Below you will find help with selected exercises from the book.

8-1,10
8-2,10
8-4,4
8-5,4
8-8,4
8-10,9
8-11,7
8-20,3

8-1, 10. Wherever there are snakes, there are frogs.

All places having snakes are places having frogs. Let this example remind you to keep an eye open for the suffix "-ever": It should make you think of "every," that is, "all."

8-2, 10. I've had days like this before.

Some prior days are days like this day; or, Some days I've had are days like today. Claims that something exists, or has existed, translate into I- or O-claims, claims that begin with "some." This shouldn't surprise you: If "some" in logic means "at least one," that word makes the best way to indicate existence.

8-4, 4. Find the converse of "Some Kurds are not Christians."

Some Christians are not Kurds; not equivalent to the original claim. When taking this converse, or doing anything else to an O-claim, remember that the word "not" is part of the frame of the claim, not part of its content. The "not" stays where it is as you move terms around.

8-5, 4. Find the contrapositive of "No students who did not score well on the exam are students who were admitted to the program."

No students who were not admitted to the program are students who scored well on the exam; not equivalent. A note on the complementary terms here: Because our universe of discourse is obviously "students," all the complementary term has to do is name those students not named by the original term, not all people who didn't score well.

8-8, 4

1. No members of the club are people who took the exam.
2. Some people who did not take the exam are members of the club.

The second claim becomes "Some members of the club are not people who took the exam." It may help to add a few words about strategy. You look at the original claims and notice that you have to rewrite the phrase "people who did not take the exam" so as to get the "not" out of it. That should make you think: complementary term. Only one procedure, contraposition, produces a complementary term in a claim's subject. And you can't use that here, with an I-claim. So move the phrase into the predicate by taking the converse: "Some members of the club are people who did not take the exam." Now take the obverse, which always gives a complementary term in the predicate, and you get "Some members of the club are not people who took the exam."

8-10, 9.

1. Not everybody who is enrolled in the class will get a grade. (True)
2. Some people who will not get a grade are enrolled in the class.

True. Your first strategy is to make these corresponding claims. This means that (a) needs to be put into standard form, and the subject term of (b) needs to be changed into its complementary term. Notice that (a) contradicts the claim "All people enrolled in the class will get a grade." So it's the contradictory of that: "Some people enrolled in the class are not people who will get a grade."

Now onward to (b). Get that problem term into the predicate by conversion: "Some people enrolled in the class are people who will not get a grade." Clean it up with obversion. "Some people enrolled in the class are not people who will get a grade." Because the two claims translate into the same thing, (b) is true

8-11, 7. All halyards are lines that attach to sails. Painters do not attach to sails, so they must not be halyards.

Valid. Two comments are worth making about this one. First, it illustrates the need to put syllogisms into standard form. Separate the second premise from the conclusion. Make the second premise, "No painters are lines that attach to sails"; make the conclusion, "No painters are halyards."

Second, think of what the conclusion's diagram will look like: As an E-claim, it will have a shaded lens-shaped area between the circles for P (painters) and H (halyards). Because the completed diagram has that shaded area, the argument is valid.

8-20, 3. This is not the best of all possible worlds, because the best of all possible worlds would not contain mosquitoes, and this world contains plenty of mosquitoes!

Valid. First, specify the terms and assign abbreviations to them, for instance as follows:

T = worlds identical to this one (i.e., this world).

B = worlds that are the best of all possible worlds.

M = worlds that contain mosquitoes.

Then identify the conclusion, which is the first clause of this compound sentence: No T are B. Make sure the two subsequent clauses are put into standard form with the same terms: No B are M; All T are M. As a syllogism, the argument then says:

No B are M.

All T are M.

No T are B.

If you are using the rules method, circle the distributed terms. In this syllogism, every term is distributed except the M in the second premise. You will see that one premise is a negative claim, as the conclusion also is (rule 1); that the middle term is M and is distributed in the first premise (rule 2); that T is distributed in the conclusion and also in the second premise, and that B is distributed in the conclusion and also in the first premise (rule 3).