Discr
e
te
Structur
es
Rules of i
n
fer
ence
Dr
.
Muh
ammad
Hu
ma
y
oun
Assis
tan
t
Pr
of
es
so
r
C
OM
S
A
T
S
Ins
ti
tut
e
of
Comput
er
Sci
e
nce,
Lahor
e.
mhuma
youn@ci
itl
ahor
e.edu.pk
h
ttp
s://
si
tes.
g
oo
gl
e.c
om/
a/
ci
itl
a
hor
e.edu.pk/ds
truc
t/
1
Rules
of
In
fer
ence
V
alid A
rgum
en
ts
in Pr
opositional
Logic
l
Assume
you
are
give
n
the following
two
statem
en
ts:
§
“i
f
yo
u
are in
thi
s
class,
th
en
yo
u
will g
et a
grad
e”
§
“y
ou
are in
thi
s
class”
Therefor
e,
§
“Y
ou
will g
et a
grad
e”
2
Modus
P
onen
s
(La
ti
n
for
“the
w
a
y th
a
t
a
ffi
rms
b
y
a
ffi
rmi
n
g
”
•
If
i
t
sno
w
s
toda
y,
the
n
w
e
wi
ll
g
o
sk
ii
ng
•
Hypothes
is
:
It
is
snowi
ng
toda
y
•
B
y
modus
ponens,
the c
oncl
us
ion
is
:
•
W
e
wi
ll
g
o sk
ii
ng
3
•
If I smok
e, then
I c
ough
•
I
Smok
e
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
I c
ough
4
Modus
Tollens
(La
ti
n
for "the w
a
y th
a
t
deni
es
by den
yi
ng")
●
Assu
m
e y
ou
a
re giv
en
th
e
fo
llo
wing
two st
atem
ents:
§
“y
ou
wi
ll
no
t g
e
t
a gr
ade”
§
“i
f
y
ou ar
e i
n thi
s
cl
a
ss,
y
ou wi
ll
g
et
a gr
ade
”
●
Let
p =
“you ar
e
in t
his
cl
ass
”
●
Let
q =
“you wil
l
get
a
gra
de”
●
By Mod
us
T
ol
lens
, you
ca
n con
clud
e
th
at
yo
u a
re
no
t
in t
his
c
la
ss
5
Addi
tio
n
•
If
yo
u
kn
ow
that
p is true,
then
p
Ú
q will
AL
W
A
YS
be true
i.e.
p
→
p
Ú
q
6
Addi
tio
n
•
If
yo
u
kn
ow
that
p is true,
then
p
Ú
q will
AL
W
A
YS
be true
i.e.
p
→
p
Ú
q
•
p : “
It
is
bel
o
w fr
ee
zi
ng
no
w
”
•
q : “
It i
s
rai
ni
ng
no
w
”
•
ש
ݍ
“It
is
bel
o
w
fr
ee
zi
ng or
rai
ni
ng
no
w
”
•
՜
ש
ݍ
“If i
t
is
bel
ow
fr
ee
zi
ng no
w th
en
it
i
s
bel
o
w fr
ee
zi
ng or r
ai
ni
ng
no
w
”
7
Simpli
fic
a
tio
n
•
If p
Ù
q is true,
then
p will
AL
W
A
YS
be
true
i
.e.
p
Ù
q
→ p
8
Simpli
fic
a
tio
n
•
If
p
Ù
q is true,
then
p will
AL
W
A
YS
be
true
i
.e.
p
Ù
q
→
p
•
p:
“It
is
bel
ow
fr
ee
zi
ng
”
•
q:
“It i
s r
ai
ni
ng
no
w
”
•
p
Ù
q
: It
is be
low
free
zing
an
d
rai
ni
ng
no
w
.
•
p
Ù
q
→
p: I
t
is
be
low
free
zing
an
d
rai
ni
ng
no
w
imp
li
es
tha
t
it
is be
low
free
zing
Hypothe
tic
al
syllogis
m
•
՜
ݍ
ר
ݍ
՜
ݎ
՜
ሺ
՜
ݎሻ
•
If i
t
rai
ns
t
oda
y,
then
w
e
wi
ll
not
ha
ve a barbecue
toda
y.
•
If
w
e
do not ha
ve a barbecu
e t
oda
y,
the
n
w
e
wi
ll
ha
ve a barbecue t
omorr
ow
.
•
Ther
e
for
e
,
if
i
t
rai
ns
t
oda
y,
then
w
e
wi
ll
ha
ve
a
barbecue
tomorr
ow
.
1
0
Dis
junctiv
e
syllo
gism
•
ש
ݍ
ר
՜
ݍ
1
1
R
esoluti
on
•
Comp
ut
er
pr
ogr
ams
ha
ve been
de
vel
oped t
o
aut
oma
te
the t
ask of
reasoni
ng and pr
ovi
ng theor
ems.
•
Man
y o
f t
hese pr
ogr
ams
mak
e use
resol
ution
1
2
Rul
es
of In
fer
ence
to Bui
ld Ar
gumen
ts
•
It
is no
t sunn
y
thi
s
a
ft
ern
oo
n and i
t i
s
col
der than
yes
ter
da
y
•
W
e wi
ll
g
o
swi
mmi
ng
onl
y i
f i
t
is
sunn
y
•
If
w
e do
no
t g
o
swi
mmi
ng
,
then w
e
wi
ll
t
ak
e
a
canoe tri
p
•
If
w
e t
ak
e
a c
anoe
tri
p, then w
e
wi
ll
be home b
y
sunse
t
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
W
e will
be h
om
e
b
y
sun
set
(Conclusion)
1
3
Rul
es
of In
fer
ence
to Bui
ld Ar
gumen
ts
•
It
is no
t sunn
y
thi
s
a
ft
ern
oo
n and i
t i
s
col
der than
yes
ter
da
y
•
W
e wi
ll
g
o
swi
mmi
ng
onl
y i
f i
t
is
sunn
y
•
If
w
e do
no
t g
o
swi
mmi
ng
,
then w
e
wi
ll
t
ak
e
a
canoe tri
p
•
If
w
e t
ak
e
a c
anoe
tri
p, then w
e
wi
ll
be home b
y
sunse
t
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
W
e will
be h
om
e
b
y
sun
set
(Conclusion)
•
p:
It
is
sunn
y
thi
s
a
ft
ern
oo
n
•
q:
It
is
col
der than y
es
ter
da
y
•
r:
W
e
wi
ll
g
o
swi
mmi
ng
•
s:
W
e wi
ll
t
ak
e
a
canoe tri
p
•
t:
W
e
wi
ll
be home b
y
sunse
t
1
4
Rul
es
of In
fer
ence
to Bui
ld Ar
gumen
ts
•
It
is no
t sunn
y
thi
s
a
ft
ern
oo
n and i
t i
s
col
der than
yes
ter
da
y
ר
ݍ
•
W
e wi
ll
g
o
swi
mmi
ng
onl
y i
f i
t
is
sunn
y
ݎ
՜
•
If
w
e do no
t g
o
swi
mmi
ng
,
then w
e
wi
ll
t
ak
e
a
canoe tri
p
ݎ
՜
ݏ
•
If
w
e t
ak
e
a c
anoe
tri
p, then w
e
wi
ll
be home b
y
sunse
t
ݏ
՜
ݐ
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
W
e will
be h
om
e
b
y
sun
set
(Conclusion)
࢚
•
p:
It
is
sunn
y
thi
s
a
ft
ern
oo
n
•
q:
It
is
col
der than y
es
ter
da
y
•
r:
W
e
wi
ll
g
o
swi
mmi
ng
•
s:
W
e wi
ll
t
ak
e
a
canoe tri
p
•
t:
W
e
wi
ll
be home b
y
sunse
t
ሾ
ר
ר
࢘
՜
ר
࢘
՜
࢙
ר
ሺ࢙
՜
࢚ሻ
ሿ՜
࢚
1
5
De
finitio
ns
●
An
Ar
gumen
t
in
pr
oposi
ti
onal
l
ogi
c i
s a
sequen
ce of p
roposi
ti
ons
tha
t end wi
th
c
concl
us
ion.
●
Al
l e
xcep
t
the
fi
nal
pr
oposi
ti
on
ar
e
cal
led
pr
emi
ses
.
●
The fi
nal
pr
oposi
ti
on
is
c
al
led
concl
us
ion
.
●
An ar
gumen
t
is
val
id
i
f the truth
of al
l
pr
emi
ses i
mpl
ies
tha
t the c
oncl
us
ion
is
true.
§
i.e.
ሺͳ
Ù
ʹ
Ù
ǥ
Ù
݊
ሻ
®
ݍ
i
s
a t
aut
ol
ogy
.
Rul
es of In
fer
ence t
o Bui
ld Ar
gumen
ts
•
ሾ
ר
ר
࢘
՜
ר
࢘
՜
࢙
ר
ሺ࢙
՜
࢚ሻሿ
՜
࢚
1
7
ר
՜
Rul
es of In
fer
ence t
o Bui
ld Ar
gumen
ts
•
ሾ
ר
ר
࢘
՜
ר
࢘
՜
࢙
ר
ሺ࢙
՜
࢚ሻሿ
՜
࢚
1
Rul
es of In
fer
ence t
o Bui
ld Ar
gumen
ts
•
ሾ
ר
ר
࢘
՜
ר
࢘
՜
࢙
ר
ሺ࢙
՜
࢚ሻሿ
՜
࢚
1
9
Rul
es of In
fer
ence t
o Bui
ld Ar
gumen
ts
•
ሾ
ר
ר
࢘
՜
ר
࢘
՜
࢙
ר
ሺ࢙
՜
࢚ሻሿ
՜
࢚
2
0
•
If
you send me an
e
-mai
l messag
e,
then
I wi
ll
fi
ni
sh
wri
ti
ng
the pr
ogr
am
•
If
you do
not
send me an e
-mai
l messag
e,
the
n
I wi
ll
g
o t
o sl
eep
earl
y
•
If
I g
o
to
sl
eep earl
y, then
I
wi
ll
w
ak
e
up
feel
in
g
re
fr
eshed
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
If
I do
not
fi
ni
sh
wri
ti
ng
the pr
ogr
am, then
I
wi
ll
w
ak
e up
feel
in
g
re
fr
eshed
2
1
•
If y
ou send
me an e
-mai
l messag
e,
then
I wi
ll
f
ini
sh
wri
ti
ng the
pr
ogr
am
՜
•
If y
ou do
not send
me an e
-mai
l messag
e,
then
I wi
ll
g
o
to sl
eep
earl
y
՜
࢘
•
If I
g
o
to sl
eep
earl
y,
then
I wi
ll
w
ak
e
up f
eel
ing
re
fr
eshed
࢘
՜
࢙
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
If I
do no
t
fi
ni
sh
wri
ti
ng th
e
pr
ogr
am,
then
I wi
ll
w
ak
e
up
feel
ing
re
fr
eshed
՜
࢙
•
p
=
Y
ou
send
me an
e
-m
a
il
•
q =
I
wi
ll
fi
ni
sh
wri
ti
ng
pr
ogr
am
•
r
=
I
wi
ll
g
o t
o sl
eep
earl
y
•
s
=
l
wi
ll
w
ak
e
up f
eel
ing
re
fr
eshed
ሾ
՜
ר
՜
࢘
ר
࢘
՜
࢙
ሿ՜
ሺ
՜
࢙ሻ
2
2
•
ሾሺ
՜
ሻ
ר
ሺ
՜
࢘ሻ
ר
ሺ࢘
՜
࢙ሻ
ሿ՜
ሺ
՜
࢙ሻ 23
•
ሾሺ
՜
ሻ
ר
ሺ
՜
࢘ሻ
ר
ሺ࢘
՜
࢙ሻ
ሿ՜
ሺ
՜
࢙ሻ 24
•
ሾሺ
՜
ሻ
ר
ሺ
՜
࢘ሻ
ר
ሺ࢘
՜
࢙ሻ
ሿ՜
ሺ
՜
࢙ሻ 25
•
ሾሺ
՜
ሻ
ר
ሺ
՜
࢘ሻ
ר
ሺ࢘
՜
࢙ሻ
ሿ՜
ሺ
՜
࢙ሻ 26
•
ሾሺ
՜
ሻ
ר
ሺ
՜
࢘ሻ
ר
ሺ࢘
՜
࢙ሻ
ሿ՜
ሺ
՜
•
Hypot
hes
es:
ר
ݍ
ש
ݎ
and
ݎ
՜
ݏ
im
pl
y
the
concl
us
ion:
ש
ݏ
•
ר
ש
࢘
ר
࢘
՜
࢙
՜
ש
࢙
•
ר
ݍ
ש
ݎ
ؠ
ש
ݎ
ר
ሺݍ
ש
ݎሻ
de
Mor
g
an’
s
la
w
•
ݎ
՜
ݏ
ؠ
ݎ
ש
ݏ
•
ש
ݎ
ש
ݎ
ש
ݏ
ؠ
ש
ݏ
resol
ut
ion
2
8
Falla
cies
•
Sev
er
al
c
ommon f
al
laci
es ari
se i
n i
nc
orr
ect
ar
gumen
ts
.
•
The pr
oposi
ti
on
՜
ݍ
ר
ݍ
՜
i
s not
a
taut
ol
ogy
,
bec
ause i
t i
s f
al
se when p i
s
fal
se and
q i
s
tru
e
•
Ther
e
ar
e
man
y
inc
orr
ect
ar
gumen
ts
tha
t
tr
ea
t
thi
s
as a
taut
ol
ogy
•
Thi
s
typ
e of i
n
corr
ect
reasoni
ng i
s c
al
led the
falla
cy
of a
ffirming
the c
onclusion
2
9
Ex
ampl
e
•
If y
ou do
e
ver
y
pr
obl
em i
n
thi
s
book, then
y
ou
wi
ll
l
earn
di
scr
e
te
ma
the
ma
ti
cs.
•
You l
earned di
scr
e
te
ma
thema
ti
cs
.
•
Ther
e
for
e
,
you di
d
e
ver
y
pr
obl
em i
n
thi
s
book
.
•
p:
You
di
d
e
ver
y
pr
obl
em i
n thi
s
book
•
q
:
You
learned
di
scr
e
te
ma
thema
ti
cs
3
0
•
If y
ou do
e
ver
y
pr
obl
em i
n
thi
s
book, then
y
ou
wi
ll
l
earn di
scr
e
te
ma
thema
ti
cs.
•
You l
earned di
scr
e
te
ma
thema
ti
cs
.
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
Ther
e
for
e
,
you di
d
e
ver
y
pr
obl
em i
n
thi
s
book
.
•
p:
Y
ou
di
d
e
very p
robl
em
i
n thi
s
book
•
q
:
You
learned
di
scr
e
te
ma
thema
ti
cs
3
1
•
If y
ou do
e
ver
y
pr
obl
em i
n
thi
s
book, then
y
ou
wi
ll
l
earn di
scr
e
te
ma
thema
ti
cs.
•
You l
earned di
scr
e
te
ma
thema
ti
cs
.
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
Ther
e
for
e
,
you di
d
e
ver
y
pr
obl
em i
n
thi
s
book
.
•
p:
Y
ou
di
d
e
very p
robl
em
in
thi
s
book
•
q
:
You
learn
ed
di
scr
e
te
ma
thema
ti
cs
•
If
՜
ݍ
and
ݍ
then
•
՜
ݍ
ר
ݍ
՜
3
2
•
If
you do
e
very p
robl
e
m
in thi
s
book, then y
ou w
il
l
le
arn di
scr
e
te
ma
the
ma
ti
cs.
•
You l
earn
ed
di
scr
e
te
ma
thema
ti
cs
.
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
•
The
re
for
e
, y
ou di
d
e
very
pr
obl
em
in
thi
s book
.
•
p: Y
ou
di
d
e
ver
y
pr
obl
em
in
thi
s
book
•
q
:
You
learned
di
scr
e
te
ma
thema
ti
cs
•
If
՜
ݍ
and
ݍ
then
•
՜
ݍ
ר
ݍ
՜
F
al
lac
y
•
It
is
possi
ble
for
y
ou t
o learn
discr
e
te
ma
thema
tics
in
some w
a
y
other
than
b
y
doi
ng
e
very
pr
obl
em
in
thi
s
book
(R
eadi
n
g
, Li
st
eni
ng
Lectur
es,
doi
ng
some
but
not
al
l pr
obl
ems).
3
3
Rul
es
of Inf
er
ence
for
Quan
ti
fi
ed St
a
temen
ts
3
4
3
5
3
Ex
ampl
e
•
Show t
ha
t
the
pr
emi
ses:
•
"E
ver
yone
in
thi
s
di
scr
e
te
ma
them
a
ti
cs
cl
ass
has
tak
en a
cour
se
in
compu
ter
sci
ence"
and
“
As
lam
is
a
studen
t
in
thi
s
cl
ass
"
•
Impl
y
the c
oncl
us
ion
“
As
lam
has
tak
en a
cour
se
in
comput
er sci
ence
.“
3
7
•
D(x
):
x
is
i
n
thi
s
di
scr
e
te
ma
them
a
ti
cs
cl
ass
•
C(x):
x has
tak
en
a c
our
se i
n c
omput
er
sci
en
ce
•
Pr
emi
ses:
x(D(x)
→
C(x
))
and
D
(As
lam
)
•
Concl
usi
on:
C
(As
lam
)
3
8
•
D(x
):
x
is
i
n
thi
s
di
scr
e
te
ma
them
a
ti
cs
cl
ass
•
C(x):
x has
tak
en
a c
our
se i
n c
omput
er
sci
en
ce
•
Pr
emi
ses:
x(D(x)
→
C(x
))
and
D
(As
lam
)
•
Concl
usi
on:
C
(As
lam
)
•
St
ep
s
R
eason
•
x(D(x)
→
C(x
))
Premi
se
•
D
(As
lam
)
→ C(
As
lam
)
Uni
ver
sal
i
ns
tan
ti
a
ti
on
3
9
•
D(x
):
x
is
i
n
thi
s
di
scr
e
te
ma
them
a
ti
cs
cl
ass
•
C(x):
x has
tak
en
a c
our
se i
n c
omput
er
sci
en
ce
•
Pr
emi
ses:
x(D(x)
→
C(x
))
and
D
(As
lam
)
•
Concl
usi
on:
C
(As
lam
)
•
St
ep
s
R
eason
•
x(D(x)
→
C(x
))
Premi
se
•
D
(As
lam
)
→ C(
As
lam
)
Uni
ver
sal
i
ns
tan
ti
a
ti
on
•
D
(As
lam
)
Pr
emi
se
4
0
•
D(x):
x
is
i
n
thi
s
di
scr
e
te
ma
them
a
ti
cs
cl
ass
•
C(x):
x has
tak
en
a c
our
se i
n c
omput
er
sci
en
ce
•
Pr
emi
ses:
x(D(x)
→
C(x
))
and
D
(As
lam
)
•
Concl
usi
on:
C
(As
lam
)
•
St
ep
s
R
eason
•
x(D(x)
→
C(x
))
Premi
se
•
D
(As
lam
)
→ C(
As
lam
)
Uni
ver
sal
i
ns
tan
ti
a
ti
on
•
D
(As
lam
)
Pr
emi
se
•
C
(As
lam
)
Modus
ponens
•
՜
ר
՜
4
1
Ex
ampl
e
•
Show t
ha
t
the pr
emi
ses “
A
studen
t
in
thi
s
cl
ass
has not
read
the book,
”
and
“E
v
er
y
one
in
thi
s
cl
ass
pass
ed the fi
rs
t
e
xam” i
m
pl
y
the c
oncl
us
ion
“S
omeone
who pass
ed
the
fi
rs
t
e
xam has
not r
ead
the book
.”
4
2
•
Show t
ha
t
the pr
emi
ses “
A
studen
t
in
thi
s
cl
ass
has not
read
the book,
”
and
“E
v
er
y
one
in thi
s
cl
ass
passed the fi
rs
t
e
xam” i
m
pl
y
the c
oncl
us
ion
“S
omeone
who pass
ed
the
fi
rs
t
e
xam has not
read t
he boo
k.
”
•
C(x
)
: “
x
is
i
n thi
s
cl
ass
”
•
B(x):
“
x
has r
ead t
he
book”
•
P(x):
“
x
pass
ed
the
fi
rs
t
e
xam”
•
Pr
emi
ses:
???
4
3
•
Show t
ha
t
the pr
emi
ses “
A
studen
t
in
thi
s
cl
ass
has not
read
the book,
”
and
“E
v
er
y
one
in thi
s
cl
ass
passed the fi
rs
t
e
xam” i
m
pl
y
the c
oncl
us
ion
“S
omeone
who pass
ed
the
fi
rs
t
e
xam has not
read t
he boo
k.
”
•
C(x
)
: “
x
is
i
n thi
s
cl
ass
”
•
B(x):
“
x
has r
ead t
he
book”
•
P(x):
“
x
pass
ed
the
fi
rs
t
e
xam”
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(C(x
) → P(
x
))
.
•
The
concl
us
ion???
4
4
•
Show t
ha
t
the pr
emi
ses “
A
studen
t
in
thi
s
cl
ass
has not
read
the book,
”
and
“E
v
er
y
one
in thi
s
cl
ass
passed the fi
rs
t
e
xam” i
m
pl
y
the c
oncl
us
ion
“S
omeone
who pass
ed
the
fi
rs
t
e
xam has not
read t
he boo
k.
”
•
C(x
)
: “
x
is
i
n thi
s
cl
ass
”
•
B(x):
“
x
has r
ead t
he
book”
•
P(x):
“
x
pass
ed
the
fi
rs
t
e
xam”
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x
)
→ P(x
) )
.
•
The
concl
us
ion:
x
( P(
x)
ר
㻀
B(x
)
)
4
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
4
6
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
4
7
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
4
8
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
4
9
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
5
0
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
5
1
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
5
2
•
Pr
emi
ses:
x(C(x)
ר
㻀
B(x))
and
x(
C(x)
→ P(x) )
.
•
The c
oncl
us
ion:
x( P(x)
ר
㻀
B(x)
)
5
