I. Exam preview

• 1.
  Make assumptions only for >I

• 2.
  Pitfalls:

• Despair (and assuming what will work): "Don't assume"
• E,I rules used on what is not the main connective
• Citing the wrong number of inputs, e.g., "3 >E".
• Random rule application
• Citing from within a terminated subderivation

• 3.
  Symbolization questions?
  • Holmes is a crook only if Watson is.
    H>W
    
    
    
  • If neither Holmes nor Watson is
    a crook then Morarity is one.
    ~(HvW)>M
    
    
    
  • At most two of the three
    are crooks.
    ~(H&(W&M))

II. Exam

(next)

III. Strategy

1. Suppose you are trying to derive [F&~(CvD)]>(Fv~D) to show it's a logical truth.
   
   • 1.
     Don't panic!
     
     
     
   • 2.
     Just look at the main connective of your goal, assume the antecedent...
     
     
     
     
     
   • 3.
     ...and try to derive the consequent, 'Fv~D', as the last line of the subderivation.
     
     
     
     
     
   • 4.
     The big point here: We have a new goal with a new main connective and we start our strategic thinking anew as we try to derive 'Fv~D'! But this is much simpler.
     
     
     
     
     
     
   • 5.
     So, now we can think about the elimination rule for the main connective of our assumption. And, guided by our intemediate term goal, the derivation is easy.
     
     
     
     
     
     
     
     
     
     
     
     
2. Let's try it (and some more)
   
   
   
   
   
   
   
   
   

           (next)

IV. Short-cuts (A.K.A. SD+)

1. We will add three further rules of inference. We don't really need to. But each will shorten proofs. Let's see how this works.
   
   
   
   
   
   
   
2. Notice:
   What we've just seen is that each of our three new rules of inference is dispensible. We can do without them. But why bother? Obviously they will make our lives easier.
   
   
   
   
   
   
   
   
3. Here are the formal statements of the rules:
   

   SD+ Rules of Inference
   DS (Disjunctive Syllogism) input 1: input 2: output: PvQ ~P Q or PvQ ~Q P	MT (Modus Tolens ) input 1: input 2: output: P>Q ~Q ~P	HS (Hypothetical Syllogism) input 1: input 2: output: P>Q Q>R P>R

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Finally, you should try some (the premises go before the slash, the goal after the slash):
   • Kv~S,~K&(JvS) / J
   • (L&M)>J, J>(TvS) / (L&M)>(TvS)
   • (J&K)>(TvS), (TvS)>K / ~K>~(J&K)
     
     
     (computer version of these three)
     
     
     
     
     
     
     
     
     
     
5. But our most powerful new rules involve what are called
   

   rules of replacement.

   These just replace one sentence or part of a sentence with something logically equivalent.

   For example...
   

   '~(AvB)' can be replaced anywhere by '~A&~B', why?
   

   Because these are our two ways of saying "neither A nor B", they clearly mean the same thing.

   We'll call this rule DM (for De Morgan). Let's see it in action before we say just what a rule of replacement is. Example:
   

   There's and easy way and a hard way to do many problems.

   ~~~~		1		~ hard way	~~~~
   Premise		2		~(AvB)
   Assumption		3		....what if	A
   3 vI		4		....then...	AvB
   2 R		5		....then...	~(AvB)
   3-5 ~I		6		~A
   Assumption		7		....what if	B
   7 vI		8		....then...	AvB
   2 R		9		....then...	~(AvB)
   7-9 ~I		10		~B
   6,10 &I		11		~A&~B
   Or: just use our new rule...
   ~~~~		12		~ EZ way ~~
   Premise		13		~(AvB)
   13 DM		14		~A&~B

    

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
6. Here's what DM looks like ; the double arrow indicates equivalence. Notice that there are two versions. We just used the second; what does the first remind you of?
   
   

   DM (De Morgan's)
   ~(P&Q)~Pv~Q or ~(PvQ)~P&~Q

   You may use this rule to...
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   • Make a replacement going in either direction. (Unlike rules of inference.)
   • Make a replacement on any component of a sentence. (Unlike rules of inference.)
   
   
   
   
   
   
   
   
   
7. More next time!

I. Strategy Review

1. Strategy Guidelines
   
   
   
2. Presentations (don't forget to do the tutorials...there's lots of explanation there!)
   
   
   
   
   
3. Concepts and Strategy for more complex derivations
   • Sometimes we need to make a subderivation within a subderivation. Think about proving that 'K>(J>(K&J))' is logically true. You'd begin by assuming what?
     

     K

     and your new goal would be...
     

     'J>(K&J)' on the last line of the subderivation.

     But this goal has main connective '>'. So to prove it we'd need to assume it's antecedent. Here is the picture:
     

     Assumption		1		....what if	K
     Assumption		2		....then...	....what if	J
     		3		....then...	....then...
     2-3 >I		4		....then...
     1-4 >I		5

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • Let's do some

(next)

II. SD+ Revisited

Remember our SD+ Rules of Inference?

SD+ Rules of Inference
DS (Disjunctive Syllogism) input 1: input 2: output: PvQ ~P Q or PvQ ~Q P	MT (Modus Tolens ) input 1: input 2: output: P>Q ~Q ~P	HS (Hypothetical Syllogism) input 1: input 2: output: P>Q Q>R P>R

1. Finish a problem set: problems 4 and 5
   
   
   
   
   
   
2. But our most powerful new rules involve what are called
   

   rules of replacement.

   These just replace one sentence or part of a sentence with something logically equivalent.

   For example...
   

   
   
   
   
   
   
   
   
   
   
   

   DM (De Morgan's)
   ~(P&Q)~Pv~Q or ~(PvQ)~P&~Q

   the double arrow indicates equivalence

   You may use this rule to...
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   • Make a replacement going in either direction. (Unlike rules of inference.)
   • Make a replacement on any component of a sentence. (Unlike rules of inference.)
   
   
   
   
   
   
   
   
   
3. There are lots of other rules of inference. They all make sense and all are dispensible. Still, they make good short-cuts. Here are three more:
   
   

   CM (Commutation) P&Q  Q&P or PvQ  QvP or P=Q  Q=P

   DN (Double Negation)   P ~~P

   AS (Association) P&(Q&R)(P&Q)&R or Pv(QvR)(PvQ)vR

   So, 'A&~B' can be replaced by '~B&A'. And you may write 'JvC' in place of '~~(JvC)'. And parentheses can be rewritten around '(A&~C)&~D' to give 'A&(~C&~D)' if you'd like.

    

   These new rules are all dispensible too: but they are great short-cuts. Let's make sure we understand this.

    

   We'll see this applied in a moment. First, think about one that is a little harder:
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   To think about this one, think about this sign (found by a reporter on an Iraqi bridge):

   Don't cross or die.

   Not a very nice sign. But effective because it means...

   If you cross, then what?

   you'll die.

   So...
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   

   The following inference scheme makes sense:

   IM (Implication)
   P>Q  ~PvQ

   The bridge inference just goes from right to left.

   Or try this:

   • If we keep them from scoring then we win. (K>W)
   • Either we fail to keep them from scoring or we do win. (~KvW)

   These two sentence seem like they mean the same thing...yes? You can do a truth table to check or do derivations to show logical equivlalence.

   Time to try some!

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Distribution
   • Suppose we know that (a) we will go to Colorado this summer and (b) we will either climb Mount Evans or raft the Arkansas. In symbols...
     

     G&(CvR)

     But, this means that

     
     
     
     
     
     

     There are two possibilites for what might happen:

     We go to Colorado and climb Mount Evans. (G&C)

     or

     We go to Colarado and raft the Arkansas. (G&R)

     In other words 'G&(CvR)' is logically equivalent to
     

     (G&C)v(G&R)
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • This helps exlain our new rule
     
     

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

     How do we memorize this one?
     

     Notice that both the left hand sides and the right start out the same way.
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
5. Homework
   
   
   
   
   
   
   
6. Here's a rule that is self explanatory: Equivalence
   
   

   EQ (Equivalence)
   P=Q (P&Q)v(~P&~Q) or P=Q (P>Q)&(Q>P)

   
   
   
   
   
   
   
   
   
   
   
   
7. And a few final rules...
   
   

   TR (Transposition) P>Q  ~Q>~P

   ID (Idempotence) P&P P or PvP P

   EX (Exportation)   P>(Q>R)(P&Q)>R

   
   
   
   
   
   
   
   
   
   
   
   
   
   
8. Homework

   5.7ex IV

   1. From the following premises:
         ~S>~T
         ~(T>S)vL
   
    derive: L
   
   2. From the following premise:
         A>(A>E)
   
    derive: A>E
   
   
   3. From the following premises:
         ~(P=Q)
         ~(P&Q)>(~R&~S)
   
    derive: ~(RvS)
   
   
   4. From the following premise:
         (~S&G)v(~S&K)
   
    derive: (KvG)&~S
   
   
   5. From the following premises:
         L&(S>T)
         ~(L&T)
   
    derive: L&~S