[推荐]The Categorical Syllogism

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the categorical syllogism
by jonathan dolhenty, ph.d.

the aim of logic is correct thinking, and thinking consists mainly of reasoning. the function of logic is to investigate the various types of arguments and the rules which govern their consistency. consistency is the very essence of correct thinking.

arguments are composed of propositions. we have seen that propositions fall into two main types: categorical and hypothetical. there are, therefore, two main types of deductive arguments, the categorical syllogism and the hypothetical syllogism. the categorical syllogism will be examined first, followed by an examination of the hypothetical syllogism in a later essay.

the nature of the categorical syllogism

the categorical syllogism can be defined as an argument in which, from two judgments which contain a common idea and one at least of which is universal, a third judgment, distinct from either of the former, follows with necessity.

as you have already learned, the syllogism consists of two premises and a conclusion. one of the premises contains a comparison of the major term (p) with the middle term (m), and the other premise contains a comparison of the minor term (s) with the middle term m). the conclusion expresses the agreement or disagreement between the minor term s) and the major term (p). the syllogism is a categorical syllogism when the premises are categorical propositions.

there is a general axiom or principle which underlies the syllogism called the law of all and none. this law states: what is affirmed of a logical whole may be affirmed of a logical part of that whole, and what is denied of a logical whole may be denied of a logical part of that whole. what does this mean? consider the following argument:

all birds have wings.
all robins are birds.
therefore, all robins have wings.
the middle term (m) in the major premise is a logical whole or universal term--"all birds." the entire comprehension of the major term (p)--"have wings"--is affirmed of the middle term--"all birds." since the minor term--"all robins"--is a logical part or member belonging to the class of the middle term--"all birds"--the entire comprehension of the major term--"have wings"--must also be affirmed of the minor term--"all robins." the conclusion, therefore, must be correct that "all robins have wings." the first part of the above law is shown to be correct.

the application of this axiom is shown here in a diagram:

p
this is the class of all things "with wings"

m
this is the class of "all birds" - a logical "whole"
 
s
this is the class of "all robins" - a logical part of the logical whole
 

what is affirmed of a logical whole may be affirmed
of a logical part of that whole.
now consider this argument: (quadruped means four-footed)

no birds are quadrupeds.
all robins are birds.
therefore, no robins are quadrupeds.
here we see that the middle term (m) "all birds," as a logical whole or universal term, includes the minor term (s) "all robins" in its extension as a logical part or member of the class. but the middle term (m) "all birds" is excluded from the extension of the major term (p) "quadrupeds." therefore, the minor term (s) "all robins," since it is a logical part or member of the class of the middle term (m) "all birds," must be excluded from the major term (p) "quadrupeds." the conclusion that "no robins are quadrupeds," must be correct. the second part of the above law is established.

this is illustrated in this diagram:

m
this is the class of "all birds" - a logical "whole"

s
this is the class of "all robins" - a logical part of the logical whole

 p
this is the class of
"quadrupeds"
from which the
class of
"all birds"
is excluded

what is denied of a logical whole may be denied
of a logical part of that whole.

in the above examples, the middle term (m) was the subject of the major premise and the predicate of the minor premise. this is not, however, the only arrangement. the middle term (m) may be changed to another position in either premise. hence there are four possible arrangements for the categorical syllogism.

m -- p
s -- m
p -- m
s -- m
m -- p
m -- s
p -- m
m -- s

s -- p s -- p s -- p s -- p
the categorical syllogism may appear in any of the above forms

the general rules for determining the validity of syllogisms

from the nature of the categorical syllogism, logicians have developed some general rules which govern every type of categorical syllogism and must be observed or the consistency of the argument will be destroyed. the conclusion of the argument, then, will either be false or will not follow logically from the premises and be invalid.

a valid standard-form categorical syllogism must contain exactly three terms, each of which is used in the same sense throughout the syllogism. furthermore, the middle term may not appear in the conclusion. the general rules governing categorical syllogisms can be divided into two sets: the first set refers to the terms of the categorical syllogism, while the second set refers to the quantity of a sentence (rules of distribution) and to the quality of a sentence (whether it is affirmative or negative).

the general rules regarding terms are:

(1) only three terms may appear in the syllogism, each of which is used in the same sense throughout the argument.
(2) neither the major nor minor term may be a universal in the conclusion, if it was only a particular term in the premises.
(3) the middle term may not appear in the conclusion.
(4) the middle term must be distributed at least once in the premises, that is, used as a universal
the general rules of quantity and quality are:

(5) if both premises are affirmative, the conclusion must also be affirmative.
(6) both premises may not be negative; one at least must be affirmative.
(7) if either premise is negative, the conclusion must be negative; if one of the premises is a particular proposition, the conclusion must b e a particular proposition.
(8) no conclusion can be drawn from two particular premises; one at least must be a universal proposition.
an understanding of and familiarization with the above rules will enable you to judge the validity of any categorical syllogism you may encounter.

the eight general rules are a direct product of the very idea and nature of the categorical syllogism. to facilitate understanding them and perceive their necessity as rules of correct thinking, each one will be explained and proved separately.

we will use various symbols to represent the parts of the categorical syllogism:

m
= middle term

s
= minor term (subject of conclusion)

p
= major term (predicate of conclusion)

pm
= particular (undistributed) middle term

um
= universal (distributed) middle term

ps
= particular minor term

us
= universal minor term

pp
= particular major term

up
= universal major term

+
= an affirmative proposition

-
= a negative proposition

justification of the general rules

for those of you who like to see the "proof of the pudding," or just want to gain further understanding of the general rules for the categorical syllogism, each rule will be explained and justified.

rule number 1

only three terms may appear in the syllogism, each of which is used in the same sense throughout the argument.
you will recall that the purpose of the argument is to discover whether two questionable ideas agree or disagree by comparing them with a third or mediating idea, which is called the middle term. the very function of the middle term excludes any other idea from being present in the premises of the syllogism. otherwise, there would be no common bond of comparison between the two ideas compared. consider this example involving four terms:

all animals are living.
all stones are minerals.
therefore, all stones are (are not?) living.
can you make sense of the argument?

it's also important to watch for syllogisms where it appears that three terms are used but actually four are employed. consider the following:

man is a universal.
john is a man.
therefore, john is a universal.
can you spot the problem here? it's true that the term "man" appears in both premises and would seem to be the middle term. but there is a problem. the term "man" is used in two different senses. in the first premise, the term means the idea "man," a logical supposition, which is indeed a universal term since it applies to the whole class of "men." in the second premise, however, the term "man" is not used to refer to the idea "man" as a logical supposition, but to a real being, a real supposition. much confusion can be caused by such ambiguities.

consider the following argument:

all men are brothers in a common fraternity.
all brothers in a common fraternity are college students.
therefore, all men are college students.
can you spot the problem? the fallacy here is one of ambiguity. the term "common fraternity" in the first premise means something quite different from "common fraternity" in the second premise. the argument is invalid because the same term is used in two different senses. we actually have four terms in the argument instead of the required three terms. this is the fallacy of equivocation and you can learn moreabout these by reading the essays on the nonsense traps.

rule number 2

neither the major nor minor term may be a universal in the conclusion, if it was only a particular term in the premises.
consider the following argument:

all dogs are mammals.
no horses are dogs.
therefore, no horses are mammals.
notice here that the term "mammals" in the conclusion is distributed. but the term "mammals" is not distributed in the major premise. the major premise is an "a" sentence and does not distribute its predicate. the conclusion, however, is an "e" sentence which does distribute its predicate. the premises do not tell us about all mammals, but the conclusion does tell us about all mammals.

the above argument is invalid because of what is called the fallacy of illicit process or illicit distribution. the fallacy of "illicit process" is one in which the conclusion tries to give us more information than is contained in the premises.

rule number 3

the middle term may not appear in the conclusion.
the middle term is the term common to both premises and, therefore, cannot occur in the conclusion. the middle term is the term which mediates between the major and the minor premises so as to relate the subject term and the predicate term in the conclusion. the comparison of the subject and predicate terms with the middle term occurs in the premises and not in the conclusion. there is no place in the conclusion for the middle term; its purpose has been fulfilled in the premises.

consider the following example:

socrates was a philosopher.
socrates was poor.
therefore, socrates was a poor philosopher.
 m + p
m + s

m + sp

this argument is invalid. the middle term (socrates) appears in the conclusion.

rule number 4

the middle term must be distributed at least once, that is, used as a universal.
study the following syllogism:

all men are human beings.
all women are human beings.
therefore, all women are men.
 up + pm
us + pm

us + pp

in the above argument the middle term is "human beings." since it is the predicate term in both premises, and since both premises are "a" sentences, neither premise distributes (uses as a universal) its predicate. the middle term is, therefore, undistributed.

even though it's true that all men are human beings and that all women are human beings, it does not follow that everything which is a man is also a woman. the two premises are not connected by the middle term.

here is an example, by the way, of something we have already discussed. both of the premises in the above argument, taken independently, happen to be true. but the argument is invalid because of what is called the fallacy of the undistributed middle. while the premises are true, the argument is, nevertheless, invalid.

we need to recall the distinction between truth or falsity on the one hand, and validity or invalidity on the other hand. to say that an argument is valid does not mean that the premises and the conclusion are true. validity depends are how one reasons, not what one reasons.

consider the following argument:

if 1 is greater than 2, and
if 2 is greater than 3,
then 1 is greater than 3.
both premises and the conclusion are false in the above syllogism. the argument, however, is valid. if the premises were true, it would be impossible for the conclusion to be false.

consider the following argument:

if 3 is greater than 2, and
if 2 is greater than 1,
then 3 is greater than 1.
here the premises are true and the conclusion is necessarily true because it logically follows from the premises. it's important to remember than truth and validity are two different things.

consider the following argument:

all oregonians are mortal.
all americans are mortal.
therefore, all oregonians are americans.
the premises in this argument are true. the conclusion is also true. the argument is, however, invalid. the middle term "mortal" is not distributed, and the argument violates the fourth rule necessary for a valid syllogism.

rule number 5

if both premises are affirmative, the conclusion must also be affirmative.
consider the following argument:

all oranges are sweet.
some fruit are oranges.
therefore, some fruit is not sweet.
 m + p
s + m

s - p

if both premises are affirmative, it is affirmed that both the major and the minor terms are identified with the middle term. the principle of identity requires that the conclusion state expressly that the major and minor terms are identified with each other. this must be done in an affirmative conclusion. a negative conclusion would state something which is not contained in the premises and would violate the principle of contradiction.

the above argument is invalid since the conclusion is a negative proposition and cannot follow from the premises which are affirmative.

rule number 6

both premises may not be negative; one at least must be affirmative.
consider the following argument:

no cats are cold-blooded animals.
no cold-blooded animals are capable of purring.
therefore, no cats are capable of purring.
in this argument we have two negative premises. when two negative premises appear in an argument, we fail to establish any connection between the terms of the argument. in order to show that no cats are capable of purring, we have to show that cats belong to the class of cold-blooded objects. this would be to assert that "all cats are cold-blooded," an affirmative premise which goes beyond the information given to us in the premises. no conclusion can follow.

consider the following argument:

no animal is a stone.
no diamond is an animal.
therefore, no diamond is a stone.
 m - p
s - m

s - p

we have two negative premises. no conclusion can follow. at least one of the premises must be affirmative so that either the major or the minor term will be identified with the middle term.

rule number 7

if either premise is negative, the conclusion must be negative; if one of the premises is a particular proposition, the conclusion must be a particular proposition.
consider the following argument:

all cannibals are primitive people,
some oregonians are not primitive people,
therefore, some oregonians are cannibals.
here we have a middle term that is distributed and no term is distributed in the conclusion which is not distributed in the premises. furthermore, at least one premise is affirmative.

the argument, however, is invalid, since the premises are true and the conclusion is false. the fallacy is in inferring that because some oregonians are excluded from a certain group, some must belong to the group. this does not follow, since even though some oregonians may be excluded from a group, all oregonians may be excluded.

rule number 8

no conclusion can be drawn from two particular premises; one at least must be a universal proposition.
consider the following argument:

some men are taxi drivers.
some americans are men.
therefore, some americans are taxi drivers.
 pm + pp
ps + pm

ps + pp

this argument appears to be legitimate but is really invalid. both premises in the argument above are particular affirmative propositions. all four terms used in the premises are particular. the two predicates are particular because both premises are affirmative and the predicates of affirmative propositions are always particular terms.

if all four terms (americans, men 1, men 2, taxi drivers) are particular, the middle term (men) is never taken as a universal. this is a violation of rule number 4. any syllogism which consists of two particular affirmative premises will always be invalid.

now consider this argument:

some writers are american citizens.
some oregonians are not writers.
therefore, some oregonians are not american citizens.
 pm - up
ps + pm

ps - up

this, of course, is false. and it is also invalid. why? of the four terms (oregonians, writers 1, writers 2, american citizens), three are particular and one is universal. the middle term (writers) must be a universal at least once according to rule number 4. consequently, the major term (american citizens) will be particular in the premise.

according to rule number 7, the conclusion must be negative, and this negative conclusion will make the major term (american citizens) universal. this, however, is a violation of rule number 2.

if we want to avoid this fallacy called the illicit process of the major, and let the major term (american citizens) be the one universal term present in the premises, there is no other universal term left in the premises. but the middle term (writers) will be used twice as a particular term and this violates rule number 4.

no matter how this argument is made, it will contain either a fallacy of illicit major or a fallacy of the undistributed middle. no conclusion can be drawn from two particular premises.

it is sometimes difficult to determine a fallacy in an argument since arguments in conversation, books, and articles are usually so cloaked with excess words that a fallacy is not easily detected. if you doubt the validity of an argument, arrange it in strict logical form and then analyze it according to the rules.

the main fallacies to be guarded against because they easily escape notice are:

the use of four terms instead of the required three;
the undistributed middle;
the illicit major or minor term;
and two particular premises used to draw a universal conclusion.