P RE DI CA TE C A L C U L U S W I T H O U T FRE E V A RI A B L E S
M. W. BUNDER
In th is paper we develop a predicate calculus i n wh ich a ll
we ll formed formulas are closed and wh ich therefore requires
no generalisation ru le . W e s h o w t h a t t h e clo su re o f a n y
standard predicate calculus theorem is provable in this system.
In t h is closed system we cannot obtain predicate calculus
formulas b y stra ig h t substitution in t o p ro p o sitio n a l ca lcu lu s
theorems. T h e a xio ms o f p ro p o sitio n a l ca lcu lu s, th e re fo re ,
have to be changed as follows:
1. 1— (x
i
where
) ( xi,
x x „ include the free variables in A and B.
„ ) :
2.
A (x
1D
where x
)...
i A
(, x
, 3.
x(x 1 ). . . (x „): — A D --B .D .B DA,
„):
„
i
where
x
A
n A,
c B, C, etc in the above stand f o r we ll formed formulas in
i D
l u
, .x o rd in a ry predicate calculus; here we could ca ll them we ll
the
d e
t B
formed
predicate expressions (wfpes).
t In t h e case wh e re t h e wfp e s h a ve n o f re e va ria b le s t h e
, D
habove axioms can reduce to the standard propositional calculus
i C
n
eaxioms.
c :l
f
u Dd
r
e :Also w e re q u ire a p re d ica te ca lcu lu s ve rsio n o f mo d u s
e
t A
ponens
("closed modus ponens"):
e
h D
If k-• ( x
v
eables
1 i n A DB, and 1— ( x i ) . . . ( x
B
a
fj .) . variables
free
..(x
in A, then ( x
rthe free
i
kr) AD
„, ) . w
A variables
h e r e of B.
a Note t h a t t h is ve rsio n o f mo d u s ponens, to g e th e r w i t h
)ex. .D
. .i ( x ,
b
lex A
B ,i
l
)vi BD
w, n wc hl eu r d e e
e
xat C
h k h ,
e
s
xr ,e i r i
i
ia e n
c
l
u
n
b x e
d
A
l i
,
e ,
B
s x
a
i „
n
n i
726
M
.
W. BU N D ER
Axioms 1 a n d 2 a llo ws u s t o p ro ve I— ( x
xp,
p i n c l u d e the free variables of A.
. (x ,
) . If. .then
( x , ) we
A have
D1— ( x i ) . . A
thei free
w
h variables
e
r
eof A, we can b y this modus ponens, conclude
)(A
x r, ) . .w
. ( xh e r e
If
xr,
xs i x, s a re p a rt o f xi, x i , th e n w e h a ve dropped
some
quantifiers. I f xi, x i are part o f xr, x
x) A , vacuous
i
have
s aw generalised
h el
s with
o respect to some variable; this quantification
also
vacuous.
, ir is
ew of
n ecourse
c
l
u
Alte
rn
a
tive
ly
we
could
state the axioms and modus ponens
dx r e
wit, h t h e va ria b le s ra n g in g o n l y o v e r t h e f re e va ria b le s o f
thex wfpes concerned. I n this case Theorem 1 b e lo w can o n ly
besp ro ve d f o r fo rmu la s co n ta in in g n o va cu o u s q u a n tifie rs.
The
i renst o f t h is w o r k ca n e a sily b e a lte re d t o sa t isf y t h is
alternative.
c
l
No
u w we state the axioms p ro p e r to th e predicate calculus.
There
d is one mo re than the usual two, th is a llo ws us to prove
theorems
o th e rwise proved using th e generalisation ru le .
e
t
4.
h 1— (xi)... (x„) .(x)A D A*,
e
where
x
ithef re su lt o f replacing a ll fre e occurrences o f x i n A b y a
, rx having some o r a ll o f x
term
however
mu st n o t be i n th e scope o f one o f th e quantifiers
in e
xex ( ix ,
,(xi),
a
s
ifn vnr e
e
ala1— r ( xi a b l e
vc) .5.
su nr ,d
where
xi, x n , x include the free variables o f A DB, x being
e
xnot)i free
:
in A.
ti (a x ) . A n
b
h D
A
6.
l (x
e B
i
fwhere
:e
xi, x , , , x , y include the free variables o f A .
)s . . .
r D
x
e :(o
We
„
) .ca ll th e predicate calculus based o n these a xio ms a n d
f
eclosed
A
(A x modus ponens, " clo se d predicate ca lcu lu s" . W e n o w
vprove:
D
a ().
r x(
i )y
)
a B
b ,A
l D
e (
s y
i )
PREDICATE CA L CUL US W I T H O U T FREE V A RI A B LE S
7 2 7
Theorem 1 : I f 1 -D i s a th e o re m o f f irs t o rd e r predicate
calculus, wh e re
y „
a re th e fre e va ria b le s o f D, th e n
(Y
1
is a theorem of closed predicate calculus.
)•
Proof: Consider o rd in a ry •first o rd e r predicate calculus based
• a xio ms simila r t o A xio ms I , 2
on th e propositional calculus
(
and 3, modus ponens, generalisation,
Y
(a) I - (x,) A (x
m
i
where A(xi) is a we ll formed) formula (wf) and t is a te rm of the
) D
A
D
system free for xi in A(xi), and
( t ) ,
(b) 1 - (xi) (A D B) D (A D (x,)B)
where A is a w f of the system, containing no free occurrences
of x
i We then p e rfo rm the p ro o f o f the theorem b y induction on
.the p ro o f o f 1- D. Consider f irst th e steps o f the p ro o f wh ich
are instances o f the axioms o f first o rd e r predicate calculus.
If th e a xio m i s a p ro p o sitio n a l ca lcu lu s a xio m, t h e n a n y
instance of it is given by A xio m I, 2 or 3.
If the a xio m is (a), th e closed ve rsio n is o u r A x io m 4. I f it
is (b), we have A xio m 5.
In the in d u ctive step assume th a t a ll the steps in the p ro o f
to a certain p o in t have corresponding statements provable in
the closed predicate calculus.
If the n e xt step is made using t wo e a rlie r statements a n d
modus ponens, i.e. A DB and A then B, then b y o u r inductive
hypothesis w e h a v e a s th e o re ms o f t h e clo se d p re d ica te
calculus:
(x
1 xi, x
where
).
r
, a(. x. rk ) e. . . ( x
( x xe
t h1
where
,
kf Then
r )n. Aeby
closed modus ponens we have:
)
.
,e x
iv aA r
i ar. b
a
l eB s
e
,
to
fh
A
e
fB
ra
728
M
.
W. BU N D ER
(xp)... (x0B,
where xp, x „ are the free variables in B.
This is the closure of 1—B.
If t h e n e xt ste p i s ma d e b y generalising
then by the inductive step we have:
A
t o 1— (x)A,
i-where xi, x p are the free variables in A.
If x is one of these, say x
and
k closed modus ponens we obtain:
, t h e n
b
y
(x
r e p e a t e d
i
which
u
is what
s
is erequired.
)
o x is not fone of xi, x „ , we can of course generalise using
If
(
closed
A
modus
x
ponens
i
and sh if t x t o th e f ro n t as before.
x
o
Thus
the
m theorem is proved in all cases.
k
6Closed predicate ca lcu lu s ca n n o w b e a p p lie d wh e n e ve r
_
ordinary first order predicate calculus could be applied. Where,
1
such as i n th e p ro o f o f Giidels Theorem (see [1]), w e use a
we ll )fo rme d fo rmu la containing a fre e va ria b le , w e mu st o f
( use a we ll fo rme d predicate expression.
course
x
Wollongong,
l
N.S.W. Au stra lia
M
.
W
.
BUNDER
"
1
BI BL I OGR APH Y
)
.
[1] MENDELSON
- I nt roduc t ion t o Mat hemat ic al Logic . N e w Y o r k 1964.
.
.
(
x
„
)
(
x
)
A
,