Possible Worlds: An introduction to Logic and Its Philosophy

1. Yes. There is a logically possible world in which Frederick has reached the age of 21 years after only 5 birthdays. It is a world in which Frederick is born on Leap Day (Feb. 29). It is not the actual world, however, since Frederick is a fictional character (and hence does not exist in the actual world).

2. Yes. A square house built at the North Pole would have all its walls facing south. Again, we are not describing the actual world, since there is no house at the North Pole.

3. No. Epimenides can certainly utter the sentence "Everything that Cretans say is false". But the proposition he is asserting in uttering that sentence cannot be true if he, himself, is a Cretan.

4. Yes.

5. No. Any two of these three propositions can be true (together); but not all three.

6. No. Statements of arithmetic are necessarily true or necessarily false. (See page 20, section 4.)

7. Yes. The Pope (like anyone else) can believe a necessary falsehood.

8. No. Only propositions which are true can be known (to be true). No one – neither the Pope nor anyone else – can know that 2 + 2 is not equal to 4.

9. Yes. Although very unlike ours, there is a possible world in which no objects are subject to the law of gravitation. The ancients (in this world) believed that the tiny parts of material objects are held together by hooks and eyes. Even though their theories were wrong (about this world), such a(n imagined) world is not self-contradictory; hence it is logically possible.

10. No. While there can be a mountain which is higher than every other mountain, there cannot be mountain which is higher than every, since that would entail its being higher than itself!

1. The answer to this question depends on how one interprets "disappears". A grin is a certain disposition (arrangement or placement) of the lips. If the rest of the cat disappears, but the lips remain, then what is being described is logically possible, but it is not physically possible. But if by "disappears" one means that the lips, too, disappear, then the grin cannot remain: the situation is logically impossible. And whatever is logically impossible is physically impossible.

2. The situation described is logically possible, but not physically possible.

3. Provided that Nat Bartlett is an ordinary human being (who never had more than two arms), and who has not grown a new arm since losing his right arm, and is not wearing a prosthetic arm when he sits down at the table, the situation described is self-contradictory and hence logically impossible. (My own personal guess is that Eugene O'Neill, by the time he wrote the later scene, had simply forgotten what he had written some six pages earlier. I doubt that he knowingly, or intentionally, gave self-contradictory stage directions.)