

 49 How many ... ? 32 50 (P Q) and ~Q 4 [ = 4a], 8b, 9b, 11c 51 Worlds-diagram 11 with an "X" in the leftmost segment, i.e. Worlds-diagram 11a. 52 yes 53 no 54 no 55 Worlds-diagram 11 with an "X" in the rightmost segment, i.e. Worlds-diagram 11c.