Instructions:
The answers to the odd numbered problems are in white.
Highlight the text to view the answers. Even numbered
problems have no answers.
PROOFS
SIMPLIFICATION (sm)
p • q
p • q
_____ or
_____
p
q
Proofs
Using Simplification Only:
(1)
1. m • u
/ m
2. m
1 sm
(2)
1. s • t
/ t
(3)
1. (m
• p)
• t / t
2. t
1 sm
(4)
1. ~x
• (u
• z) / ~x
(5)
1. (m
• p)
• t / m • p
2. (m • p)
1 sm
(6)
1. (m
≡ p) • t / m ≡ p
(7)
1. x • (u
•
z) / u • z
2.
(u • z) 1 sm
(8)
1. t • (m
•
p) / m • p
(9)
1. (r • u) •
(p • t) / r • u
2. r • u
1 sm
(10)
1. (r • u)
• ~(p • t) / ~(p • t)
(11)
1. [(m v p)
> t] • s / s
2.
s
1 sm
(12)
1. [(m ≡ n)
• y] • s / (m ≡ n) • y
(13)
1. u • [(x •
y) • z] / (x • y) • z
2.
(x • y) • z 1 sm
(14)
1. (u • z) •
r / r
(15)
1. (u • z) •
~r / z
2. u • z
1 sm
3. z
2 sm
(16)
1. (~u • z)
• r / ~u
(17)
1. m • (x
•
z) / z
2. x •
z 1 sm
3.
z 2 sm
(18)
1. m • (x •
z) / x
(19)
1. [(m • p)
• t] • s / s
2.
s 1 sm
(20)
1. [(m • p)
• t] • s / t
(21)
1. [(t • p)
• u] • ~r / p
2. (t • p)
• u 1 sm
3. t •
p 2 sm
4.
p 3 sm
(22)
1. [(t
• r)
• x] • z / t
(23)
1. s • [(m
• p) • t] / s
2.
s 1 sm
(24)
1. s • [(m
• p) • t] / m
(25)
1. s • [(m
• p) • t] / p
2. (m
• p) • t 1 sm
3. m
• p 2 sm
4.
p 3 sm
(26)
1. s • [(m
• p) • t] / t
(27)
1. {[(m
•
p) • t] • s} • q / s
2. [(m •
p) • t] • s 1 sm
3.
s
2 sm
(28)
1. {[(m • p)
•
t] • s} • q / t
(29)
1. {[(m • p)
•
t] • s} • q / p
2. [(m • p) •
t] • s 1 sm
3. (m • p) •
t 2 sm
4. m •
p 3 sm
5.
p 4 sm
(30)
1. {[(m • p)
•
t] • s} • q / m
(31)
1. {[(m > p)
•
t] • s} • q / m > p
2. [(m > p) •
t] • s 1 sm
3. (m > p) •
t 2 sm
4. m >
p 3 sm
(32)
1. w • {[(m
•
n) • s] • y} / y
(33)
1. w • {[(m v
~n) • s] • y} / s
2.
[(m v ~n) • s] • y 1 sm
3.
(m v ~n) • s
2 sm
4.
s
3 sm
(34)
1. w • {[(m
•
n) • s] • y} / n
(35)
1. w • {[(m
•
n) • s] • y} / m
2. [(m
•
n) • s] • y 1 sm
3. (m
•
n) • s 2
sm
4. m
•
n
3 sm
5.
m
4 sm
(36)
1. [(x > p)
•
(z ≡ s)] • q / z ≡ s
(37)
1. [(x v p)
•
(z • s)] • q / x v p
2. (x v p) •
(z • s) 1 sm
3. x v p
2 sm
(38)
1. {[n • (o
•
p)] • q} • r / q
(39)
1. {[n • (o •
p)] • q} • r / o • p
2. [n • (o •
p)] • q 1 sm
3. n • (o •
p) 2 sm
4.
o • p 3 sm
(40)
1. {[n • (o •
p)] • q} • r / p
(41)
1. {[n • (o •
p)] • q} • r / o
2. [n • (o •
p)] • q 1 sm
3. n • (o •
p) 2 sm
4.
o • p 3 sm
5.
o 4 sm
(42)
1. {[n • (o •
p)] • q} • r / n
(43)
1. r • {[(x ≡ y)
• z] • q} / x ≡ y
2.
[(x ≡ y) • z] • q 1 sm
3.
(x ≡ y) • z 2 sm
4.
x ≡ y 3 sm
(44)
1. ~(s v t) •
(~u > ~x) / ~u > ~x
(45)
1. (x > m) • [(r
• ~s) • q] / ~s
2.
(r • ~s) • q 1 sm
3.
r • ~s 2 sm
4. ~s
3 sm
(46)
1.{[(~w ≡ c) •
n] • m} • (~u v y) / n
(47)
1. (x > y) • [(q
v ~q) • y] / y
2. (q v ~q) •
y 1 sm
3.
y 2 sm
(48)
1. [x • (~x >
y)] • (q v ~q) / x
(49)
1. {[y • (~x >
y)] • ~q} • q / ~q
2. [y • (~x >
y)] • ~q 1 sm
3.
~q 2 sm
CONJUNCTION (cj):
p
p
q
q
_____
_____
p • q
q • p
Proofs Using
Conjunction Only:
(1)
1. u
2. t /
u • t
3. u • t
1, 2 cj
(2)
1. u
2. t /
t • u
(3)
1. u v g
2. t v r /
(u v g) • (t v r)
3. (u v g) • (t
v r) 1, 2 cj
(4)
1. u v g
2. t v r /
(t v r) • (u v g)
(5)
1. p
2. u
3. t /
(p • u) • t
4. p •
u 1, 2 cj
5. (p • u) •
t 3, 4 cj
(6)
1. p
2. u
3. t /
(t • u) • p
(7)
1. p
2. u
3. t / (p
• t) • u
4. p •
t 1, 3 cj
5. (p • t) •
u 2, 4 cj
(8)
1. ~t
2. ~x
3. u
4. y /
u • y
(9)
1. ~t
2. ~x
3. u
4. y /
~x • ~t
5. ~x • ~t
1, cj
(10)
1. t
2. x
3. u
4. y / (t
•
x) • (y • u)
(11)
1. t
2. x
3. u
4. y / (y
•
t) • (x • u)
5. y •
t 1, 4 cj
6. x • u
2, 3 cj
7. (y • t) • (x
• u) 5, 6 cj
(12)
1. (w v y)
2. ~w /
(w v y) • ~w
(13)
1. o
2. y ≡ t
3. x / o
• x
4. o • x 1,
3 cj
(14)
1. o
2. y ≡ t
3. x /
(o • x) • (y ≡ t)
(15)
1. o ≡ (u v t)
2. q
3. w / (w
•
q) • [o ≡ (u v t)]
4. w •
q
2, 3 cj
5. (w • q) • [o
≡ (u v t)] 1, 4 cj
(16)
1. {[x > (y >
t)] • r}
2. q / {[x >
(y > t)] • r} • q
(17)
1. n • m
2. t
3. x / [(n
•
m) • t] • x
4. (n • m) •
t 1, 2 cj
5. [(n • m) • t]
• x 3, 4 cj
(18)
1. z • y
2. u
3. w / [u
•
(z • y)] • w
(19)
1. x ≡ z
2. t • y
3. w / (x
≡ z) • w
4. (x ≡ z) •
w 1, 3 cj
(20)
1. (x v r)
2. (y v t)
3. w / [(y
v t) • w] • (x v r)
SIMPLIFICATION (sm):
p • q
p • q
_____ or
_____
p
q
CONJUNCTION (cj):
p p
q
q
_____
_____
p • q q
•
p
Proofs Using
Simplification
&
Conjunction:
(1)
1. x v t
2. x • r
3. q
/ q • x
4.
x 2 sm
5. q •
x 3, 4 cj
(2)
1. x • t
2. r •
u / r • x
(3)
1. x ≡ z
2. t • y
3. w / (x
≡ z) • (w • y)
4.
y 2 sm
5. w •
y 3, 4 cj
6. (x ≡ z) • (w
• y) 1, 5 cj
(4)
1. x • t
2. r •
u / u
• t
(5)
1. (x > y) • t
2. r • (u ≡ r)
/ (x > y) • (u ≡ r)
3. u ≡
r 2 sm
4. x >
y 1 sm
5. (x > y) • (u
≡ r) 3, 4 cj
(6)
1. o v p
2. p • q
3. ~w / (p
•
~w) • (o v p)
(7)
1. w • {[(m • n)
• s] • y} / y • w
2.
[(m • n) • s] • y 1 sm
3.
y 2 sm
4.
w
1 sm
5. y •
w 3, 4
cj
(8)
1. w • {[(m • n)
• s] • y} / n • w
(9)
1. [w • (m • n)]
• y / (y • w) • n
2. w • (m • n)
1 sm
3.
(m • n) 2 sm
4.
n 3 sm
5.
y 1
sm
6.
w 2
sm
7. y •
w 5, 6 cj
8. (y • w) •
n 7, 4 cj
(10)
1. x ≡ z
2. t • y
3. w / (x
≡ z) • (w • y)
(11)
1. x • z
2. t • y
3. w / (x
• y) • (t • z)
4.
x 1 sm
5.
y 2 sm
6. x •
y 4, 5 cj
7.
t 2 sm
8.
z 1 sm
9. t •
z 7, 8
cj
10. (x • y) • (t
• z) 6, 9 cj
(12)
1. {[n • (o •
p)] • q} • r
2. z • ~y
3. x • ~u
4. m • w /
(n • x) • ~y
DOUBLE
NEGATION (dn):
p ≡ ~~p
Proofs Using
Double Negation:
(1)
1. ~~x / x
2.
x 1 dn
(2)
1. ~~z v s /
z v s
(3)
1. z v ~~x / z
v x
2. z v x
1 dn
(4)
1. ~~z v
~~s / ~~z v s
(5)
1. ~~z v
~~x / z v ~~x
2. z v
~~x 1 dn
(6)
1. ~~z v ~~x
/ z v x
(7)
1. p v z /
~~p v z
2. ~~p v z
1 dn
(8)
1. m v u /
~~m v ~~u
(9)
1. ~~(t ≡ u)
/ (t ≡ u)
2.
t ≡ u 1 dn
(10)
1. x v ~~(z •
y) / x v (z • y)
(11)
1. m > (~~n ≡
y) / m > (n ≡ y)
2. m > (n ≡ y)
1 dn
(12)
1. x > (z >
~~y) / x > (z > y)
(13)
1. ~~~z v
~~x / ~z v x
2. ~z v
~~x 1 dn
3. ~z v
x 2 dn
or
1. ~~~z v
~~x / ~z v x
2. ~z v
x 1 dn
(14)
1. ~~~(t ≡
u) / ~(t ≡ u)
DOUBLE
NEGATION (dn):
p ≡ ~~p
SIMPLIFICATION (sm):
p • q
p • q
_____ or
_____
p
q
Proofs Using
Double Negation
&
Simplification:
(1)
1. [(r ≡ ~~t) •
x] • u / r ≡ t
2. (r ≡ ~~t)
•
x 1 sm
3. r
≡
~~t 2 sm
4. r
≡
t 3 dn
(2)
1. ~x • ~~~u
/ ~u
(3)
1. (y > ~~w)
•
~~z / y > w
2. y >
~~w 1 sm
3. y >
w 2 dn
(4)
1. q • ~~~(t v
m) / ~(t v m)
(5)
1. x • (y v
~~y) / y v y
2. y v
~~y 1 sm
3. y v
y 2 dn
(6)
1. (u • ~~z)
•
~~r / z
(7)
1. {[x • (o •
~~p)] • y} • t / p
2. [x • (o •
~~p)] • y 1 sm
3. x • (o •
~~p) 2 sm
4.
o • ~~p 3 sm
5.
~~p 4 sm
6. p
5 dn
DOUBLE
NEGATION (dn):
p ≡ ~~p
CONJUNCTION (cj):
p
p
q
q
_____
_____
p • q
q • p
Proofs Using
Double Negation
&
Conjunction:
(1)
1. r
2. ~~w
3. ~~~y
/ ~y • (w • r)
4.
~y 3 dn
5.
w 2 dn
6. w •
r 5, 1 cj
7. ~y • (w •
r) 4, 6 cj
(2)
1. ~~~x
2. ~~q v y
3. u / (~x
• u) • (q v y)
(3)
1. q ≡ ~~q
2. ~~y ≡ y /
(y ≡ y) • (q ≡ q)
3. q ≡
q 1 dn
4. y ≡
y 2 dn
5. (y ≡ y) • (q
≡ q) 3, 4 cj
(4)
1. ~~q • ~p
2. ~~x / x
•
(q • ~p)
DOUBLE
NEGATION (dn):
p ≡ ~~p
SIMPLIFICATION (sm):
p • q
p • q
_____ or
_____
p
q
CONJUNCTION (cj):
p
p
q
q
_____
_____
p • q
q • p
Proofs Using
Double Negation,
Simplification, & Conjunction:
(1)
1. ~t
2. x • ~~z / z
• ~t
3.
~~z 2 sm
4.
z 3 dn
5. z • ~t
1, 4 cj
(2)
1. (t • ~~q) •
x
2. s /
q • s
(3)
1. ~~q • y
2 ~~s / s
•
q
3. s
2 dn
4. ~~q 1
sm
5.
q 4 dn
6. s • q
3, 5 cj
(4)
1. ~~y • [(~~u •
z) • x] / y • u
(5)
1. y • ~~t
2. x • ~~u
3. s • ~~m / (m
• t) • u
4.
~~m 3 sm
5.
m 4 dn
6.
~~t 1 sm
7.
~~u 2 sm
8.
t 6 dn
9.
u 7 dn
10. m •
t 5, 8 cj
11. (m • t) •
u 10, 9 cj
(6)
1. [(x • ~~q) •
t] • ~y
2. ~~r • ~~z
/ (r • q) • z
ADDITION (ad):
p
p
_____
_____
p v q
q v p
Proofs Using
Addition Only:
(1)
1. q /
q v p
2. q v p
1 ad
(2)
1. q /
p v q
2. p v q
1 ad
(3)
1. p / p v
(x • z)
2. p v (x •
z) 1 ad
(4)
1. p /
(x • z) v p
(5)
1. s /
s v [(t v z) ≡ n]
2. s v [(t v z)
≡ n] 1 ad
(6)
1. s / [(t v
z) ≡ n] v s
(7)
1. (z ≡ x) /
(z ≡ x) v r
2. (z ≡ x) v
r 1 ad
(8)
1. (z ≡ x)
/ r v (z ≡ x)
(9)
1. (t v z) ≡ n
/ [(t v z) ≡ n] v q
2. [(t v z) ≡ n]
v q 1 ad
(10)
1. (t v z)
/ (t v z) v (q v w)
(11)
1. q /
(q v s) v (t v w)
2. q v
s 1 ad
3. (q v s) v (t
v w) 2 ad
(12)
1. ~w / (q v
s) v (t v ~w)
(13)
1. t /
(q v s) v (t v ~w)
2. t v
~w 1 ad
3. (q v s) v (t
v ~w) 2 ad
(14)
1. s /
(q v s) v (t v ~w)
(15)
1. s / [(s
v t) v y] v r
2. s v
t 1 ad
3. (s v t) v
y 2 ad
4. [(s v t) v y]
v r 3 ad
(16)
1. z > x
/ (z > x) v (t • b)
(17)
1. p
/ [(p v t) v q] v z
2. p v t
1 ad
3. (p v t) v q
2 ad
4. [(p v t) v
q] v z 3 ad
ADDITION
(ad):
p
p
_____
_____
p v q
q v p
SIMPLIFICATION (sm):
p • q
p • q
_____ or
_____
p
q
Proofs Using
Addition
&
Simplification:
(1)
1. s • r / s
v n
2. s
1 sm
3. s v n
2 ad
(2)
1. s • r / s
v (t v z)
3. s v (t v z)
2 ad
(3)
1. s • r / s
v [(t v z) ≡ n]
2.
s 1 sm
3. s v [(t v z)
≡ n] 2 ad
(4)
1. r • t
2. w • m /
(m • r) v (o ≡ p)
6. (m • r) v (o
≡ p) 5 ad
(5)
1. [(x • p) • y]
• w / p v ~z
2. (x • p) •
y 1 sm
3. x •
p 2 sm
4.
p 3 sm
5.
p v ~z 4 ad
(6)
1. q • (~n •
m) / (m v r) v y
5. (m v r) v
y 4 ad
(7)
1. (n • m) • q
/ (m ≡ y) v m
2. n • m
1 sm
3.
m 2 sm
4. (m ≡ y) v
m 3 ad
(8)
1. w • s
/ [(y v s) v t] v q
4. (y v s) v
t 3 ad
5. [(y v s) v t]
v q 4 ad
(9)
1. [(n • y) • w]
• x / x v n
2.
x 1 sm
3. x v n
2 ad
or
1. [(n • y) • w]
• x / x v n
2. (n • y) •
w 1 sm
3. n •
y 2 sm
4. n
3 sm
5. x v
n 4 ad
ADDITION
(ad):
p
p
_____
_____
p v q
q v p
CONJUNCTION (cj):
p
p
q
q
_____
_____
p • q
q • p
Proofs Using
Addition
&
Conjunction:
(1)
1. z ≡ m
2. x / (z
≡
m) v (r ≡ t)
3. (z ≡ m) v (r
≡ t) 1 ad
(2)
1. ~w
2. z
/ (~w • z) v r
4. (~w • z) v
r 3 ad
(3)
1. r
2. y
3. t
/ [t • (r • y)] v y
4. [t • (r • y)] v y
2 ad
or
1. r
2. y
3. t
/ [t • (r • y)] v y
4. r • y
1, 2 cj
5. t • (r •
y) 3, 4 cj
6. [t • (r • y)]
v y 5 ad
(4)
1. r
2. t
3. x
4. z
/ {[(x • z) • r] • t} v w
5. x • z
3, 4 cj
6. (x • z) •
r
5, 1 cj
7. [(x • z) • r]
• t
2, 6 cj
8. {[(x • z) •
r] • t} v w 7 ad
(5)
1. z
2. x /
[(x • z) v t] v y
5. [(x • z) v t]
v y 4 ad
(6)
1. r
2. y
3. t /
t v (r • y)
4. t v (r •
y) 3 ad
or
1. r
2. y
3. t /
t v (r • y)
4. r • y
1, 2 ad
5. t v (r • y) 4 ad
(7)
1. r
2. y
3. t / t
v (r • y)
ADDITION
(ad):
p
p
_____
_____
p v q
q v p
SIMPLIFICATION (sm):
p • q
p • q
_____ or
_____
p
q
CONJUNCTION (cj):
p
p
q
q
_____
_____
p • q
q • p
Proofs Using
Addition,
Simplification, & Conjunction:
(1)
1. (~q ≡ u)
2. w /
[w • (~q ≡ u)] v x
3. [w • (~q ≡
u)] 1, 2 cj
4. [w • (~q ≡
u)] v x 3 ad
(2)
1. s • r
2. t /
(s • t) v q
(3)
1. s • t
2. q / w v
(s • q)
3.
s 1 sm
4. s • q
2, 3 cj
5. w v (s •
q) 4 ad
(4)
1. [(r • x) • z]
• t
2. q /
(q • x) v t
(5)
1. ~q v y
2. s • z
/ [s • (~q v y)] v t
3.
s 2 sm
4. s • (~q v
y) 1, 3 cj
5. [s • (~q v
y)] v t 4 ad
(6)
1. (~r • ~t) • y
2. z
/ (z • y) v (u ≡ T)
ADDITION
(ad):
p
p
_____
_____
p v q
q v p
SIMPLIFICATION (sm):
p • q
p • q
_____ or
_____
p
q
CONJUNCTION (cj):
p
p
q
q
_____
_____
p • q
q • p
DOUBLE
NEGATION (dn):
p ≡ ~~p
Proofs Using
Addition, Simplification
Conjunction,
& Double Negation:
(1)
1. ~~y
2. (x v t) •
u / t v u
3.
u 2 sm
4. t v u
3 ad
(2)
1. ~~q • t
2. r /
(q • r) v (y > u)
(3)
1. (~~y • u) • x
2 ~~(z • t) /
(y • t) v (~q • w)
3. ~~y •
u 1 sm
4.
~~y 3 sm
5.
y 4
dn
6. z •
t 2 dn
7.
t
6 sm
8. y •
t
5, 7 cj
9. (y • t) v
(~q • w) 8 ad
(4)
1. ~~r
2. ~~y
3. ~~(u • t) /
o v [(t • r) • y]
4.
r 1
dn
5.
y 2
dn
6. u •
t 3 dn
7.
t 6 sm
8. t •
r 7,
4 cj
9. (t • r) •
y 8 cj
10. o v [(t • r)
• y] 9 ad
(5)
1. {[(w ≡ ~~r) •
y] • t} • u
2. ~~x
/ (y • x) v w
3. [(w ≡ ~~r) •
y] • t 1 sm
4. (w ≡ ~~r) •
y 3 sm
5.
y
4 sm
6.
x
2 dn
7. Y •
x
5, 6 cj
8. (y • x) v
w 7 ad
(6)
1. ~~o • ~b
2. m • n
3. y /
(o • m) v y
(7)
1. ~~[(y v t) •
u]
2. y /
[(y v t) v u] v z
3. y v
t 2 ad
4. (y v t) v
u 3 ad
5. [(y v t) v u]
v z 4 ad
or
1. ~~[(y v t) •
u]
2. y / [(y v
t) v u] v z
3. (y v t) •
u 1 dn
4. y v
t 3 sm
5. (y v t) v
u 4 ad
6. [(y v t) v u]
v z 5 ad
COMMUTATION
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
Proofs Using
Commutation Only:
(1)
1. s v t /
t v s
2. t v s
1 cm
(2)
1. ~q • y / y
• ~q
(3)
1. (r • y) • x / (y
• r) • x
2. (y • r) • x
1 cm
(4)
1. (r • y) • x
/ x • (r • y)
(5)
1. (m v z) v u
/ (z v m) v u
2. (z v m) v u
1 cm
(6)
1. (m v z) v u
/ u v (m v z)
(7)
1. (p v x) • t
/ (x v p) • t
2. (x v p) • t
1 cm
(8)
1. t • (x v p)
/ (x v p) • t
(9)
1. t • (x v p)
/ (p v x) • t
2. (x v p) • t
1 cm
3. (p v x) •
t 2 cm
(10)
1. u v (m • z)
/ (z • m) v u
(11)
1. (r • z) v (t •
p) /
(p • t) v (z • r)
2. (t • p) v (r
• z) 1 cm
3. (p • t) v (r
• z) 2 cm
4. (p • t) v (z
• r) 3 cm
Commutation
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p
q
Proofs Using
Commutation
&
Simplification:
(1)
1. o • (~m v
r) / r v ~m
2. ~m v
r 1 sm
3. r v
~m 2 cm
(2)
1. (y v t) • (x
• z) / t v y
(3)
1. t • [(x • t)
• r] / t • x
2. (x • t) •
r 1 sm
3. x •
t 2 sm
4. t •
x 3 cm
(4)
1. [w • (y v t)]
• u / t v y
(5)
1. [(x • z) • t]
• u / z • x
2. (x • z) •
t 1 sm
3. x •
z 2 sm
4. z •
x 3 cm
(6)
1. (n v ~s) • (r
> u) / ~s v n
(7)
1. {[(y v p) •
t] • u} • r / p v y
2. [(y v p) •
t] • u 1 sm
3. (y v p) •
t 2 sm
4. y v
p 3 sm
5. p v
y 4 cm
Commutation
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p q
CONJUNCTION
(cj):
p p
q q
_____
_____
p • q q
•
p
Proofs Using
Commutation
Simplification, & Conjunction:
(1)
1. r v u
2. t • y /
y • (u v r)
3.
y 2 sm
4. u v
r 1 cm
5. y • (u v
r) 3, 4 cj
(2)
1. x • (z v t)
2. ~q / ~q
• (t v z)
(3)
1. r • ~s
2. t v y
3. u / (r
•
u) • (y v t)
4.
r 1 sm
5. r •
u 4, 3 cj
6. y v
t 2 cm
7. (r • u) • (y
v t) 5, 6 cj
(4)
1. y • u
2. r
3. r v y / (y
v r) • (r • y)
(5)
1. {(m • n)
•
[(y v r) • t]} • u
2. ~s • t
/ (r v y)
• ~s
3.
~s 2 sm
4. (m • n) • [(y
v r) • t] 1 sm
5. (y
v r) • t 4 sm
6. y v r 5 sm
7.
r v y 6 cm
8. (r v y) •
~s 3, 7 cj
Commutation
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p q
CONJUNCTION
(cj):
p p
q q
_____
_____
p • q q
•
p
DOUBLE
NEGATION (dn):
p ≡ ~~p
Proofs Using
Commutation,
Simplification, Conjunction,
& Double
Negation:
(1)
1. s • t
2. ~~x / (t
•
s) • x
3.
x 2 dn
4. t •
s 1 cm
5. (t • s) •
x 3, 4 cj
(2)
1. s v y
2. ~~r / r
•
(y v s)
(3)
1. s • ~~y
2. r v u / (u
v r) • y
3. u v
r 2 cm
4.
~~y 1 sm
5.
y 4 dn
6. (u v r) •
y 3, 5 cj
(4)
1. (y v r) • ~~u
2. ~~m • n / (n
• m) • (r v y)
(5)
1. ~~(x v ~~t) •
r
2. s / (t
v x) • s
3. (x v ~~t) •
r 1 dn
4. x v
~~t 3 sm
5. x v
t 4 dn
6. t v
x 5 cm
7. (t v x) •
s 2, 6 cj
(6)
1. (t v y) • (x
v ~~z)
2. u / (z v
x) • u
(7)
1. s • ~~t
2. r
3. y v ~~s / [(s v y)
• t] • r
4. y v
s 3 dn
5. s v
y 4 cm
6.
~~t 1 sm
7.
t 6 dn
8. (s v y) •
t 5, 7 cj
9. [(s v y) • t]
• r 8, 2 cj
COMMUTATION (cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
SIMPLIFICATION (sm):
p • q
p •
q
_____ or
_____
p q
CONJUNCTION
(cj):
p p
q q
_____
_____
p • q q
•
p
DOUBLE
NEGATION (dn):
p ≡ ~~p
ADDITION
(ad):
p
p
_____
_____
p v q q v
p
Proofs Using
Commutation,
Simplification, Conjunction,
Double
Negation, & Addition:
(1)
1. ( r v s) • u
2. ~~t
/ (t • u) v z
3. u
1 sm
4. t
2 dn
5. t • u
4,3 cj
6. (t • u) v z
5 ad
(2)
1. t • u
2. ~~u /
u v t
(3)
1. ~~t • y
2. u
/ (u v t) v x
3. u v
t 2 ad
4. (u v t) v
x 3 ad
(4)
1. ~~(x v y) •
z / (y v x) • z
(5)
1. ~~(x v y) •
z / (y v x) v z
2. (x v y) •
z 1 dn
3. x v
y 2 sm
4. y v
x 3 cm
5. (y v x) v
z 4 ad
(6)
1. u v ~~y
2. x • p / [(y v
u) • p] v (s ≡ t)
(7)
1. ~~w • u
2. ~~t / (u v
y) v (t • w)
3.
u 1 sm
4. u v
y 3 ad
5. (u v y) v (t
• w) 4 ad
(8)
1. (y • u) v (t
v ~~w)
2. ~m / [(w
v t) v (u • y)] v ~m
(9)
1. [(x • ~~y) •
(x > z)] • t
2. ~~r / [r
•
(y • x)] v t
3.
t 1 sm
4. [r • (x •
~~y)] v t 3 ad
(10)
1. {[(~~x v y) •
u] • t} • z
2. ~~r
/ (s ≡ o) v (u • r)
(11)
1. ~~(u • ~~y) •
[m v (z ≡ ~x)]
2. o / {[(o
• y) v t] v z} v r
3. ~~(u •
~~y) 1 sm
4. u
•
~~y 3 dn
5.
~~y 4 sm
6.
y 5 dn
7. o •
y 2, 6 cj
8. (o • y) v
t 7 ad
9. [(o • y) v
t] v z 8 ad
10. {[(o • y) v
t] v z} v r 9 ad
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
Proofs Using
Tautology:
(1)
1. q v q /
q
2. q 1 ta
(2)
1. p • p
/ p
(3)
1. p / p v
p
2. p v p 1 ta
(4)
1. ~s / ~s
•
~s
(5)
1. (y > ~t) • (y
> ~t) / y > ~t
2. y >
~t 1 ta
(6)
1. (r ≡ s) v (r
≡ s) / r ≡ s
(7)
1. (o • o) v o
/ o
2. o v
o 1 ta
3. o 2 ta
(8)
1. p v p / p
• p
(9)
1. o • o /
o v o
2. o 1 ta
3. o v
o 2 ta
(10)
1. (q • q) ≡ z
/ (q v q) ≡ z
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p
q
Proofs Using
Tautology
&
Simplification:
(1)
1. (~y • ~y) •
t / ~y
2. ~y •
~y 1 sm
3.
~y 2 ta
(2)
1. [y • (m • m)]
• t / m
(3)
1. {u • [s • (o
v o)]} • r / s v s
2. u • [s • (o
v o)] 1 sm
3. s
• (o
v o) 2 sm
4.
s 3 sm
5. s v
s 4 ta
or ad
(4)
1. r • [(t v y)
• (t v y)] / t v y
(5)
1. (y v y) • (u
v u) / y • y
2. y v
y 1 sm
3. y 2 ta
4. y •
y 3 ta
(6)
1. x • [(o • o)
v (o • o)] / o
(7)
1. [(s v s) v (s
v s)] • u / s
2. (s v s) v (s
v s) 1 sm
3. s v (s
v s) 2 ta
4. s v
s 3 ta
5.
s 4 ta
(8)
1. t • [x • (o v
o)] / o • o
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p q
CONJUNCTION
(cj):
p p
q q
_____
_____
p • q q
•
p
Proofs Using
Tautology,
Simplification, & Conjunction:
(1)
1. w
2. (t v t) • s
/ (t • s) • w
3.
t • s
2 ta
4. (t • s) •
w 1, 3 cj
(2)
1. (z v x) • t
2. [(t • y) • r]
• s / t • t
(3)
1. x • z
2. y v y /
(x • z) • y
3.
y 2 ta
4. (x • z) •
y 1, 3 cj
(4)
1. ~t v ~t
2. x • x
3. y • u
/ (u • ~t) • x
(5)
1. ~r
2. s • t
3. x v x / (x
• s) • ~r
4.
s 2 sm
5.
x 3 ta
6. x •
s 4, 5 cj
7. (x • s) •
~r 1, 6 cj
(6)
1. (x v x) v (x
v x)
2. z • t
/ x • z
(7)
1. [z v (x v x)]
• y
2. m • t
/ t • (z v x)
3. t 2 sm
4. z v (x
v x) 1 sm
5. z v
x 4 ta
6. t • (z v
x) 3, 5 cj
(8)
1. m • (x • z)
2. [(u • y) • s]
• m
3. ~r / [(z
v z) • ~r] • u
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p
q
CONJUNCTION
(cj):
p p
q q
_____
_____
p • q q
•
p
DOUBLE
NEGATION (dn):
p ≡ ~~p
Proofs Using
Tautology,
Simplification, Conjunction,
& Double
Negation:
(1)
1. ~s • y
2. t v ~~t /
t • y
3. t v t 2 dn
4.
t 3 ta
5.
y 1 sm
6. t • y
4, 5 cj
(2)
1. ~~(r v r)
2. t • z
/ r • z
(3)
1. ~~q • t
2. ~~s / (s
v s) • t
3. s 2 dn
4.
t 1 sm
5. s v
s 3 ta
6. (s v s) •
t 5, 4 cj
(4)
1. ~~z
2. y / (y
•
z) v (y • z)
(5)
1. ~~(t v y) • u
2. r / r
•
[(t v y) • (t v y)]
3. ~~(t v y)
1 sm
4. t v
y 3 dn
5. (t v y) • (t
v y)] 4 ta
6. r • [(t v y)
• (t v y)] 2, 5 cj
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p
q
CONJUNCTION
(cj):
p p
q
q
_____
_____
p • q q
•
p
DOUBLE
NEGATION (dn):
p ≡ ~~p
Commutation
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
Proofs Using
Tautology,
Simplification, Conjunction,
Double
Negation, & Commutation:
(1)
1. x v z
2. s v s /
(z v x) • s
3. s 2 ta
4. (x v z) • s
1, 3 cj
5. (z v x) •
s 4 cm
(2)
1. (z v z) v (x
• y) / (x • y) v z
(3)
1. ~~r v ~~r
2. x • u /
(u • x) • r
3. u •
x 2 cm
4.
~~r 1 ta
5. (u • x) •
r 3, 4 cj
(4)
1. ~~(t v t) • y
2. r v u /
t • (u v r)
(5)
1. ~~(y v y) •
(x v z)
2. w
• u /
[(z v x) • y] • w
3. (y v y) •
(x v z) 1 dn
4. y
•
(x v z) 3 ta
5. (x v z) •
y 4 cm
6. (z v x) •
y 5 cm
7.
w 2 sm
8. [(z v x) •
y] • w 6, 7 cj
(6)
1. m • (u v ~~u)
2. p /
(p • u) • m
(7)
1. (r v r) • (n
v m)
2. ~~y • z / (y
• r) • (m v n)
3.
~~y 2 sm
4.
y 3 dn
5. r v
r 1 sm
6.
r 5 ta
7. n v
m 1 sm
8. m v
n 7 cm
9. y •
r 4, 6 cj
10. (y • r) • (m
v n) 8, 9 cj
(8)
1. (r v r) v
(~~p • ~~p)
2. ~~x • u
3. m • n /
(x • m) • (p v r)
ASSOCIATION
(as):
[p • (q • r)] ≡
[(p • q) • r]
[p v (q v r)] ≡
[(p v q) v r]
Proofs Using
Association Only:
(1)
1. p v (q v
x) / (p v q) v x
2. (p v q) v
x 1 as
(2)
1. p • (q
•
x) / (p • q) • x
(3)
1. (r v s) v
~t / r v (s v ~t)
2. r v (s v
~t) 1 as
(4)
1. (r • s) • ~t
/ r • (s • ~t)
ASSOCIATION
(as):
[p • (q • r)] ≡
[(p • q) • r]
[p v (q v r)] ≡
[(p v q) v r]
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
Proofs Using
Association &
Tautology:
(1)
1. p v (p v q)
/ p v q
2. (p v p) v q
1 as
3. p v
q 2 ta
(2)
1. (s v p) v p
/ s v p
(3)
1. (t • t) v (x
v z) / (t v x) v z
2. t v (x v
z) 1 ta
3. (t v x) v
z 2 as
(4)
1. (u • u) v [(t
• t) v m] /
(u v t) v m
DISJUNCTIVE
SYLLOGISM (ds):
p v
q p v q
~p ~q
______ _____
q
p
Proofs Using
Disjunctive Syllogism:
(1)
1. s v t
2. ~s
/ t
3. t 1, 2 ds
(2)
1. m v u
2. ~u /
m
(3)
1. (x • p) v t
2. ~t / x
• p
3. x • p 1, 2 ds
(4)
1. (x v u) v t
2. ~(x v u)
/ t
(5)
1. y v (t > p)
2. ~(t >
p) / y
3. y 1, 2 ds
(6)
1. (u v t) v (x
> z)
2. ~(u v t)
/ x > z
(7)
1. (u v t) v (x
> z)
2. ~(x > z)
/ u v t
3. u v t
1, 2 ds
(8)
1. m v ~t
2. ~m
3. t v y
/ y
(9)
1. o v ~u
2. u v r
3. ~o
/ r
4. ~u 1, 3 ds
5. r 2, 4 ds
(10)
1. y v m
2. t v ~y
3. ~m /
t
DISJUNCTIVE
SYLLOGISM (ds):
p v
q p v q
~p ~q
______ _____
q p
CONJUNCTION
(cj):
p p
q
q
_____
_____
p • q q
•
p
Proofs Using
Disjunctive
Syllogism &
Conjunction:
(1)
1. ~w
2. s v p
3. ~p
/ s • ~w
4.
s 2, 3 ds
5. s •
~w 4, 1 cj
(2)
1. y v ~t
2. u
3. t
/ u • y
(3)
1. x v y
2. t v ~(x v y)
3. u
/ t • u
4.
t 2, 1 ds
5. t •
u 4, 3 cj
(4)
1. m > n
2. p
3. x v ~(m > n)
4. ~o
/ (p • ~o) • x
(5)
1. ~z ≡ u
2. p v y
3. ~p / y
•
(~z ≡ u)
4.
y 2, 3 ds
5. y • (~z ≡
u) 4, 1 cj
(6)
1. ~q v t
2. u v ~t
3. x v ~u
4. o
5. q / (o
• x) • u
MODUS PONENS
(MP):
p > q
p
_____
q
Proofs Using
Modus Ponens:
(1)
1. p > q
2. p
/ q
3. q 1, 2 mp
(2)
1. p
2. p > q
/ q
(3)
1. r > t
2. r
/ t
3. t 1, 2 mp
(4)
1. (y ≡ u) > o
2. y ≡ u
/ o
(5)
1. x > p
2. (x > p) > r
/ r
3. r 2, 1 mp
(6)
1. u v y
2. (u v y) > (x
v t)
3. w • s
/ x v t
(7)
1. [(o ≡ p) •
x] > (s • u)
2. (o ≡ p) •
x
3. ~y v t
/ s • u
4. s • u 1, 2 mp
(8)
1. m v ~m
2. [(m v y) v
t] > [(s > p) • x]
3. (m v y) v
t / (s > p) • x
(9)
1. p > [(x v y)
v z]
2. ~r • y
3. ~u
4. p
/ (x v y) v z
5. (x v y) v z 1, 4 mp
MODUS PONENS
(MP):
p > q
p
_____
q
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p q
Proofs Using
Modus Ponens & Simplification:
(1)
1. t • p
2. t > ~w /
~w
3. t 1 sm
4. ~w 2, 3
mp
(2)
1. o > m
2. t • o /
m
(3)
1. (q • r) • y
2. q > u /
u
3. q • r 1 sm
4. q 3 sm
5. u 2, 4
mp
(4)
1. o • ~p
2. ~p > y
3. y > u
/ u
(5)
1. y > o
2. o > u
3. y • m /
u
4. y 3 sm
5. o 1, 4
mp
6. u 2, 5
mp
(6)
1. u > o
2. u • p
3. o > n
4. n > ~q /
~q
MODUS PONENS
(MP):
p > q
p
_____
q
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p
q
DOUBLE
NEGATION (dn):
p ≡ ~~p
Proofs Using
Modus Ponens,
Simplification, & Double Negation:
(1)
1. x > (y • ~~u)
2. x
/ u
3. y • ~~u
1, 2 mp
4. ~~u
3 sm
5. u 4 dn
(2)
1. p > q
2. ~~p • u / q
(3)
1. ~~s > p
2. s
• y /
p
3. s > p 1 dn
4. s 2 sm
5. p
3, 4
mp
(4)
1. (~~q • ~r) •
u
2. q > ~~y
/ y
(5)
1. p > q
2. q > ~~y
3. y > (n •
m)
4. p
/ m
5. q 1, 4 mp
6. ~~y 2, 5 mp
7. y 6 dn
8. n • m 3, 7 mp
9. m
8 sm
MODUS PONENS
(MP):
p > q
p
_____
q
ADDITION
(ad):
p
p
_____ _____
p v q q v p
Proofs Using
Modus Ponens
& Addition:
(1)
1. u
2. u > ~o
/ ~o v y
(2)
1. (m v t) > s
2. m
/ s
3. m v t 2
ad
4. s
1, 3 mp
(3)
1. u
2. (u v y) > (p
• x) / p • x
(4)
1. m
2. (m v n) >
~o / ~o v y
3. m v
n 1 ad
4.
~o 2, 3 mp
5. ~o v
y 4 ad
(5)
1. [(m v y) v t]
> [(s > n) • r]
2. m / (s > n)
• r
MODUS PONENS
(MP):
p > q
p
_____
q
Commutation
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
Proofs Using
Modus Ponens
&
Commutation:
(1)
1. x > (w v y)
2. x
/ y v w
3. w v
y 1, 2 mp
4. y v
w 3 cm
(2)
1. p
2. p > u
3. u > (x •
o) / o • x
(3)
1. u • m
2. (m • u) > p / p
3. m •
u 1 cm
4. p 2, 3 mp
(4)
1. p v o
2. (o v p) > y
3. y > (m •
n) / n • m
(5)
1. (x • z) > (t
v n)
2. z • x
3. (n v t) > p
/ p
4. x •
z 2 cm
5. t v n 1, 4 mp
6. n v
t 5 cm
7. p 3, 6 mp
(6)
1. p > q
2. (p > q) > (x
v y)
3. (y v x) > (w
• t)
4. (t • w) >
u / u
MODUS
TOLLENS (MT):
p > q
~q
______
~p
Proofs Using
Modus Tollens:
(1)
1. p > q
2. ~q
/ ~p
3. ~p 1, 2 mt
(2)
1. ~y
2. x > y
/ ~x
(3)
1. m > ~w
2. ~~w
/ ~m
3. ~m 1, 2 mt
(4)
1. ~u > p
2. ~p
/ ~~u
(5)
1. x > (o • m)
2. ~(o • m)
/ ~x
3. ~x 1, 2 mt
(6)
1. (p ≡ o) > u
2. ~u /
~(p ≡ o)
(7)
1. (p ≡ o) > (o
• m)
2. ~(o • m)
/ ~(p ≡ o)
3. ~(p ≡ o) 1, 2 mt
(8)
1. x > t
2. ~t
3. s > x
/ ~s
(9)
1. ~r
2. u > ~~r
3. x > p
4. p > u /
~x
5. ~u 2, 1 mt
6. ~p 4, 5 mt
7. ~x 3, 6 mt
MODUS TOLLENS
(MT):
p > q
~q
______
~p
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
Proofs Using
Modus Tollens & Tautology:
(1)
1. ~p v ~p
2. o > p
/ ~o
3. ~p
1 ta
4. ~o
2, 3 mt
(2)
1. y > u
2. ~u • ~u /
~y
(3)
1. ~(r ≡ p) v
~(r ≡ p)
2. q > (r ≡ p) / ~q
3. ~(r ≡ p)
1 ta
4. ~q
2, 3 mt
(4)
1. ~y v ~y
2. p > y
3. u > p / ~u
(5)
1. p > (q v
q)
2. ~q
/ ~p
3. p > q 1 ta
4. ~p
3, 2 mt
(6)
1. x v x
2. y > ~x
/ ~y
(7)
1. m > (~y v ~y)
2. y / ~m
3. m > ~y 1 ta
4. ~m 2, 3 mt
MODUS TOLLENS
(MT):
p > q
~q
_____
~p
MODUS PONENS
(MP):
p > q
p
_____
q
Proofs Using
Modus Tollens
& Modus
Ponens:
(1)
1. ~x
2. p > x
3. ~p > q /
q
4. ~p 2, 1 mt
5. q 3, 4
mp
(2)
1. z > q
2. ~q
3. ~z > p /
p
(3)
1. p > ~q
2. s > q
3. p /
~s
4. ~q 1, 3
mp
5. ~s 2, 4 mt
(4)
1. x • z
2. (x • z) > r
3. r > ~y
4. q > y /
~q
(5)
1. ~t
2. x > t
3. q > x
4. ~q > r /
r
5. ~x 2, 1 mt
6. ~q 3, 5 mt
7. r 4, 6
mp
(6)
1. ~m > w
2. m > z
3. z > t
4. ~t /
w
|
MATERIAL
IMPLICATION (mi):
(p > q) ≡ (~p v
q)
|
NOTE:
If you don't study, than you'll fail (~S > F ). This is
the same as: Study or you'll fail (S v F).
If you don't pay me, than I'll kill you (~P > K). This
is the same as: Pay me or I'll Kill you (P v K).
Your money or your life (M v L). This is the same as:
If you don't give me your money, than I'll take your
life (~M > L). |
|
|
p |
> |
q |
≡ |
~ |
p |
v |
q |
|
T |
T |
T |
T |
F |
T |
T |
T |
|
T |
F |
F |
T |
F |
T |
F |
F |
|
F |
T |
T |
T |
T |
F |
T |
T |
|
F |
T |
F |
T |
T |
F |
T |
F |
|
Proofs using
Material Implication:
(1)
1. s > r /
~s v r
2. ~s v r
1 mi
(2)
1. ~p > ~t
/ p v ~t
(3)
1. p v y
/ ~p > y
2. ~p >
y 1 mi
(4)
1. ~x v t
/ x > t
(5)
1. (u v y) > p
/ ~(u v y) v p
2. ~(u v y) v
p 1 mi
(6)
1. ~(m ≡ t) v p / (m
≡ t) > p
(7)
1. s v (t v w)
/ ~s > (~t > w)
2. ~s > (t v
w) 1 mi
3. ~s > (~t >
w) 2 mi
(8)
1. (x v u) >
n / ~(~x > u) v n
(9)
1. (r v t) v (z
v x) / (~r > t) v (~z > x)
2. (~r > t) v (z
v x) 1 mi
3. (~r > t) v
(~z > x) 2 mi
(10)
1. (m > u) > (p
> x) /
~(~m v u) v (~p v x)
MATERIAL
IMPLICATION (mi):
(p > q) ≡ (~p v
q)
Commutation
(cm):
(p v q) ≡ (q v
p)
(p • q) ≡ (q •
p)
Proofs using
Material Implication
&
Commutation:
(1)
1. p > q / q v
~p
2. ~p v q 1
mi
3. q v ~p 2
cm
(2)
1. ~x v ~y /
y > ~x
(3)
1. (z • x) >
p / ~(x • z) v p
2. (x • z) > p
1 cm
3. ~(x • z) v
p 2 mi
(4)
1. (z > m) v p / ~p > (~z v m)
(5)
1. x v ~(t v z) / ~(~t > z) v x
2. x v ~(~t >
z) 1 mi
3. ~(~t > z) v
x 2 cm
(6)
1. ~m > (~q >
s) / m v (s v q)
MATERIAL
IMPLICATION (mi):
(p > q) ≡ (~p v
q)
TAUTOLOGY
(ta):
(p • p) ≡ p
(p v p) ≡ p
Proofs Using
Material Implication
& Tautology:
(1)
1. ~t > t
/ t
2. t v
t 1 mi
3.
t 2 ta
(2)
1. p • (y v u) / (p
• p) • (~y > u)
(3)
1. (y v y) > x / ~y v x
2. y >
x 1 ta
3. ~y v
x 2 mi
(4)
1. p > ~p
/ ~p
(5)
1. (p > u) ≡ (n
v n) / (~p v u) ≡ n
2. (~p v u) ≡ (n
v n) 1 mi
3. (~p v u) ≡ n 2 ta
(6)
1. x / ~x >
x
(7)
1. (o • o) • (~o
> o) / o
2. o • (~o >
o) 1 ta
3. o • (o v
o) 2 mi
4. o •
o 3 ta
5.
o 4 ta
or
1. (o • o) • (~o
> o) / o
2. o • o
1 sm
3.
o
2 ta / 2 sm
CONDITIONAL
PROOF (cp)
.
.
│p ap
│.
│.
│.
│q
└──────────
p > q cp
Proofs Using
Conditional
Proof, Double
Negation, &
Simplification:
(1)
1. x
• y /
p > y
┌─2. p
ap
│ 3. y
1 sm
└──────────────────────
4. p > y
2-3 cp
(2)
1. ~~y
• t
/ s > y
5. s > y
2-4 cp
(3)
1. (y
• ~~u) •
m / t > u
┌─2.
t ap
│ 3. y
•
~~u 1 sm
│ 4.
~~u 3 sm
│ 5.
u 4 dn
└───────────────────────────
6. t >
u 2-5 cp
(4)
1. ~~(u v r)
•
y / x > (u v r)
(5)
1. p
• ~~y
/ x > (u > y)
┌──2.
x ap
│┌─3.
u ap
││
4. ~~y 1 sm
││
5. y 4 dn
│└───────────────────────────
│ 6. u > y 3-5 cp
└────────────────────────────
7. x > (u >
y) 2-6 cp
CONDITIONAL
PROOF (cp)
.
.
│p ap
│.
│.
│.
│q
└──────────
p > q cp
MODUS PONENS
(MP):
p > q
p
_____
q
ADDITION
(ad):
p p
_____ _____
p v q q v p
Proofs Using
Conditional
Proof, Modus
Ponens, & Addition:
(1)
1. p > q
2. q > t /
p > t
┌─3. p
ap
│ 4. q 1, 3
mp
│ 5. t 2, 4
mp
└────────────────────
6. p > t 3-5 cp
(2)
1. x
2. (x v u) >
p / u > p
(3)
1. p > q
2. q > r / p
> (o v r)
┌─3. p
ap
│ 4. q 1, 3 mp
│ 5. r 2, 4 mp
│ 6. o v r
5 ad
└────────────────────────
7. p > (o v
r) 3-6 cp
(4)
1. s > m
2. (x v t) >
s / x > m
(5)
1. A > B
2. B > C
3. C > D / Q
> (C > D)
┌──4.
Q ap
│┌─5. C
ap
││ 6. D
3, 5 mp
│└───────────────────────
│ 7. C > D
5-6 cp
└────────────────────────
8. Q > (C >
D) 4-7 cp
(6)
1. x > z
2. (z v t) >
r / x > (u > r)
(7)
1. (y > w) >
u
2. y > x
3. x > w / u v r
┌─4. y
ap
│ 5. x 2, 4 mp
│ 6. w 3, 5 mp
└───────────────────────
7. y >
w 4-6 cp
8.
u 1, 7 mp
9. u v
r 8 ad
(8)
1. {p > [x >
(r v y)]} > t
2. r / t
|
DEMORGAN'S
RULES (dm):
~(p v q) ≡ (~p •
~q)
~(p • q) ≡ (~p v
~q)
|
|
~ |
(P |
v |
q) |
≡ |
(~ |
p |
• |
~ |
q) |
|
|
F |
T |
T |
T |
T |
F |
T |
F |
F |
T |
|
|
F |
T |
T |
F |
T |
F |
T |
F |
T |
F |
|
|
F |
F |
T |
T |
T |
T |
F |
F |
F |
T |
|
|
T |
F |
F |
F |
T |
T |
F |
T |
T |
F |
|
|
|
NOTE:
A denial of a conjunction is equivalent to a
disjunction of denials. A denial of a disjunction is equivalent to a
conjunction of denials.
|
|
~ |
(P |
• |
q) |
≡ |
(~ |
p |
v |
~ |
q) |
|
|
F |
T |
T |
T |
T |
F |
T |
F |
F |
T |
|
|
T |
T |
F |
F |
T |
F |
T |
T |
T |
F |
|
|
T |
F |
F |
T |
T |
T |
F |
T |
F |
T |
|
|
T |
F |
F |
F |
T |
T |
F |
T |
T |
F |
|
|
Proofs Using
Demorgan's Rules Only:
(1)
1. ~(p v q) /
~p • ~q
2. ~p • ~q
1 dm
(2)
1. ~(m v n) • p / (~m
• ~n) • p
(3)
1. p > ~(q v s) / p > (~q
• ~s)
2. p > (~q • ~s)
1 dm
(4)
1. ~s • ~r /
~(s v r)
(5)
1. p > (~q • ~s) / p > ~(q v s)
2. p > ~(q v
s) 1 dm
(6)
1. ~(p • q) / (~p v ~q)
(7)
1. s v ~(m • u)
/ s v (~m v ~u)
2. s v (~m v
~u) 1 dm
(8)
1. ~s v ~r /
~(s • r)
(9)
1. x • (~s v
~r) / x • ~(s • r)
2. x • ~(s • r)
1 dm
(10)
1. ~(p v ~q) / ~p
• ~~q
DEMORGAN'S
RULES (dm):
~(p v q) ≡ (~p •
~q)
~(p • q) ≡ (~p v
~q)
SIMPLIFICATION (sm):
p • q p
•
q
_____ or
_____
p q
DISJUNCTIVE
SYLLOGISM (ds):
p v
q p v q
~p ~q
______ _____
q p
CONDITIONAL
PROOF (cp)
.
.
│p ap
│.
│.
│.
│q
└──────────
p > q cp
Proofs Using
Demorgan's Rules, Double Negation,
Simplification, Disjunctive
Syllogism,
& Conditional
Proof:
(1)
1. ~(p v
q) /
~ p
2. ~ p •
~q 1 dm
3. ~ p 2 sm
(2)
1. ~(~s • r) •
~s / ~r
(3)
1. ~(~x v r)
•
u
2. r v ~y / t > ~y
┌─3. t
ap
│ 4. (~~x
• ~r) • u
1 dm
│ 5.
~~x
• ~r
4 sm
│ 6. ~r
5 sm
│ 7. ~y 2, 6 ds
└───────────────────────
8. t >
~y 3-7 cp
(4)
1. ~(~p • q) •
~(p v s) / ~q
(5)
1. ~(s
• q)
2. s
• u
/ w > ~q
┌─3.
w ap
│ 4. s
2 sm
│ 5. ~s v ~q
1 dm
│ 6. ~q
5, 4 ds
└──────────────────────────
7. w >
~q 3-6 cp
INDIRECT
PROOF (ip):
.
.
│~p
│ .
│ .
│ .
│ q • ~q
└──────────
p ip
Indirect
Proofs Using
Modus Ponens, &
Simplification:
(1)
1. p
• y
2. p >
(~s > s)
/ s
┌─3. ~s ap
│ 4. p 1 sm
│ 5.
~s > s 2, 4
mp
│ 6. s 5, 3 cj
|
7. s • ~s
6, 3 cj
└───-───-───-───────────
8. s
3-7 ip
(2)
1. u > p
2. (x
• ~t) •
t / p
(3)
1. (x > z)
• x
2.
x > ~z / w
┌─3. ~w
ap
│ 4. x > z 1 sm
│ 5. x 1 sm
│ 6. z 4, 5 mp
│ 7.
~z 2,
5 mp
│ 8.
z
• ~z 6, 7 cj
└─────────────────────
9. w 3-8 ip
(4)
1. p
• q
2. u > t
3. p > (u
•
~t) / s
Indirect
Proofs Using Modus
Ponens, Modus
Tollens, &
Disjunctive
Syllogism:
(1)
1.
~a
2.
a v s
3.
s > (~p > a) / p
┌─4. ~p
ap
│ 5. s
1, 2 ds
│ 6.
~p > a
3, 5 mp
│ 7. a
4, 6 mp
|
8. a • ~a 1, 7 cj
└───────────────────────
9. p
4-8 ip
(2)
1. p v [f • ~(d v f)] /
p
(3)
1.
p > ~(b v ~d)
2.
p v s
3.
s > p
/ d
┌─4. ~d
ap
│ 5. b v ~d
4 ad
│ 6. ~p
1, 5 mt
│ 7. ~s
3, 7 mt
│ 8. s
2, 7 ds
│ 9. s • ~ s
9, 8 cj
└────────────────────
10. d
4-10 ip
(4)
Can you solve this by not using Indirect Proof?
1. (x
• p) > t
2. ~t v
(u • ~ u)
3. x
4.
p /
s
(5)
1. s v t
2. ~x
3. x v ~t
4. s > p
/ p
┌─5. ~p ap
│ 6. ~s 4, 5 mt
│ 7. t 1, 6 ds
│ 8. ~t 3, 2 ds
│ 9. t
• ~t 7, 8 cj
└────────────────────
10. p
5-9 ip
|
MATERIAL
EQUIVALENCE (me):
(p ≡ q)
≡ [(p > q) • (q > p)]
(p ≡ q)
≡ [(p • q) v (~p • ~q)]
|
|
(P |
≡ |
q) |
≡ |
(p |
> |
q) |
• |
(q |
> |
P) |
|
|
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
|
|
T |
F |
F |
T |
T |
F |
F |
F |
F |
T |
T |
|
|
F |
F |
T |
T |
F |
T |
T |
F |
T |
F |
F |
|
|
F |
T |
F |
T |
F |
T |
F |
T |
F |
T |
F |
|
|
(P |
≡ |
q) |
≡ |
[(p |
• |
q) |
V |
(~ |
p |
• |
~ |
q)] |
|
T |
T |
T |
T |
T |
T |
T |
T |
F |
T |
F |
F |
T |
|
T |
F |
F |
T |
T |
F |
F |
F |
F |
T |
F |
T |
F |
|
F |
F |
T |
T |
F |
F |
T |
F |
T |
F |
F |
F |
T |
|
F |
T |
F |
T |
F |
F |
F |
T |
T |
F |
T |
T |
F |
|
Proofs Using
Material Equivalence,
Simplification, Conjunction, & Modus
Ponens:
(1)
1. p ≡
q / q > p
2. (p > q) • (q
> p) 1 me
3. q >
p 2 simp
(2)
1. (p > q) • (q
> p) / p ≡ q
(3)
1. (s • r) v (~s
• ~r) / s ≡ r
2. s ≡ r
1 me
(4)
1. r ≡ q
2. r
/ q
(5)
1. p ≡ q
2. q
3. p >
s / s
4. (p > q) • (q
> p) 1 me
5. q >
p 4 sm
6.
p 2, 5 mp
7.
s 3, 6 mp
(6)
1. [(s • z) v
(~s • ~z)] > x
2. (s ≡
z) • (x > p) / p
(7)
1. (p > q)
2. (q > p)
3. (p ≡
q) > (s ≡ z) / s > z
4. (p > q) • (q
> p) 1, 2 cj
5. p ≡
q 4 me
6. s ≡
z 3, 5 mp
7. (s > z) • (z
> s) 6 me
8. s >
z 7 sm
|
DISTRIBUTION
(dist):
[p v (q • r)]
≡ [(p v q) • (p v r)]
[p • (q v r)]
≡ [(p • q) v (p • r)]
|
|
[p |
v |
(q |
• |
r) |
≡ |
[(p |
v |
q) |
• |
(P |
v |
r)] |
|
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
|
T |
T |
T |
F |
F |
T |
T |
T |
T |
T |
T |
T |
F |
|
T |
T |
F |
F |
T |
T |
T |
T |
F |
T |
T |
T |
T |
|
T |
T |
F |
F |
F |
T |
T |
T |
F |
T |
T |
T |
F |
|
F |
T |
T |
T |
T |
T |
F |
T |
T |
T |
F |
T |
T |
|
F |
F |
T |
F |
F |
T |
F |
T |
T |
F |
F |
F |
F |
|
F |
F |
F |
F |
T |
T |
F |
F |
F |
F |
F |
T |
T |
|
F |
F |
F |
F |
F |
T |
F |
F |
F |
F |
F |
F |
F |
|
[p |
• |
(q |
v |
r) |
≡ |
[(p |
• |
q) |
v |
(P |
• |
r)] |
|
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
|
T |
T |
T |
T |
F |
T |
T |
T |
T |
T |
T |
F |
F |
|
T |
T |
F |
T |
T |
T |
T |
F |
F |
T |
T |
T |
T |
|
T |
F |
F |
F |
F |
T |
T |
F |
F |
F |
T |
F |
F |
|
F |
F |
T |
T |
T |
T |
F |
F |
T |
F |
F |
F |
T |
|
F |
F |
T |
T |
F |
T |
F |
F |
T |
F |
F |
F |
F |
|
F |
F |
F |
T |
T |
T |
F |
F |
F |
F |
F |
F |
T |
|
F |
F |
F |
F |
F |
T |
F |
F |
F |
F |
F |
F |
F |
|
|
The first formula is a
disjunction; one or both sides are true. If p is true, then (p v
q) is true and (p v r) is true. If q and r are true, then (p v
q) is true and (p v r) is true.
The second formula is a conjunction; both sides are true; p is
true, and q or r or both are true. Both sides of the v are
conjuncts [(p • q) v (p • r)]. At least one or both sides are
true. Since p is on both sides, it is true; q or r or both have
to be true: p • (q v r). |
Proofs Using
the Rules of Distribution,
Simplification, Modus Ponens, &
conjunction:
(1)
1. p v (q • r)
2. (p v q) >
s / s
3. (p v q) • (p
v r) 1 dist
4. q v
p 3 sm
5.
s 2, 4 mp
(2)
1. p v q
2. p v r
/ p v (q • r)
(3)
1. s
2. q v t / (s
• q) v (s • t)
3. s • (q v
t) 1, 2 cj
4. (s • q) v (s
• t) 3 dist
(4)
1. (p • q) v (p
• r)
2. (q v r) >
x / x
Proofs Using
the Rules of
Inference
Only:
(1)
1. (o v y) > m
2. y /
m
3. o v y 2
ad
4. m 1, 3
mp
(2)
1. w > (o
•
u) / w > u
(3)
1. m v ~n
2. n
• u
3. m v t /
~m > t
┌─4. ~m ap
│ 5. t 3, 4 ds
└────────────────────
6. ~m > t
4-5 cp
(4)
1. p • r
2. p > ~q
3. s > q
/ ~s
(5)
1. ~w
2. x v w
3. r
4. (x
• r) >
q / y > q
┌─5.
y ap
│ 6. x
1, 2 ds
│ 7. x
• r
6, 3 cj
│ 8. q
4, 7 mp
└─────────────────────────
9. y >
q 5-8 cp
(6)
1. p > w
2. ~w • r
3. r > p
/ q
(7)
1. x
• z
2. z > ~u
3. s > u
4. s v t / z > t
┌─5. z
ap
│ 6. ~u 2, 5 mp
│ 7. ~s 3, 6 mt
│ 8. t 4, 7 ds
└───────────────────────
9. z >
t 5-8 cp
Proofs Using
All the Rules:
(1)
1. J > T
2. J • (B v ~T)
3. / (J
• T) • B
4. J
1
sm
5. T
1, 4 mp
6. B v ~T
2 sm
7. B
5, 6 ds
8. J • T
4, 5 cj
9. (J • T) • B
8, 7 gj
(2)
1. Q > (A v ~B)
2. B
3. ~A
/ ~Q
(3)
1. a
2. ~a
/ s
3. a v s 1
ad
4. s 2, 3 ds
(4)
1. (x v p) > t
2. ~t v u
3. x
4.
p / u
(5)
1. ~(z • b) > t
2. ~b
/ t
3. ~z v
~b 2 ad
4. ~(z •
b) 3 dm
5. t
1, 4 mp
(6)
1. A • B
2. (B • A) > (T
V C)
3. (C V T) > P
/ P
(7)
1. (p > q) > (s
v ~t)
2. ~p v q / ~s
> ~t
3. p > q 2 mi
4. s v ~t 1, 3 mp
5. ~s > ~t
4 mi
(8)
1. A • (B • D)
2. (A • B) > Q
3. Q > (P v Z) / Z v P
(9)
1. p > q
2. q > s
3. s >
~p / ~p
┌─4. p
ap
│ 5. q
1, 4 mp
│ 6. s
2, 5 mp
│ 7. ~p
3, 6 mp
│ 8. p
• ~p 4, 7 cj
└────────────────────────
9.
~p 4-8 ip
(10)
1.
~a
2.
a v s
3.
s > ~s /
p
(11)
1. s v r
2. r > y
3. ~s • p
/ s v y
┌─4. ~(s v y)
ap
│ 5. ~s • ~y
4 dm
│ 6. ~s
5 sm
│ 7. r 1, 6 ds
│ 8. y 2, 7 mp
│ 9. ~y
5 sm
│ 10. y • ~y
8, 9 cj
└───────────────────────
11. s v y 4-10 ip
or
1. s v r
2. r > y
3. ~s • p / s v y
4. ~s
3 sm
5. r
1, 4 ds
6. y
2, 5 mp
7. s v y
6 ad
(12)
1. ~P > (B >
E)
2. P >
~C
3. ~(C • A) >
R
4. ~R • S
/ (B > E) v P
(13)
1. J > X
2. (X v ~W) >
R
3. ~B > ~(R v
~W)
4. ~(C v B) / ~R > ~J
┌─5. ~R
ap
│ 6. ~(X v ~W)
2, 5 mt
│ 7. ~X
• W
6 dm
│ 8. ~X
7 sm
│ 9. ~J 8, 1 mt
└───────────────────────
10. ~R >
~J 5-9 cp
(14)
1. ~A v B
2. (A • ~B) v C
3. ~C v ~F
4. ~F > (Q •
B) / Q
(15)
1. ~[D v (R
•
A)]
2. ~D > R
3. (B • T) > A / ~B v ~T
4. ~D • ~(R •
A) 1 dm
5.
~D 4 sm
6.
R 5, 2 mp
7. ~(R •
A) 4 sm
8. ~R v ~A
7 dm
9.
~A 6, 8 ds
10. ~(B •
T) 3, 9 mt
11. ~B v ~T
10 dm
(16)
1. Q > ~P
2. ~X > P
3. A • ~(C v
X) / ~Q
(17)
1. (T • X) > B
2. (Z ≡ B) >
(~E ≡ T)
3. (~X v Z) >
(Z ≡ B)
4. ~B • T /
~E ≡ T
5.
~B 4 sm
6. T
4 sm
7. ~(T • X) 1, 5 mt
8. ~T v
~X 7 dm
9. ~X 6, 8 ds
10. ~X v
Z 9 ad
11. Z ≡ B 10, 3 mp
12. ~E ≡ T 2, 11 mp
(18)
1. A >
S
2. (A > S) >
~(P > ~Q)
3. (P
• T) >
~(Q v ~S)
4. T
/ X
(19)
1. P > S
2. (E v R) >
~S
3. [~T v (C
•
X)] > E
4. T > (~A
•
B)
5. A > R /
A > ~P
┌─6. A
ap
│ 7. R 5, 6 mp
│ 8. E v R 7 ad
│ 9. ~S 8, 2 mp
│ 10. ~P 9, 1 mt
└──────────────────────
11. A > ~P
6-10 cp
(20)
1. D v Q
2. ~(D > ~C)
> ~Q
3. ~S v ~W
4. ~Q > W /
~S v (D > ~C)
┌──5.
S ap
│┌─6.
D ap
││ 7. ~W
3, 5 ds
││ 8. Q
4, 7 mt
││ 9. D > ~C
2, 8 mt
││ 10. ~C
6, 9 mp
│└─────────────────────────
│ 11. D > ~C
6-10 cp
└──────────────────────────
12. S > (D >
~C) 5-11 cp
13. ~S v (D >
~C) 12 mi
or
1. D v Q
2. ~(D > ~C)
> ~Q
3. ~S v ~W
4. ~Q > W /
~S v (D > ~C)
┌──5.
S ap
│ 6. ~W
3, 5 ds
│ 7.
Q
4, 6 mt
│ 8.
D > ~C
2, 7 mt
└──────────────────────────
9. S > (D >
~C) 5-8 cp
10. ~S v (D >
~C) 9 mi
(21)
1. (S • ~P) > (X
> A)
2. ~S > ~A
3. (~B • ~C) >
~P
4. ~[A > (B v
C)] / ~(X • ~A)
5. ~[~A v (~B v
C)] 4 mi
6. ~~A • ~(B v
C) 5 dm
7.
~~A 6 sm
8. ~(B v
C) 6 sm
9.
S 2, 8 mt
10. ~B •
~C 8 dm
11.
~P 3, 10 mp
12. S •
~P 9, 11 cj
13. X >
A 1, 12 mp
14. ~X v
A 13 mi
15. ~(X •
~A) 14 dm
(22)
1. S > T
2. B v ~A
3. R > (T • ~X)
4. [(A > B) v X]
> (R • X)
5. ~(A • ~B) / (T v M) v ~W
BRAIN TWISTERS:
(1)
1. (T
≡ A) >
~(M v ~w)
2. (A
• ~X) >
[(T ≡ A) v B]
3. (~B
• X) >
(B • ~C)
4. ~(A > B) / ~(M
• X)
┌─5. M
• X
ap
│ 6. X
5 sm
│ 7. ~(~A V B) 4 mi
│ 8. A
• ~B
7 dm
│ 9. ~B
8 sm
│ 10. ~B • X 9, 6 cj
│ 11. B
• ~C 3, 10 mp
│ 12. B
11 sm
│ 13. B
• ~B
12, 9 cj
└─────────────────────────
14. ~(M •
X) 5-13 ip
or
1. (T ≡ A) >
~(M v ~w)
2. (A • ~X) >
[(T ≡ A) v B]
3. (~B • X) >
(B • ~C)
4. ~(A > B)
/ ~(M • X)
5. ~(~A v
B) 4 mi
6. ~~A •
~B 5 dm
7.
~B 6 sm
8. ~B v
C 7 ad
9. ~(B •
~C) 8 dm
10. ~(~B •
X) 3, 9 mt
11. ~~B v
~X 10 dm
12.
~X 7, 11 ds
13. ~M v ~X
12 ad
14. ~(M •
X) 13 dm
(2)
1. (~P v ~T) >
S
2. ~X v (R
•
~S)
3. (~A > C) >
X
4. C > A
5. ~(A
• C) >
C / P
┌─6. ~P
ap
│ 7. ~P v ~T
6 ad
│ 8. S 1, 7 mp
│ 9. ~R v S
8 ad
│ 10. ~(R • ~S)
9 dm
│ 11. ~X 2, 10 ds
│ 12. ~(~A > C) 3, 11 mt
│ 13. ~(A v C)
12 mi
│ 14. ~A • ~C 13 dm
│ 15. ~C
14 sm
│ 16. A • C 5, 15 mt
│ 17. C 16 sm
│ 18. C • ~C 17, 15 cj
└───────────────────────
19. P 6-18 ip
(3)
1. ~C v R
2. ~R v (P
•
~Q)
3. ~Q > Z
4. (~S > D) >
~(A > ~C)
5. ~[A • (P
•
~Q)] / ~S
┌─6.
A ap
│ 7. ~A v ~(P
•
~Q) 5 dm
│ 8. ~(P • ~Q)
7, 6 ds
│ 9. ~R
2, 8 ds
│10. ~C
1, 9 ds
└──────────────────────────
11. A > ~C
6-10 cp
12 ~(~S >
D) 4, 11 mt
13. ~(S v
D) 12 mi
14. ~S •
~D 13 dm
15.
~S 14 sm
(4)
1. ~M v T
2. R > (M
• T)
3. (~Q v B) >
R
4. A > B /
(~A v T) v C
┌─5. A
ap
│ 6. B
4, 5 mp
│ 7. ~Q v B
6 ad
│ 8. R
3, 7 mp
│ 9. M • T
2, 8 mp
│10. T
9 sm
└─────────────────────────
11. A > T
5-10 cp
12. ~A v T
11 mi
13. (~A v T) v
C 12 ad
(5)
1. ~(Q > ~A)
> ~A
2. ~R v
Q
3. S v R
4. ~(S
• ~T) / ~T > ~A
┌─5. ~T
ap
│ 6. ~S v T
4 dm
│ 7. ~S
6, 5 ds
│ 8. R
3, 7 ds
│ 9. Q
2, 8 ds
│10. (Q > ~A) v
~A 1 mi
│11. (~Q v ~A) v
A 10 mi
│12. ~Q v (~A v
~A) 11 as
│13. ~A v ~A
12, 9 ds
│14.
~A 13 ta
└───────────────────────────
15. ~T >
~A 5-14 cp
(6)
1. ~R > (C >
P)
2. ~[~D v (C >
P)]
3. ~R v ~(~R >
~D) / B v L
4. D • ~(C >
P) 2 dm
5. ~(C >
P) 4 sm
6.
~~R 1, 5 mt
7. ~(~R >
~D) 3, 6 ds
8. ~(R v
~D) 7 mi
9. ~R •
D 8 dm
10.
~R 9 sm
11. ~R v (B v
L) 10 ad
12. B v
L 6, 11 ds
(7)
1. ~(~Q > ~P) v
~Q
2. ~[(A > P) v
~ B]
3. ~Q > (A >
P) / P v M
4. ~(A > P) •
B 2 dm
5. ~(A >
P) 4 sm
6.
~~Q 3, 5 mt
7. ~(~Q >
~P) 1, 6 ds
8. ~(Q v
~P) 7 mi
9. ~Q •
P 8 dm
10.
P 9 sm
11. P v
M 10 ad
(8)
1. M > (S
•
T)
2. B >
~S
3. ~(~R v ~W)
> M
4. ~(R
• W) >
~C / A > (B >~C)
┌──5.
A ap
│┌─6.
B ap
││ 7.
~S 2, 6 mp
││ 8. ~S v
~T 7 ad
││ 9. ~(S
•
T) 8 dm
││ 10. ~M
1, 9 mt
││ 11. ~R v
~W 3, 10 mt
││ 12. ~(R
•
W) 11 dm
││ 13.
~C 4, 12 mp
│└────────────────────────────────
│ 14. B >
~C 6-13 cp
└─────────────────────────────────
15. A > (B >
~C) 5-14 cp
(9)
1. T > ~(~E v
Q)
2. ~(~M
• T) >
R
3. R > (~A
•
S) / A > [C > (D > E)]
┌───4. A
ap
│┌──5.
C ap
││┌─6.
D ap
│││ 7. A v
~S 4 ad
│││ 8. ~(~A
•
S) 7 dm
│││ 9.
~R 3, 8 mt
│││ 10. ~M
•
T 2, 9 mt
│││ 11.
T 10 sm
│││ 12. ~(~E v
Q) 1, 11 mp
│││ 13. E
•
~Q 12 dm
│││ 14.
E 13 sm
││└────────────────────────────────
││ 15. D >
E 6-14 cp
│└─────────────────────────────────
│ 16. C > (D >
E) 5-15 cp
└──────────────────────────────────
17. A > [C >
(D > E)] 4-16 cp
(10)
1. (~R v ~X) >
~A
2. (R • X) > (B
v ~A)
3. (A • ~B) v
~(C v B)
4. A
/ ~A
5. ~(~R v
~X) 1, 4 mt
6. R •
X 5 dm
7. B v
~A 2, 6 mp
8. ~A v B 7 cm
9. ~(A •
~B) 8 dm
10. ~(C v B)
3, 8 ds
11. ~C •
~B 10 dm
12.
~B 11 sm
13. ~A 7, 12 ds
(11)
1. (a v ~a) >
t / t
┌─2. ~t
ap
│ 3. ~(a v ~a) 1, 2 mt
│ 4. ~a
• a
3 dm
└───────────────────────
5.
t 2-4 ip
or
1. (a v ~a) >
t / t
┌─2. ~a
ap
│ 3. ~a v ~a
2 ad
│ 4. ~a
3 ta
└────────────────────────
5. ~a >
~a 2-4 cp
6. a v
~a 5 mi
7.
t 1, 6 mp
(12)
1. ~C > A
2. ~(W > ~B)
> ~A
3. ~B > ~M
4. M > W / W
> (M > C)
┌──5. W
ap
│┌─6. M
ap
││ 7. B
3, 6 mt
││ 8. W
• B
5, 7 cj
││ 9. ~(~W v ~B)
8 dm
││10. ~(W > ~B)
9 mi
││11. ~A 2, 10 mp
││12. C
1, 11 mt
│└────────────────────────
│ 13. M > C 6-12 cp
└─────────────────────────
14. W > (M >
C) 5-13 cp
(13)
1. X > [~(~D
> ~B) > D]
2. ~X > (D v
~B)
3. ~B > D
/ D
┌─4.
~D ap
│ 5. B
3, 4 mt
│ 6. ~D
•
B 4, 5 cj
│ 7. ~(D v
~B) 6 dm
│ 8. X
2, 7 mt
│ 9. ~(~D > ~B) >
D 1, 8 mp
│10. ~D >
~B 9, 4 mt
│11.
~B 10, 4 mp
│12. B •
~B 5, 11 cj
└───────────────────────────
13. D
4-12 ip
(14)
1. R > (C > T)
2. T
3. B > (R v ~T)
4. A > X / (A >
B) > (C > T)
5. ~C v
T 2 ad
6. C >
T 5 mi
7. ~(A > B) v (C
> T) 6 ad
8. (A > B) > (C
> T) 7 mi
or
1. R > (C > T)
2. T
3. B > (R v ~T)
4. A > X / (A >
B) > (C > T)
┌─-5.
(A > B)
ap
|┌─6.
C
ap
|└────────────────────────
|
7. C > T
6-2 cp
└─────────────────────────
8. (A >
B) > (C > T)
5-7 cp
(15)
1. ~(x • t) >
~(x v w)
2. (z > ~m) >
x
3. n > ~(m v
p)
4. ~(z v
~n) / t
┌─5. ~t
ap
│ 6. ~x v ~t
5 ad
│ 7. ~(x • t)
6 dm
│ 8. ~(x v w)
1, 7 mp
│ 9. ~x • ~w
8 dm
│10. ~x
9 sm
│11. ~(z > ~m) 2, 10 mt
│12. ~(~z v ~m) 11 mi
│13. z
• m
12 dm
│14. z
13 sm
│15. ~z • n
4 dm
│16. ~z
15 sm
│17. z
• ~z 14, 16 cj
└─────────────────────────
18.
t 5-17 ip
or
1. ~(x • t) >
~(x v w)
2. (z > ~m) > x
3. n > ~(m v p)
4. ~(z v ~n)
/ t
5. ~z • n
4 dm
6. ~z
5 sm
7. ~z v ~m
6 ad
8. z > ~m
7 mi
9. x
2, 8 mp
10. x v w
9 ad
11. x • t
1, 10 mt
12. t
11 sm
|
