This lesson is from Bachhuber, Andrew H., S.J. 1957. "Chapter 7: The Simple Categorical Syllogism," An Introduction to Logic. New York: Appleton-Century-Crofts, Inc. Pp. 89-136.
The simple categorical syllogism is the most important elementary type of syllogism. It consists of three categorical, or attributive, propositions so put together that the subject (t) and predicate (T) of the conclusion are united or separated through the intermediacy of a middle term (M). For instance, in the following example “dog” and “mortal” are united through the union of each of them with animal.
Every animal is mortal; but every dog is an animal; therefore every dog is mortal.
The first proposition of this example is the major premise; the second proposition is the minor premise; and the third is the conclusion. “Mortal,” the predicate of the conclusion, is the major term; “dog,” the subject of the conclusion, is the minor term; and “animal,” which occurs in both the premises but not in the conclusion, is the middle term.
a. Major Term
The major term is the predicate of the conclusion. The major term must occur in the conclusion and in one of the premises, generally the first, which is therefore called the MAJOR PREMISE. We shall designate the major term by T, or, to display the structure of a syllogism more graphically, by a rectangle.
b. Minor Term
The minor term is the subject of the conclusion. The minor term must occur in the conclusion and in the premise in which the major term does not occur. This MINOR PREMISE is often introduced by the adversative conjunction “but” (because in controversy it introduces a turn of thought contrary to the expectations of an opponent). We shall designate the minor term by t, or, to display the structure of a syllogism more graphically, by an ellipse.
c. Middle Term
The middle term occurs in each of the premises but not in the conclusion. In the major premise it occurs in conjunction with the major term; and in the minor premise, in conjunction with the minor term. It is the medium through which the major and minor terms are united in the affirmative syllogism and separated in the negative syllogism. As opposed to the middle term, the minor and major terms are called the EXTREMES.
[Note: These preliminary definitions of major, minor, and middle term are practical working definitions that are suitable for our present purposes but need some qualification. Strictly speaking, in an affirmative syllogism, the major term is the term with the greater extension and the minor term with less extension (major and minor are the Latin words for “greater” and “lesser,” respectively). In the so-called fourth figure the major term is the subject and the minor term the predicate of the conclusion. However, in the practical working out of syllogisms we can pass over these subtle distinctions and call the predicate of the conclusion the major term and its subject the minor. In an affirmative syllogism of the first figure (which is the “perfect” figure), the middle term gets its name not only because it is a medium for uniting the major and minor terms but also because it is midway between them in extension.]The relationship of the terms of a syllogism towards one another—and consequently the validity of a syllogism—can often be made evident through the use of diagrams. We shall now examine two syllogisms and their accompanying diagrams, first an affirmative syllogism and then a negative syllogism. Consider the following affirmative syllogism and the accompanying diagram:
Every animal is mortal; but every dog is an animal; therefore every dog is mortal.
“Dog” (t) is drawn into the extension of “animal” (M), and “animal” (M) is drawn into the extension of “mortal (being)” (T). Since every dog is an animal, what is true of every animal must also be true of every dog. Hence, since every animal is mortal, every dog must be mortal too.
Now consider the following negative syllogism and the accompanying diagram:
No animal is an angel; but every dog is an animal; therefore no dog is an angel.
“Dog” (t) is drawn into the extension of “animal” (M); “animal” (M) is completely excluded from the extension of “angel” (T). Since every dog is an animal, what is denied of every animal must also be denied of every dog. Hence, just as no animal is an angel, so too no dog is an angel.
In analyzing a syllogism, first pick out the conclusion, noting its subject (t) and predicate (T). Then, if you are analyzing a categorical syllogism, look for the premise in which the minor term (t) occurs; this is the minor premise and should contain the minor (t) and middle (M) terms. Then look for the premise in which the major term (T) occurs; this is the major premise and should contain the major (T) and middle (M) terms. At first, most of our examples will be arranged in the order of major premise, minor premise, conclusion, which is the order required by the logical form of the syllogism. But very few syllogisms in newspapers, magazines, and books—with the exception of logic books—are arranged in this order. Sometimes the minor premise comes first; perhaps even oftener the conclusion is placed first. Indeed, the latter is a very natural order, for it first centers our attention on what is to be proved and then on the proof itself.
We shall now state and explain the general rules of the categorical syllogism. Pay special attention to the headings that are supplied to help you see the order of the rules. Note that the first four rules are rules of the terms—Rules 1 and 2 treating of their number and arrangement, Rules 3 and 4 treating of their quantity; and that the remaining rules are rules of the propositions—Rules 5, 6, and 7 treating of the quality of the propositions. Rules 8 and 9 of their quantity, and Rule 10 of their existential import.
The following outline will be of great help both in remembering the rules and in seeing their relationship to one another:
a. The Rules of the Terms
1. THEIR NUMBER AND ARRANGEMENT
(1) Their Number: ...
(2) Their Arrangement: ...
2. THEIR QUANTITY, OR EXTENSION
(3) The Quantity of the Minor and Major Terms: ...
(4) The Quantity of the Middle Term: ...
b. The Rules of the Propositions
1. THEIR QUALITY
(5) If both premises are affirmative, ...
(6) If one premise is affirmative and the other negative, ...
(7) If both premises are negative, ...
2. THEIR QUANTITY
(8) At least one premise must be ...
(9) If a premise is particular, the conclusion must be ...
3. THEIR EXISTENTIAL IMPORT
(10) If the actual real existence of a subject has not been asserted in the premises, ...
a. The Rules of the Terms
1. THEIR NUMBER AND ARRANGEMENT
Rule 1. There must be three terms and only three--the major term, the minor term, and the middle term.
The necessity of having only three terms follows from the very nature of a categorical syllogism, in which a minor (t) and a major (T) term are united or separated through the intermediacy of a third term, the middle term (M).
The terms must have exactly the same meaning and (except for certain legitimate changes in supposition) must be used in exactly the same way in each occurrence. A term that has a different meaning in each occurrence is equivalently two terms. We must be especially on our guard against ambiguous middle terms.
[Note: Supposition of terms is a property that terms acquire in propositions and by which they stand for a definite one of the various things they can stand for. Notice that in Examples 1 and 2 the supposition of "man" changes; the word stands for one thing in the major premise and for a different thing in the minor premise.]This rule is violated in each of the following examples.
1. Men must eat; but the picture on the wall is a man; therefore the picture on the wall must eat.
In Example 1 the pseudo middle term ("men" and "man") has two meanings and is therefore really two terms. In its first occurrence it signifies men of flesh and blood; however, the picture on the wall is not a man of flesh and blood but is merely called a man by extrinsic denomination because of its resemblance to a real man.
2. "Man" rimes with "ban"; but you are a man; therefore you rime with "ban."
In Example 2 "man" likewise stands for something different in each occurrence. The word "man" rimes with "ban"; however, you are not the word "man" but a being having a human nature.
3. Wisconsin is next to Illinois; but Illinois is next to Missouri; therefore Missouri is next to Wisconsin.
Example 3 has six terms: "Wisconsin," "next to Illinois," "Illinois," "next to Missouri," "Missouri," and "next to Wisconsin."
4. A short and thin man cannot weigh 250 pounds; but John is short; therefore John cannot weigh 250 pounds.
In the major premise the term "a short and thin man" means "a man who is both short and thin." But the argument proceeds as though the term meant "a man who is either short or thin." On account of this ambiguity, the term "a short and thin man" is equivalent to two terms, and the syllogism incurs the fallacy of four terms.
Rule 2. Each term must occur in two propositions. The major term must occur in the conclusion, as predicate, and in one of the premises, which is therefore called the major premise. The minor term must occur in the conclusion, as subject, and in the other premise, which is therefore called the minor premise. The middle term must occur in both premises but not in the conclusion. Hence, there must be three propositions.
The necessity of having three terms arranged in this way in three propositions also follows from the very nature of a categorical syllogism. Two propositions (the premises) are required for the middle term to fulfill its function of uniting or separating the minor and major terms, and a third proposition (the conclusion) is required to express the union or separation of the minor and major terms.
2. THE QUANTITY, OR EXTENSION, OF THE TERMS
Rule 3. The major and minor terms may not be universal (or distributed) in the conclusion unless they are universal (or distributed) in the premises.
The reason for this rule is that we may not conclude about all the inferiors of a term if the premises have given us information about only some of them. The conclusion is an effect of the premises and must therefore be contained in them implicitly; but all are not necessarily contained in some—at least not by virtue of the form of argumentation alone.
Violation of this rule is called either extending a term or an illicit process of a term. There is an illicit process of the major term if the major term is particular in the premise but universal in the conclusion; and an illicit process of the minor term, if the minor term is particular in the premise but universal in the conclusion.
Note that there is no illicit process if the major or minor term is universal in the premises and particular in the conclusion. To go from a particular to a universal is forbidden—just as on the square of opposition; but to go from a universal to a particular is permissible
We shall now examine five examples of syllogisms in which this rule is violated. The conclusions of some of them are true by accident; that is, the conclusions do not actually flow from the premises but are true for some other reason. It will be helpful to display the logical form of these syllogisms and to mark the quantity of each proposition and of the predicate terms as well. An arrow is used to indicate an illicit process of a term.
1. All dogs are mammals; but no men are dogs; therefore no men are mammals.
A consideration of the following diagram will help us see why this syllogism is invalid and how the rules of the categorical syllogism are intimately related to the rules governing eduction and oppositional inferences.
All dogs are mammals; therefore some mammals are dogs. If I is true, O is doubtful; hence, there might be some mammals that are not dogs, and men might be among them. In other words, it might be possible not to be a dog and still be a mammal. From the mere fact, then, that man is not a dog, you cannot tell whether or not he is a mammal.
The conclusion of Example 1 (“therefore no men are mammals”) is false. Example 2 has exactly the same form as Example 1; the conclusion, however, is true by accident.
2. Horses are irrational animals; but men are not horses; therefore men are not irrational animals.
3. Every circle is round; but every circle is a figure; therefore every figure is round.
Example 3 has an illicit process of the minor term. The minor term “figure” is particular, or undistributed, in the minor premise where it is the predicate of an affirmative proposition, but universal, or distributed, in the conclusion where it is universalized by the quantifier “every.”
What is wrong with Examples 4 and 5?
4. Some round things are circles; but some figures are not circles; therefore some figures are not round.
5. A good stenographer is a good typist; but Mary is not a good stenographer; therefore Mary is not a good typist.
Compare Example 5 with Examples 1 and 2. From the fact that every good stenographer is a good typist, it does not follow that every good typist is also a good stenographer, but only that some good typist is a good stenographer. Hence, from the mere fact that Mary is not a good stenographer, you cannot tell whether or not she is a good typist. She may be one of the good typists—if there are any—who are not good stenographers.
Note that an illicit process of the minor term is never incurred if the conclusion is particular and that an illicit process of the major term is never incurred if the conclusion is affirmative.
Rule 4. The middle term must be universal, or distributed, at least once. [Note: A syllogism is valid if the middle term is singular in both occurrences.]
The reason for this rule is that when the middle term is particular in both premises it might stand for a different portion of its extension in each occurrence and thus be equivalent to two terms, and therefore fail to fulfill its function of uniting or separating the minor and major terms.
We shall now examine five examples of syllogisms in which this rule is violated.
1. A dog is an animal; but a cat is an animal; therefore a cat is a dog.
A consideration of the diagram shows us that animal stands for a different portion of its extension in each of the premises and thus does not unite “cat” and “dog.” Both of them are animals but not the same animals.
2. Many rich men oppress the poor; but Jones is a rich man; therefore Jones oppresses the poor.
In the major premise “rich men” stands only for the rich men enclosed within the big circle representing all those who oppress the poor; but for all we know, “rich man” in the minor premise stands for a rich man outside the big circle. Hence, we cannot tell whether Jones is one of the rich men who oppress the poor or one of those (if there are any) who do not oppress the poor. [Note: Of course, there are rich men who do not oppress the poor, but you cannot infer this from the fact that some rich men oppress the poor--if I is true, O is doubtful.]
The same diagram will throw light on Example 3.
3. Many rich men do not oppress the poor; but Jones is a rich man; therefore Jones does not oppress the poor.
Example 4 note the inverted order of the subject and predicate in the premises.
4. Blessed are the poor in spirit; but blessed are the meek; therefore the meek are poor in spirit.
5. A sick man needs medicine; but castor oil is medicine; therefore a sick man needs castor oil.
Notice that the middle term may be universal in both occurrences, but has to be universal only once.
Violation of this rule is often called the fallacy of the undistributed middle.
b. The Rules of the Propositions
1. THE QUALITY OF THE PROPOSITIONS
Rule 5. If both premises are affirmative, the conclusion must be affirmative.
The reason for this rule is that affirmative premises either unite the minor and major terms, or else do not bring them into relationship with one another at all—as when there is an undistributed middle. In neither case may the major term be denied of the minor term. Hence, to get a negative conclusion you must have one—and only one—negative premise.
The following four examples illustrate either real or apparent violations of this rule.
1. All sin is detestable; but some pretense is sin; therefore some pretense is not detestable.
“Some pretense is detestable” is a valid conclusion of the premises; note how we tend to proceed invalidly from an implicit I to an O. This example, besides violating Rule 5, also illustrates an illicit process of the major term.
As soon as you see that both premises are affirmative but the conclusion negative, you can be sure that your syllogism is invalid. Be on your guard, however, against apparent affirmative or negative propositions. The following syllogism is valid although it seems to violate this rule.
2. Animals differ from angels; but man is an animal; therefore man is not an angel.
Example 2 is valid because "differ from" is equivalent to "are not."
3. Man is two-legged; but a horse is four-legged; therefore a man is not a horse.
Example 3 is also valid—at least materially. When we say that man is two-legged, we do not mean that he has at least two legs (and maybe more) but that he has two and only two. Hence we imply that he is not four-legged.
4. A lion is an animal; but a fox is an animal; therefore--since the middle term "animal" is undistributed--a fox is not a lion.
Rule 6. If one premise is affirmative and the other negative, the conclusion must be negative.
The reason for this rule is that the affirmative premise unites the middle term with one of the extremes (that is, with either the minor or the major term) and the negative premise separates the middle term from the other extreme. Two things, of which the one is identical with a third thing and the other is different from that same third thing, cannot be identical with one another. Hence, if a syllogism with a negative premise concludes at all, it must conclude negatively. Thus, Example 1 is invalid.
1. Every B is a C; but some A is not a B; therefore some A is a C.
From the fact that some A is not a B you cannot tell whether or not some A is a C. It is possible that no A is either a B or a C.
There are apparent exceptions to this rule, but they will cause no difficulty if we keep in mind that many negative propositions are equivalent to affirmative propositions and can be changed into them by one or the other kinds of immediate inference. Number 2, for instance, is a valid syllogism.
2. Dogs are not centipedes; but hounds are not dogs; therefore hounds differ from centipedes.
The conclusion is equivalently negative, since "differ from" is here equivalent to "are not."
Rule 7. If both premises are negative--and not equivalently affirmative--there is no conclusion at all.
To fulfill its function of uniting or separating the minor and the major term, the middle term must itself be united with at least one of them. But if both premises are negative, the middle term is denied of each of the extremes and we learn nothing about the relationship of the extremes towards one another. Some examples and a diagram will make this clear.
1. A stone is not an animal; but a dog is not a stone; therefore a dog is not an animal.
The non-animals, represented by the horizontal lines, and the non-stones, represented by the vertical lines, overlap; hence, from the fact that a dog is not a stone, you cannot tell whether or not a dog is an animal. Some non-stones are animals, and others are not.
Example 2 has the same form as Example 1; but the conclusion is true by accident.
2. A dog is not a cat; but a rat is not a dog; therefore a rat is not a cat.
3. The poor do not have security; but these men are not poor; therefore these men have security.
4. No murderer shall enter into the kingdom of heaven; but John is not a murderer; therefore John shall enter into the kingdom of heaven.
5. Insulators are not conductors of electricity; but glass is not a conductor of electricity; therefore glass is an insulator.
The following examples illustrate apparent but not real violations of this rule. Note how the mind sometimes spontaneously substitutes for a proposition its obverse or some other kind of immediate inference.
1. No B is not a C; but no A is not a B; therefore no A is not a C (or: therefore every A is a C).
The major premise of Example 1 is equivalent to “Every B is a C”; the minor premise is equivalent to “Every A is a B”; and the conclusion, to “Every A is a C.”
2. What is not material is not mortal; but the human soul is not material; therefore the human soul is not mortal.
In Example 2, the term “mortal” in the major premise is denied of the term “what is not material,” and in the minor premise the human soul is said to be “something that is not material” (obversion). Thus, if what is not material is not mortal and if the human soul is something that is not material, it must also be something that is not mortal.
3. Non-voters are not eligible; but John is not a voter; therefore John is ineligible.
4. No man is not mortal; but no American citizen is not a man; therefore no American citizen is immortal.
5. What is not metallic is not magnetic; but carbon is not metallic; therefore carbon is not magnetic.
2. THE QUANTITY OF THE PROPOSITIONS
The rules on the quantity of the propositions are corollaries of the rules on the quantity of the terms.
Rule 8. At least one premise must be universal.
[Note: An expository syllogism (that is, a syllogism whose middle term is singular) may have two particular premises if the middle term is the predicate of each of them.We shall consider every possible arrangement of the terms in categorical syllogisms in which both the premises are particular propositions and see how in every arrangement either Rule 3 or Rule 4 is violated.
1—If both premises are affirmative, the middle term is particular in each occurrence, and Rule 4 is violated. Let a, e, i, and o indicate the quality and quantity of the propositions and t, M, and T the minor, middle, and major terms, respectively. We shall now diagram the four possible arrangements.
If the middle term (M) is subject, it is particular, because we are here dealing with particular propositions. If the middle term is predicate, it is also particular, because the predicate of an affirmative proposition is particular.
2—If one premise is affirmative and the other negative, either Rule 3 or Rule 4 is violated. If the middle term is the predicate of the negative premise, there will always be an illicit process of the major term and thus Rule 3 will always be violated, as illustrated in the first and second diagrams. Rule 4, which requires that the middle term be universal at least once, is violated in two cases: first, if the middle term is the subject of both premises and, second, if the middle term is the subject of the negative premise and the predicate of the affirmative premise, as is illustrated in the third and fourth diagrams.
Rule 9. If a premise is particular, the conclusion must be particular.
According to Rule 3, the minor term may not be universal in the conclusion unless it is universal in the minor premise. But an examination of cases reveals that in a valid syllogism having a particular premise the minor term can never be universal in the minor premise. Let us consider syllogisms whose major and minor premises, respectively, are I-E, A-O, and I-A.
The combination, I-E, is always invalid because, as the following diagram shows, it always contains an illicit process of the major term.
A-O has an illicit process of the major term if the major term is the predicate of the major premise and an undistributed middle whenever the middle term is predicate of the major premise and subject of the minor premise.
I-A has an undistributed middle if the middle term is the predicate of the minor premise.
We have examined the arrangements which give us a minor term that is universal in the minor premise, and have discovered that all of them contain an illicit process of the major term or an undistributed middle.
Hence, in syllogisms having a particular premise, we can conclude validly only when the minor term is particular in the premises; and, according to Rule 3, when it is particular in the premises, it must also be particular in the conclusion. If the minor premise is I, the minor term may be either the subject or the predicate; if the minor premise is A, the minor term must be the predicate; if it is O, the minor term must be the subject. As we have already seen, the minor premise may never be E. The following schema gives us a synopsis of all the valid forms of syllogisms having a particular premise. We can see at a glance that in all of them the conclusion must be particular.
3. THE EXISTENTIAL IMPORT OF THE PROPOSITIONS
Rule 10. The actual real existence of a subject may not be asserted in the conclusion unless it has been asserted in the premises.
The reason for this rule is the general principle that nothing may ever be asserted in the conclusion that has not been asserted implicitly in the premises. This rule takes us out of the domain of formal logic, which does not consider existence except incidentally. We mention it only as a practical aid to argumentation.
A Note on the Expository Syllogism. An expository syllogism differs from an ordinary syllogism in that its middle term is singular in both premises. It is not an inference at all in the strictest sense of the word but rather an appeal to experience. As its name suggests, it “exposes” a truth to the senses by setting an example before the mind. An expository syllogism is useful for refuting A and E propositions by establishing their contradictories. For instance, the statement “All wood floats” can be refuted as follows:
This does not float; but this is wood; therefore not all wood floats.
Or if someone were to assert "No Greek was a philosopher," he could be refuted as follows:
Socrates was a philosopher; but Socrates was a Greek; therefore some Greek was a philosopher.
The expository syllogism follows the general rules of the categorical syllogism except on two counts: the first is that the middle term is singular and not universal; the second is that both premises may be particular if the middle term is the predicate of each of them. For instance, in the following example, both premises are particular but the syllogism is nevertheless valid:
One of the largest cities in the world is Bombay; but one of the largest port cities is Bombay; therefore one of the largest port cities is one of the largest cities in the world.
Obviously, if the middle term were not singular in this example, it would have to be particular since it is the predicate of an affirmative proposition, and the syllogism would be invalid. But this fallacy is avoided because it is singular.
Logical form, as we have seen, is the basic structure, or the basic arrangement of the parts, of a complex logical unit. Now the categorical syllogism is a complex logical unit having as its parts (a) terms and (b) propositions in which these terms are affirmed or denied of one another. The logical form, then, of the categorical syllogism includes (a) the arrangement of the terms—which is called figure—and (b) the arrangement of the propositions according to quality and quantity—which is called mood.
A study of the logical forms of the categorical syllogism will serve several purposes. At present its chief fruit will be to deepen our understanding of the general rules of the syllogism and to give us practice in applying them. Later on it will serve as a background for the consideration of the principles underlying the syllogism and for the reduction of syllogisms of the second, third, and fourth figures to syllogisms of the first figure.
First we shall explain the general nature of the figures and moods; then we shall derive the valid moods of each of the figures; finally we shall show why the first figure is the perfect figure.
a. General Nature of the Figures and Moods
1) FIGURE. The figure of a categorical syllogism consists of the arrangement of the terms in the premises. There are four figures and each is defined by the position of the middle term. In the first figure, the middle term is the subject of the major premise and the predicate of the minor premise (sub-pre). In the second figure, the middle term is the predicate of both premises (pre-pre). In the third figure, the middle term is the subject of both premises (sub¬ sub). In the fourth figure, the middle term is the predicate of the major premise and the subject of the minor premise (pre-sub). The so-called fourth figure differs only accidentally from the first—it is an inverted first figure—but for practical purposes we shall treat of it separately.
2) MOOD. The mood of a categorical syllogism consists of the disposition of the premises according to quality and quantity. There are sixteen possible arrangements of the premises according to qual¬ ity and quantity. The possible arrangements are:
By applying the general rules we shall see that only eight of these arrangements are ever valid. Rule 7 (“If both premises are negative, there is no conclusion”) excludes e-e, e-o, o-e, and o-o. Rule 8 (“at least one of the premises must be universal”) excludes i-i, i-o, o-i, and o-o—the last of which was already excluded by Rule 7. Rule 3 ( The major and minor terms may not be universal in the conclusion unless they are universal in the premises”) excludes i-e, for the major term would be universal in the conclusion but particular in the premise. The moods indicated by the capital letters remain:
But not all of these are valid in every figure.
b. The Valid Moods of Each Figure
We shall now apply the general rules of the categorical syllogism to determine the valid moods of each figure.
1) THE FIRST FIGURE. In the first figure the middle term is the subject of the major premise and the predicate of the minor premise (sub-pre).
As we saw above, the eight possible moods are:
By experiment we shall find that of these eight moods only the four indicated by the capital letters are valid.
Rule 3 (“The major and minor terms may not be universal in the conclusion unless they are universal in the premises”) excludes a-e and a-o. In the premise, the major term is the predicate of an affirmative proposition and therefore particular; but in the conclusion it is the predicate of a negative proposition and therefore universal. Rule 4 (“The middle term must be universal at least once”) excludes i-a and o-a. As subject of an I or O proposition, the middle term is particular; and as predicate of an A proposition, it is also particular in its second occurrence.
Only four moods remain:
They conclude in:
An inspection of these moods enables us to draw up the following RULES OF THE FIRST FIGURE:
The major premise must be universal (A or E).
The minor premise must be affirmative (A or I).
2. THE SECOND FIGURE. In the second figure, the middle term is the predicate of both premises (pre-pre).
T M
t M
Beginning with the eight possible moods, we shall proceed, just as with the first figure, by applying the general rules to each of them. The eight possible moods are:
a A a A E E i o
a E i O A I a a
Whenever both premises are affirmative, the middle term will be particular in each occurrence; hence. Rule 4 (“The middle term must be universal at least once”) excludes a-a, a-i, and i-a. Rule 3 (“The major and minor terms may not be universal in the conclusion unless they are universal in the premises”) excludes o-a; as the subject of O, the major term is particular in the premise but, as the predicate of a negative proposition, is universal in the conclusion.
Only four moods remain:
They conclude in
An inspection of these moods enables us to draw up the following RULES OF THE SECOND FIGURE:
The major premise must be universal (A or E).
One premise must be negative.
3. THE THIRD FIGURE. In the third figure the middle term is the subject of both premises (sub-sub).
We shall proceed just as with the first and second figures. The eight possible moods are:
An inspection of the forms given above reveals that Rule 3 (“The major and minor term may not be universal in the conclusion unless they are universal in the premises”) excludes a-e and a-o. Note that every conclusion is particular.
There remain six moods:
They conclude in:
An inspection of these moods and conclusions enables us to draw up the following RULES OF THE THIRD FIGURE:
The minor premise must be affirmative.
The conclusion must be particular.
4. THE FOURTH FIGURE. In the fourth figure, the middle term is the predicate of the major premise and the subject of the minor premise (pre-sub).
The eight possible moods are:
Rule 4 (“The middle term must be universal at least once”) excludes a-i and a-o. Rule 3 (“The major and minor terms may not be universal in the conclusion unless they are universal in the premises”) excludes o-a.
There remain five moods
They conclude in
An inspection of these moods and conclusions enables us to draw up the following RULES OF THE FOURTH FIGURE (inverted first figure):
If the major premise is affirmative, the minor premise must be universal.
If the minor premise is affirmative, the conclusion must be particular.
If a premise (and the conclusion) is negative, the major premise must be universal.
Violation of the first rule involves an undistributed middle; of the second, an illicit process of the minor term; and of the third, an illicit process of the major term.
c. The Perfect Figure
The first figure is considered the perfect figure. In the first place, as we shall see later, the principles underlying the categorical syllogism regulate syllogisms of the first figure most directly and most obviously. Secondly, only the first figure can have a universal affirmative conclusion, which is the kind of conclusion with which science is principally concerned. Thirdly, the first figure is the only figure in which the middle term gives, or at least can give, the reason why what is signified by the major term belongs to what is signified by the minor term. [Note: In the first figure the middle term can give not only the ratio cognoscendi (the reason for knowing) of the conclusion but also the ratio essendi (the reason for being). It can give not only the reason why we know that the conclusion is true but also the reason why it actually is true—that is, the reason why what is signified by the minor term has the attribute signified by the major term.] Examine, for instance, the following syllogism:
A spiritual substance is immortal; but the human soul is a spiritual substance; therefore the human soul is immortal.
In this syllogism, the middle term “spiritual substance” contains the reason why immortality belongs to the human soul—it is that the human soul is a spiritual substance, and a spiritual substance, as such, must be immortal. For these reasons the first figure is the figure of scientific and philosophical demonstration.
a. The Problem and a Brief Answer
Now that we are familiar with the mechanics of the categorical syllogism and have made a study of the various figures, we are ready to take up the principles underlying the logical movement of the categorical syllogism. We shall endeavor to penetrate more deeply into the nature of the syllogism by trying to grasp the principles that are operative every time a minor and major term are united (or separated) through the intermediacy of a middle term. Previously we grasped the validity of individual syllogisms and of various types of syllogisms but without adverting to the most general principles illustrated in each; in the present section we shall try to dis¬ engage these principles from the examples in which they are illustrated and to enunciate them explicitly. For instance, from our previous study we know that the following syllogism is valid:
A dog is an animal; but a hound is a dog; therefore a hound is an animal.
We can tell that this syllogism is valid because, aided by the rules of the syllogism, we clearly grasp the relationship of the terms “hound,” “dog,” and “animal”—we clearly see that if a dog is an animal and a hound is a dog, a hound must also be an animal. We see this directly and need not appeal to any principle at all. Yet, on reflection, we see that this argument fulfills certain basic conditions —that certain basic principles underlie its logical movement—and that the fulfillment of these conditions is the reason for its validity. To discover what these conditions and principles are is the aim of our present inquiry.
A PRINCIPLE is something that is first and from which something else either is or becomes or is known. A PRINCIPLE OF KNOWLEDGE is knowledge from which other knowledge flows or on which other knowledge somehow depends. The premises of a syllogism, for instance, are a principle of the conclusion because the conclusion flows from them and because (at least in some cases) our knowledge of the conclusion is dependent on our knowledge of the premises.
Notice that a principle of knowledge is not necessarily known first chronologically. Chronologically, we may first know particular exemplifications of the principle; then, by generalization, we work back to the principle itself (as is the case with the principles of the syllogism).
Some principles are first, or ultimate, principles; others are derived principles. A FIRST PRINCIPLE is first absolutely and is not dependent on any broader principle. A DERIVED PRINCIPLE is first in its own order but not absolutely. It is either a particularization of some broader principle (as are the principles of the identifying and separating third), or a conclusion deduced from premises (as a conclusion of mathematics may be a principle in physics).
In the present section we are concerned with the principles of knowledge of the categorical syllogism. We are concerned not with the principles serving as premises from which the conclusion is deduced but with the principles underlying the logical movement itself.
The first question that we shall endeavor to answer is this: What are the special principles on which the validity of the categorical syllogism (but of no other type of syllogism) ultimately depends? The answer (as we shall soon see) is the PRINCIPLE OF THE IDENTIFYING THIRD in an affirmative syllogism and the PRINCIPLE OF THE SEPARATING THIRD in a negative syllogism. If one of these principles underlies the logical movement of a categorical syllogism, the conclusion must be true if the premises are true.
But we must also answer a second question: How are we to know whether or not the principles of the identifying or separating third underlie a syllogism? Two other principles give the answer. They are the DICTUM DE OMNI (“law of all”) for the affirmative syllogism and the DICTUM DE NULLO (“law of none”) for the negative syllogism. The various rules governing the syllogism are nothing but practical aids for telling whether or not these principles are operative in a syllogism.
The principles of the identifying and separating third are particularized formulae of the more general principles of identity and contradiction (which are the basic principles of absolutely all judgment and inference), phrased in such a way as to be more immediately applicable to the syllogism. Hence, first of all (under b), we shall treat of the principles of identity and contradiction. Next (under c), we shall treat of the principles of the identifying and separating third. Finally (under d), we shall treat of the dictum de omni and the dictum de nullo.
[Note: In a certain sense, sense experience is the principle of all human knowledge, since, as will be explained in the philosophy of human nature, there is nothing in the intellect that was not somehow or other in the senses first. In the order of apprehension (but not in the orders of judgment and inference), being is the principle of all knowledge. ]b. The Principles of Identity and Contradiction
We shall now state the principles of identity and contradiction, which are the absolutely first principles of all judgment and inference, in their most general formulae.
1) THE PRINCIPLE OF IDENTITY has several formulae. The two that seem most correct philosophically are: “What is, is” and “Everything is what it is.” Notice that this principle is true of things as they are in themselves and independently of their being thought of. For this reason it is not only a logical principle but also a metaphysical principle.
2) THE PRINCIPLE OF CONTRADICTION also has several formulae. Sometimes it is enunciated as a metaphysical principle in a formula that is true of things as they are in themselves; sometimes, again, it is enunciated as a purely logical principle in a formula that is not applicable to things as they are in themselves but only to things as they exist in the mind as a result of being known.
Insofar as the principle of contradiction is a metaphysical principle, it has two common formulae. The first and broadest is: “A thing cannot be and not be in the same respect.” The second formula, which is narrower inasmuch as it does not extend to all existence but only to the presence or absence of attributes, is as follows: “A thing cannot both have and not have the same attribute in the same respect.” Notice that the principle does not assert that a thing cannot be at one time and not be at another, or have an attribute in one respect and not have it in another. For instance, there is no contradiction in John’s being good at basketball but not good at dominoes, in his being heavy in comparison with George but not heavy in comparison with Jim, or in his having a puppy as a pet one day but not having it the next. Inasmuch as this principle, as stated in both of these formulae, is true of things as they are in themselves and independently of our thought of them, it is a metaphysical principle.
As a purely logical principle, the principle of contradiction is also expressed in two formulae corresponding to the formulae given above. The first and broadest formula is: “The same thing cannot be both affirmed and denied in the same respect.” The second is: “The same attribute cannot be both affirmed and denied of the same subject in the same respect.” Thus expressed, the principle is a purely logical principle because it is not applicable to things as they are in themselves but only to things insofar as they are known, or mentally reproduced.
The purely logical principle of contradiction is grounded on the metaphysical principle enunciated above—contradictory propositions, in other words, cannot both be true because things cannot be and not be in the same respect. Yet the logical principle is not deducible from the metaphysical principle since it brings in a new element, the impossibility of knowingly asserting each of two contradictories at the same time.
A little reflection will show that, although we have not stated the principle of contradiction explicitly, we have nevertheless made constant use of it throughout our study of inference.
Why, for instance, is the partial conversion of an A proposition (A to I) a formally valid inference, whereas the simple conversion of A (A to A) is invalid? Let us examine an example of the conversion of A to I:
Every dog is an animal; therefore some animal is a dog.
Now, to affirm the antecedent (“every dog is an animal”) and to deny the consequent (“some animal is a dog”) is to affirm and deny the same thing in the same respect and thus to run counter to the principle of contradiction. Let us now examine an example of the conversion of A to A:
Every dog is an animal; therefore every animal is a dog.
This example is obviously invalid; but why? Because to affirm the antecedent and to deny the consequent (or pseudo consequent) does not involve affirming and denying the same thing in the same respect and does not run counter to the principle of contradiction: The fact that every dog is an animal does not exclude the possibility of animals that are not dogs.
An examination of any example of valid inference will show that it is valid precisely because the admission of the antecedent and the denial of the consequent involve a contradiction. Our use of the principle of contradiction is perhaps most obvious in contradictory opposition. The principle of contradiction also governs the categorical syllogism, the conditional syllogism, and so on.
c. The Principles of the Identifying and Separating Third
We shall now treat of the principles of the identifying and separating third, which are specialized, or particularized, statements of the principles of identity and contradiction phrased in a way that is directly applicable to the categorical syllogism and only to it.
1) THE PRINCIPLE OF THE IDENTIFYING THIRD. The principle of the identifying third is stated as follows: “Two things that are identical with the same third thing are identical with one another.”
Notice that in this formula the words “two things” and “third thing” do not refer to three really distinct existing things, but to one thing (or one kind of thing) that is grasped in three distinct concepts. The “three things,” therefore, are three only in the mind, as will be explained below. In the real order, for instance, the very same being is at once a hound, a dog, and an animal.
The following example and diagram are not presented as a proof of the principle but merely as an illustration. If we examine the example and consider the relationship of the terms as displayed in the diagrams, we shall, by insight into this example, clearly understand the principle itself.
Moreover, the explanation given here presupposes the dictum de omni (“law of all”), which will be explained in the next section. Let us now examine the following syllogism and the accompanying diagrams and explanations:
Every dog is an animal; but every hound is a dog; therefore every hound is an animal.
For convenience is diagramming, we shall invert the order of the premises, giving the minor premise first and the major premise second.
What is presented to the mind under the formality of “hound” is the very same thing that is presented under the formality of “dog”; and what is presented under the formality of “dog” is the same as that presented under the formality of “animal.” Let us unite the two parts of our diagram.
What is signified by “hound” has been shown to be identical with what is signified by “animal” because what is signified by each of them is identical with what is signified by “dog.” The very same reality is at once hound, dog, and animal. The middle term “dog” is the identifying third because, through it, the minor term “hound” and the major term "animal" have been identified in the sense explained above.
The principle of the identifying third can be grasped directly through insight into an example and without any appeal to the broader principles of identity or contradiction. However, we can easily show how a denial of the principle of the identifying third implies a denial of the principle of contradiction. If, for instance, after admitting that every dog is an animal and that a hound is a dog, you nevertheless deny that a hound is an animal, you implicitly deny what you have explicitly affirmed, simultaneously asserting both an A and an O proposition. By denying that every hound is an animal you are implicitly asserting that some dog (namely a hound) is not an animal (O), thus denying the universal proposition “every dog is an animal” (A), which you already admitted as a premise.
Notice that we did not say that a “hound” is equal to a “dog” and an “animal,” nor that “hound” is similar to “dog” and “animal”; but that a hound is a dog and therefore is an animal. [Note: The principle of the identifying third should not be confused with the mathematical principle "Two things equal to the same third thing are equal to one another."]
2) THE PRINCIPLE OF THE SEPARATING THIRD. The principle of the separating third is stated thus: “Two things of which the one is identical with the same third thing but the other is not are not identical with one another.”
Notice that one of the two things must be identical with the same third thing and the other not. It is not enough if neither of the two is identical with the same third thing. From the fact, for instance, that neither a cow nor a horse is a man, it is impossible to tell whether or not a cow is a horse. (Recall the rule that a syllogism cannot conclude if both of its premises are negative.)
Let us examine the following syllogism together with the diagrams given below.
Every dog is an animal; but no animal is an angel; therefore no dog is an angel.
What is presented to the mind under the formality of “dog” is the very same thing that is presented under the formality of “animal.” But what is presented to the mind by “animal” is not the same as is presented by “angel.” Let us unite the two parts of the diagram.
What is signified by “dog” has been shown not to be identical with what is signified by “angel,” because it is identical with what is signified by “animal,” whereas what is signified by “angel” is not. The middle term “animal” is the separating third because, through its union with the minor term “dog” and its separation from the major term “angel,” “dog” has been separated from “angel,” as explained above.
The principle of the separating third, just like the principle of the identifying third, is grasped directly through insight into an example and without any appeal to the broader principle of contradiction; yet its denial, like the denial of the principle of the identifying third, implies a denial of the principle of contradiction. If, for instance, after affirming that every dog is an animal and denying that any animal is an angel, you nevertheless affirm that a dog is an angel, you implicitly affirm what you have already explicitly denied, simultaneously asserting both an E and an I proposition. By affirming that some dog is an angel (or denying that no dog is an angel), you are implying that some animal (namely, a dog) is an angel—which is an I proposition; but you have already explicitly asserted that no animal is an angel—which is an E proposition and the contradictory of the preceding I.
d. The "Dictum de Omni" and the "Dictum de Nullo"
How are we to know that the middle term actually fulfills its function of identifying or separating the minor and major terms? How are we to know, for instance, that the same beings are referred to in both occurrences of the term 'dog” in the example used to illustrate the principle of the identifying third, and that the same beings are referred to in both occurrences of the term animal in the example illustrating the principle of the separating third? The dictum de omni and the dictum de nullo (“law of all and law of none”) are the principles that give the ultimate answer to this question.
We are by no means unfamiliar with these principles although we have never stated them explicitly. Indeed, whenever we used diagrams to display the quantitative relationship of the terms of a syllogism, our diagrams were actually a visual aid to telling whether or not the conditions required by these principles were fulfilled. Besides, many of the examples and diagrams we have already used can serve as sufficient evidence for the grasping of these principles. We shall repeat an example and a diagram that we have already used in Section 1 of the present chapter to display the quantitative relationship to one another of the syllogistic terms.
Notice that these principles are not metaphysical principles at all but purely logical principles because they are not applicable to things as they are in themselves but only to things as they are reproduced in the mind: namely, to the relationship of a logical whole to its inferiors.
1) THE DICTUM DE OMNI (“LAW OF ALL”). The dictum de omni (or “law of all”) is the principle operative in the affirmative syllogism. It is stated as follows: “What is predicated of a logical whole may be predicated distributively of each of its inferiors.”
For instance, if “mortal” is predicated of animal as such (and therefore of every animal), it can also be predicated of “dog,” since a dog is an animal.
An inspection of the following diagrams will make this perfectly clear.
“Dog” is drawn into the extension of “animal,” and “animal” is drawn into the extension of “mortal (being).” Since every dog is an animal, what is true of every animal must also be true of every dog. Hence, since every animal is mortal, every dog must be mortal too.
2) THE DICTUM DE NULLO (“LAW OF NONE”). The dictum de nullo (or “law of none”) is the principle operative in the negative syllogism. It is stated as follows: “What is denied of a logical whole may be denied distributively of each of its inferiors.”
For instance, if “angel” is denied of animal as such (and therefore of every animal), it can also be denied of “dog,” since a dog is an animal.
An inspection of the following diagrams will make this perfectly clear.
“Dog” is drawn into the extension of “animal”; “animal” is completely excluded from the extension of “angel.” Since every dog is an animal, what is denied of every animal must also be denied of every dog. Hence, since no animal is an angel, so too no dog is an angel.
The first figure has special demonstrative force because the principles of the categorical syllogism regulate syllogisms of the first figure most directly and most obviously. Consider, for instance, the following affirmative syllogism of the first figure and reflect on the way in which the principles of the syllogism underlie it:
Every dog is an animal; but every hound is a dog; therefore every hound is an animal.
The dictum de omni states that what is predicated of a logical whole may be predicated distributively of each of its inferiors. Now in the major premise, the major term (“animal”) is predicated of a logical whole (the middle term, “dog”); and what is affirmed of this logical whole in the major premise is affirmed of its inferior (“hound”) in the conclusion.
Next consider the following negative syllogism of the first figure and again reflect on the way in which the principles of the syllogism underlie it:
No animal is an angel; but every dog is an animal; therefore no dog is an angel.
The dictum de nullo states that what is denied of a logical whole may be denied distributively of each of its inferiors. In the major premise, the major term (“angel”) is denied of a logical whole (the middle term, “animal”); and what is denied of this logical whole in the major premise is denied of its inferior (“dog”) in the conclusion.
Thus, syllogisms of the first figure are very obviously regulated by the dictum de omni and the dictum de nullo, which are the principles that regulate the syllogism most immediately and which enable us to see whether or not the principles of the identifying third and separating third underlie its movement.
REDUCTION consists in changing a syllogism of an imperfect figure to a syllogism of the first figure.
Its PURPOSE is twofold. Its first purpose is to show that the imperfect figures participate in the demonstrative force of the first figure. In scholastic disputation, if an opponent admitted the truth of the premises of a syllogism but denied its validity, he could be confuted by the reduction of the syllogism to a syllogism of the first figure whose validity all would immediately admit. Its second purpose is to render our knowledge of the categorical syllogism scientific by enabling us to see more clearly how the same set of principles regulates all categorical syllogisms of all figures.
Notice, however, that reduction is not required for recognizing the validity of the so-called imperfect figures. Indeed, we recognized their validity before we studied either the figures or the principles underlying the syllogism. But our knowledge was natural, not scientific.
Reduction is either DIRECT or INDIRECT.
DIRECT REDUCTION consists in changing a syllogism of an imperfect figure to one of the first figure that is the exact equivalent of the original syllogism. For instance, the syllogism “No angel is a dog; but every hound is a dog; therefore no hound is an angel,” which is a syllogism of the second figure, is changed by direct reduction to the following syllogism of the first figure: “No dog is an angel; but every hound is a dog; therefore no hound is an angel.”
INDIRECT REDUCTION rests on the principle “if the consequent is false, the antecedent is false” and consists in showing by a syllogism of the first figure that the denial of the conclusion of a syllogism of an imperfect figure leads to a denial of one of its premises. In the syllogism of the first figure, one premise is the contradictory of the original conclusion; the other premise is one of the premises of the original syllogism; the conclusion is the contradictory of the other premise of the original syllogism. For instance, the following example is a valid syllogism of the second figure:
All hounds are dogs; but some animals are not dogs; therefore some animals are not hounds.
To reduce it to a syllogism of the first figure by indirect reduction, you must: (a) retain the original major premise; (b) use the contradictory of the conclusion as the minor premise; and (c) use the contradictory of the original minor premise as your conclusion—as follows:
All hounds are dogs; but all animals are hounds; therefore all animals are dogs.
If the original syllogism were not valid, the denial of its conclusion would not involve the denial of a premise.
In all, there are nineteen valid moods of the categorical syllogism if you do not count the subalterns of the five concluding in A and E: there are four moods of the first figure, four of the second, six of the third, and five of the fourth (or inverted first). To indicate these nineteen moods, as well as the way of reducing each mood of the imperfect figures to a mood of the first figure, logicians have composed some of the most ingenious mnemonic verses ever written. The verses are Latin hexameters. There are many variants, but the following arrangement, which is found in many English works on logic, is as convenient as any:
Barbara, Celarent, Darii, Ferioque prioris;
Cesare, Camestres, Festino, Baroco secundae;
Tertia Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison habet; quarta insuper addit:
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
“Barbara, Celarent, Darii, Ferio” indicate the four moods of the first figure; “Cesare, Camestres, Festino, Baroco” indicate the four moods of the second figure; and so on. The vowels of every word (except of the italicized words) indicate the quantity and quality of the propositions (that is, whether they are a, e, i, or o) in the order of their occurrence in the syllogism—that is, the first vowel stands for the major premise, the second for the minor premise, and the third for the conclusion. Thus “Celarent” signifies a syllogism of the first figure whose major premise is an E proposition, whose minor is A, and whose conclusion is E.
All the key words in these verses begin with the letters B, C, D, or F. In the words indicating the moods of the second, third, and fourth figures, the first letter indicates the mood of the first figure to which the mood of the imperfect figure is to be reduced. Thus, for instance, “Camestres,” which begins with the letter C, is to be reduced to “Celarent,” which is the mood of the first figure beginning with C.
Of the remaining consonants, s, p, m, and small c have a special meaning:
S signifies that the proposition indicated by the preceding vowel is to be converted by simple conversion.
P signifies that the proposition indicated by the preceding vowel is to be converted per accidens—that is, by changing the quantity from universal to particular, or from particular to universal.
M means that the premises are to be interchanged so that the original major premise becomes the minor and vice versa.
Small c, when it occurs in the body of a word, means that the mood cannot be reduced directly but only indirectly. The contradictory of the conclusion is to be substituted for the premise indicated by the vowel after which the c is placed. It occurs twice: in “Bocardo” and “Baroco.” Notice, however, that all moods can be reduced by indirect reduction and that these two receive special mention because they can be reduced in no other way.
We shall now give some examples of reduction. Examples 1 and 2 illustrate direct reduction; Examples 3 and 4 illustrate indirect reduction.
Example 1 is a syllogism in “Camestres”—that is, a syllogism of the second figure whose major premise is an A proposition, whose minor is E, and whose conclusion is E.
All true democracies are free countries; but no totalitarian states are free countries; therefore no totalitarian states are true democracies.
The C of “Camestres” indicates that the syllogism is to be reduced to “Celarent.” The m indicates that the premises are to be interchanged, the original major becoming the minor and the original minor becoming the major. The first s indicates that the original minor premise must be converted by simple conversion, and the second s indicates that the original conclusion must be converted by simple conversion. Thus, following the instructions contained in the word “Camestres,” we get this syllogism of the first figure:
No free countries are totalitarian states; but all true democracies are free countries; therefore no true democracies are totalitarian states.
Example 2 is a syllogism in “Darapti”—that is, a syllogism of the third figure whose major premise is an A proposition, whose minor is also A, and whose conclusion is I.
2. Potassium floats on water; but potassium is a metal; therefore some metal floats on water.
Only one change need be made; it is signified by the p in “Darapti.” The minor premise must be converted per accidens—that is, with a change in quantity. Thus, Example 2 is reduced to the following syllogism of the first figure, a syllogism in “Darii”:
Potassium floats on water; but some metal is potassium; therefore some metal floats on water.
Example 3 is a syllogism in “Baroco”—that is, a syllogism of tbje second figure whose major premise is an A proposition, whose minor is O, and whose conclusion is O. The B in “Baroco” indicates that it is to be reduced to “Barbara.” The small c indicates that this can be done only indirectly. The position of the c after the second vowel of “Baroco” shows that the contradictory of the conclusion should be used as the minor premise.
3. All birds have feathers; but some flying animals do not have feathers; therefore some flying animals are not birds.
In order to reduce this syllogism, (a) keep the original major premise; (b) as the minor premise use the contradictory of the conclusion; and (c) as the conclusion use the contradictory of the minor premise—as follows:
All birds have feathers; but all flying animals are birds; therefore all flying animals have feathers.
Example 4 illustrates a syllogism in “Bocardo” and its indirect reduction to “Barbara.”
4. Some birds cannot fly; but all birds have feathers; therefore some feathered animals cannot fly.
The position of the c indicates that the contradictory of the conclusion must be substituted for the original major premise:
All feathered animals can fly; but all birds have feathers; therefore all birds can fly.