most argument analysis. That tradition, called Aristotelian logic, was very broad,
and in the Middle Ages—especially from about 1100 to 1400—it was made into a
very subtle tool of analysis of reasoning.
In the late 1500s scholars became more interested in studying informal
reasoning, inspired also by the work of Aristotle. They ignored the complexities of
the formal logic of the medievals and were content with just the rules and forms of
Aristotelian logic, rote exercises and puzzles for students. That simplified tradition
of Aristotelian logic, current since about 1600, is what I'll present here. It is worth
studying because many writers from that time to today have used its terminology.
It also makes a contrast with modern formal logic. But it is only in the work of the
medievals, which has begun to be rediscovered, translated, and discussed only in the
last hundred years, that the Aristotelian tradition can offer us much in the way of
a serious study of arguments in terms of their form.
373
374 APPENDIX: Aristotelian Logic
B. Categorical Claims
Categorical claims A categorical claim is one that can be rewritten as
an equivalent claim that has one of the following standard forms:
All S are P. Some S is P. No S is P. Some S is not P.
For example,
All dogs are mammals.
No nurse is a doctor.
Some newspaper is written in Arabic.
Some snow is not white.
Most of the claims we reason with in daily speech aren't in any of these forms.
But, Aristotelians suggest, we can rewrite many of them to show that they are
categorical. For example, using " = ' to stand for "is equivalent to", we can rewrite: All dogs bark. = All dogs are things that bark.
No horse eats meat. = No horse is a thing that eats meat.
Some cats eat birds. = Some cat is a thing that eats birds.
Some dogs don't chase cats. = Some dog is a thing that doesn't chase cats.
Somewhat more colloquially, or at least avoiding the constant use of the phrase
"thing that," we might rewrite these as:
All dogs are barkers.
No horse is a meat eater.
Some cat is a bird eater.
Some dog is not a cat chaser.
It might seem that categorical claims are concerned only with things and
collections of things. But the following argument uses only categorical claims:
All snow is white.
All that is white is visible.
So, all snow is visible.
And snow, whatever it is, isn't a thing or collection of things, like dogs or pencils.
Snow is spread out everywhere across many times and places. It is a mass, like gold
or mud, and Aristotelian logic is useful for reasoning about masses, too.
It's often difficult to rewrite claims to "show" their categorical form, and there
are no general rules for how to do so. That's because so many different kinds of
words for so many different kinds of things and substances and classes can be used
EXERCISES for Section B 375
for the S or P in the forms. In this appendix we'll concentrate on words that stand
for classes or collections of things in order to make the discussion easier. We'll also
adopt the Aristotelian assumption that the S and P stand for things that actually
exist. So "All dodos are flightless birds" is not a categorical claim, because there are no dodos.
Recall (Chapter 8) that "All S is not P" is equivalent to "No S is P." So claims
of the form "All S are P" and "No S is P" are called universal claims. Aristotelians call claims of the form "Some S is P" and "Some S is not P" particular claims, since they are about some particular things, even if those are not picked out. In order
to make their logic more applicable, they also say that claims of the form "a is P" or
"a is not P" are universal categorical claims, where "a" stands for a name, as in: Maria is Hispanic.
Spot is not a cat.
Claims of the form "All S are P" and "Some S is P" are called affirmative, and claims of the form "No S is P" and "Some S is not P" are called negative. So, for example, "All dogs are mammals" is a universal affirmative claim, while "No dog is
a feline" is a negative universal claim. Whether a claim is universal or particular
denotes its quantity; whether a claim is affirmative or negative denotes its quality.
In a categorical claim, the term (word or phrase) that replaces the letter S is
called the subject of the claim. The term that replaces the letter P is called the
predicate of the claim. These words are not used in the way we use them in
grammar. In "All dogs are mammals" your English teacher would say that the
predicate is "are mammals," while in Aristotelian logic the predicate is "mammals."
Exercises for Section B
1. What is a categorical claim?
2. What assumption about the existence of things do we make about the terms used in
categorical claims?
3. What is a universal categorical claim?
4. What is a particular categorical claim?
5. What is an affirmative categorical claim?
7. What does the quantity of a categorical claim designate?
8. What does the quality of a categorical claim designate?
On the following two pages are some of Tom's exercises, as graded by Dr. E.
376 APPENDIX: Aristotelian Logic
All students are employed.
Categorical? Yes. Already in standard form.
Subject: Students.
Predicate: Employed.
Quantity: Universal.
Quality: Affirmative.
Good, except that since we've decided to view all subjects and predicates as either
things or collections of things, let's take, the predicate here to be "employed people."
Not even one art student is enrolled in calculus.
Categorical? Yes. "No art student is enrolled in calculus."
Subject: Art students.
Predicate: Enrolled in calculus.
Quantity: Universal.
Quality: Negative.
Good, except take the predicate here to be "people enrolled in calculus" or
"calculus enrollees."
Someone who likes Picasso also likes Monet.
Categorical? Yes. "Some people who like Picasso are people who
like Monet."
Subject: People who like Picasso.
Predicate: People who like Monet.
Quantity: Particular.
Quality: Affirmative.
G ood work.
Dr. E's students all pass.
Categorical? Yes. "Al students of Dr. E pass."
Subject: Students of Dr. E.
Predicate: Pass.
Quantity: Universal.
Quality: Affirmative.
Almost. "But you haven't given a categorical form for the claim. Where is "is a"
or "is not a " or "are"? We need "All students of "Dr. £ are people who pass."
Then the predicate is "people who pass."
EXERCISES for Section B 377
Very few dogs chase mice.
Categorical? Yes. "No dog chases mice."
Subject: Dogs.
Predicate: Mice chasers.
Quantity: Universal.
Quality: Negative.
No. "Very fezv" does not mean the same as "No," which means the same as "None."
Don't try to force every claim into one of these forms.
Some football players don't take steroids.
Categorical? Yes. "Some stude
nt who is a football player is not
someone who takes steroids."
Subject: Students who are football players.
Predicate: People who take steroids.
Quantity: Particular.
Quality: Negative.
Almost— -just delete the words "student who is a": "Some football player is not someone who takes steroids." Your claim isn't equivalent, because it could be true
and the original false if a professional football player takes steroids.
Some student at this school is majoring in football or there is a student who will
not get a degree.
Categorical? No. This is a compound claim, and I can't figure out how to
get it into a standard form.
Subject: Students.
Predicate: Football players and people who get degrees.
Quantity: Particular.
Quality: Affirmative and negative.
I don't think this exercise is very funny, Dr. E. We football players work hard
at school and sport.
You're right that if it's a compound it isn't a categorical claim. But then why
did you fill in after the other parts? Were you on automatic pilot? Only
categorical claims have subjects and predicates, quantity and quality.
You're also right that I should be more sensitive about the examples. In the
future I'll talk about basketball players.
Categorical? (If yes, and it is not already in one of the standard forms, rewrite it.)
Subject:
Predicate:
Quantity:
Quality:
378 APPENDIX: Aristotelian Logic
9. All dogs are carnivores.
10. Some cat is not a carnivore.
11. Tom is a basketball player.
12. No fire truck is painted green.
13. Donkeys eat meat.
14. There is at least one chimpanzee who can communicate by sign language.
15. Every border collie likes to chase sheep.
16. No one who knows critical thinking will ever starve.
17. Nearly every college graduate is employed at a full-time job.
18. All dogs bark or Spot is not a dog.
19. There is a teacher of critical thinking at this school who gives all A's to her students.
20. Heroin addicts cannot function in a 9-5 job.
21. Some people who like pizza are vegetarians.
22. Not every canary can sing.
23. Dr. E does not have a cat.
24. If Zoe does the dishes, then Dick will take Spot for a walk.
25. Of all the teachers at his school, none is as good as Dr. E.
26. Maria has a part-time job.
27. Waiters in Las Vegas make more money than lecturers at the university there.
28. In at least one instance a professor at this school is known to have failed all the students
in his class.
29. Make up five claims, three of which are categorical and two of which are not. Give
them to a classmate to classify.
C. Contradictories, Contraries, and Subcontraries
Recall that two claims are contradictory if in every possible circumstance they have
opposite truth-values. We say that two claims are contrary if there is no way in
which they both could be true. So if two claims are contradictory, they're also
contrary, but not vice-versa. For example, "All dogs bark" and "No dogs bark" are contrary (they can't both be true), but they're not contradictory: Since "dogs" must
refer to some object when it's used here, they can both be false.
We say that two claims are subcontrary if there is no possible way for them
both to be false. So contradictories are also subcontraries, but not vice-versa. For
example, "Some dogs bark" and "Some dogs don't bark" can't both be false, since to use the term "dogs" in a categorical claim is to assume there are such things. But
SECTION C Contradictories, Contraries, and Subcontraries 379
both of these claims could be true.
In order to discuss these relationships when they apply to pairs of categorical
claims, it is traditional to name the forms with letters:
"All S are P." A
"No S is P." E
"Some S is P." I
"Some S is not P." O
From Chapter 8 we already know that "All S are P" and "Some S is not P" are
contradictory. So any A claim and O claim using the same S and P are contradictory.
Also "No S is P" and "Some S is P" are contradictory: any E claim and I claim using the same S and P are contradictory.
On the other hand, "All S are P" and "No S is P" are contraries (they can't both
be true). And "Some S is P" and "Some S is not P" are subcontraries (they can't
both be false), since to use S as a subject term there must be something that is an S.
There is a further relationship that Aristotelians noted. From "All dogs bark"
we can conclude "Some dogs bark." Since using a term S as subject in a categorical
claim requires that there be at least one thing that is an S, we have generally:
• If an A claim is true, the I claim using the same S and P is true.
Similarly, from "No S is P" we can conclude "Some S is not P," because "No S is P"
is equivalent to "All S is not P," and the use of S comes with the assumption that
there is at least one S. That is:
• If an E claim is true, the O claim using the same S and P is true.
Going the other direction works, too, except that it's falsity that's inherited:
• If an I claim is false, then the corresponding A claim is also false
• If an O claim is false, then the corresponding E claim is false.
The Aristotelians summarized these relationships by saying that A and I claims using
the same subject and predicate are subalternates, and E and O claims using the
same subject and predicate are subalternates. Here is how they diagrammed these
relationships:
The Square of Opposition
380 APPENDIX: Aristotelian Logic
For nearly a thousand years students were expected to commit this diagram to
memory. But don't bother. Even if you don't remember the definitions, it's not hard
to spot that "All basketball players at this school are on scholarship" and "Some
basketball player at this school is not on scholarship" are contradictory, or that "No
employee of this school is enrolled in a health-care plan" and "All employees at this
school are enrolled in a health-care plan" can't both be true.
Exercises for Section C
1. What is the contradictory of a claim?
2. a. What does it mean to say that two claims are contrary?
b. Give an example of two claims that are contrary but not contradictory.
3. a. What does it mean to say that two claims are subcontrary?
b. Give an example of two claims that are subcontrary but not contradictory.
4. a. What does it mean to say that "All dogs bark" and "Some dogs bark" are
subalternate?
b. What does it mean to say that "No cats bark" and "Some cats do not bark" are
subalternate?
5. a. What is an A claim? Give an example.
b. What is an E claim? Give an example.
c. What is an I claim? Give an example.
d. What is an O claim? Give an example.
6. Show that for claims that use the same subject and predicate:
a. If the I claim is false, then the A claim is false.
b. If the O claim is false, then the E claim is false.
For each pair of claims below state which of the following terms apply:
contradictory contrary subcontrary subalternate none
7. All dogs bark.
Some dogs do not bark.
8. No Russians are Communists.
All Russians are Communists.
9. Maria is a widow.
Maria was never married.
10. No animals with horns are carnivores.
Some animals with horns are carnivores.
11. All uranium isotopes are highly unstable substances.
Some uranium isotopes are highly unstable substances.
12. Some assassinations are morally justifiable.
Some assassinations are not morally justifiable.
SECTION D Syllogisms 381
13. Dick and Tom are friends.
Dick and Tom can't stand to be in the same room together.
14. Not even one zebra can be trained to jump through fire.
Every zebra can be trained to jump through fire.
15. Homeless people don't like to sleep on the street.
Some homeless people don't like to sleep on the street.
16. Dick almost always washes the dishes after dinner.
Dick almost never washes the dishes after dinner.
17. Very few cats will willingly take a bath.
Very few cats won't willingly take a bath.
D. Syllogisms
We said that the arguments for which Aristotelian logic was devised contain only
categorical claims. Many of those can be reduced to arguments of a special kind.
Categorical syllogism A categorical syllogism is an argument composed
of three categorical claims (two premises and a conclusion). The three
claims use three terms as subject or predicate, each of which appears in
exactly two of the claims.
The first argument we considered in this chapter is a categorical syllogism:
No police officers are thieves.
Some thieves are sent to prison.
So no police officers are sent to prison.
Here the terms are "police officers," "thieves," "people sent to prison." Each appears in exactly two of the claims.
Aristotelians identify the predicates and subjects in syllogisms by the roles they
play in determining whether the argument is valid.
Major, minor, and middle terms of a categorical syllogism
major term = predicate of the conclusion
minor term = subject of the conclusion
middle term - the term that appears in both premises
major premise = premise that contains the major term
minor premise - premise that contains the minor term
382 APPENDIX: Aristotelian Logic
For example, in the last argument the major term is "people sent to prison."