Richard L Epstein
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The minor term is "police officers." The middle term is "thieves." The major
premise is "Some thieves are sent to prison." The minor premise is "No police
officers are thieves."
The main focus of Aristotelian logic, as traditionally presented, is to show that
we can mechanically determine of any given categorical syllogism whether it is valid
or invalid. One way to do that is by inspecting its form. We list all possible forms of
syllogisms in standard form: All the claims are in standard form, and the major
premise comes first, then the minor premise, then the conclusion. For example,
"No S is M; All M are P; so No S is P" has form EAE. We determine for each form
whether it is valid or invalid; this one is valid. Given any categorical syllogism, we
can first rewrite it in standard form and then check whether it is one of the valid
forms.
But instead of listing all the forms, Aristotelians have shown how we can start
with knowing whether a few are valid or invalid, and then convert any other form
into one of those by a detailed reduction procedure.
Alternatively, we can take any categorical syllogism, rewrite it in standard
form, and then use the method of diagrams presented in Chapter 8 to determine
whether it is valid. Or we can use one of several other well-known diagram
methods, similar to but distinct from the methods of Chapter 8.
Once we've checked for validity, we still have to decide whether the syllogism
is a good argument. We know that a valid argument need not be good, for a premise
could be false, or the premises may not be more plausible than the conclusion.
Indeed, many valid Aristotelian syllogisms beg the question. For example, with "All
dogs eat meat. Spot is a dog. So Spot eats meat.", it's more plausible that Spot eats
meat than that all dogs do. Categorical syllogisms, as originally used by Aristotle,
are really a logic of explanations, not arguments. In an explanation the conclusion is
supposed to be more plausible than the premises, as when someone tries to explain
why "The sky is blue" is true. (The Science Workbook for this text teaches how to
reason about explanations.)
In any case, in ordinary speech we first have to decide how the person giving
the argument intends "all" and "some" to be understood, and many times those
readings won't be compatible with the assumptions of Aristotelian logic. Even if
those readings are compatible, we often have to do a lot of work to rewrite the claims
into standard categorical form. Then we have to check against a (memorized?) list of
valid Aristotelian forms. Then we have to ask about the plausibility of the premises
to determine whether the syllogism is a good argument. Even then, many simple
arguments using "some" or "all" can't be analyzed as categorical syllogisms, such as
"Some dogs like cats; some cats like dogs; so some dogs and cats like each other."
For hundreds and hundreds of years students and scholars preoccupied
themselves with the methods of Aristotelian logic as the primary focus of their
EXERCISES for Section D 383
analysis of reasoning. They could rely on standard methods and checkable rules.
But that tradition missed most of the important work in critical thinking that has been
incorporated into the foundations of reasoning analysis only in the last 150 years,
even though much of that can also be traced to Aristotle.
For reasoning in your daily life, being able to listen and analyze as you read
and speak, the methods and work we did in Chapter 8 will be more useful than the
formal methods of Aristotelian logic. To decide whether a categorical syllogism is
valid, do what we've always done: See if there is a possible way for the premises to
be true and the conclusion false.
Exercises for Section D
1. What is a categorical syllogism?
2. What is the major term of a categorical syllogism?
3. What is the minor term of a categorical syllogism?
4. What is the middle term of a categorical syllogism?
5. What is the major premise of a categorical syllogism?
6. What is the minor premise of a categorical syllogism?
7. What is the standard form for a categorical syllogism?
Which of the forms of categorical syllogisms in Exercises 8-15 are forms of arguments that
must be valid? The forms are presented by giving the letter name of the standard form of the
major premise, then the minor premise, then the conclusion.
8. EAE (No S is M; all M are P; so no S is P.)
9. AAA
10. AEO
11. IAO
12. Ill
13. AEE
14. AOO
15. AAI
For each of the following arguments, either rewrite it in the standard form of a categorical
syllogism and identify the form, or explain why it cannot be rewritten that way. In either
case, determine if the argument is valid.
16. All students at this school pay tuition. Some people who pay tuition at this school will
fail. So some students at this school will fail.
17. There aren't any wasps that will not sting. Some bumblebees will not sting. So some
bumblebees aren't wasps.
384 APPENDIX: Aristotelian Logic
18. Badly managed businesses are unprofitable. No oyster cultivating business in North
Carolina is badly managed. So some oyster cultivating business in North Carolina is
profitable.
19. Most critical thinking books do not teach Aristotelian logic. Chemistry textbooks never
teach Aristotelian logic. So most chemistry books are not critical thinking textbooks.
20. Nothing that's smarter than a dog will cough up hair balls. Cats cough up hair balls.
So cats are not smarter than dogs.
21. Dick will not visit Tom tonight if Zoe cooks dinner. Zoe didn't cook dinner. So Dick
visited Tom tonight.
22. No pacifists will fight in a war. Dick is a pacifist. So Dick will not fight in a war.
23. Police chiefs who interfere with the arrest of city officials are always fired. People who
are fired collect unemployment. So some police chiefs who interfere with the arrest of
city officials collect unemployment.
24. Some temporary employment agencies do not give employee benefits. All employees of
Zee Zee Frap's restaurant get employee benefits. So no employee of Zee Zee Frap's is
hired through a temporary employment agency.
Key Words categorical claim A claim
standard form of a E claim
categorical claim I claim
universal categorical claim O claim
particular categorical claim subalternate
affirmative categorical claim Square of Opposition
negative categorical claim categorical syllogism
quantity of a categorical claim major term
quality of a categorical claim minor term
subject of a categorical claim middle term
predicate of a categorical claim major premise
contradictory minor premise
contrary standard form of a
subcontrary categorical syllogism
Further Study There are many textbooks that present the "traditional" Aristotelian
logic with lots of diagrams and a listing of all valid and invalid forms of categorical
syllogisms. But to see the real power of the Aristotelian tradition, you need to study
m
edieval logic in the work of Buridan, Duns Scotus, Peter of Spain, and others.
There are some good translations and expositions of the work of those logicians, but
you're best off taking a philosophy course on the history of logic.
Diagramming Arguments
A. Diagrams 385
• Exercises for Section A 388
B. Counterarguments 389
• Exercises for Section B 390
A. Diagrams
This appendix is a supplement to the section Complex Arguments for Analysis. It
provides a way to visualize the structure of complex arguments. For example:
Spot chases rabbits. 1
Spot chases squirrels. 2
Therefore, Spot chases all small animals. 3
To picture this argument, we number the premises and conclusion. Then we
ask which claim is meant to support which other. Here support just means that it's
a reason to believe the other claim.
The conclusion will have to be at the bottom, since all the premises are supposed to
support it. And both do. The picture we'll draw is:
Neither 1 supports 2, nor does 2 support 1. So there is no arrow from one to the other. But both support 3, so we have arrows there. That's simple.
Now consider:
Dogs are mammals. 1
Cats are mammals. 2
Some dogs hate cats. 3
Therefore, some dogs hate mammals. 4
385
386 APPENDIX: Diagramming
We number the claims. It's easy to see which is the conclusion (it's labeled
with the word "therefore"). Which claims are meant to support which others? We
need 2 and 3 to get the conclusion 4. But what's 1 doing? Nothing. The
argument doesn't get any better by adding it, since it doesn't support any of the
other claims. So our picture is:
We also need a way to represent premises that are dependent, that is, they are
meant together to support another claim, in the sense that if one is false, the other(s)
do not give support.
In a diagram we indicate that premises are dependent by putting '+'
between them and drawing a line under them.
Dogs are loyal. 1
Dogs are friendly. 2
Anything that is friendly and loyal makes a great pet. 3
Hence, dogs are great pets. 4
Recall now the argument discussed on pp. 221-222:
Whatever you do, don't take the critical thinking course from Dr. E. 1
He's a really tough grader 2, much more demanding than the other
professors that teach that course. 3 You could end up getting a bad
grade. 4
We rewrote 1 as "You shouldn't take the critical thinking course from Dr. E."
And we rewrote 3 as "He's much more demanding than the other professors that
teach that course." It wasn't clear which claim was supposed to support which other.
We had two choices:
We chose to repair this argument with:
SECTION A Diagrams 387
If you take critical thinking from someone who's more demanding than
other professors who teach that course and who is a really tough grader,
then you could end up getting a bad grade, a
That makes the second diagram a better choice, though we still need to get from 4
to 1. We can use:
You shouldn't take any course where you might get a bad grade, b
We can see that the argument is only as good as the unsupported premise b.
Let's see how adding a series of unstated premises can affect the picture.
Consider:
My buddies John, Marilyn, and Joe all took Dr. E's critical thinking class and
did well. 1 So I'm going to sign up for it, too. 2 I need a good grade. 3
First, we need to rewrite 2 as a claim "I should sign up for Dr. E's critical thinking
class." I take this to be the conclusion (try the other possibilities, asking where you
could put "therefore" or "because"). Initially we might take the diagram:
But we need some glue for this to be even moderately strong. To begin with, why do
1 and 3 yield 2 ? A (fairly weak) assumption might be:
Usually if John, Marilyn, and Joe all do well in a class, I'll do well, a
But even that plus 3 won't give us 2. We need some further assumption like:
I should sign up for classes in which I know I'll get a good grade, b
Then the argument becomes:
Still, there's something missing. We need:
I'll do well in Dr. E's course, c
And that changes the picture entirely:
388 APPENDIX: Diagramming
We have a strong argument, in which we see a dependence between 3 and
what we get from 1. Whether this is a good argument depends on whether the
premises are plausible.
Exercises for Section A
For each of the following, if it is an argument, diagram it, repairing as necessary.
1. Dr. E is a teacher. All teachers are men. So Dr. E is a man.
2. No one under sixteen has a driver's license. So Zoe must be at least sixteen.
3. Sheep are the dumbest animals. If the one in front walks off a cliff, all the rest will
follow him. And if they get rolled over on their backs, they can't right themselves.
4. I'm on my way to school. I left five minutes late. Traffic is heavy. Therefore, I'll be
late for class. So I might as well stop and get breakfast.
5. Pigs are very intelligent animals. They make great pets. They learn to do tricks as
well as any dog can. They can be housetrained, too. And they are affectionate, since
they like to cuddle. Pigs are known as one of the smartest animals there are. And if
you get bored with them or they become unruly, you can eat them.
6. Smoking is disgusting. It makes your breath smell horrid. If you've ever kissed
someone after they smoked a cigarette you feel as though you're going to vomit.
Besides, it will kill you.
7. You're good at numbers. You sort of like business. You should major in accounting—
accountants make really good money.
8. Inherited property such as real estate, stocks, bonds, etc. is given a fresh start basis
when inherited. That is, for purposes of future capital gains tax computations, it is
treated as though it were purchased at its market value at the time of inheritance. Thus,
when you sell property which was acquired by inheritance, tax is due only on the
appreciation in value since the time it was inherited. No tax is ever paid on the increase
in value that took place when the property belonged to the previous owner.
1994 Tax Guide for College Teachers
SECTION B Counterarguments 389
B. Counterarguments
Recall the conversation between Dick and Zoe we looked at in Chapter 7:
We ought to get another dog. 1
(objection) We already have Spot. 2
The other dog will keep Spot company. 3
(objection) Spot already has us for company. 4
We are gone a lot. 5
He is always escaping from the yard. 6
He's lonely. 7
We don't give him enough time. 8
He should be out running around more. 9
(objection) It will be a lot of work to have a new dog. 10
(objection) We will have to feed the new dog. 11
(objection) It will take a lot of time to train the new dog. 12
Dick will train him. 13
We can feed him at the same time as Spot. 14
Dog food is cheap. 75
We can diagram this if we have a way to r
epresent that a claim is an objection, not
support, for another claim.
To diagram the argument, then, note that it seems that Dick intends but never says:
Spot needs company, a
That with 3 will be what gets the conclusion.
14 15 13
11 12 9 8 7 6 5 4
V 3 ^
v. 3 + a
Claim 4 is an objection to a. That is, it's an attempt to show that a crucial
premise of Dick is false. It must be answered. And Dick answers it by amassing
390 APPENDIX: Diagramming
enough other evidence for a. Claim 10 is a direct challenge to the conclusion. If it is true, the conclusion is in doubt. So it must be answered. Dick doesn't try to show
that it is false directly. Rather he shows that the two claims Zoe uses to support 10
are false. So there is no reason to believe 10.
When we finish diagramming we can see at a glance whether the argument has
left some objection to a premise or objection to the conclusion unanswered. Either
the objection is knocked off with a counterclaim above the support for it (as with
13-15 against 10) or other claims are amassed as evidence (as with 5-9 against 4). Of course you'll still need to evaluate whether the various claims are plausible.
Exercises for Section B
Diagram and evaluate the following arguments:
1. You should not take illegal drugs. They can kill you. If you overdose, you can die.
If you share a needle, you could get AIDS and then die. If you don't die, you could end
up a vegetable or otherwise permanently incapacitated. By using drugs you run the risk
of getting arrested and possibly going to jail. Or at least having a hefty fine against you.
Although some think the "high" from drugs is worth all the risks, the truth is that they
are addicted and are only trying to justify supporting their habit.
2. Zoe: I think sex is the answer to almost everyone's problems.
Dick: How can you say that?
Zoe: It takes away your tensions, right?
Dick: Not if you're involved with someone you don't like.
Zoe: Well, anyway, it makes you feel better.
Dick: Not if it's against your morals. Anyway, heroin makes you feel good, too.
Zoe: But it's healthy, natural, just like eating and drinking.
Dick: Sure, and you can catch terrible diseases. Sex should be confined to marriage.
Zoe: Is that a proposal?