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