The Categorical syllogism reference article from the English Wikipedia on 24-Jul-2004
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Categorical syllogism

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A categorical syllogism is a valid argument of the following form:

.
.
Therefore, .

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.

Table of contents
1 Quality, Quantity and Distribution
2 Mood and Figure
3 Validity

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 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 quanitfier is called a particular. It is an indifinative number, which could mean five, twenty or, perhaps, all, but always at least one. From quality and quantity are 4 types of categorical propositions designated alphanumerically:

A, E, I and O propositions have different distribution properties. Distribution referes to what can be infered from the proposition. An A proposition distributes the subject to the predicate, but not the reverse however. 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 distrubute bidirectionally between the subject and predicate. From the categorical proposition--No beatles are mammals--we can infere that no beatles are animals and likewise, no mammals are beatles. Both terms in a I proposition are undistributed. For example, some Americans are conservatives. Neither term in tthe proposition can be entirely distributed to the other term. From this proposition its not possible to say that all Americans are conservatives or all conservatives are Americans. In an O proposition only the predicate term is distributed. Consider the following: Some hardware are nails. By knowing screws are considered hardware it can stated that there exists some members outside the class of nails that are members of the class of hardware. However, it can be infered 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 primise and the conclusion, then state them in that order. In the catergorical syllogism

All A is B

All C is A

Therefore, all C is B

the mood would be AAA, seeing that all propositions are universial 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 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 numberically 1-4

As such, the appropriate mood and figure of the aforementioned categorical syllogism is AAA-1. The combination of mood and figure is known as form.

Validity

It would be rather tedious to pounder the validity of various categorical syllogisms. Luckily, people have already done this and as a result they have devised three alternate methods of finding validity. The first is to memorize the various forms. Here are a few of the fifteen valid forms:

You can obtain the remaining valid forms via the other methods. One method is to construct a Venn diagram. Since there three terms, a Venn diagram will require three over-laping circles which represent each class. First, construct a circle for the major term. Adjacent to the circle for the major term will be an over-laping circle for the minor term. Beneath those two will be the cirlce for the minor term. It should over-lap at three places: the major term, the minor term and the place at which the major term and minor term over-lap. If the syllogism is valid it would necessitate the truth of the conclusion by diagraming the premises. Never diagram the conclusion, for the conclusion must be infered 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 non-membership. So in the premise All A is B shade in all areas in which A does not over-lap with B, including where A over-laps with C. Then repeat the same procedure for the second premise. From those two premises we can infere that all members in the class of C also have membership in the class of B. However, we can not infere if all members of the class of B have membership in the class of C.Image:Venn1222.jpg


The last method is to memorize six rules using the information presented thus far.  While Venn Diagrams are good tools for illustrative purposes, it may be preferable for some to test validity with the following rules:

See also: syllogistic fallacy.

Other forms of syllogism: hypothetical syllogism, disjunctive syllogism.