L1 W (L2)
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Exam...full length! One Week!!!
I. Stragegy Review and Homework
 Think about soundness and validity and the truth of an argument's conclusion.
 Informally we can prove that a sound argument has a true conclusion:
Proof:
Suppose that an argument A is sound.
A is valid and has only true premises.
A is valid so couldn't have a false conclusion if it has true premises.
A does have all and only true premises. So it must have a true conclusion. Q.E.D.
But...
So ...
Then...
 So that we can transform this proof into an SD one think about our Strategy Guidelines (Review)
 Let's do this one formally in SD (no shortcut rules)!
 Homework from 5.6
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II. PL Introduction
 Suppose we have an argument like
Bill and Ann are O.U. law students.
All O.U. law students receive loans.
Therefore, Bill and Ann receive loans.
With SL, we can only symbolize this with compound sentences. We could
use an interpretation like so:
S_{1}: Ann is a law student.
S_{2}: Bill is a law student.
O: All O.U. law students receive loans.
L_{1}: Ann receives a loan.
L_{2} : Bob receives a loan.
and we'll....
Be able to symbolize the argument only as:
S_{1}&S_{2 }
O
L_{1}&L_{2}
the relationships between types of people  here between people who are students and those who are indebted  these don't show up in our symbols.
We need a way to say that both Bob and Ann share the property of being students AND SO they share the property of indebtedness!
And to do this in our argument we...
must be able to talk about ALL O.U. law students. This is to express the quantity (all) who receive loans.
But this is clearly not valid
we've left out...
 Names and Properties

Names...
We'll use lower case letters to stand for particular objects.
So, 'a' may stand for Ann, and 'b' for Bob.

Predicates...
We'll let our upper case letters do double duty. They will continue to stand for whole sentences. But
also
will stand for descriptions or predicates.
So, 'S' may stand for "is a student" and
'R' may stand for "receives loans".
Then how will we say
"Ann receives loans"?
aR might seem right. BUT we'll do it this way:
Ra
 Quantifiers
These are used to talk about the quantity of things: All or Some

Existential
We'll say that SOMETHING has a property by using this funny symbol:
%
And we'll say that someone receives a loan this way.
(%x)Rx
There is someone (namely x) and he or she, x, receives a loan.

Universal
When we talk about ALL or EVERY object, the quantity is universal and we use this symbol:
^
And to symbolize that everyone receives a loan, we'd write
(^x)Rx
Read this as "Everyone x is such that x receives a loan".
 Problems
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IV. SD+
 Review
 Don't forget the SINGLE QUOTE and SLASH typing aids. These save a tremendous amount of time.
 SINGLE QUOTE: just type for reiteration or automatic assumption (works only on the final goal)
 SLASH: read about this one in T6!!!

 The two main types of rule are
 1. rules of
???
2. rules of replacement.
Here's a rule of inference:
DS 
(Disjunctive Syllogism) 

input 1:
input 2:
output: 
PvQ
~P
Q 
or 
PvQ
~Q
P 



And a rule of replacement:

P>Q ~PvQ 

Rules of replacement are more powerful. Rules of inference are one way (input to output)
while rules of replacement
work in both directions.
But just as important, rules of inference work on main connectives, while rules of replacement may be used to
replace
any component of a sentence.
 Problems...
 For you to do... (problems 13)
 Av~B,B / A
 (A&C)vD,[(C&A)vD]>L / L
 ~(A&B) / A>~B
 Homework Questions? (from 5.7)
 Final Rules (from W8 homework)
Now try some (Problems 4 on)
 ~(A=B),~A / B
 ~(J&K)>~L / L>J (try this one w/o doing a subderivaiton)
 (AvA)>A is a logical truth
 (~FvK)&(~FvJ) / F>(K&J)
L2
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EXAM! Chapter 5 and a little of 4.
I. Review 5.7 stuff (just above).
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II. 5.8 SD+ Strategy and Harder Derivations
 Don't forget to do the tutorials. This one is only four pages long. But you still should spend a good 2040 minutes making sure you get the hints and strategy. Let's begin working on one page.

P&(QvR)(P&Q)v(P&R) 
or 
Pv(Q&R)(PvQ)&(PvR) 


Our example was...
"We'll go and climb or raft" which is logically equivalent to "We'll go and climb or we'll go and raft".
Or...
G&(CvR) (G&C)v(G&R)

 Demonstrations: make sure you work through these...
 Think about
DI
(distribution). Here's a similar application of DI. Suppose you have
this sentence...
and
this one:
then, if you think about he second DI equivalence, you should notice that P is '~G' and Q is 'M&T', and R is '~F'. These two fit the DI pattern.
So, we can move from one to the other by DI.
Then
~Gv[ (M&T)& ~F]
[ ~Gv (M&T)]&[ ~Gv ~F]
 So, let's do that demonstration (T5.8 p. 2)

Stratgy Guidelines
First, here are the basic SD startegy guidelines:
 Keep your goal in mind. Often you will have intermediate goals as well as the final goal, i.e., the last sentence of the derivation.
 See if there is any obvious way to get to your goal.
 Consider using Irules for the main connective of your goal and Erules for accessible statements.
 If all else fails, use ~E to prove your goal.
SD+...
a) Once you get used to the new rules, they will sometimes allow you to quickly see how to move toward a goal in just one or two steps. Thus, the use of SD+'s new rueles will be obvious and comes in at step 2 of the strategy guidelines.
Then...
b) These rules allow one to analyze complex negations. For example,
consider a premise '~(P>Q)'.
In SD one must usually reiterate this premise for contradiction; there is no way in SD to break this sentence into parts. (For example, think about how you would use SD to derive '~Q' from this single premise.)
But,
in SD+ one may first apply IM inside the parentheses (to derive '~(~PvQ)') and then DM (to derive '~~P&~Q'). This then can be further simplified with &E or DN.
Finally
c) don't get lost! There are now so many rules that one can spend all day (literally!) going around in circles, getting no nearer one's goal sentence.
Remember, that "obvious" in step two of the strategy means "obviously helps move toward your goal". Do not apply the rules at random!
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V. Mock Exam
(cafe derivations )
