I. Review?

1. Tables with n-atomic sentences.
2. Concepts
3. only if, neither, etc.
4. Homework?

II. Exam

1. Do all problems...go for extra credit.
2. All answers go on the white paper.
3. There are different exam versions...don't copy. No books, no notes...
4. In the one "like 3.1", do one row tables.
5. In 7, the handout doesn't show metavariables P and Q well.

next...

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

1. The first step in the reasoning puts two ideas together:
   
   
   
   If ____ then ~~~~.
   And _____ is true.
   So, ~~~~~~.
   
   
   
   
   
   
   
2. 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.
   
   
   
   
   
   
   
   
   
3. We can see formulate this rule as follows:
   

   >E
   input 1: input 2: output:	P>Q P Q

   
   
   
   
   
   
   
   
   
   
4. 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...

I. Review:

1. rules? What was that rule name... > what?
   
   

   >E
   input 1: input 2: output:	P>Q P Q

   
   
   
   
   
   
   
   
   
2. How do we know this is valid?
   
   
   
   
   
   
   
   
   
   
   
   
   
   
3. Other rules?
   

   &E input: output: P&Q P or P&Q Q

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

   
   
   
   
   
   
   
   
   
   
   
   
4. Let's do some.
   
   
   
   
   
   
   
   
   
   
   
5. You Do:
   • From K>(J&L) and K, derive L
   • From K>J and (K>R)&F derive J&F

II. More rules...

1. = is called the biconditional because?
   
   It's equivalent to TWO conditionals?
   
   
   
2. So the following little arguments are valid.
   

   A=B B    A

   A>B B>A A=B

   
   
   
   
   
   
   
   
   
3. 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

   
   
   
   
   
   
   
   
   
   
   
   
4. So, for examples.
   
   
   
   
   
   
   
   
   
   
5. v-rules
• 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 v-I???
  

  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...
  

  D>S
  G>S
  DvG
    S

  OK?
  

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

  vE
  input 1: input 2: input 3: output:	PvQ P>R Q>R R

  
  
  
  

 

 




 

6. More Problems...
   
   
   
   
   
   
   
   
   
   
   
7. 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

1. How do we prove a conditional in informal proofs? We begin with what word?
   
   suppose
   
   
   
   
   
   
   
2. 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
   
   
   
   
   
   
   
   
   
3. Informal proof:
   
   If an argument is sound, then it does not have a false premise .      fill in...
   

   
   

4. How will we put this "suppose" into our derivations? Let's do some...
   
   
   
   
   
   
   
   
   
   
5. Oh and here's that rule...
   

   >I
   input: output:	P Q P>Q