Possible Worlds: An introduction to Logic and Its Philosophy
Copyright © Raymond Bradley and Norman Swartz, 2010
Answers to selected exercises on pages 2728

 Let "A" stand for the proposition that Canada is north of Mexico.
Translate each of the following expressions into English prose.
 A
Canada is north of Mexico.
 ~A
It is false that Canada is north of Mexico.
 A
In some possible world Canada is north of Mexico.
 ~A
In some possible world it is false that Canada is north of Mexico.
 ~A
It is false that in some possible world Canada is north of Mexico.
 ~~A
It is false that in some possible world it is false that Canada is north of Mexico.
 A
In all possible worlds Canada is north of Mexico.
 ~A
In all possible worlds it is false that Canada is north of Mexico.
 ~A
It is false that in all possible worlds Canada is north of Mexico.
 ~~A
It is false that in all possible worlds it is false that Canada is north of Mexico.
 A
It is contingent that Canada is north of Mexico.
 ~A
It is contingent that it is false that Canada is north of Mexico.
 ~A
It is false that it is contingent that Canada is north of Mexico.
 ~~A
It is false that it is contingent that it is false that Canada is north of Mexico.
 A
It is noncontingent that Canada is north of Mexico.
 ~A
It is noncontingent that it is false that Canada is north of Mexico.
 ~A
It is false that it is noncontingent that Canada is north of Mexico.
 ~~A
It is false that it is noncontingent that it is false that Canada is north of Mexico.
 For each of the cases (c) through (r) above, say whether the proposition
expressed is true or false.
 T
 T
 F
 F
 F
 F
 T
 T


 T
 T
 F
 F
 F
 F
 T
 T

 Letting "A" now stand for the proposition that all squares have four
sides, say for each of the expressions (a)  (r) in question 1, whether the
proposition is true or false.
 T
 F
 T
 F
 F
 T
 T
 F
 F


 T
 F
 F
 T
 T
 T
 T
 F
 F

 Question:
(i) Explain why "P" is not to be translated
as "P is noncontingently true" but as "it is noncontingent that P is true."
(ii) Find a proposition of which it is true that it is noncontingent that it is
true, but of which it is false that it is noncontingently true.
Answer:
(i) The expression "P is noncontingently true" means
"P is noncontingent and P is true." Thus the expression "P is
noncontingently true" would be the translation into English of
"P & P" which is not the expression
under consideration.
(ii) The question asks that we find a proposition of which two claims are
true, namely:
(a) it is true that it is noncontingent that it is true,
i.e., P
and
(b) it is false that it is noncontingently true,
i.e., ~(P & P).
In short, what we are looking for is a proposition that is both (a)
noncontingent but (b) not both noncontingent and true. Clearly, then, the
soughtfor proposition is necessarily false. For example, that some
square has six sides (or, that 2 + 2 = 7, or that Sally's
sister is an only child, etc.).
 Say of each of the following which is true and which is false. (Note:
it is actually true that some cows are infertile.)
s. It is contingent that some cows are infertile. 

i.e. C 

True 
t. It is contingent that it is not the case that some cows are infertile. 

i.e. ~C 

True 
u. It is contingently true that some cows are infertile. 

i.e. C & C 

True 
v. It is contingently false that some cows are infertile. 

i.e. C & ~C 

False 
 Answers:
 T
 T
 T
 F
 Question: Explain the difference in meaning in the two phrases
"noncontingently true" and "not contingently true".
Answer: "Noncontingently true" means "noncontingent and true", i.e.
necessarily true (or, equivalently, true in all possible worlds).
"Not contingently true" means "~(contingent and true)", i.e. not both
contingent and true. There are three different kinds of propositions
which satisfy this latter description (i.e. which are not both contingent
and true). They are (a) propositions which are contingent and false; (2)
propositions which are noncontingent and true; and (3) propositions which
are noncontingent and false.

