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

Answers to exercises on pages 30, 35 and 40-41

Exercise on page 30

Answer: No. Suppose that G and H are contraries of one another. Then (according to the definition of "contrariety" [see top of page 29]) they cannot both be true (i.e. ~(G & H)) but they can both be false (i.e. (~G & ~H)). But contradictories cannot both be false (that is, there is no possible world in which they are both false; in each possible world, they have opposite truth-values). Thus no propositions which are contraries of one another are also contradictories of one another.

Exercise on page 35
1. "Q is a false implication of P" means that Q is false and that Q is an implication of P.
In symbols:
~Q & (P Q)
"It is false that P implies Q" means that P does not imply Q.
In symbols:
~(P Q)
2. Let
"J" = "Jean Chretien [the former Prime Minister of Canada] has more than 29 children."
"K" = "Jean Chretien [the former Prime Minister of Canada] has more than 18 children."
K is a false implication of J. That is, K is false, and J implies K, i.e.
~K & (J K)
3. Let
"M" = "Jean Chretien owns a bicycle."
"N" = "Margaret Thatcher likes to arrange flowers."
It is false that the former implies the latter, i.e.
~(M N)
Exercises on pages 40-41

1.
 Inconsistency a is inconsistent with iv c is inconsistent with i c is inconsistent with ii c is inconsistent with iii c is inconsistent with iv c is inconsistent with v Consistency a is consistent with i a is consistent with ii a is consistent with iii a is consistent with v b is consistent with i b is consistent with ii b is consistent with iii b is consistent with iv b is consistent with v d is consistent with i d is consistent with ii d is consistent with iii d is consistent with iv d is consistent with v e is consistent with i e is consistent with ii e is consistent with iii e is consistent with iv e is consistent with v Implication a implies i a implies ii a implies iii b implies i c implies i c implies ii c implies iii c implies iv c implies v d implies i e implies i e implies v Equivalence a is equivalent to iii b is equivalent to i d is equivalent to i
2a.   A is inconsistent with B.

2b.   C is consistent with D.

2c.   E is self-inconsistent.

3a.   Inconsistency is a modal relation: it has to do with the way in which the truth-values of a pair of propositions are distributed across the members of the sets of all possible worlds. It is not a feature that is restricted to (or unique to) the actual world.

3b.   Implication is a modal relation: it has to do with the way in which the truth-values of a pair of propositions are distributed across the members of the sets of all possible worlds. It is not a feature that is restricted to (or unique to) any particular world (e.g. to the world described in Time Enough for Love).