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...

   Proof:

   
   
   
   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
   
   terminating the subderivation.
   
   
   
2. When a subderivation is terminated, we say that its assumption is
   
   discharged.
   
   
   
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?
   

   cited.

   No line in a terminated subderivation can be used to justify a later line.

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

   Assume 2+(3x7)=(2+3)x7

   2 + 21 = 5 x 7

   23 = 35

   But this is absurd! Hence our original assumption must be wrong.

    

    

   
   
   
   
   

    

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.

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

I. Review:

1. Subderivations and >I. (repeated from above)
   
   
   
   
2. If you need to prove ~P, you assume
   
   P, the opposite of your goal!
   
   
   
   
3. Homework (try 5,4,3 in that order)

II. Concepts

B. Logical Equivalence

1. Two sentences are logically equivalent if and only if it is not possible that...
   one is true and the other false.
   
   
   
   
   
2. So, if we set up a derivation, from one sentence as premise to the other as conclusion, that shows
   
   if one is true then the other must also be true.
   
   
   
   
   
3. So, if we use derivations to test for logical equivalence, then we will use how many derivations? Answer:
   

   Two.

   To show that 'A&B' and 'B&A' are logically equivalent, do this:

   1. 2. 3. 4.

   1. 2. 3. 4.

    

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Let's do some...

C. Logically True Sentences

1. A sentence is logically true if and only if WHAT?
   
   it is impossible for the sentence to be false.
   
   
   
   
   
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?
   

   None!

   And here's the point: you need NO evidence and so NO PREMISES to prove logical truths.

   (Of course, this means that you'll need an assumption to get anywhere.)

   
   
   
   
   
   
   
   
   
   
   
   
   
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 _____ ___?
   

   possibly true.

   A logically false statement is one that in and of itself could not be true. It's a kind of self-contradiction. Like 'A&~A'.

   A logically inconsistent set of sentences is
   

   a set that couldn't all be true together. Like {J, J=K, ~K}.

   So,
   

    

    

   Our test is to take the sentence or sentences AS PREMISES OF A DERIVATION and see if they lead to a contradiction: any P and ~P (derived on two distinct lines of the derivation).

    

    

    

    

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   
   
   
   
   
   
   
   
   
   

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:

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

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: