1. A v B
~ A
B > C ~C
B > (C . P) B
/ ~ B mt 1,2
/ B ds 1,2
/ C . P mp 1,2
~~S
R > ~S
A v B
C > D
/ ~ R mt 1,2
/ (A v B) . (C > D) cn 1,2
A > (B. C)
(B . C) > K
1. (L > M) . (D v N)
/ L > M sm 1
/ A > K hs 1,2

2. R . T premise
R > S premise
R ___
S ___
1 sm
2,3 mp
B > P premise
P > X premise
B > X ___
(B > X) . (B > P) ___
1,2 hs
3,1 cn

3. ~ (A > P) > ~S premise
2, dn
1,3 mt
4, dn
(D v S) > (P . Q) premise
S v D premise
D v S ___
P . Q ___
Q . P ___
Q ___
2, cm
3,1 mp
4, cm
5 sm

4. P . (Q . R) premise
(P . Q) . R ___
R . (P . Q) ___
R ___
1, as
2, cm
3 sm
S > (A . (B v C)) premise
S premise
A . (B v C) ___
B v C ___, ___
1,2 mp
3, cm , sm
(L v X) > (A . B) premise
M premise
M > (X v L) premise
(X v L) > (A . B) ___
M > (A . B)
A . B ___
A ___
B ___
1 cm
3,4 hs
2,5 mp
6 sm
6 cm, sm

5. J > (K > L) premise
L v J premise
~L premise
J ___
K > L ___
~K ___
2,3 ds
1,4 mp
3,5 mt
~G > ( G v (S > G)) premise
(S v L) > ~G premise
S v L premise
~ G ___
G v (S > G) ___
S > G ___
~S ___
L ___
2,3 mp
1,4 mp
4,5 ds
4,6 mt
3,7 ds

6. (K . B) v ( ~L > E)
~( B . K)
~E / L
(K . O) > (N v T) O
K / T v N
(R > F) > ((R > ~G) > (S > Q))
(Q > F) > (R > Q)
~G > F
Q > ~G / S > F
If a tenth planet exists, its orbit is perpendicular to that of the other
planets. Either a tenth planet is responsible for the death of the
dinosaurs or its orbit is not perpendicular to that of the others. A tenth
planet is not responsible for the death of the dinosaurs, so there is
no tenth planet.

7. How do you approach doing a proof?
First, you look to see if the conclusion –at least a piece of it—can be located in
one of the premises: here you spot it in line 1, as the consequent.
Unfortunately, the order of the letters is wrong.
But that tells us two things: Commutation will be the final step of the proof; and
MP will be the one before that. How do we know?
1. (K . O) > (N v T)
2. O
3. K / T v N
Because MP is the rule that delivers a consequent, and CM is the rule that
switches the order of letters around dots and wedges.
4. K . O CN 2,3
5. N v T MP 4, 1
6. T v N CM 5

8. 1. (K . B) v ( ~L > E)
2. ~( B . K)
3. ~E / L
Find the conclusion in the premise:
It’s in line 1, but it’s negated!
Not only that, but it’s in the location of an antecedent.
Negated antecedent?
Sounds like MT.
Can we use line 3 with line 1 to get ~~L (which is equivalent to L by DN)?
No! MT –and all the rules of inference—only works at the main operator of
a line; the main operator of 1 is a wedge. You’ll have to use DS to break the
right-hand disjunct out of 1 and put it on a line by itself. To that line you could
do MT.
CM will change the order of B and K; you can either change line 1 or change
line 2 –one of them has to be changed so that they match up exactly for DS.

9. 1. (K . B) v ( ~L > E)
2. ~( B . K)
3. ~E / L
4. ~( K . B) CM 2
5. ~L > E DS 1,4
6. ~~L MT 5,3
7. L DN 6

10. As you might expect, if you are just guessing,
If you don’t start out by looking for the conclusion
–or a piece of it at least– in the premises,
then you are just guessing.
As you might expect, if you are just guessing,
it’s just a matter of luck.
That’s not how to succeed in Logic, though.