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(Belvedere by 1 The Theory of Deduction
M.C. Escher)
2 Classes and Categorical Propositions
3 The Four Kinds of Categorical Propositions
4 Quality, Quantity, and Distribution
5 The Traditional Square of Opposition
6 Further Immediate Inferences
7 Existential Import and the Interpretation of Categorical
Propositions
8 Symbolism and Diagrams for Categorical Propositions
1 The Theory of Deduction
We turn now to the analysis of the structure of arguments. We have dealt main-
ly with the language in which arguments are formulated. In this chapter we
explore and explain the relations between the premises of an argument and its
Deductive argument
conclusion.
An argument whose
premises are claimed to This section of the chapter is devoted to deductive arguments. Adeductive
provide conclusive argument is one whose premises are claimed to provide conclusive grounds for
grounds for the truth of
the truth of its conclusion. If that claim is correct—that is, if the premises of the
its conclusion.
argument really do assure the truth of its conclusion with necessity—that deduc-
Validity
tive argument is valid. Every deductive argument either does what it claims, or it
A characteristic of any
deductive argument does not; therefore, every deductive argument is either valid or invalid. If it is
whose premises, if they valid, it is impossible for its premises to be true without its conclusion also being
were all true, would true.
provide conclusive
The theory of deduction aims to explain the relations of premises and conclu-
grounds for the truth of
its conclusion. Such an sion in valid arguments. It also aims to provide techniques for the appraisal of
argument is said to be deductive arguments—that is, for discriminating between valid and invalid de-
valid.
ductions. To accomplish this, two large bodies of theory have been developed.
Classical or The first is called classical logic (or Aristotelian logic, after the Greek philoso-
Aristotelian logic
pher who initiated this study). The second is called modern logic or modern
The traditional account of
syllogistic reasoning, in symbolic logic, developed mainly during the nineteenth and twentieth cen-
which certain turies. Classical logic is the topic of this chapter.
interpretations of
Aristotle (384–322 BCE) was one of the towering intellects of the ancient world.
categorical propositions
After studying for twenty years in Plato’s Academy, he became tutor to Alexander
are presupposed.
the Great; later he founded his own school, the Lyceum, where he contributed
Modern ormodern
symbolic logic substantially to nearly every field of human knowledge. His great treatises on rea-
The account of soning were collected after his death and came to be called the Organon, meaning
syllogistic reasoning
literally the “instrument,” the fundamental tool of knowledge.
accepted today. It differs
The word logicdid not acquire its modern meaning until the second century
in important ways from
the traditional account. CE, but the subject matter of logic was long understood to be the matters treated in
(cid:20)(cid:25)(cid:27)
Categorical Propositions
Aristotle’s seminal Organon. Aristotelian logic has been the foundation of ration-
al analysis for thousands of years. Over the course of those centuries it has been
very greatly refined: its notation has been much improved, its principles have
been carefully formulated, its intricate structure has been completed. This great
system of classical logic, set forth in this chapter, remains an intellectual tool of
enormous power, as beautiful as it is penetrating.
2 Classes and Categorical Propositions
Classical logic deals mainly with arguments based on the relations of classes of
objects to one another. By a class we mean a collection of all objects that have
some specified characteristic in common. Everyone can see immediately that two
classes can be related in at least the following three ways:
1. All of one class may be included in all of another class. Thus the class of all
dogs is wholly included(or wholly contained) in the class of all mammals.
2. Some, but not all, of the members of one class may be included in an-
other class. Thus the class of all athletes is partially included(or partially
contained) in the class of all females.
3. Two classes may have no members in common. Thus the class of all tri-
angles and the class of all circles may be said to excludeone another.
These three relations may be applied to classes, or categories, of every sort. In
a deductive argument we present propositions that state the relations between
one category and some other category. The propositions with which such argu-
ments are formulated are therefore called categorical propositions. Categorical
propositions are the fundamental elements, the building blocks of argument, in
the classical account of deductive logic. Consider the argument
No athletes are vegetarians.
All football players are athletes.
Therefore no football players are vegetarians. Class
The collection of all
This argument contains three categorical propositions. We may dispute the objects that have some
truth of its premises, of course, but the relations of the classes expressed in these specified characteristic
in common.
propositions yield an argument that is certainly valid: If those premises are true,
that conclusion must be true. It is plain that each of the premises is indeed cate- Categorical
gorical; that is, each premise affirms, or denies, that some class S is included in some proposition
Aproposition that can
other classP, in whole or in part. In this illustrative argument the three categorical
be analyzed as being
propositions are about the class of all athletes, the class of all vegetarians, and the about classes, or
class of all football players. categories, affirming or
denying that one class,
The critical first step in developing a theory of deduction based on classes,
S, is included in some
therefore, is to identify the kinds of categorical propositions and to explore the
other class, P, in whole
relations among them. or in part.
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Categorical Propositions
3 The Four Kinds of Categorical Propositions
There are four and only four kinds of standard-form categorical propositions.
Here are examples of each of the four kinds:
1. All politicians are liars.
2. No politicians are liars.
3. Some politicians are liars.
4. Some politicians are not liars.
We will examine each of these kinds in turn.
1. Universal affirmative propositions. In these we assert that the whole of
one class is included or contained in another class. “All politicians are liars”
is an example; it asserts that every member of one class, the class of
politicians, is a member of another class, the class of liars. Any universal
affirmative proposition can be written schematically as
All S is P.
where the letters Sand Prepresent the subjectand predicateterms, re-
spectively. Such a proposition affirmsthat the relation of class inclusion
holds between the two classes and says that the inclusion is complete, or
universal. All members of Sare said to be also members of P. Proposi-
tions in this standard form are called universal affirmative propositions.
They are also called Apropositions.
Categorical propositions are often represented with diagrams, using
two interlocking circles to stand for the two classes involved. These are
called Venn diagrams, named after the English logician and mathemati-
Standard-form cian, John Venn (1834–1923), who invented them. Later we will explore
categorical
these diagrams more fully, and we will find that such diagrams are ex-
proposition
Any categorical ceedingly helpful in appraising the validity of deductive arguments. For
proposition of the form the present we use these diagrams only to exhibit graphically the sense
“All Sis P” (universal of each categorical proposition.
affirmative), “No Sis P”
(universal negative), We label one circle S, for “subject class,” and the other circle P, for
“Some Sis P” (particular “predicate class.” The diagram for the Aproposition, which asserts that
affirmative), or “Some S all Sis P, shows that portion of Swhich is outside of Pshaded out, indi-
is not P” (particular
cating that there are no members of Sthat are not members of P. So the
negative). Respectively,
these four types are Aproposition is diagrammed thus:
known as A, E, I, and O
propositions.
S P
Venn diagram
Iconic representation of
a categorical proposition
or of an argument, used
to display their logical
forms by means of
overlapping circles. All S is P.
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Categorical Propositions
2. Universal negative propositions. The second example above, “No
politicians are liars,” is a proposition in which it is denied, universally,
that any member of the class of politicians is a member of the class of
liars. It asserts that the subject class, S, is wholly excluded from the
predicate class, P. Schematically, categorical propositions of this kind
can be written as
No S is P.
where again Sand Prepresent the subject and predicate terms. This
kind of proposition deniesthe relation of inclusionbetween the two
terms, and denies it universally. It tells us that no members of Sare
members of P. Propositions in this standard form are called universal
negative propositions. They are also called Epropositions.
The diagram for the Eproposition will exhibit this mutual exclusion
by having the overlapping portion of the two circles representing the
classes Sand Pshaded out. So the Eproposition is diagrammed thus:
S P
No S is P.
3. Particular affirmative propositions. The third example above, “Some
politicians are liars,” affirms that some members of the class of all politi-
cians are members of the class of all liars. But it does not affirm this of
politicians universally. Only some particular politician or politicians are
said to be liars. This proposition does not affirm or deny anything about
the class of all politicians; it makes no pronouncements about that entire
class. Nor does it say that some politicians are not liars, although in
some contexts it may be taken to suggest that. The literal and exact in-
terpretation of this proposition is the assertion that the class of politi-
cians and the class of liars have some member or members in common. That
is what we understand this standard-form proposition to mean.
“Some” is an indefinite term. Does it mean “at least one,” “at least
two,” or “at least several”? How many does it mean? Context might affect
our understanding of the term as it is used in everyday speech, but logi-
cians, for the sake of definiteness, interpret “some” to mean “at least one.”
Aparticular affirmative proposition may be written schematically as
Some S is P.
which says that at least one member of the class designated by the sub-
ject term Sis also a member of the class designated by the predicate
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Categorical Propositions
term P. The proposition affirmsthat the relation of class inclusionholds,
but does not affirm it of the first class universally—it affirms it only par-
tially; that is, it is affirmed of some particularmember, or members, of
the first class. Propositions in this standard form are called particular
affirmative propositions. They are also called Ipropositions.
The diagram for the Iproposition indicates that there is at least one
member of Sthat is also a member of Pby placing an xin the region in
which the two circles overlap. So the Iproposition is diagrammed thus:
S P
x
Some S is P.
4. Particular negative propositions. The fourth example above, “Some
politicians are not liars,” like the third, does not refer to politicians uni-
versally, but only to somemember or members of that class; it is
particular. Unlike the third example, however, it does not affirm the in-
clusion of some member or members of the first class in the second
class; this is precisely what is denied. It is written schematically as
Some S is not P.
which says that at least one member of the class designated by the sub-
ject term Sis excluded from the whole of the class designated by the
predicate term P. The denial is not universal. Propositions in this stan-
dard form are called particular negative propositions. They are also called
Opropositions.
The diagram for the Oproposition indicates that there is at least one
member of Sthat is not a member of Pby placing an xin the region of S
that is outside of P. So the Oproposition is diagrammed thus:
S P
x
Some S is not P.
The examples we have used in this section employ classes that are simply
named: politicians, liars, vegetarians, athletes, and so on. But subject and predi-
cate terms in standard-form propositions can be more complicated. Thus, for ex-
ample, the proposition “All candidates for the position are persons of honor and
(cid:20)(cid:26)(cid:21)
Categorical Propositions
integrity” has the phrase “candidates for the position” as its subject term and the
phrase “persons of honor and integrity” as its predicate term. Subject and predi-
cate terms can become more intricate still, but in each of the four standard forms
a relation is expressed between a subject class and a predicate class. These four—
A, E, I, and Opropositions—are the building blocks of deductive arguments.
This analysis of categorical propositions appears to be simple and straightfor-
ward, but the discovery of the fundamental role of these propositions, and the ex-
hibition of their relations to one another, was a great step in the systematic
development of logic. It was one of Aristotle’s permanent contributions to human
knowledge. Its apparent simplicity is deceptive. On this foundation—classes of
objects and the relations among those classes—logicians have erected, over the
course of centuries, a highly sophisticated system for the analysis of deductive ar-
gument. This system, whose subtlety and penetration mark it as one of the great-
est of intellectual achievements, we now explore in the following three steps:
A. In the remainder of this chapter we will examine the features of standard-
form categorical propositions more deeply, explaining their relations to
one another, and what inferences may be drawn directlyfrom these cate-
gorical propositions. Much of deductive reasoning can be mastered with
no more than a thorough grasp of A,E,I,and Opropositions and their
interconnections.
B. In this text, we will examine syllogisms, the arguments that are commonly
constructed using standard-form categorical propositions. We will explore
the nature of syllogisms, and show that every valid syllogistic form is
uniquely characterized and is therefore given its own name. We will then
develop powerful techniques for determining the validity (or invalidity)
ofsyllogisms.
C. In this text we integrate syllogistic reasoning and the language of argu-
ment in everyday life. Some limitations of reasoning based on this founda-
tion will be identified, but the wide applicability that this foundation
makes possible will be demonstrated.
overview
Standard-Form Categorical Propositions
Proposition Form Name and Type Example
All Sis P. A Universal affirmative All lawyers are wealthy
people.
No Sis P. E Universal negative No criminals are good
citizens.
Some Sis P. I Particular affirmative Some chemicals are
poisons.
Some Sis not P. O Particular negative Some insects are not pests.
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