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Chapter 9
Deductive Arguments I
Categorical Logic
Some students find the “logic” section of courses much more to their liking than the rest. Others
suffer paroxysms and loss of breath when they open the book to the pages on derivations. The
presumed reason some like it is that many of the answers are pretty cut-and-dried: when they
have basic knowledge of the material, they ought to get most of the answers right. This contrasts
with much of the “critical thinking” part of the course, where the material can be a good bit
Students are often surprised to learn that so much can be said by using nothing but the four
standard-form categorical types of claims. Much of what can be said must be said awkwardly, of
course—in “logician’s English”—but it can be said nonetheless.
If one is especially careful (and a little less worried about confusing students unnecessarily), one
would say more about the distinction between a categorical claim and a categorical claim form.
“All Xs are Ys” is not a claim at all, but a claim form (or an open sentence, or a propositional
function), since it contains variables for the subject and predicate terms. Such expressions can’t
be true or false; only their instantiations can have truth values. It took a bit of audacity to step on
the distinction in the way people do—people don’t know of another text that treats it this way—
but are willing to fly in the face of tradition if teachability benefits from it. It was discovered
that, technically correct or not, the distinction between claims and forms can be glossed over
An understanding of the usual categorical inferences, both immediate and syllogistic, is valuable
to a student. But it is believed that learning to translate from informal English claims to standard-
form claims serves an equally valuable purpose. One can hardly help but wonder how deeply
students are thinking about the claims they hear and make themselves when they have great
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difficulty deciding whether a claim should be translated as “All Xs are Ys” or “All Ys are Xs.”
Requiring such translations must, at a minimum, get students to think a little harder about the
kinds of sentences they use all the time.
Regarding the square of opposition, sharp students may notice that contraries could both be true
if the subject class was empty and that subcontraries could both be false under the same
view is counterintuitive to most students. Hence, there’s no point in beginning their study of
formal logic with a gross counterintuition. (Every edition of the book produces a letter or two,
usually from students, who fail to notice the discussion of this point).
Another note about the square of opposition: subalternation is not discussed (the relationship
between an A-claim and its corresponding I-claim and between an E-claim and its corresponding
O-claim). This is just more terminology to learn, and students find that going from an A-claim to
an E-claim and then to an I-claim—that is, going across the top of the square and then
Chapter Recap
The following list of topics covers the basics of categorical logic as discussed in this chapter:
• The four types of categorical claims include A, E, I, and O.
• There are Venn diagrams for the four types of claims.
• Ordinary English claims can be translated into standard-form categorical claims. Some
rules of thumb for such translations are as follows:
“only” introduces predicate term of A-claim
claims about individuals are treated as A- or E-claims
• The square of opposition displays contradiction, contrariety, and subcontrariety among
corresponding standard-form claims.
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Education.
• Conversion, obversion, and contraposition are three relations that result from operations
performed on standard-form claims; some are equivalent to the original, and some are not.
• Categorical syllogisms are standardized deductive arguments; we can test them for validity
by the Venn diagram method or by the rules method—the latter relies on the notions of
distribution and the affirmative and negative qualities of the claims involved.
Answers to Text Exercises
Exercise 9-1
1. ▲All senators are politicians.
2. Some senators are not politicians.
3. All senators are politicians.
14. All scholars are philosophers.
15. Some scholars are not philosophers.
Exercise 9-2
When we ask students to turn in exercises like these, we make them put parentheses or brackets
around the subject and predicate terms—doing this helps students keep their answers straight,
and it helps us while reading them.
1. ▲All salamanders are lizards.
2. Some lizards are not salamanders.
3. All lizards are reptiles.
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Exercise 9-4
4. ▲Translation: Some allergies are things that can kill you. (True)
Corresponding A-claim: All allergies are things that can kill you. (Undetermined)
Corresponding E-claim: No allergies are things that can kill you. (False)
Corresponding O-claim: Some allergies are not things that can kill you. (Undetermined)
5. Translation (A-claim): All woodpeckers are birds that sing really well. (False)
Corresponding E-claim: No woodpeckers are birds that sing really well. (Undetermined)
Corresponding O-claim: Some woodpeckers are not birds that sing really well. (True)
Corresponding I-claim: Some woodpeckers are birds that sing really well. (Undetermined)
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Education.
Corresponding E-claim: No colorful frogs are poisonous frogs. (Undetermined)
Corresponding I-claim: Some colorful frogs are poisonous frogs. (Undetermined)
Exercise 9-5
We looked it up: The prefix “non” is combined with other words with a hyphen only when the
other word is capitalized or is itself a compound word (e.g., “non–Greek” and “non–ablebodied,”
but “nonreligious”).
5. No Muslims are Hindus. (Equivalent)
6. Some non-Hindus are not non-Indians. (Equivalent)
Exercise 9-6
1. ▲Some students who scored well on the exam are not students who didn’t write poor
essays. (Equivalent)
2. All students who wrote poor essays are students who were not admitted to the program.
(Equivalent)
7. ▲All people whose automobile ownership is not restricted are people who don’t live in the
dorms. (Equivalent)
8. All people whose automobile ownership is restricted are noncommuters. (Equivalent)
9. Some students who do not do poorly (or who do all right) in history classes are students
with no short-term memory problems. (Not equivalent)
10. ▲All first basemen are people who aren’t right-handed. (Equivalent)
Exercise 9-7
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1. Some percussion instruments are clarinets.
Exercise 9-8
Translations of the lettered claims:
a. Some people who have been tested are not people who can give blood.
e. Logically equivalent to: “All people who have been tested are people who cannot give
blood.” Logically equivalent to: “No people who have been tested are people who can give
blood” [obverse].
1. ▲Equivalent to: “Some people who have not been tested are people who can give blood,”
which is equivalent to: “Some people who can give blood are people who have not been
Exercise 9-9
1. ▲Equivalent to (b).
Exercise 9-10
Note: There is often more than one way to do problems like this.
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1. ▲Obvert (a) to get “Some Slavs are not Europeans.”
2. Contrapose (a) to get “All non-Westerners are non-Europeans.”
5. Contrapose claim (a) and they correspond.
6. Obvert claim (a) and they correspond.
Exercise 9-11
1. ▲Invalid (This would require the conversion of an A-claim.)
7. ▲Valid (The premise is the obverse of the conclusion.)
8. Invalid (The premise and conclusion are contrapositives of each other, but they are both I-
claims and thus do not imply their contrapositives.)
Exercise 9-12
Some students find these among the most difficult exercises in this chapter. Work on such
problems needs to be divided into two parts: making the claims correspond, if possible, and then
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1. ▲The converse of (a) is the contradictory of (b), so (b) is false.
2. Obvert (a) and convert (b), and they are contradictories, s (b) is false.
6. (a) becomes an I-claim and (b) the corresponding O-claim. So the truth of (a) leaves (b)
undetermined.
7. ▲Translate (a) as “Some of GB’s novels are novels in which the hero gets killed,” and (b)
as “Some novels in which the hero gets killed are GB’s novels.” Converting the latter, we
have two subcontraries. If one subcontrary, (a), is false, the other, (b), must be true.
Exercise 9-13
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2. Invalid:
3. Valid:
G
Sound
arguments
Interesting
arguments
x
Mathematicians
Statisticians
x
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5. Invalid:
6. Valid:
Chords in the
mixolydian mode
E
Drugs that can be taken
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4. ▲All tobacco products are substances damaging to people’s health.
Some tobacco products are addictive substances.
Some addictive substances are substances damaging to people’s health.
Valid:
5. Some digital players are machines that use 3.0 transfer rates.
No machines that use 3.0 transfer rates are players that cost at least twenty dollars.
Some digital players are not players that cost at least twenty dollars.
6. All things that Bob won are things that Pete won.
All things that Bob won are junk.
All things that Pete won are junk.
Invalid:
Addictive
substances
Substances damaging to
people’s health
Tobacco products
x
Players that cost at least
twenty dollars
Players that use
3.0 transfer rate
x
Things that Bob won
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7. ▲All people who may vote are stockholders in the company.
No people identical with Mr. Hansen are people who may vote.
No people identical with Mr. Hansen are stockholders in the company.
People who may vote
One has to remember that claims with individuals as subject terms are treated as A- or E-
claims.
8. No off-road vehicles are vehicles allowed in the . . . park.
9. Some people affected by the drainage tax are residents of the county.
Off-road vehicles
10. ▲After converting, then obverting the conclusion:
No arguments with false premises are sound arguments.
Residents of
the county
People paying
the sewer tax
x