Categorical syllogism
From Academic Kids

A categorical syllogism is a valid argument of the following form:
 <math>a \subset b<math>.
 <math>b \subset c<math>.
 Therefore, <math>a \subset c<math>.
In other words, this kind of argument states that if all a's are b's, and if all b's are c's, then all a's are c's.
A categorical syllogism contains two premises followed by a conclusion, both of which are generically known as propositions. There can be only three terms predicated through out the entirety of the syllogism. Each proposition must be in the following form: A subject linked to a predicate by a copula. A premise might go as follows: All S is P, where S is the subject, 'is' is the copula, and P is the predicate.
Contents 
Quality, quantity and distribution
Categorical propositions can be categorized on the basis of their quality, quantity, and distribution qualities. Quality refers to whether the proposition affirms or denies the inclusion of a subject to the class of the predicate. The two qualities are affirmative and negative. On the other hand, quantity refers to the amount of subjects in one class which are included in the other class. The first quantifier is the universal, "all". This means that every subject of one class has membership in the predicated class. The other quantifier is called a particular. It is an indefinite number, which could mean five, twenty or, perhaps, all, but always at least one. From quality and quantity are four types of categorical propositions designated alphabetically:
 A proposition is a universial affirmative: All S is P
 E proposition is a universial negative: No S is P
 I proposition is a particular affirmative: Some S is P
 O proposition is a particular negative: Some S is not P
All four types have different distribution properties. Distribution refers to what can be inferred from the proposition. An A proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: all dogs are mammals. All dogs are indeed mammals but it would be false to say all mammals are dogs. E propositions do distribute bidirectionally between the subject and predicate. From the categorical proposition, no beetles are mammals, we can infer that no mammals are beetles. Both terms in an I proposition are undistributed. For example, some Americans are conservatives. Neither term in the proposition can be entirely distributed to the other term. From this proposition it is not possible to say that all Americans are conservatives or that all conservatives are Americans. In an O proposition only the predicate term is distributed. Consider the following: some hardware are not nails. Knowing that screws are considered hardware, it can be stated that there exist some members outside the class of nails that are members of the class of hardware. However, it can be inferred that all nails are hardware. Thus, only the predicate term is distributed in an O proposition.
Mood and figure
Now that we can differentiate between the various types of categorical propositions, we can easily identify the mood of the syllogism. To do so, simply identify the types of propositions in the first premise, the second premise and the conclusion, then state them in that order. In the categorical syllogism
 All A is B
 All C is A
 Therefore, all C is B
the mood would be AAA, seeing that all propositions are universal affirmative. Next to be discussed is the figure of a categorical syllogism. However, in order to comprehend the figure, one must be able to identify the three different types of terms: major term, minor term, and the middle term. The term occurring as the predicate of the conclusion is the major term. In the aforementioned syllogism B is the major term. The minor term is the term that occurs as the subject of the conclusion; C is the minor term. Finally, by process of elimination, it can be deduced that the middle term is the term which does not occur in the conclusion, but instead once in each premise. Accordingly, A is the middle term. The figure of a categorical syllogism can be known by identifying the four possible arrangements of the middle term. The figures are represented numerically 14
 1 the middle term occupies the subject of the first premise and the predicate of the second premise
 2 the middle term occupies the predicate of both the first premise and second premise
 3 the middle term occupies the subject of both the first premise and second premise
 4 the middle term occupies the predicate of the first premise and the subject of the second premise
As such, the appropriate mood and figure of the aforementioned categorical syllogism is AAA1. The combination of mood and figure is known as form.
Validity
It would be rather tedious to ponder the validity of various categorical syllogisms. Luckily, people have already done this and as a result they have devised three alternative methods of finding validity. The first is to memorize the various forms. Here are a few of the fifteen valid forms:
 AAA1
 AEE4
 AEE2
 OAO3
 EIO1
You can obtain the remaining valid forms via the other methods. One method is to construct a Johnston diagram. Since there are three terms, a Johnston diagram will require three overlapping circles which represent each class. First, construct a circle for the major term. Adjacent to the circle for the major term will be an overlapping circle for the minor term. Beneath those two will be the circle for the middle term. It should overlap at three places: the major term, the minor term and the place at which the major term and minor term overlap. If the syllogism is valid it would necessitate the truth of the conclusion by diagramming the premises. Never diagram the conclusion, for the conclusion must be inferred from the premises. Always diagram the universal propositions first. This is accomplished by shading the areas in which one class does not have membership in the other class. In other words, shaded is equated with nonmembership. So in the premise All A is B shade in all areas in which A does not overlap with B, including where A overlaps with C. Then repeat the same procedure for the second premise. From those two premises we can infer that all members in the class of C also have membership in the class of B. However, we can not infer that all members of the class of B have membership in the class of C.
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AAA1
As another example of this method, consider a syllogism of the form EIO1. Let its first premise be "No B is an A", its second premise be "Some Cs are Bs" and its conclusion be "Some Cs are not As." This syllogism's major term is A; its minor term is C, and its middle term is B. The first premise is shown on the diagram by shading the intersection A ∩ B. The second premise cannot be represented by shading any area. Instead, we may use the ∃ (existence) symbol in the nonshaded portion of the intersection B ∩ C in order to signify that "Some Cs are Bs." (N.B. Shaded areas and existentially quantified areas are mutually exclusive.) Then, since this existence symbol lies within C but outside of A, then it is correct to conclude that "There exist some Cs which are not As."
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EIO1
The last method is to memorize six rules using the information presented thus far. While Johnston Diagrams are good tools for illustrative purposes, it may be preferable for some to test validity with the following rules:
 As noted before, categorical syllogisms must contain exactly three terms, no more, no less (cp. fallacy of four terms). As a cautionary note, beware that synonyms and antonyms can create the illusion of invalidity, but can sometimes be rectified by substituting the interexchangeable terms for one of choice.
 If either premise is negative, then the conclusion must be negative (cp. affirmative conclusion from a negative premise).
 Both premises cannot be negative (cp. fallacy of exclusive premises).
 Any term distributed in the conclusion must be distributed in either premise.
 The middle term must be distributed once and only once (cp. fallacy of the undistributed middle).
 You cannot draw a particular conclusion with two universal premises (cp. existential fallacy).
List of syllogisms
The following is a list of fourteen syllogisms whose names were given to them during the middle ages, but which are all based on Aristotle's Analytics. For the names, see term logic.
 Barbara
Every B is an A.
Every C is a B.
∴ Every C is an A.
 Celarent
No B is an A.
Every C is a B.
∴ No C is an A.
 Darii
Every B is an A.
Some Cs are Bs.
∴ Some Cs are As.
 Ferio
No B is an A.
Some Cs are Bs.
∴ Some Cs are not As.
 Cesare
No B is an A.
Every C is an A.
∴ No C is a B.
 Camestres
Every B is an A.
No C is an A.
∴ No C is a B.
 Festino
No B is an A.
Some Cs are As.
∴ Some Cs are not Bs.
 Baroco
Every B is an A.
Some Cs are not As.
∴ Some Cs are not Bs.
 Darapti
Every C is an A.
Every C is a B.
∴ Some Bs are As.
(This form needs the assumption that some Cs do exist.)
 Datisi
Every C is an A.
Some Cs are Bs.
∴ Some Bs are As.
 Disamis
Some Cs are As.
Every C is a B.
∴ Some Bs are As.
 Felapton
No C is an A.
Every C is a B.
∴ Some Bs are not As.
(This form needs the assumption that some Cs do exist.)
 Ferison
No C is an A.
Some Cs are Bs.
∴ Some Bs are not As.
 Bocardo
Some Cs are not As.
Every C is a B.
∴ Some Bs are not As.
See also: syllogistic fallacy.
Other forms of syllogism: hypothetical syllogism, disjunctive syllogism.
External links
 Categorical syllogisms (http://philosophy.lander.edu/logic/syllogism_topics.html) by the p.l.e Introduction to Logic.