A v B

1

1. A v B
2. A

1. B gt C
2. C

1. B gt (C . P)
2. B

/ B mt 1,2
/ B ds 1,2
/ C . P mp 1,2

1. S
2. R gt S

1. A v B
2. C gt D

/ R mt 1,2
/ (A v B) . (C gt D) cn 1,2

1. A gt (B. C)
2. (B . C) gt K

1. (L gt M) . (D v N)
/ L gt M sm 1
/ A gt K hs 1,2
2

1. R . T premise
2. R gt S premise
3. R ___
4. S ___

1 sm
2,3 mp

1. B gt P premise
2. P gt X premise
3. B gt X ___
4. (B gt X) . (B gt P) ___

1,2 hs
3,1 cn
3

1. (A gt P) gt S premise
2. S premise
3. S ___
4. (A gt P) ___
5. A gt P ___

2, dn
1,3 mt
4, dn

1. (D v S) gt (P . Q) premise
2. S v D premise
3. D v S ___
4. P . Q ___
5. Q . P ___
6. Q ___

2, cm
3,1 mp
4, cm
5 sm
4

1. P . (Q . R) premise
2. (P . Q) . R ___
3. R . (P . Q) ___
4. R ___

1, as
2, cm
3 sm

1. S gt (A . (B v C)) premise
2. S premise
3. A . (B v C) ___
4. B v C ___, ___

1,2 mp
3, cm , sm

• (L v X) gt (A . B) premise
• M premise
• M gt (X v L) premise
• (X v L) gt (A . B) ___
• M gt (A . B)
• A . B ___
• A ___
• B ___

1 cm
3,4 hs
2,5 mp
6 sm
6 cm, sm
5

1. J gt (K gt L) premise
2. L v J premise
3. L premise
4. J ___
5. K gt L ___
6. K ___

2,3 ds
1,4 mp
3,5 mt

1. G gt ( G v (S gt G)) premise
2. (S v L) gt G premise
3. S v L premise
4. G ___
5. G v (S gt G) ___
6. S gt G ___
7. S ___
8. L ___

2,3 mp
1,4 mp
4,5 ds
4,6 mt
3,7 ds
6

1. (K . B) v ( L gt E)
2. ( B . K)
3. E / L

1. (K . O) gt (N v T)
2. O
3. K / T v N

1. (R gt F) gt ((R gt G) gt (S gt Q))
2. (Q gt F) gt (R gt Q)
3. G gt F
4. Q gt G / S gt 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 itcan 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) gt (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 gt E) 2. ( B . K) 3. E
/ L
Find the conclusion in the premise
Its in line 1, but its negated!
Not only that, but its 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 inferenceonly
works at the main operator of a line the main
operator of 1 is a wedge. Youll 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 gt E) 2. ( B . K) 3. E
/ L
4. ( K . B) CM 2
5. L gt E DS 1,4
6. L MT 5,3
7. L DN 6
10
If you dont 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,
its just a matter of luck. Thats not how to
succeed in Logic, though.