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

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

~ (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

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

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

(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.

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

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.

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

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.