1. IMPORTANT!!
As one member of our class recognized, there is a major mistake on page 180 of the text where the rule schemas for SD are laid out.
It symbolizes a rule it calls E2 – there is no such rule! – as
P  Q Q P
Such a rule is NOT truth preserving and not in SD

2. IMPORTANT!!
There is only one rule for eliminating the horseshoe (E). And it is symbolized properly inside the front cover of the text and used throughout the chapter.
P  Q P Q

3. Less important…
I did not notice that this new edition has us add the relevant rule following an auxiliary assumption that starts a subderivation.
This is useful when you’re trying to go back to fill in line numbers especially if the derivation contains a lot of subderivations and auxiliary assumptions.

4. Proving SD notions Using derivations to prove that
a sentence of SL is a theorem in SD
a sentence P is derivable in SD from a set  of sentences of SL
an argument of SL is valid in SD
a set of sentences of SL is inconsistent in SD
sentences P and Q are equivalent in SD

5. Show that ⊦ A  (B  A)
A A/I
B A/I
A 1 R
B  A I
A  (B  A) I

6. Can we show that ⊦ A  (B  C)
A A/I
B A/I
----- C
B  C I
A  (B  C) I

7. Show that the following argument is valid in SD: ~A v ~B A -----------
An argument is valid in SD IFF its conclusion is derivable from the set consisting of its premises
Show that the following argument is valid in SD:
~A v ~B A ~B

8. 1 ~A v ~B A
2 A A
3 ~A A/vE ~B
~B A/vE
~B vE

9. 1 ~A v ~B A
2 A A
3 ~A A/vE
B A/~I
~B ~I
~B A/vE
~B R
~B vE

10. 1 ~A v ~B A
2 A A
3 ~A A/vE
B A/~I
A 2R
~A 3R
7 ~B ~I
8 ~B A/vE
9 ~B 8R
10 ~B 1, 3-7, 8-9 vE

11. There’s more than one way to derive a sentence, but some are easier…
1 ~A v ~B A
2 A A
~B how about ~I?

12. There’s more than one way to derive a sentence, but some are easier…
1 ~A v ~B A
2 A A
3 B A/~I
4 A 2 R ~A
~B how about ~I?

13. 1 ~A v ~B A
2 A A
3 B A/~I
4 A 2 R
~A A/vE
6 ~A 5R
7 ~B A/vE
A A/~I
B 3 R
~B 7R
~A ~I
~A 5-6, 7-11 vE
13 ~B ~I

14. One special case of validity…
Show that the following argument is valid in SD:
A  B
A  ~B A
M  R

15. Special cases…
1 A  B A
2 A  ~B A
3 A A
M  R

16. Special cases…
1 A  B A
2 A  ~B A
3 A
4 ~(M  R) A/~E B ~B
M  R ~E

17. Special cases… 1 A  B A 2 A  ~B A 3 A 4 ~(M  R) A/~E 5 B 1, 3  E
M  R 4-6, ~E

18. P and Q are equivalent in SD IFF Q is derivable in SD from {P} and P is derivable in SD from {Q}
Show that the following pair of sentences is equivalent in SD: A
~~A
So we need 2 derivations

19. Demonstrating equivalence
1 A A
~A A/~I
~A 2R
4 A 1R
5 ~~A 2-4,~I

20. Demonstrating equivalence
1 ~~A A A

21. Demonstrating equivalence
1 ~~A A
2 ~A A/~E
3 ~A 2 R
4 ~~A 1 R
5 A ~E

22. Demonstrating that a set is inconsistent in SD
A set  is inconsistent in SD IFF there is some sentence P such that both P and ~P are derivable from .
A set  is consistent in SD IFF there is no sentence P such that both P and ~P are derivable from 

23. Show that {A  B, B  ~A, A} is inconsistent in SD
1 A  B A
2 B  ~A A
3 A A
4 A 3R ~A

24. Show that {A  B, B  ~A, A} is inconsistent in SD
1 A  B A
2 B  ~A A
3 A A
4 A 3R
5 B , 4  E
6 ~A 2, 5 E