More Reductio, Translations with 'v', Logical Equivalence

Agenda

1. Quiz
2. Questions about homework?
3. Law of non-contradiction and law of the excluded middle. 
4. Translating sentences with 'v'.
5. Logical Equivalence.
6. RA proofs.

Quiz
A. 

1. /:. (P&Q)>~(P>~Q)

B. 

1. Either you'll get this RIGHT or you won't unless CONTRADICTIONS are true.

2. You can have a sticker only if you either make no MISTAKES or bring me a COOKIE. 

Law of Non-Contradiction

Definition: The negation of a contradiction is always true.

Symbolized: ~(P&~P) = True

E.g., 

1. The fact that I'm both alive and not alive is false. This whole sentence is true.
2. The fact that the room is both empty and not empty is false. This whole sentence is true.
3. The fact that I like cookies and don't like cookies is false. This whole sentence is true.

The Law of the Excluded Middle

Definition: The assumption that any sentence is either true or false is always true.

Symbolized: Pv~P = True

E.g.,

1. Either I'm going to study or I'm not going to study. The whole sentence is true.
2. Either you are wearing pants or you are not wearing pants. This whole sentence is true.
3. Either Bob is here or he isn't here. This whole sentence is true. 

Translations with 'v'
A.

1. Either I'm not going to STUDY or I'm not going to WATCH  a movie. 
2. Unless you WORK hard you'll either make your MOTHER cry or your FATHER angry. 
3. You can have either COOKIES or DONUTS but you can't have both. (Careful!)
4. You can have either COOKIES or DONUTS or both.
5. You can have neither COOKIES nor DONUTS unless you HELP me. 
6. I would HIKE Death Valley only if either my BROTHER or JEREMY came with me.
7. You can have neither COOKIES nor DONUTS unless you EAT your dinner.

Logical Equivalence 

A. 

1. /:. (P>Q)<>(~Q>~P)

Reductio Proofs (left over from last week)

D. 

1. ~D>(A&C)
2. (B&D)>E
3. (DvF)>~E  /:. ~(A&C)>~(Bv~D)

E. 

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

F.

1. (~D&~E)>F
2. A&~D
3. ((F&~B)vG)>~(A>~B)    /:.  (A>~B)>~(Dv~E)