w6
L1 (L2)
EXAM: Soon
Class will resume after 1 hour (we'll take roll and do important stuffa big step):
Derivations
I. Review?
 Tables with natomic sentences.
 Concepts
 only if, neither, etc.
 Homework?
II. Exam
 Do all problems...go for extra credit.
 All answers go on the white paper.
 There are different exam versions...don't copy. No books, no notes...
 In the one "like 3.1", do one row tables.
 In 7, the handout doesn't show metavariables P and Q well.
next...
w6 (continued)
roll
next exam: two weeks! We need to get busy!
Exam Answers...
III. Derivations
Who are Holmes and Moriarity? What's Holmes so famous for? (Brilliant .... Elementary my dear... )
Think about our longer argument (oops, I removed a tilde):
(A&~B)>C
A&~B
C&A
Maybe this is:
If Arthur is a student and Barb is not, then she's been cheated. Since it turns out that Arthur is a student and Barb is not, it follows that she's been cheated as well as that he's a student.
C. Rules
 The first step in the reasoning puts two ideas
together:
If ____ then ~~~~.
And _____ is true.
So, ~~~~~~.
 Lots of easy inferences have this
form:
If there's fire, then there must be oxygen present.
There is fire.
Thus...
There must be oxygen present.
 We can see
formulate this rule as follows:
>E 
input 1:
input 2:
output: 
P>Q
P
Q 
 So we can name rules , look
again...
1. Premise that (A&~B) >C
2. And that A&~B
3. So, we have C (from 1 and 2 by > E)
4. And we have ~B (from 2 alone by
???)
5. Conclusion C&~B (from 3 and 4 by
???)
&E 
input:
output: 
P&Q
P 
or 
P&Q
Q 
&I 
input 1:
input 2:
output: 
P
Q
P&Q 
5. Let's do this one more formally...
L2
roll
exam return 5 = A, 10 = B, 15 = C, 20 = D
next exam: 12 days! We need to get busy!
Office Hours: T 121:30, Th 23 and by appointment. My Tuesday OH sometimes run a little late because I'm in another class.
I. Review:
 rules? What was that rule name... > what?
>E 
input 1:
input 2:
output: 
P>Q
P
Q 
 How do we know this is valid?
 Other
rules?
&E 
input:
output: 
P&Q
P 
or 
P&Q
Q 

&I 
input 1:
input 2:
output: 
P
Q
P&Q 

 Let's do some.
 You Do:
 From K>(J&L) and K, derive L
 From K>J and (K>R)&F derive J&F
II. More rules...
 = is called the biconditional
because?
It's equivalent to TWO conditionals?
 So the following little arguments are
valid.
 Our new rules simply
follow these validities:
=E 
input 1:
input 2:
output: 
P=Q
P
Q 
or 
P=Q
Q
P 

=I 
input 1:
input 2:
output: 
P>Q
Q>P
P=Q 

 So, for examples.
 vrules
 What would you need to know, in order to be certain that at least one of Ken and Beth is from Minnesota?
What ONE thing about Beth would you need to know to be sure that the disjunction is true?
That she's from Minnesota? Yes, of course!
Also, if you knew that Ken was from Minnesota, that would do the trick as well.
So, to prove a disjunction, one need only know that at least one of the two disjuncts is
true...
vI 
input:
output: 
P
PvQ 
or 
Q
PvQ 
 Now, how should we think about
vI???
Suppose that if Dirk wins the starting center position, then our team will have a 7 footer.
But, Gene is also 7' tall. So, as well, if Gene wins out, then we'll have a 7 footer on the team.
Now, as it happens, either Dirk or Gene will win the position.
So...
Either way, then, we'll have a seven footer.
The reasoning is
this...
vE 
input 1:
input 2:
input 3:
output: 
PvQ
P>R
Q>R
R 
 More Problems...
 Your Problems
A. A=B,B>C,A&L / C&L (i.e., from the first three derive 'C&L')
B. (AvB)=C,B&L / C
C. (AvB)=C,C&L / AvB
D. AvB,A>J,(B>J)&R / J
next...
III. Conditional Introduction
 How do we prove a conditional in informal proofs? We begin with
what word?
suppose

Example:
1. Premise: If an argument is sound, then it has true premises. S>T
2. Premise: If an argument has true premises, then it premises are not false. T>~F
3. Conclusion: If an argument is sound, then its premises are not false. S>~F
 Informal proof:
If an argument is sound, then it does not have a false premise . fill in...
 How will we put this "suppose" into our derivations? Let's do some...
 Oh and here's
that rule...
>I 
input:
output: 

P
Q 

P>Q 

