Syllogism

A syllogism is simply a three-line argument, that is, an argument that consists of exactly two premises and a conclusion. Each of the premises has one term in common with the conclusion: the major term in the major premise, which forms the predicate of the conclusion, and the minor term in the minor premise, which forms the subject of the conclusion. The categorical term in common in the premises is called the "middle term". This takes the general form:

Major premise: A general statement.
Minor premise: A specific statement.
Conclusion: based on the two premises. 

There are three major types of syllogism:
Conditional Syllogism: If A is true then B is true (If A then B).
Categorical Syllogism: If A is in C then B is in C.
Disjunctive Syllogism: If A is true, then B is false (A or B).

Also of note for syllogisms is:
Categorical Propositions: Statements about categories.
Enthymeme: a syllogism with an incomplete argument.
Modus Ponens: If X is true then Y is true. X is true. Therefore Y is true.
Modus Tollens: If X is true then Y is true. Y is false. Therefore X is false.
Set Theory: The basics of overlapping groups. 

For example:

Major premise: All birds are animals.
Minor premise: All parrots are birds.
Conclusion: All parrots are animals.

In this example, "animal" is the major term and predicate of the conclusion, "parrot" is the minor term and subject of the conclusion, and "bird" is the middle term.

Syllogisms are particularly interesting in persuasion as they include assumptions that many people accept which allow false statements or (often unspoken) conclusions to appear to be true. There is a difference between truth and validity in syllogisms. A syllogism can be true, but not valid (i.e. make logical sense). It can also be valid but not true.