L1 W7        (L2)

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Exam: 1 week!

 

 

 

 

I. Warm-up

  1. Problems 4 and 5...




  2. Subderivation idea again...by informal argument.



    B. If an argument is sound, then it is valid.      fill in...




    Let's do this one formally.






  3. Homework
    • Note the quizzes!
    • Problems?

 

II. Subderivation Ideas and Rules

  1. When we finish a subderivation (and move back to the originating column) this is called


  2. When a subderivation is terminated, we say that its assumption is


  3. A discharged assumption, and any other individual line in a terminated subderivation can no longer be _____. What's that word to fill in the blank?






  4. When a line can no longer be cited, it is called what?




  5. Let's see this in action.

 

 

III. Negation Rules

  1. These rules just use the reductio ad absurdum strategy.
  2. Example: 1. 2. 3.

     

     





     

  3. To formalize this, we just make an assumption...but we assume the opposite of what we are trying to prove. Let's do some.








  4. And, now you do:
    • A>L,~L / ~A
    • V&~C , V>~(P&~C) / ~P
                 (note: I'm cheating a bit to use 'V' for "an argument is valid". The premises can be read as "The argument is valid and has a false conclusion" and "if an argument is valid, then it doesn't have true premises and a false conclusion". Makes sense, yes?)

 

 

IV. Symbolizations

  1. L only if R
  2. L if and only if R
  3. L if R
  4. R is a sufficient condition for L
  5. L unless R
  6. Both neither L nor R and Q.
  7. Not both of L and R.
  8. If neither L nor R then Q if S.

 

 

 

 

 

 

L2

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EXAM next time

Office Hours...by appointment (tomorrow? I'll be around in the early afternoon.)

 

 

 

 

 

I. Review:

  1. Subderivations and >I. (repeated from above)



  2. If you need to prove ~P, you assume



  3. Homework (try 5,4,3 in that order)

 

 

 

II. Concepts

A. Validity: Two show that an argument is valid, we can just take its premises and derive what?



B. Logical Equivalence

  1. Two sentences are logically equivalent if and only if it is not possible that...



  2. So, if we set up a derivation, from one sentence as premise to the other as conclusion, that shows




  3. So, if we use derivations to test for logical equivalence, then we will use how many derivations? Answer:



















  4. Let's do some...

 

C. Logically True Sentences

  1. A sentence is logically true if and only if WHAT?




  2. Now, an example of a logical truth is "Rv~R". Thinking about 'R' as standing for "it's raining", what kind of evidence do you need to know that "Rv~R" is true?












  3. Let's try some.




  4. Now, try these:
  • Show that 'A=(AvA)' is logically true.
  • Show that 'A=B' and 'B=A' are logically equivalent.

           (On the computer...)

D. Logical falsehood and inconsistency:

  1. What do these two concepts have in common?...on their definitions, a sentence or a collection of sentences is not _____ ___?


























  2. Finally, let's do some more...

 

 

 

 

 

III. Review for exam: Mock Exam (online) and a mini-mock exam (below).

A. Fill in the following:

Premise 1 (K&J)&(J>R)
1 &E 2
1 &E 3
4
5
6
7
8
??? 9 R???

 

Premise 1 L>(K&W)
Premise 2 (L>(K&W))>L
3
4
5
6
7
8
9

 

B. Do derivations to show that the following arguments are valid.

1.

J
L&[(JvR)>T]
T

2.

S>L
T>L
L>S
S>T
S=L

3.

L>(KvT)
L>R
LvO
O>R
RvL
R

4.

B=(K&R)
K&S 
R>B

 

C. Symbolize (using the symbolization key of the last exam)

  1. Moriarity is a crook; moreover so is Holmes!
  2. Neither Holmes nor Watson is a crook.
  3. Unless Holmes is a crook, Moriarity is a crook.
  4. If exactly one of the three is a crook, then it's Moriarity.

Answers: