This script is a syllogisms tutor. It will train you in recognizing valid categorical syllogisms, as well as in identifying the mood and figure of a syllogism and the various formal fallacies that can make a syllogism invalid. A valid argument is one whose premises entail its conclusion. A syllogism is an argument with two premises and a conclusion. A categorical syllogism is one whose premises and conclusion are all categorical statements. A categorical statement is a statement about the relationship between categories, and there are four basic relationships two categories can have. One category can be a subset of the other or not, and they can intersect or not. The four forms of categorical statement that represent these four relationships are normally designated as A, E, I, and O. Here they are illustrated with Venn diagrams, where the left side is the subject category (S), and the right side is the predicate category (P).
A: All S is P
E: No S is P
I: Some S is P
O: Some S is not P
|A||All S is P||One category is a subset of another|
|E||No S is P||The two categories do not intersect|
|I||Some S is P||The two categories intersect|
|O||Some S is not P||One category is not a subset of another|
They are illustrated above in a pattern called the square of opposition. Universal forms appear at the top, existential forms at the bottom. Universal statements merely express relations between categories without asserting that they have any members. Existential statements assert that something actually exists. Because of this, an existential conclusion will not validly follow when the only premises are universal. This would commit the Existential Fallacy. Positive forms appear on the left side, and negative forms appear on the right. A positive statement positively asserts something about its subject, and a negative statement denies something about its subject. With categorical statements, what is asserted or denied is inclusion of the subject in the predicate category. When both premises are negative, there is no connection between them, making the syllogism invalid. This commits the Fallacy of Exclusive Premises. When either premise is negative, only a negative conclusion can logically follow. When a syllogism has a positive conclusion when one of the premises is negative, it commits the Fallacy of Drawing an Affirmative Conclusion from a Negative Premise. One more thing to note about the square of opposition is that statements in opposite corners contradict each other if the subject and predicate are the same. For example, "All cats are mammals" contradicts "Some cats are not mammals," and "Some cats are mothers" contradicts "No cats are mothers." This is why it gets the name square of opposition.
The Venn diagrams illustrate a property known as distribution. A category is said to be distributed in a statement if that statement tells us something about every member of that category. In an A statement, the subject is distributed, for it tells us that every member of the subject belongs to the predicate. In an O statement, which is the contradiction of an A statement, the predicate is distributed, for it says that every member of the predicate is not part of the subject. For example, if some cats are not calicos, it implies that every calico fails to be part of the group of cats identified in the subject. In an E statement, both subject and predicate are distributed, for it tells us that they are mutually exclusive. Every member of the subject category is not part of the predicate category, and every member of the predicate category is not a member of the subject category. In an I statement, which is the contradiction of an E statement, no category is distributed. If some cats are mothers, there may still be cats who are not mothers and mothers who are not cats.
Distribution is an important factor in determining validity. A syllogism has three terms. These are the major term, the minor term, and the middle term. The major term is shared by the first premise and the conclusion, the minor term by the second premise and the conclusion, and the middle term by the two premises. When the middle term is never distributed, the argument is invalid, committing the Fallacy of the Undistributed Middle. When a term is distributed in the conclusion, it must also be distributed in the premise it appears in. When the major term is distributed in the conclusion but not in the major premise, the syllogism commits the Fallacy of Illicit Process of the Major Term. When it is the minor term, the syllogism commits the Fallacy of Illicit Process of the Minor Term. These are known more briefly as Illicit Major and Illicit Minor.
A valid syllogism is merely one whose conclusion logically follows from its premises. It can be valid without having true premises or a true conclusion. An argument is sound when it is valid, and its premises are all true. In that case, its conclusion must be true. Validity is worth knowing about, because it helps us find out what logically follows from our premises, but it is not the whole picture. To emphasize the difference between a valid argument and a sound argument, all premises and conclusions are randomly generated, such that many will be false. Very few of the randomly generated syllogisms will be sound, but a fair number will be valid.
Each syllogism is identified by a mood and a figure. Its mood is represented by the three letters identifying the type of categorical statement made in its major premise, its minor premise, and its conclusion, in that order. So a syllogism with the mood EIO has an E statement for a major premise, an I statement for a minor premise, and an O statement for a conclusion. Its figure is a number from 1 to 4, and it identifies the placement of the middle term in each premise, as shown in this table:
|Figure||Major Premise||Minor Premise|
Here are some hints on memorizing them. Old MacDonald had a farm, e-i-e-i-o. All the EIO forms are valid. This is due in part to E and I statements being completely symmetrical.
No S is P means the same as
No P is S, and
Some S is P means the same as
Some P is S. Since major and minor terms have fixed places in the conclusion, and their positions in the premises don't change the meanings of the premises, changing the figure of an EIO mood syllogism doesn't change the argument being made. Every other valid form includes an A statement with the other two being the same as each other. AAA-1 is first, it is the only one with its form, and it is the only one where the conclusion is an A statement.
Even starts with E, and only the AEE forms with even figures are valid. I is the Roman numeral 1, which is odd, and only the odd figures are valid for the AII form. A and E appear alphabetically before I and O, and for EAE, only the first two figures are valid. While the first two figures of EAE are valid, it is the last two figures of IAI that are valid. Finally, OAO-3 is the last valid form. The only valid form beginning with a O ends with an O, and three is the number of statements in a syllogism. If you've seen the TV series Lexx, you may remember the Brunnen-G song Yo Way Yo, which sounds like OAO.
For each syllogism, fill in the fields for mood, figure, validity, and fallacies. When you're finished, press "Check Answers" to find out what the correct answers are. Press "New Syllogism" to do a new syllogism.
The first premise in a categorical syllogism
The second premise in a categorical syllogism
The category mentioned in both the major premise and the conclusion. The second term in the conclusion.
The category mentioned in both the minor premise and the conclusion. The first term in the conclusion.
The category mentioned in both premises but not the conclusion. It is what links major term and minor term together in the syllogism.
The mood of a categorical syllogism is a matter of what kind of categorical statement each statement is, and it is represented by a three letter acronym. The first letter represents the form of the first premise; the second represents the form of the second premise; and the third represents the form of the conclusion. The letters used are A, E, I, and O, as described above.
The figure of a categorical syllogism is the position of its major, minor, and middle terms. There are four figures. The major and minor terms have standard positions in the conclusion, which are the same for all figures. Each figure is disinguished by the placement of the middle term.
|Figure||Major Premise||Minor Premise|
A mistake in reasoning which makes an argument invalid.
A category is distributed in a statement when the statement refers to every members of the category. The first term is distributed in A statements; the second is distributed in O statements; both are distributed in E statements; and none are distributed in I statements.
Fallacy of the Undistributed Middle
When neither premise refers to every member of the middle term, the middle term fails to connect the two premises, and nothing can follow from them. This makes the argument invalid.
Fallacy of Illicit Process of the Major Term
When the conclusion is about every member of the major term, the major premise must also be about every member of the major term. The argument is otherwise invalid.
Fallacy of Illicit Process of the Minor Term
When the conclusion is about every member of the minor term, the minor premise must also be about every member of the minor term. The argument is otherwise invalid.
Fallacy of Exclusive Premises
When both premises are negative (E or O), there is no connection between them, and nothing follows from them. This makes the argument invalid.
Fallacy of Drawing an Affirmative Conclusion from a Negative Premise
When either premise is negative (E or O), only a negative conclusion can follow. When there is an affirmative conclusion (A or I) with a negative premise, the argument is invalid.
The existential statements (I and O) imply the existence of their subject, but the universal statements (A and E) do not. It is true, for example, that all Vulcans are frogs, because there are no Vulcans, making this statement vacuously true. Since Vulcans aren't real, the set of all Vulcans is the empty set. The empty set is a subset of every set. In saying that all Vulcans are frogs, I am merely saying that the empty set is a subset of the set of frogs, which is true, and I am not asserting that any Vulcans exist, which would be false. Since universal statements do not imply the existence of anything, all that follows from two universal statements is another universal statement. If a conclusion is existential but both premises are universal, the syllogism is invalid.