Syllogisms/Modus Ponens/Modus Tollens

A Syllogism has three statements and two terms in each statement. The first statement is known as the major premise, which is more general, where as the second statement is the minor premise, which is more specific, and then the conclusion states the result of these two statements. A formula of these is like this: A=B, B=C, then A=C

An example:
(B) (A)

minor premise: All humans need to drink water.
(C) (B)
major premise: Aaron is a human.
(C) (A)
conclusion: Therefore, Aaron needs to drink water.

A modus ponen is like a syllogism in a way that they both have three statements, but they are all positive.

The rule for this is:

If p, then q.
p.
Therefore q.

An example:

If I am hungry, I will eat a whole pizza.
I am hungry.
Therefore, I will eat a whole pizza.

A modus tollen also has three statements with two terms each, however, unlike the Modus Ponen, all three statements are negative.

The rule for this is:

If not p, then not q.
Not p.
Therefore, not q.

An example:
If I do not run, I will not make the Track team.
I did not run.
Therefore, I did not make the Track team.
(that one is fairly untrue!)