1. Solutions
Homework 4
Problems 18-20

2. Problem 18
Give a proof of a = c from the premises a = b and b = c using only the indiscernability of identicals.
Indiscernability of Identicals: If a = b, then whatever we can say about a must be true of b as well

3. Problem 18 Continued Suppose that a = b and b = c.
If a = b, then, anything we can say about b must be true of a as well.
Since we know that b = c, a = c.
Therefore, a = c.

4. Problem 19 1. Premise: LeftOf(a, b); Conclusion: RightOf (b, a)
Because LeftOf and RightOf are complimentary, the conclusion follows.
We know this because there is no way of constructing a world in which LeftOf(a,b) is true and RightOf(b, a) is false.

5. Problem 19 Continued
2. Premises: LeftOf(a, b), b = c; Conclusion: RightOf(c, a).
If b = c, anything we can say about b must be true of c as well, so we know
LeftOf(a, c) is true.
Because LeftOf(a, c) is true, RightOf(c, a) must be true as well.

6. Problem 19 Continued
3. Premises: LeftOf(a, b), RightOf(c, a); Conclusion: LeftOf(b, c)
If we draw this on a sheet of paper or use Tarski’s world, we see that both premises could be true and the conclusion false because
c could be between a and b.
Two possibilities:
a b c (conclusion is true)
a c b (conclusion is false)
A conclusion is not a logical consequences of the premises if there is at least one case where the premises could be true and the conclusion false.

7. Problem 19 Continued
4. Premises: BackOf(a, b), FrontOf(a, c); Conclusion: FrontOf(b, c)
Because BackOf(a, b) means the same thing as FrontOf(b, a)
and because if FrontOf(b, a) and FrontOf(a, c)
we know that FrontOf(b, c)

8. Problem 19 Continued
5. Premises: Between(b, a, c), LeftOf(a, c); Conclusion: LeftOf(a, b)
Recall the definition of Between…
If a and c are in adjacent rows and if we draw a line from the midpoint of c to the midpoint of a, b is between if the line passes through it.

9. Problem 19 Continued Between(b, a, c) is true a LeftOf(a, c) is true
LeftOf(a, b) is false
So LeftOf(a, b) does not follow from our premises.
Remember: Our conclusion does not follow if there is at least one case in which the premises are true but the conclusion can be false. a b c

10. Files for Problem 19 2-19-3.WLD 2.19-5.WLD
Right click and choose Save As to download.

11. Problem 20
1. Folly was Claire’s disk at 2 pm.
2. Silly was Max’s disk at 2 pm.
3. Silly was not Claire’s disk at 2 pm.
Does 3 from 1 and 2?
No. Consider: Owns(Claire, Silly, 2:00) and Owns(Claire,Folly, 2:00).
(Nothing says she can’t own two disks at the same time).

12. Problem 20 Does 2 follow from 1 and 3? Owns(Claire, Folly, 2:00)
1. Folly was Claire’s disk at 2 pm.
2. Silly was Max’s disk at 2 pm.
3. Silly was not Claire’s disk at 2 pm.
Does 2 follow from 1 and 3?
Owns(Claire, Folly, 2:00)
~ Owns(Claire,Silly, 2:00)
Do either of the propositions tell us anything about Max?
No, because there is no rules that says that every student must own a disk.

13. Problem 20 Does 1 follow from 2 and 3? Owns(Max, Silly, 2:00)
1. Folly was Claire’s disk at 2 pm.
2. Silly was Max’s disk at 2 pm.
3. Silly was not Claire’s disk at 2 pm.
Does 1 follow from 2 and 3?
Owns(Max, Silly, 2:00)
~ Owns(Claire,Silly, 2:00)
Do either of the propositions tell us anything about Claire owning Folly?
2 does not follow because there’s no rule that says every disk must be owned by someone.