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01521--Write a note on Categorical Syllogisms.

A categorical syllogism consists of three categorical propositions—two premises and a conclusion. To be tested for validity, it must be stated in standard form. Standard-form categorical syllogisms can be sorted in terms of mood and figure into 256 possible arrangements, only some of which pass muster as valid in the system. To be valid (that is, for its conclusion to “follow” from its premises), it must satisfy certain formal restrictions on the number of terms that may occur in it; on the positions in which those terms may occur; on the “distribution” of the middle, major, and minor terms; and on the occurrence of negative statements


A categorical syllogism consists of three standard-form categorical statements: two serving as premises and the third as the conclusion.  Each of the individual statements must be unambiguous.   Example: “All undergraduates aren’t philosophy majors” might be either a universal negative claim that “No undergraduate students are philosophy majors” or a much less sweeping particular negative that “Some undergraduates are not philosophy majors.”

To be in standard form, a syllogism must have exactly three terms, each of which occurs in two of its three propositions.

1. The predicate term of the conclusion (called the major term) will also occur as either the subject or predicate of the premise that is stated first (hence, the major premise).

2. The subject term of the conclusion (called the minor term) will also occur as either the subject or predicate of the premise that is stated second (hence, the minor premise).

3. The third term (called the middle term) will occur in both of the premises (it can be the subject or the predicate term of either one) and will not occur in the conclusion.

Major premise:                                   middle term, major term (in either order)
Minor premise:                                   middle term, minor term (in either order)
Conclusion:                                          minor term, major term (in this order only)

Some nonstandard-form syllogisms can be reduced to standard form by reducing their number of terms to three—for example, if the syllogism seems to have more than three terms because of synonymy or because of the use of complementary terms—by manipulating their constituent propositions by means of immediate inferences, or by placing their premises and conclusion in proper order.

 Example: The argument “All Athenians are mortal because they are all Greeks and no Greeks are immortal” can be reduced to standard form by obverting “No Greeks are immortal” to “All Greeks are mortal,” by specifying the reference of “they,” and by placing the statements in proper order (major premise, minor premise, conclusion): 

All Greeks are mortal. 
All Athenians are Greeks. 
Therefore, all Athenians are mortal.

Note:  A syllogism with irreducibly more than three terms is not valid in this system.

For a standard-form categorical syllogism to be valid, it must comply with rules that restrict its structure in terms of (a) the “distribution” of the middle, major, and minor terms and (b) the occurrence of negative statements.

Restrictions on distribution: The distribution of a term has to do with whether or not the proposition in which it occurs conveys some information about all, or only part of, the class it names. No term can be distributed in the conclusion that is not distributed in the premise in which it occurs; that is, a conclusion cannot say more than the premises support. 

1. If the major term is distributed in the conclusion but not in the major premise, the argument fails due to “illicit process of the major term.” If the minor term is distributed in the conclusion but not in the minor premise, the argument fails due to “illicit process of the minor term.”

2. The middle term must be distributed in at least one of the two premises. If it is not, the argument fails due to “undistributed middle.”

There are also restrictions on negative propositions. 

1. If one of the premises is negative, the conclusion must be negative.
2. If both of the premises are negative, no valid conclusion can be drawn. 
Again, there must be exactly three terms and exactly three propositions, not four or more.

Standard-form categorical syllogisms display both mood and figure.  The mood of a syllogism is captured by listing the quality/quantity of each of its propositions in order (AAA, AEO, EIA, and so on).

The figure of a syllogism depends on where its middle term resides. 
Figure One: The middle term (M) is the subject of the major premise and predicate of the minor.  Figure Two: The middle term (M) is the predicate of both premises. 
Figure Three: The middle term (M) is the subject of both premises. 
Figure Four: The middle term (M) is the predicate of the major premise and subject of the minor.

In all four figures, the subject of the conclusion (S) appears in the minor premise (the second premise) of the syllogism and is known as the minor term. The predicate of the conclusion (P) appears in the major premise (the first premise) of the syllogism and is known as the major term. C. There are 256 possible moods and figures for standard form syllogisms, from AAA-1 to OOO-4. Very few are valid. 
AAA-1 (Valid). 
All Greeks are mortal. 
All Athenians are Greeks 
Therefore, all Athenians are mortal.
AAA-2 (Invalid, undistributed middle). 
All Rastafarians are bearded. 
All billy goats are bearded. 
 Therefore, all billy goats are Rastafarians.

By Aristotelian standards, 24 of the 256 possible categorical syllogisms are valid.
Problems with null classes remain, as will be seen.

[Courtesy: Professor James Hall]


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