5.5 Selected Answers

5.5ex I

Problem Two.  
Assumption 1 ....what if A&~D
1 &E 2 ....then... A
2 vI 3 ....then... AvX
1-3 >I 4 (A&~D)>(AvX)  

 

5.5ex II

Problem 2.  
Assumption 1 ....what if A
1 R 2 ....then... A
1-2 >I 3 A>A  
3,3 =I 4 A=A  
Problem 5.  
Premise 1 A
  2  
  3  
1,1 &I 4 A&A
~~~~ 5 ~part II~~
Premise 6 A&A
  7  
  8  
6 &E 9 A

5.5ex III

Problem One.
Assumption 1 ....what if A&B  
Assumption 2 ....then... ....what if ~J
1 &E 3 ....then... ....then... B
2-3 >I 4 ....then... ~J>B  
1-4 >I 5 (A&B)>(~J>B)    
Problem Three.  
Premise 1 J&(~F>T)  
Premise 2 ~(FvT)  
Assumption 3 ....what if F
3 vI 4 ....then... FvT
2 R 5 ....then... ~(FvT)
3-5 ~I 6 ~F  
1 &E 7 ~F>T  
6,7 >E 8 T  
2 R 9 ~(FvT)  
8 vI 10 FvT  
Problem Five.  
Assumption 1 ....what if A=B  
Assumption 2 ....then... ....what if B
1,2 =E 3 ....then... ....then... A
2-3 >I 4 ....then... B>A  
Assumption 5 ....then... ....what if A
1,5 =E 6 ....then... ....then... B
5-6 >I 7 ....then... A>B  
4,7 =I 8 ....then... B=A  
1-8 >I 9 (A=B)>(B=A)    

      5.5ex IV
(problems taken from The Logic Book.)

2f.
Premise 1 (~Bv~H)>M         
Premise 2 K&~M  
Assumption 3 What if.. ~B
3 vI 4 then... ~Bv~H
1,4 >E 5 then... M
2 &E 6 then... ~M
3-6 ~E 7 B  
3c.
Premise 1 R>S         
Premise 2 S>T  
Assumption 3 What if.. R
1,3 >E 4 then... S
2,4 >E 5 then... T
3-5 >I 6 R>T  
Note for those who've read to 5.7: this one could be done in SD+ more quickly. What new rule does the job?
3g.
Premise 1 A=B         
Premise 2 B=C  
Assumption 3 What if.. A
1,3 =E 4 then..... B
2,4 =E 5 then...... C
3-5 >I 6 A>C  
Assumption 7 What if.. C
2,7 =E 8 then... B
1,8 =E 9 then... A
7-9 >I 10 C>A  
6,10 =I 11 A=C  
4c.
Assumption 1 What if..  A=B                 
Assumption 2 What if.. What if... A
1,2 =E 3 then... then.... B
2-3 >I 4 then... A>B  
1-4 >I 5 (A=B)>(A>B)    
5d.
Premise  1 A>B           
Assumption  2 What if... ~B  
Assumption  3 then... What if... A
1,3 >E  4 then... then... B
2 R  5 then... then... ~B
3-5 ~I  6 then... ~A  
2-6 >I  7 ~B>~A    
~~~~~~~~   ~new~~~~~ ~~~~~~~ ~~~~~
Premise  1 ~B>~A           
Assumption  2 What if... A  
Assumption  3 then... What if... ~B
1,3 >E  4 then... then... ~A
2 R  5 then... then... A
3-5 ~E  6 then... B  
2-6 >I  7 A>B    
I've started renumbering after 7. You may make the second premise be number 9.
6b.
Premise 1 P>~P         
Premise 2 ~P>P  
Assumption 3 What if.. P
1,3 >E 4 then... ~P
3-4 ~I 5 ~P  
2,5 >E 6 P