5.6 Selected Answers

      5.6ex I

2.
Premise 1 A=(Lv~S)          
Premise 2 L&(~A=T)  
Assumption 3 What if.. T
2 &E 4 then...... L
4 vI 5 then...... Lv~S
1,5 =E 6 then...... A
2 &E 7 then...... ~A=T
3,7 =E 8 then...... ~A
3-8 ~I 9 ~T  

The goal is '~T', so one may think of ~I and the assumption of 'T'.

 
3.
Premise 1 SvT         
Premise 2 S>(W&Z)  
Premise 3 T>Z  
Assumption 4 What if.. S
2,4 >E 5 then...... W&Z
5 &E 6 then...... Z
4-6 >I 7 S>Z  
1,3,7 vE 8 Z  
8 vI 9 ZvL  

The giveaway here is the main connective of line 1. One needs to set up for vE. That requires two conditionals. One is line 3. The other you need to derive at line 7.

Thus you have a new, intermediate goal: 'S>Z'. It needs to be derived by >I.

 
5.
Premise  1 ~X>(Y&T)           
Premise  2 ~(LvX)  
Assumption  3 What if.. X
3 vI  4 then...... LvX
2 R  5 then...... ~(LvX)
3-5 ~I  6 ~X  
1,6 >E  7 Y&T  
Assumption   8 What if.. Y
7 &E  9 then...... T
8-9 >I 10 Y>T  
Assumption 11 What if.. T
7 &E 13 then......

Y
11-13 >I 14 T>Y  
10,14 =I 15 Y=T  

     5.6ex II

1.
Premise  1 ~(O&J)           
Assumption  2 What if.. O  
Assumption  3 then...... What if.. J
2,3 &I  4 then...... then...... O&J
1 R  5 then...... then...... ~(O&J)
3-5 ~I  6 then...... ~J  
2-6 >I  7 O>~J    
3.
Premise  1 AvB           
Premise  2 ~B    
Assumption  3 What if.. A  
3 R  4 then...... A  
3-4 >I  5 A>A    
Assumption  6 What if.. B  
Assumption  7 then...... What if.. ~A
6 R  8 then...... then...... B
2 R  9 then...... then...... ~B
7-9 ~E 10 then...... A  
6-10 >I 11 B>A    
1,5,11 vE 13 A    
5.
Premise  1 Z           
Assumption  2 What if.. ~M  
Assumption  3 then...... What if.. W&~K
1 R  4 then...... then...... Z
3-4 >I  5 then...... (W&~K)>Z  
2-5 >I  6 ~M>[(W&~K)>Z]    

     5.6ex III

2.
Premise  1 (AvB)>C           
Assumption  2 What if.. A  
Assumption  3 then...... What if.. ~C
2 vI  4 then...... then...... AvB
1,4 >E  5 then...... then...... C
3 R  6 then...... then...... ~C
3-6 ~I  7 then...... ~~C  
2-7 >I  8 A>~~C    

The strategy here is straightforward: one's ultimate goal has main connective >, so line 2 assumes its antecedent. This >I application needs '~~C' on the last line of its subderivation: thus one thinks of ~I and assumes '~C' (line 3).

 
4.
Premise  1 ~(A>B)           
Assumption  2 What if.. ~B  
Assumption  3 then...... What if.. A
2 R  4 then...... then...... ~B
3-4 >I  5 then...... A>~B  
1 R  6 then...... ~(A>B)  
2-6 ~E  7 B    

Notice how Reiteration is used in line 6. It's the only thing we could do to line 1.

      5.6ex VII (Derivations showing logical truth. These make better sense if you read them while thinking about the goal: bottom up rather than top down!)

Problem 1.  
Assumption 1 ....What if L  
Assumption 2 ....then... ....What if J
1 R 3 ....then... ....then... L
2-3 >I 4 ....then... J>L  
1-4 >I 5 L>(J>L)    
Problem 2.  
Assumption 1 ....What if A
1,1 &I 2 ....then... A&A
1-2 >I 3 A>(A&A)  
Assumption 4 ....What if A&A
4 &E 5 ....then... A
4-5 >I 6 (A&A)>A  
3,6 =I 7 A=(A&A)  
Problem 3.  
Assumption 1 ....What if L&S  
Assumption 2 ....then... ....What if L
1 &E 3 ....then... ....then... S
2-3 >I 4 ....then... L>S  
Assumption 5 ....then... ....What if S
1 &E 6 ....then... ....then... L
5-6 >I 7 ....then... S>L  
4,7 =I 8 ....then... L=S  
1-8 >I 9 (L&S)>(L=S)    
Problem 4.  
Assumption 1 ....what if E  
Assumption 2 ....then... ....What if ~E
1 R 3 ....then... ....then... E
2-3 ~I 4 ....then... ~~E  
1-4 >I 5 E>~~E    

5.6ex VIII

Problem 1.  (showing logical falsehood)
Premise 1 ~(Jv~T)&(~T&L)
1 &E 2 ~T&L
2 &E 3 ~T
3 vI 4 Jv~T
1 &E 5 ~(Jv~T)
Problem 2.   (showing logical falsehood)
Premise 1 J&~~~J  
Assumption 2 ....What if ~~J
1 &E 3 ....then... ~~~J
2-3 ~E 4 ~J  
1 &E 5 J  
Problem 4.   (showing logical inconsistency)
Premise 1 F=(A&~A)  
Premise 2 ~F>F  
Assumption 3 ....What if ~F
2,3 >E 4 ....then... F
3-4 ~E 5 F  
1,5 =E 6 A&~A  
6 &E 7 A  
6 &E 8 ~A  
Problem 5.   (showing logical inconsistency)
Premise 1 ~CvA    
Premise 2 C&~A    
Assumption 3 ....What if ~C  
Assumption 4 ....then... ....what if ~A
2 &E 5 ....then... ....then... C
3 R 6 ....then... ....then... ~C
4-6 ~E 7 ....then... A  
3-7 >I 8 ~C>A    
Assumption 9 ....what if A  
9-9 >I 10 A>A    
1,8,10 vE 11 A    
2 &E 12 ~A    

     5.6ex IX (Derivations to show logical equivalece.)

2.   Notice that the "new" derivation starts after line 10.
Premise 1 L>W    
Assumption 2 ....what if ~W  
Assumption 3 ....then... ....What if L
1,3 >E 4 ....then... ....then... W
2 R 5 ....then... ....then... ~W
3-5 ~I 6 ....then... ~L  
  7      
  8      
2-6 >I 9 ~W>~L    
~~~~ 10 ~ part II ~~~~ ~~~~
Premise 11 ~W>~L    
Assumption 12 ....what if L  
Assumption 13 ....then... ....What if ~W
11,13 >E 14 ....then... ....then... ~L
12 R 15 ....then... ....then... L
13-15 ~E 16 ....then... W  
  17      
12-16 >I 18 L>W    
3.  
Premise 1 J  
Assumption 2 ....what if ~J
1 R 3 ....then... J
2-3 ~I 4 ~~J  
~~~~ 5 ~ part II ~ ~~~~
Premise 6 ~~J  
Assumption 7 ....what if ~J
6 R 8 ....then... ~~J
7-8 ~E 9 J  
4.  
Premise 1 ~(A>A)    
Assumption 2 ....what if ~(A&~A)  
Assumption 3 ....then... ....What if A
3-3 >I 4 ....then... A>A  
1 R 5 ....then... ~(A>A)  
2-5 ~E 6 A&~A    
~~~~ 7 ~ part II ~ ~~~~ ~~~~
Premise 8 A&~A    
Assumption 9 ....what if A>A  
8 &E 10 ....then... A  
8 &E 11 ....then... ~A  
9-11 ~I 12 ~(A>A)    

5.6ex X

Problem 1.  
Premise 1 A>B  
Premise 2 ~B  
Assumption 3 ....What if A
1,3 >E 4 ....then... B
2 R 5 ....then... ~B
3-5 ~I 6 ~A  
Problem 2.  
Premise 1 ~(LvS)&(~L>S)  
Assumption 2 ....What if L
2 vI 3 ....then... LvS
1 &E 4 ....then... ~(LvS)
2-4 ~I 5 ~L  
1 &E 6 ~L>S  
5,6 >E 7 S  
7 vI 8 LvS  
1 &E 9 ~(LvS)  
Problem 4.  
Premise 1 A=B  
Premise 2 ~(A>B)  
Assumption 3 ....What if A
1,3 =E 4 ....then... B
3-4 >I 5 A>B  
Problem 5.  
Premise1A&~C   
Assumption2 ....What if A>C  
1 &E3....then...A  
2,3 >E4....then...C  
1 &E5....then...~C  
2-5 ~I6~(A>C)   
~~~~7 ~part II~ ~~~~~~~~~~~~
Premise8~(A>C)   
Assumption9 ....What if ~A  
Assumption10....then... ....What if A 
Assumption11....then.......then... ....What if ~C
10 R12....then.......then.......then...A
9 R13....then.......then.......then...~A
11-13 ~E14....then.......then...C 
10-14 >I15....then...A>C  
8 R16....then...~(A>C)  
9-16 ~E17A   
Assumption18....what ifC  
Assumption19....then... ....What if A 
18 R20....then.......then...C 
19-20 >I21....then...A>C  
8 R22....then...~(A>C)  
18-22 ~I23~C   
17,23 &I24A&~C