Modus Ponens and Modus Tollens

Modus Ponens
1. Modus ponens refers to particular underlying structure of an argument that uses a conditional (if-then statement) as one premise and the antecedent (the part following the 'if) of the conditional as the other premise. The conclusion is the consequent (the part following the 'then') of the conditional.

Otis Ponens

Here's an example:
P1. If [I put Money in the machine] then [I'll get a Snickers bar].
P2. [I put Money in the machine].
C.  [I got a Snickers bar]. 

We can symbolize the underlying structure using a capital letter to represent each clause.
1.  M>S ('>' means 'if___then___'
2. M
3. S

Modens Ponens is a valid argument form. This means that no matter what you substitute for the variables (even if they are false in the 'real' world) the argument is valid. Validity refers to the structure of an argument NOT its truth (i.e., soundness). We can define validity two different ways:

(a) An argument is valid if and only if if all its premises are true and it's impossible for its conclusion to be false.

(b) An argument is valid if and only if if all its premises are true then its conclusion must also be true. 

Modus Tollens
Modus tollens is like a modus ponens in reverse. It has the following structure: one premise is a condidtional and the other premise is the negation of the consequent. The conclusion is the negation of the antecedent. 

Here's an example:
P1. If [I put Money in the machine] then [I'll get a Snickers bar].
P2. [I don't have a Snickers bar].
C.  [I didn't put Money in the machine].

Symbolized, modus tollens looks like this:
1. M>S
2. ~S ('~' means 'not')
3. ~M

Proofs :
(Modus Ponens Only)

A. 

1. A>(B>C)
2. A
3. B    /C

B. 

1. (A>B)>(C>D)
2. A>B
3. C   /D

C. 

1. A>(B>(C>(D>(E>F))))
2. A
3. B
4. C
5. D
6. E   /F

D. 

1. (A>B)>(C>(D>E))
2. A>(A>B)
3. A
4. A>C   /D>E

E.

1. (A>(B>C))>(A>B)
2. A>(B>C)
3. A   /C

F. 

1. (P>Q)>(R>((R>Q)>P))
2. R
3. P>Q
4. R>Q
5. P>S   /S