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

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