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)

Conditional Proofs 2

Agenda

1. Quiz
2. Please help me! help you! during office hours if you couldn't do the quiz question.
3. Reset/Taking stock
4. Questions from HW?
5. Affirming the consequent/Denying the antecedent
6. Practice proofs

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



















How to set up a conditional proof:
If you have to solve for a conditional,
1. write the conditional at the bottom of your proof;
2. write the antecedent of the conditional you're trying to prove on the line immediately below the last given premise/assumption OR if there are no premises/assumptions, write it at the top of the proof. Write 'ACP' in the justification column.
3. write the consequent on the line above the conditional you're trying to prove.

Example: 
I'm asked to solve for (P&Q)>Q
Step one: write the conditional at the bottom of your proof
.
.
.
.
.
.
.
(P&Q)>Q           CP _____

Step two: write the antecedent of the conditional you're trying to prove on the line immediately below the last given premise/assumption OR if there are no premises/assumptions, write it at the top of the proof. Write 'ACP' in the justification column.

P&Q                  ACP
.
.
.
.
.
.
(P&Q)>Q          CP

Step three: write the consequent on the line above the conditional you're trying to prove.

P&Q                 ACP
.
.
.
.
.
Q                    ______
(P&Q)>Q       CP

Now that the CP proof has been set up, you solve it the way you'd solve any proof. All you're trying to do is justify Q.


Proofs with CP rule
A.

1. ~C>~A /:. (A&~B)>(~B>C)

B.

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

C. 

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

D.

      /:.  P>(~Q>(R>(~S>(R&~S))))

E.  

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

Modus Tollens and Conjunction Rules

Class
1. Review:

• Validity
• Translation
• Modus tollens (MT)

2. Conjunction Rules (&In and &Out)
3. Bonus Office Hours This Week: 4:20pm 3rd Floor Shatzel Hall, Seminar Room

Conjunction Rules: &In and &Out

A.  &In

1. P
2. Q
3. P&Q

B.  &Out

1. P&Q
2. P/Q

Proofs with DN and MP Only

A.

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

Proofs 

MT Only

B. 

1. ~A>(B>C)
2. ~(B>C)
3. A>~B  /:. ~B

C. 

1. ~S>(Q>~R)
2. P
3. (Q>~R)>~P  /:. S

MP, MT, DN

D. 

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

E. 

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

F. 

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

MP, MT, DN, &In, &Out

H. 

1. ~E>~A
2. A>B
3. (A>B)>(A>~(D&E)
4. A>D
5. A

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