PART I (9 points): Perform the following operations.
1. Change the quality but not the quantity of the proposition “Some movies are not comedies”. _____________________________________________________________________
ANSWER: “Some movies are not comedies” is an O proposition. Because it is an O, it is negative in quality and particular in quantity. You are asked to change the quality but not the quantity; in other words, your answer will leave the quantity (particular) the same but change the quality from negative to affirmative. The affirmative particular proposition is I, so the answer is the I proposition “Some movies are comedies.”
2. Change the quantity but not the quality of the proposition “All WVC students are logical thinkers”. ______________________________________________________
ANSWER: “All WVC students are logical thinkers” is an A proposition. Because it is an A, it is affirmative in quality and universal in quantity. You are asked to change the quantity but not the quality; in other words, your answer will leave the quality (affirmative) the same but change the quantity from universal to particular. The affirmative particular proposition is I, so the answer is the I proposition “Some WVC students are logical thinkers.”
3. Change both the quality and the quantity of the proposition “No politicians are honest”. ______________________________________________
ANSWER: “No politicians are honest” is an E proposition. Because it is an E it is negative in quality and universal in quantity. You are asked to change both the quantity and the quality; in other words, your answer will change the universal negative to the particular affirmative. The particular affirmative proposition is I, so the answer is the I proposition “Some politicians are honest.”
Part II (16 points): In the following problems you are given a statement, its truth value in parentheses, and an operation to be performed on that statement. You supply the new statement and the truth value of the new statement (true, false, or undetermined).
Given Statement Operation New Statement truth value
1. No S are non-P. (T) Contraposition _No P are non-S ___ ____U_
ANSWER: The contrapositive flips the subject and predicate terms and replaces both with their term complements. In the statement “No S are Non-P” the subject term is S and the predicate term is non-P. Flipping and negating both we get “No non-non-P are non-S”, or “No P are S”. The contrapositive of E does not necessarily have the same truth value as the E; therefore the truth value is undetermined.
2. All non-S are non-P.(F) Contradictory _ Some non-S are not non-P _ __T___
ANSWER: The contradictory changes both quality and quantity. This is an A proposition, the universal affirmative. To form the contradictory you change it to the negative particular proposition O: Some non-S are not non-P. The truth value of the contradictory is always the opposite of the the truth value of the original statement, so the truth value is T.
3. No S are non-P. (T) Obversion __All S are P_________ __T___
ANSWER: The obverse is formed by changing the quality and replacing the predicate term with its term complement: All S are P. The truth value of the obverse is always the same as the truth value of the original statement.
4. Some non-S are not P. (F) Conversion _ Some P are not non-S ___ __U___
The converse flips the subject and predicate terms. In “Some non-S are not P” (an O proposition) the subject term is non-S and the predicate term is P. So the converse is Some P are not non-S. The truth value of the converse of O is undetermined.
Part III (10 points): Determine the validity or invalidity of the following arguments using your knowledge of the logical relationships among the categoricals. Show in detail how your determined your answer by some method you have learned in this class (Venn diagram method, conversion, obversion, contraposition, square of opposition, counterexample method, etc.). It is unacceptable to justify your answer by saying “I just know” or “It’s obvious”.
1. It is false that no murders are happy events. Thus, some murders are not happy events.
ANSWER: The argument is:
It is false that no murders are happy events.
Some murders are not happy events.
The premise is not a categorical as it stands, because a categorical has to begin with a quantifier. However, since the sentence is the negation of a categorical, it is equivalent to the contradictory of the statement negated. The contradictory of E (no murders are happy events) is I (some muders are happy events).
So the argument is from I to O.
It is invalid.
By Venn diagrams: premise puts an X in area 2; conclusion puts an X in area 1.
By square of opposition: illicit subalternation.
By logical analogy: Some apples are fruit. (T, since all apples are fruit and apples
Some apples are not fruit. (F)
2. No lunatics are professors, since some non-professors are not non-lunatics.
word “since” is a premise
Some non-professors are not non-lunatics.
No lunatics are professors.
You can eliminate the term complements in the premise by contraposition (which is legal for O): Some lunatics are not professors.
So the argument becomes:
Some lunatics are not professors.
No lunatics are professors.
It is an argument from O to E. It is invalid.
By Venn diagrams: premise puts an X in area 1; conclusion shades area 2. New information in conclusion.
By traditional square of opposition: illicit superalternation
By logical analogy:
Some women are not pregnant (T).
No women are pregnant. (F)
PART IV (26 points): True or False?
1. F Subject terms of both affirmative categoricals are undistributed.
ANSWER: Affirmative categoricals are A and I. Subject of A is distributed.
2. T Predicate terms of both negative categoricals are distributed.
ANSWER: Negatives are E and O.
3. T If “All S are non-P” is false, then “No S are P” must also be false.
ANSWER: These two statements are the obverses of each
4. F The A proposition is affirmative in quantity.
ANSWER: A is affirmative in quality.
5. F The middle term is in the predicate position in the minor premise of Figure 4.
ANSWER: Look at the lay-out of terms in syllogisms.
6. T To determine the mood and figure of a categorical syllogism, the syllogism must first be put into standard form.
7. F If a categorical syllogism violates one of the first four rules, it may still be valid.
8. F If a categorical syllogism has an I statement as its conclusion, it never violates Rule 2.
9. F The quality of a categorical statement is always determined by the copula.
ANSWWER: The copula is the word “are”.
10. T If a categorical proposition makes an assertion about every member of a class denoted by a term, that term is said to be distributed.
11. T “No mermaids are non-swimmers” follows from “All mermaids are swimmers”.
ANSWER: These two statements are obverses, which always have the same truth value. Hence it can’t be the case that one is true and the other false.
12. F The obverse of a categorical statement is formed by interchanging the subject and predicate terms and negating the predicate term.
13. F If “No Ss are Ps” is false, the truth value of its converse is undetermined.
ANSWER: The converse of E always has the same truth value as the
PART V (14 points): Translate the following statements into standard-form categorical statements.
1. A few students work in the library.
Some students are students who work in the library.
2. Only music students are allowed in the practice rooms.
All students allowed in the practice rooms are music students.
3. There are oil wells in
All places identical to
4. The only people allowed in this locker room are women.
All persons allowed in this locker room are women.
5. Dogs are
No dogs are vegetarians.
6. It’s false that all politicians are honest.
Some politicians are not honest people.
7. If a person doesn’t play the Lottery, a person can’t win the Lottery.
All persons who
don’t play the Lottery are persons who don’t win the
PART VI (8 points). Symbolize the following syllogism using the designated letters to represent the terms, reduce the number of terms, put into standard form, name mood and figure, and solve for validity or invalidity. Use the Venn diagram method.
No cars that are not fuel-efficient are cars environmentalists purchase. Hence, all Hummers are cars environmentalists do not purchase, since no fuel-efficient cars are Hummers. (E- cars environmentalists purchase; F- fuel-efficient cars; H- Hummers)
“Hence” is a conclusion
No non-F are E. by conversion: No E are non-F. Then by obversion: All E are F.
No F are H. no simplification necessary No F are H.
All H are non-E. by obversion: No H are E.
Argument is AEE-4.
By Venn diagrams: (assuming F on top, H on bottom left, E on bottom right)
Premise 1: Areas 6 and 7 empty.
Premise 2: Areas 2 and 3 empty.
Conclusion: Areas 3 and 6 empty. (No new information.)
PART VII (8 points). Symbolize the following syllogism using the designated letters to represent the terms, reduce the number of terms, put into standard form, name mood and figure, and solve for validity or invalidity. Use the rule method.
No cars purchased by environmentalists are not fuel-efficient. Hence, Hummers must not be fuel-efficient, since environmentalists purchase only non-Hummers.
(E- cars environmentalists purchase; F- fuel-efficient cars; H- Hummers)
Only H are E. Becomes All E are H.
All E are non-F. By obversion, No E are F.
No H are F. No simplification required No H are F.
Putting premises in correct order,
All E are F.
No E are H.
Therefore, No H are F.
Invalid by rules: violates rule 2. F is distributed in the conclusion, but not in the premise.
PART IX. (9 points). Translate the following arguments into standard form categorical syllogisms. Write the premises and conclusion in English, without using letters to symbolize the terms.
1. A few lazy students do not prepare for class. Steve prepares for class. We can conclude that Steve is not a lazy student.
Some students who do not prepare for class are lazy students.
No students identical to Steve are students who do not prepare for class.
Therefore, no students identical to Steve are lazy students.
2. Crazy people never eat broccoli. Anyone who pays full price for airline tickets is crazy. So all broccoli-eaters pay full price for airline tickets.
All persons who pay full price … are crazy people.
No crazy people are broccoli-eaters.
Therefore, all broccoli-eaters are persons who pay full price …
3. Whenever suicide rates decline, we can infer that people’s lives are better adjusted. Accordingly, since suicide rates have been declining in recent years, we can infer that people’s lives have been better adjusted in recent years.
All time suicide rates decline are times people’s lives are better adjusted.
All times identical to recent years are time suicide rates decline.
Therefore, all time identical to recent years are times people’s lives are better adjusted.