Deck 14: Syllogistic Logic

Question 1

General Theory

-Assuming existential import, explain what the differences are between (a) contradictories; (b) contraries; and (c) subcontraries.

Answer 1

a. Two propositions are contradictories if both cannot be true and both cannot be false. A and O, and E and I propositions are contradictories.
b. Assuming existential import, two propositions are contraries if both can be false but both cannot be true. A and E propositions are contraries.
c. Again assuming existential import, two propositions are subcontraries if both can be true but both cannot be false. I and 0 propositions are subcontraries.

Question 2

General Theory
-Prove that no valid syllogism in the second figure can have an affirmative conclusion.

Answer 2

The general form of a syllogism in the second figure is
$P _ M$
$S _ M$
$S _ P$
If the conclusion is affirmative, no premises may be negative (by rule 4 for valid syllogisms). So the premises must be either A or I propositions. But neither A nor I propositions distribute their predicates. Hence any syllogism in the second figure containing only A or I premises violates the rule (rule 1) requiring the middle term to be distributed at least once.

Question 3

General Theory

-a. If we interpret the sentence "All scientists are philosophers" so that it is correctly symbolized by , then why will "All scientists are philosophers" not imply "Some scientists are philosophers"?

b. If we interpret “All scientists are philosophers” so that it has existential import, then does “All scientists are philosophers” imply “Some scientists are philosophers” or not? (Defend your answer.)

Answer 3

a. Because when so interpreted, the truth of "All scientists are philosophers" might follow from the fact that the class scientists is empty, in which case "Some scientists are philosophers" will be false.
b. If we interpret the sentence to have existentential import, then we require the classes scientists and philosophers to have members (so "All scientists are philosophers" is not correctly symbolized as , then "All scientists are philosophers" does imply "Some scientists are philosophers", because if all scientists are philosophers, and there are scientists, then obviously some scientists are philosophers.

Question 4

General Theory
-Assume for the moment (whether it actually is the case or not) that the four types of categorical propositions are adequately represented by the four formulas below, and show (a) why A and E propositions will not have the properties that contraries are traditionally supposed to have; and (b) why I and O propositions will not have the properties that subcontraries are traditionally supposed to have.

Question 5

True-False

-The predicate term of "Some scientists are not philosophers" is not distributed.

Question 6

True-False

-The subject term of "All scientists are philosophers" is not distributed.

Question 7

True-False

-The subject term of "Some scientists are philosophers" is not distributed.

Question 8

True-False

-"All scientists are philosophers" and "No scientists are philosophers" are contradictories.

Question 9

True-False

-"Some scientists are not philosophers" and "All scientists are philosophers" are contradictories.

Question 10

True-False

-"Some scientists are not philosophers" and "Some scientists are philosophers" are contradictories.

Question 11

True-False

-We can validly infer from "All scientists are philosophers" to its contrapositive.

Question 12

True-False

-We can validly infer from "Some scientists are philosophers" to its contrapositive.

Question 13

True-False

-We can validly infer from "No scientists are philosophers" to its obverse.

Question 14

True-False

-We can validly infer from "Some scientists are not philosophers" to its obverse.

Question 15

Multiple choice:
Assume existential import

-The proposition "All scientists are philosophers" entails:

A) No nonscientists are philosophers
B) No scientists are nonphilosophers
C) No nonphilosophers are scientists
D) Some scientists are philosophers
E) none of these

Question 16

Multiple choice:
Assume existential import

-The proposition "Some scientists are not philosophers" follows from:

A) Some scientists are nonphilosophers
B) Some nonscientists are philosophers
C) No nonphilosophers are scientists
D) Some scientists are philosophers
E) none of these

Question 17

Multiple choice:
Assume existential import

-The proposition "Some nonscientists are nonphilosophers" is equivalent to:

A) Some nonphilosophers are scientists
B) Some scientists are not nonphilosophers
C) No nonphilosophers are scientists
D) Some scientists are philosophers
E) none of these

Question 18

Multiple choice:
Assume existential import

-The contrapositive of "No scientists are nonphilosophers" is:

A) No philosophers are nonscientists
B) No nonphilosophers are scientists
C) No philosophers are scientists
D) No scientists are philosophers
E) none of these

Question 19

Multiple choice:
Assume existential import

-The obverse of "No nonscientists are philosophers" is:

A) All nonphilosophers are scientists.
B) No nonphilosophers are scientists
C) All nonscientists are nonphilosophers
D) All nonphilosophers are scientists
E) none of these

Question 20

Inference:
Answer the following questions, not assuming existential import

-Suppose it is true that "All logicians are mathematicians." Then what can be said about the truth values of the following:
a. Some logicians are not mathematicians
b. Some logicians are mathematicians
c. No logicians are nonmathematicians
d. All mathematicians are logicians
e. All nonmathematicians are nonlogicians
f. No logicians are mathematicians
g. Some nonmathematicians are logicians
h. Some logicians are nonmathematicians

Question 21

Inference:
Answer the following questions, not assuming existential import
-Suppose it is false that "No logicians are mathematicians." Then what can be said about the truth value of the following sentences?
a. Some logicians are not mathematicians
b. Some logicians are mathematicians
c. All logicians are mathematicians
d. All logicians are nonmathematicians
e. All mathematicians are nonlogicians
f. No mathematicians are logicians
g. Some mathematicians are logicians
h. Some logicians are nonmathematicians

Question 22

Inference:
Answer the following questions, not assuming existential import
-Suppose it is true that "Some logicians are not mathematicians." Then what can be said about the truth value of the following sentences?
a. Some mathematicians are not logicians
b. Some logicians are mathematicians
c. All logicians are mathematicians
d. No logicians are nonmathematicians
e. Some nonmathematicians are logicians
f. No logicians are mathematicians
g. Some nonmathematicians are not nonlogicians
h. Some logicians are nonmathematicians

Question 23

Inference:
Answer the following questions, not assuming existential import
-Suppose it is false that "Some logicians are mathematicians." Then what can be said about the truth value of the following sentences?
a. Some logicians are not mathematicians
b. No logicians are mathematicians
c. All logicians are mathematicians
d. All logicians are nonmathematicians
e. All mathematicians are nonlogicians
f. No mathematicians are logicians
g. Some mathematicians are logicians
h. Some mathematicians are not nonlogicians

Question 24

Syllogisms
Put the following into standard syllogistic form and test for validity, using either the five rules for valid syllogism or Venn diagrams:
-Some logicians are nonphilosophers, so some mathematicians are logicians, since all mathematicians are nonphilosophers.

Question 25

Syllogisms
Put the following into standard syllogistic form and test for validity, using either the five rules for valid syllogism or Venn diagrams:
-All nonlogicians are nonphilosophers, because all logicians are mathematicians, while no philosophers are nonmathematicians.

Question 26

Syllogisms
Put the following into standard syllogistic form and test for validity, using either the five rules for valid syllogism or Venn diagrams:
-All logicians live in the United States, but some philosophers dwell elsewhere; so some philosophers are nonlogicians.

Question 27

Syllogisms
Put the following into standard syllogistic form and test for validity, using either the five rules for valid syllogism or Venn diagrams:
-It follows from the fact that no mathematicians are logicians that Art is not a mathematician, since Art is a logician.

Question 28

Syllogisms
Put the following into standard syllogistic form and test for validity, using either the five rules for valid syllogism or Venn diagrams:
-Due to the fact that logicians are brilliant but scientists are not brilliant, some scientists are nonlogicians.

Question 29

Sorites.
Translate the following sorites into standard form and test for validity:
-No teachers are good teachers.
Art is a logician.
No one who is not a good teacher is a logician.
$/ \therefore$ Art is not a teacher.

Question 30

Sorites.
Translate the following sorites into standard form and test for validity:

-No freshmen are nonstudents.
Some coeds are not ladies.
All noncoeds are nonstudents.
$/ \therefore$ Some freshmen are ladies.

Question 31

Sorites.
Translate the following sorites into standard form and test for validity:

-Anyone who is a nonstudent ought to study logic.
There are some athletes who are nonstudents.
There are some studious types that ought to study logic.
$/ \therefore$ There are some athletes who are studious.

Question 32

Sorites.
Translate the following sorites into standard form and test for validity:

-There are dogs that are cats.
There are no cats that are nonfelines.
All monkeys are nonfelines.
$/ \therefore$ There are dogs that aren't monkeys.