Modus Ponens, Negation, and Double Negation

Today's Class Content
1. Review:

• Validity
• Translation of conditionals

2. Homework Questions/Problems? 

3. New content:
    (a) Translating with negations.
    (b) Double Negation rule (DN). 
    (c) Modus Tollens.

4. Basic proofs with negation.

Negatins and Double Negation (DN)
Translations
1.  If you Study you won't Fail.
2. Mark will be disappointed if you don't know who Rain man is.
3. I wouldn't leave my Nuts uncovered for winter if I were a Squirrel.
4. If you don't use your Cellphone in class I won't have to Judo chop you.


Proofs
MP rule says if I have the antecedent of a conditional I can write down the consequent. However, in order to apply MP I have to have the exact antecedent. Even if I have an antecedent that is logically equivalent, I can't apply the rule.

Example:
1. P>Q
2. ~~P /:. Q

WRONG:
1. P>Q       A
2. ~~P        A
3. Q           MP 1, 2

In order to use MP I need P because P not ~~P is the antecedent. ~~P will not work. However, I can change ~~P into P by applying double negation rule (DN).

CORRECT:
1. P>Q      A
2. ~~P       A
3. P           DN 2
4. Q          MP 1,3

DN and Parenthesis

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

Exercises
MP + DN Only
A.

1. (A>B)>(C>D)
2. A>B
3. ~~C  /:. D

B. 

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

C. 

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

D.

1. ~~(P>~~~Q)>~~~~S
2. S>(~~Q>P)
3. P>~~~Q
4. (Q>P)>~~~T
5. ~T>Q  /:. ~~Q

MT Only

E. 

1. ~A>(B>C)
2. ~(B>C)
3. A>~B  /:. ~B

F. 

1. ~S>(Q>~R)
2. P
3. (Q>~R)>~P  /:. S

MP, MT, DN

G. 

1. ~(P>Q)>(R>~S)
2. R
3. (P>Q)>~R  /:. ~S

H. 

1. T>U
2. ~(~P>~Q)>(~R>~S)
3. (~R>~S)>~(T>U)
4. ~P   /:.  ~Q