Syllogisms

Questia defines syllogisms quite well.

Every syllogism is a sequence of three propositions such that the first two imply the third, the conclusion. There are three basic types of syllogism: hypothetical, disjunctive, and categorical.

The rest of this post expounds on their entry.

Hypothetical

The hypothetical syllogism, modus ponens, has as its first premise a conditional hypothesis:

If P then Q. P. Therefore, Q.

P → Q
P
∴ Q

Disjunctive

The disjunctive syllogism, modus tollens, has as its first premise a statement of alternatives. If one of them can be found not to be true, then the other must be true.

Either P or Q. Not Q. Therefore, P.
P or Q
Not Q
∴ P

Categorical

The categorical syllogism comprises three categorical propositions, which must be statements of the form :

Type	Form	Distributed
A	all S are P	S
E	no S is P	S P
I	some S is P
O	some S is not P	P

A term is said to be distributed when it refers to all members of the denoted class. "A term in a categorical proposition is distributed if and only if the proposition implies every proposition that results from replacing the term with a more specific term." -- Fallacy Files. The term S is distributed in "all S are P" and "no S is P". [ See Fallacy Files]

A categorical syllogism contains precisely three terms:

• the major term, which is the predicate of the conclusion
• the minor term, the subject of the conclusion
  
• the middle term, which appears in both premises but not in the conclusion

Thus: All philosophers are men (middle term); all men are mortal; therefore, All philosophers (minor term) are mortal (major term). The premises containing the major and minor terms are named the major and minor premises, respectively. Aristotle noted five basic rules governing the validity of categorical syllogisms:

1. The middle term must be distributed at least once
   
2. A term distributed in the conclusion must be distributed in the premise in which it occurs
3. Two negative premises imply no valid conclusion
4. If one premise is negative, then the conclusion must be negative
5. Two affirmatives imply an affirmative

Philosophy Pages has another introduction.

Bruce Thompson has a nice table.