I. Stragegy Review and Homework

1. 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 short-cut rules)!
  
  
  
  
  
  
2. Homework from 5.6

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II. PL Introduction

1. 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:
   

   S₁: Ann is a law student.
   S₂: Bill is a law student.
   O: All O.U. law students receive loans.
   L₁: Ann receives a loan.
   L₂ : Bob receives a loan.            and we'll....

   
   
   
   
   
   
   
   
   
   

   Be able to symbolize the argument only as:

   S₁&S₂
      O  
   L₁&L₂

   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...
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. 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

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
3. 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".

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
4. Problems

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IV. SD+

1. 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!!!
     
     
2. The two main types of rule are
•   1. rules of
  inference and...
           ???    
  
  
  

  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: IM (Implication) 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.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

   

3. Problems...
   
   
   
   
   
   
4. For you to do... (problems 1-3)
   
   • Av~B,B / A
   • (A&C)vD,[(C&A)vD]>L / L
   • ~(A&B) / A>~B

     
     
     
     
     
     
     

5. Homework Questions? (from 5.7)
   
   
   
   
   
   
   
   
6. 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)

I. Review 5.7 stuff (just above).

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II. 5.8 SD+ Strategy and Harder Derivations

1. Don't forget to do the tutorials. This one is only four pages long. But you still should spend a good 20-40 minutes making sure you get the hints and strategy.     Let's begin working on one page.
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   DI (Distribution) 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)

   
   
   
   
   
   
   
   
   
   
   
2. 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
     

     as a check on your work notice that these two start out in the same way (except for a swap of main connectives from '&' to 'v').

     ~Gv[(M&T)&~F]

     [~Gv(M&T)]&[~Gv~F]

     ~Gv[(M&T)&~F]

     
     
     
     

     [~Gv(M&T)]&[~Gv~F]
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • So, let's do that demonstration (T5.8 p. 2)
     
     
     
     
     
     
     
     
     
     
   • Stratgy Guidelines
     
     Strategy in SD+ is very similar to that in SD. Just a couple of minor amendments:
     

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V. Mock Exam

(cafe derivations )