Intermediate Translations for '&' and '>'

Agenda

1. If you don't understand anything up to and including CP rule you need to come to my office hours so I can help you. Coming for help after Spring Break will be too late since we're going to be learning even more rules...and the old ones will still be in play.
2. Any questions about the HW?
3. Intermediate translations with '&' and '>'. 
4. Proofs.

Logically Equivalent to '&'
1. The following words are all translated as '&': 

• Although
• However
• But
• Nevertheless
• Yet
• Despite
• As well as

Example:

a. I want ABS but I really want to eat DONUTS. = A&D

b. Bob thinks he's a GOOD student despite never DOING well. 

c. Although I understand the HOMEWORK, I didn't do well on the TEST.

d. Despite getting his MACROS right Bob couldn't make any GAINS.

e. She works hard for the MONEY but he never treats her RIGHT. 

2. Fancy stuff: Neither P nor Q= ~P&~Q

Example: 

a. Neither BOB nor ALICE are going downtown. = ~B&~A

b. Although the students PROMISED they'd do it they neither brought me COOKIES nor DONUTS. 

c. Neither MARK nor AMI will study for me however I'm a BIG boy/girl and I can STUDY by myself.

Logical Cousins with 'If'
If P then Q; Q if P = P>Q. Rule: Whatever immediately follows the 'if' comes first.
P only if Q; Only if Q, P = ~Q>~P. Rule: Whatever follows 'only if' is negated then put first, the other proposition is negated and put after the '>'.
Unless P, Q; Q unless P = ~P>Q. Rule: Whatever follows 'unless' is negated and put first. Think of 'unless' as 'if not P'.

Easy Examples
a. I'll HELP you with your logic only if you bring me COOKIES.
b. Unless you go to the CHOPPA you'll get shot.
c. Only if you give me a large COFFEE will I go to LOGIC.
d. He won't stop DANCING unless we turn off the MUSIC.
e. Dicapprio will be HAPPY if he wins an ACADEMY Award.
f. If I close my EYES it will all remain UNCHANGED.
g. Don't TURN yourself around unless you do the HOKEY-pokey.
h. My ANACONDA don't want none unless you got BUNS hun.

Intermediate Translations with '&' and '>'.
a. I'll HELP you with your logic only if you bring me COOKIES and DONUTS.
b. If neither MARK nor AMI understand something then it's UNINTELLIGIBLE.
c. Unless I understand how to do CP proof I should should go see AMI for help and if I don't understand MP I'm not going to do well on the TEST.
d. My ANACONDA don't want none unless you got BUNS hun and if you don't got BUNS then you should SQUAT.
e. If you have neither BUNS nor DONUTS my ANACONDA don't want none.

Translate the Argument and Solve the Proof

A species has the capacity for EMPATHY only if it can take the PERSPECTIVE of others and unless you have a concept of SELF you can't take another's PERSPECTIVE. Therefore, if a species is capable of EMPATHY is will have a concept of SELF as well as the ability to take the PERSPECTIVE of others.

Proof

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

How to Translate:

Translation song: 

Two step, two step,

Two step, two step,

Now go on and two step, now go on and two step
Now go on and two step, now go on and two step
Now get jiggy wit' it, now get jiggy wit' it
Now get jiggy wit' it, now get jiggy wit' it

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))

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