The Distribution of Terms in Categorical Statements
The subject or predicate can either be distributed or undistributed in the given premise. If the subject or predicate refers to all the things for which it stands, it is called distributed, otherwise, it is undistributed.In a premise, â€˜All birds are mammalsâ€™, the subject term â€˜birdsâ€™ is distributed, while the predicate term â€˜mammalsâ€™ is undistributed. In the premise, â€˜No birds are reptilesâ€™, both the subject and predicate terms are distributed. In the premise, â€˜Some birds are sparrowsâ€™, neither the subject nor the predicate term is distributed. Finally, in the statement â€˜Some birds are not carnivoresâ€™, only the predicate term is distributed.
The following table shows the distribution pattern of the subject and predicate.
Middle Term
The term which appears in both the given statements is called middle term.
Type of Propositions

Subject

Predicate

AUniversal affirmative

Distributed

Undistributed

EUniversal negative

Distributed

Distributed

IParticular affirmative

Undistributed

Undistributed

OParticular negative

Undistributed

Distributed

Example:
 All boys are students.
 All students are smart.
The term â€˜studentsâ€™ appears in both the statements. Hence, it is the middle term.
Major Proposition and Minor Proposition
The proposition in which the middle term appears as a subject is called major proposition. On the other hand, the proposition in which the middle term appears as a predicate is called as minor proposition.Mediate and Immediate Inferences: Syllogism is actually a problem of mediate inference. In mediate inference, conclusion is drawn from the two given statements.
For example, consider the following two statements:
 All birds are beautiful.
 All beautiful objects are precious.
Then the conclusion could be drawn that â€˜All birds are preciousâ€™.
On the other hand, if a conclusion is drawn from only one given proposition, such conclusion is called an immediate inference.
For example, consider the statement â€˜All students are playersâ€™. Then based on this statement a conclusion could be drawn that â€˜Some players are studentsâ€™.
Implication of a Given Proposition: Suppose we are given a proposition, â€˜All birds have feathersâ€™. Then this proposition naturally implies that the conclusion â€˜Some birds have feathersâ€™ must be true (some is only a part of all). The above example implies that if a given proposition is â€˜Aâ€™ type, then it also implies that the â€˜Iâ€™ type conclusion must be true.
Example:
â€˜No businessmen are honestâ€™. If this statement is true then the conclusion â€˜Some businessmen are not honestâ€™ also must be true. The above example shows that an â€˜Eâ€™ type proposition also implies an â€˜Oâ€™ type conclusion.
From the table, it is clear that if a statement is of type â€˜Oâ€™ then the type â€˜Aâ€™ conclusion is definitely false, while the type â€˜Eâ€™ and type â€˜Iâ€™ conclusions are both doubtful, that is they may or may not be true.
If the Given Statement is of Type

Then the Conclusion of the Type


A is

E is

I is

O is


A

â€“

False

True

False

E

False

â€“

False

True

I

Doubtful

False

â€“

Doubtful

O

False

Doubtful

Doubtful

â€“

Thus, if a statement given as â€˜Some girls are not hard working (O)â€™, then we have the following conclusions:
 All girls are hardworking (A)â€”False
 No girls are hardworking (E)â€”doubtful (may or may not be true)
 Some girls are hardworking (I)â€”doubtful (may or may not be true)
Rules for Deriving the Conclusion (Mediate inferences)
 No term can be distributed in the conclusion unless it is distributed in any of the premises
Statement (i) is an â€˜Iâ€™ type proposition where neither the subject nor the predicate is distributed. Statement (ii) is an â€˜Aâ€™ type proposition where the subject â€˜dogsâ€™ is distributed. In conclusion A, the term â€˜goatsâ€™ is distributed. Since the term â€˜goatsâ€™ is not distributed in any of the premises, conclusion A does not follow.  The middle term should be distributed at least once in the given premises. Otherwise, the conclusion will not follow.
In the premises, the middle term is â€˜watchesâ€™. It is not distributed in the first premise which is an â€˜Aâ€™ proposition as it does not form its subject. In addition, it is not distributed in the second premise which is an â€˜Iâ€™ proposition. Since the middle term is not distributed at least once in the premises, no conclusion follows.  If both the premises are particular, no conclusion follows.
Since both the premises are particular, neither conclusion A nor conclusion B follows.  If both the premises are negative, no conclusion follows.
Since both the premises are negative, conclusion A does not follow.  If the middle term is distributed twice, the conclusion cannot be universal.
Here, the first premise is an â€˜Aâ€™ proposition and so the middle term â€˜radiosâ€™ forming the subject is distributed. The second premise is an E proposition and so the middle term â€˜radiosâ€™ forming the predicate is distributed. Since the middle term is distributed twice, the conclusion A, being universal, does not follow.  If one premise is particular, the conclusion is particular.
Since one premise is particular, the conclusion must also be particular. Hence, only â€˜Aâ€™ follows.  If one premise is negative, the conclusion must be negative.
Since one premise is negative, the conclusion also must be negative, so conclusion â€˜Aâ€™ follows.  If both the premises are affirmative, the conclusion must be affirmative.
Here, conclusion B does not follow since it is not affirmative. Conclusion A follows since it is affirmative.
Rules for Deriving Immediate Inferences
 A term cannot be distributed in the conclusion (immediate inference), unless it is distributed in the premise.
 If the premise is particular, the conclusion should also be particular.
 If the premise is particular negative (â€˜Oâ€™ type), no conclusion follows.
 If the premise is universal negative, the conclusion should be negative.
 If the premise is affirmative, the conclusion should also be affirmative.