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
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(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
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(P&Q)>Q          CP

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

P&Q                 ACP
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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))

Conditional Proof

Me in my office when students don't come in and I know some don't understand what's going on.
Agenda
1. Notes on the test.

• DN Rule 
• Assumption column 

2. CP Rule

Extra practice with DN, MP, MT, &In, &Out
You should be able to do these.

Conditional Proofs (CP rule)
How to do conditional proof:
Sing the following:
Started from the bottom, now we're here.
Started from the bottom, put the antecedent up here.

Proofs are solved not only from the top down (e.g., DN, MP, MT, &In, &Out) but also bottom up. We work from both ends. CP proof is what's known as a "bottom up" rule.

Practice (CP only)
A.

1. Q /:. P>Q

B.

1. R /:. P>(Q>R)

C. 

1. T /:. P>(Q>(~R>(S>(~T>(U>V)))))

D. (This proof isn't *only* CP)

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

Putting It All Together: Proofs with CP, DN, MT, MP, &In, &Out
A.

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

Extra Practice Proofs with DN, MP, MT, &In, &Out

A.

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

B.

1. (A&~B)&(C&~D)
2. A>E
3. (E&~B)>F
4. G>B
5. (~G&C)>H
6. (H&F)>I  /:. I

C.

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

Practice Questions for Test 1: MP, MT, and DN

A.

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

B. There are two ways to solve this prove. One short and one long. Both are correct.

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

C.

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

D. 

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

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

Modus Ponens 2: More Translation Practice

Hot dog hot dog

Lesson:
1. Review validity.
2. Review translation.
3. Simple proofs.

Translation
Here's how we do a basic conditional translation:

E.g. 1, If you're happy and you know it then clap your hands. = H>C

First write down the first letter of a key word in the antecedent (e.g., 'H' from happy) then write down the symbol '>' and finally write down the first letter of a key word in the antecedent (e.g., 'C' from clap).

Order doesn't matter!
When we translate, the symbol for antecedent always goes first followed by '>', followed by the symbol the consequent.

E.g. 2, Clap your hands if you're happy and you know it. = H>C.

Notice that even though the words of the sentence are ordered differently from e.g., 1, we still symbolize it the same as e.g., 2. Doing so preserves the logical structure. That is, "If you're happy and you know it then clap your hand" and "Clap your hand if you're happy and you know it" both have the same logical structure. To see why, let's look at a similar example but with a different logical structure.

E.g. 3, If you clap your hands then you're happy and you know it. = C>H.
or
E.g. 4, You're happy and you know it if you clap your hands. = C>H.

Notice that e.g. 1 and 2 mean something different than e.g. 3 and 4. If we had translated e.g., 2 as C>H it would mean the same thing as e.g., 3.

Translation Practice
A.

1. If I were a Boy just for a day I'd Roll out of bed in the morning.
2. I'd be Coming right back home to you if I could Fly.
3. If I could Find a way I'd take back those Words that hurt you and you'd stay.
4. You shoulda put a Ring on it if you Liked it.
5. If I Eat too much I feel Tired. 
6. If you Lose yourself in the music, the moment then you Own it, you better never let it go. 
7. You only get one Shot, do not miss your chance to blow if this opportunity Comes once in a lifetime.