Disjunctive Syllogism

Agenda

1. Exam: 
1. If you didn't do as well as you would have liked 
2. (a) what was your attendance record? Did you take advantage of office hours and review sessions?
3. (b) What percentage of the homework did you complete?
4. (c) What can I do to help? 
2. Disjunctive Syllogism and 'v' Introduction.
3. Proofs

Exam

Surprising Results!!!!

Students who scored less than 80% on the test also had 80% or lower attendance rates. The lower the score, the lower the attendance rate. It's almost as though the two are related...

1. If you didn't do as well as you would have liked: 
2. (a) what was your attendance record? Did you take advantage of office hours and review sessions?
3. (b) What percentage of the homework did you complete?
4. (c) What can I do to help? 

Disjunctive Syllogism 

Basic Rule: 

1. PvQ              A
2. ~P/~Q          A
3. Q/P              DS

In natural language: 

1. Either you want PIZZA or you want QUOLA.
2. You don't want PIZZA/You don't want QUOLA.
3. You want QUOLA/You want PIZZA.

Adding complexity (but same rule):

A.

1. (P&Q)v(P>Q)  A
2. ~(P>Q)             A
3. P&Q                 DS

B. 

1. ((A<>~B)>(C>~B))v~((~DvC)>(C<~~D))      A
2. what would I need in order to derive                A
3. ((A<>~B)>(C>~B))                                          DS

Fun Activity!!!

In your groups create at least two natural language arguments using DS then symbolize it.

'v' Introduction

Basic Rule:

A. 

1. P                                                  A
2. PvQ (where Q can be anything) v Int

B. Let's kick it up a notch....BAAAM!!

1. P                                        A
2. Pv(P<>((~Q>R)&(SvT))  vInt

In natural language:

1. I'm going to study LOGIC tonight.                 A
2. I'm going to study LOGIC tonight or I'm going to order a PIZZA, drink BEER, and WATCH Trailer Park Boys.                                                       vInt

Fun Activity!!!

In your groups create at least 2 vInt arguments in natural language then symbolize it.

Proofs

A. 

1. P
2. (PvQ)>R   /R

B. 

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

C. 

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

D. 

1. (A>B)v(~C&D)
2. ~(A>B)   /:.  ~C

Introduction to Biconditionals and Disjunctions

Agenda

1. Quiz
2. Biconditional Proofs

Quiz

A.

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

B. 

1. Unless you BRUSH your teeth and SHOWER you shouldn't WRESTLE with me.
2. If you neither did the HOMEWORK nor came to OFFICE hours you're going to have a BAD time.
3. You should PANIC only if you don't know how to do MODUS ponens and modus TOLLENS.

Setting Up Proofs with Biconditional (Biconditional Strategy)
You will use this method anytime a biconditional is the bottom line unjustified line of your proof. Think of <> as & for two >s. In other words, I going to have to do two proofs: one proof for one direction and one for the other.
Suppose you have to solve for P<>Q. P<>Q is logically equivalent to P>Q & Q>P. This means in order to prove P<>Q I'm going to have to do two proofs: one for P>Q and one for Q>P then I can put them together using <>Int.

Step 1: Write P<>Q at the bottom of your proof.

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x. P<>Q

Step 2: I know that P<>Q is the same as P>Q and Q>P. Scan the given assumptions above to see if I'm given either of the conditionals. If I'm not given any then I'm going to set up two proofs: one for P>Q and one for Q>P. If one of the two conditionals is given to me in the assumptions then I only need to set up a proof for the conditional not given in the assumptions.

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e. Q>P
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m. P>Q
x. P<>Q        <>Int e, m

Step 3: Now that I've set up my proof I go about independently solving for both P>Q and Q>P beginning with CP as we've learned from previous lessons.

a.    Q    ACP
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.
.
d.    P     ____
e. Q>P  CP a, d
f.    P    ACP
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l.    Q  ____
m. P>Q  CP f, l
x. P<>Q  <>Int. e, m.


Proofs with <> + MP, MT, DN, &Int, &Elim, CP
A.

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

B.   

1. P&~Q  /:. P<>~Q

C. 

1. ~(C&~B)>~A    /:. (A&~B)<>A

D.

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

E. 

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

F. 

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

Review: Proofs with MP, MT, &Int, &Elim, DN, and CP

Agenda

1. If you're having trouble with anything we've done so far please see me during office hours so I can help you. Shatzel Hall Rm 340, M&F after class. 
2. Quiz
3. Review of MP, MT, Denying the Antecedent, Affirming the Consequent.
4. Review of Translations.
5. Review of Proofs.

Quiz

A. Translate the argument, name the argument structure and say whether it's valid. If it's invalid, make it valid.

Argument 1

1.  If Jerry DRINKS too much he won't WAKE up for work on time.
2.  Jerry didn't WAKE up for work on time.
3.  Therefore, Jerry DRANK (translate same as DRINKS) too much. 

Argument 2

1.  Unless you're the JESUS of logic then you should PRACTICE a bit of logic every day and see AMI for help.
2.  You're not the JESUS of logic.
3.  You should PRACTICE a bit of logic every day and see AMI for help.

B. Proof

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

Review of MP, MT, Denying the Antecedent, and Affirming the Consequent

A. Construct a conditional argument and give that argument in each of the above formats.

Review of Translations

A. You'll get this RIGHT only if you STUDIED.

B. Unless you STUDIED you won't get this RIGHT.

C. If you get this RIGHT then you STUDIED.

D. If you don't get this RIGHT then you neither STUDIED nor UNDERSTOOD it, moreover you should be CONCERNED if you didn't get it RIGHT.

Do You Haz Proofs?

A.

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

B. 

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

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
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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