A SIMPLE NATURAL DEDUCTION SYSTEM
R. ROUTLEY
A natural deduction system N D is introduced and proofs in
it illustrated. This system is much simpler and more flexible
than familiar natural deduction systems: it avoids completely such
complicating devices as subscripting, flagging, ordering of variables, and distinctions of several sorts of variables or parameters.
Furthermore proofs are valid line by line.
ND is presented as a formal system. It still needs emphasizing
that natural deduction systems can be made just as formal as
Hilbert-type systems. The precise conditions on substitution —
quite important for natural deduction but often suppressed —
are made explicit in the rules of ND.
The correctness of N D is shown by comparing N D with a
Hilbert-type system H. H has as formal theorems effectively the
theorems of restricted predicate logic and of Hilbert's &system.
Both N D and H have as primitive symbol the term-forming indefinite description operator 'C'. ' E ' i s for mally s imilar t o
Hilbert's &symbol, (9 but it is intended to have a quite different
interpretation, 'Ext(x)' is read 'any x which is f', not 'an x which
is f provided there exists an x which is f'. An item x may not
exist or even be possible. Likewise the quantifiers ' A' and 'S',
though formally similar to 'Y' and 'a', have different intended
interpretations. T h e non-ontological quantifier ' S ' reads ' f o r
some', and not 'there exists a'.
A well-formed formula A of ND or of H is (-free if no E-term
does occur in A. I t is shown that the E-free theorems o f N D
coincide formally with the theorems of restricted predicate logic.
In the final section ND and a simpler system N D
gated
1 independently
a r e i n v of
e classical
s t i - predicate logics. Interpretations
(
1
thematik,
v ol. 11, Berlin (1939), §1.
)
F
o
r
d
e
t
a
i
l
s
130
R
.
RO UTLE Y
and validity are defined for these systems; consistency, completeness and Skolem-Leiwenheim theorems are easily reached.
The primitive frame of N D
Vocabulary • - • - •
x y
f g
A S e ( )
z x ' y' z '
h f g' h '
r p'
Formation rules
I. A sentential variable p is a wff.
2. I f / is an n-ary predicate and z
i zables
„ or awf rterms,
e then f(z
3.
n I if zA is a wff, —A is a wff.
i nI n
i v Bi are
d wff,
u athen
l (A D B) is a wff.
4.
f dA and
v5. I )f aA isi ra sw f fi and- xa an individual variable, then ExA is a
w term.
wf
f f .
6. I f A is a wff and x an individual variable, then (Ax)A and
(Sx)A are wff.
Transformation rules
1. Hypothesis introduction:
A
IA
,l
last hypothesis of A .
D
A
A
3. A-introduction:
(Ax) B 22
2. Hypothesis elimination:
(
A
A
)
—(A
/
),
w
h e
provided variable x is not free in anr hypothesis
of B.
e
A
I
i
s
t
h
e
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
4. A-elimination:
1
3
1
(Ax) B
Sx B 1
where z an individual variable or wf term.
5. S-introduction:
Sx BI
(Sx) B
where z an individual variable or wf term.
6. S-elimination:
(Sx) B
B, (e,B) •
A
7 Sentential derivation:
I
,
provided A
A logic.
tof aD tautology of sentential
2
(By 2
admitting the case where
n = 0 some trivial steps could
A
be
removed
from
later
derivations.)
D
D
ND has no axioms.
A
A deduction in N D is a finite sequence of wff each of which
„
is derived
from preceding w f f in the sequence (including null
wff)B according to the transformation rules of ND. A deduction
is ai completed deduction if all hypotheses (introduced by hypothesis
s introduction) have been eliminated (by hypothesis elimination)
a by the end of the sequence.
sA w
u ff bA is a theorem of N D if f A is the last member of a
s t i deduction.
completed
t u t
The
notation is explained as follows:
i osubstitution
n
1.S- A I is the w ff resulting from substituting variable, wf term
i ffn N sfor all free occurrences of variable u in A, provided
or w
a free
n occurrence o f u in A is i n a w f part (Aw)C or
thatt no
(Sw)C
c eor cwC of A, where w is a free variable of N (or is N
if N is a variable); and otherwise is A.
2. S'uA I is the w f f resulting fr om substituting variable y for
all (free) occurrences of variable u in w ff A, provided y is free
in the resulting w ff where and only where u is free in A; and
otherwise is A.
132
R
.
3. Bx (cyC) = Sx
B
cyS';',C
RO UTL E Y
I
4. A is a substitution-instance of w f f B o f sentential logic if f
A results fr om B by simultaneous substitution o f c-variants
Clx, C
B,
1 is an (-variant of Sx A F o r example,
u where SxA
cyS'zC1
c
z
C
x
f(cyg(y))
f( c z g( z ) ) is a substitution-instance of p p .
o f
w
Sample theorems of ND
f f
C
1. ( Sy ) ( Ax ) f ( x , y ) D (Ax)(Sy)f(x,y)
I
Proof:
(Sy)(Ax)f(x,y)
((Sy)(A
,
(Ax)1(x,Ey(Ax)1(x,y))
x
C
f(x,EY(Ax)f(x,y))
)f(x,y))
(Sy)f(x,Y)
i
(Ax)(Sy)f(x,y)
,
(Sy)(Ax)f(x,y)D (AX)(SY)f(X,Y)
—((Sy)(Ax)f(x,y))
f
o
2. ( A x ) g ( x ) D (Sx)g(x)
rProof: (Ax)g(x)
((Ax)g(x))
d
g(x)
i
(Sx)g(x)
s
(Ax)g(x)D (SX),g(X)
—((Ax)g(x)).
t
i Proofs can be presented more perspicuously by using a lining
ntechnique to show the extent of hypotheses. I n place of exprescsions written to the right of a (vertical) proof in hypothesis inttroduction and elimination, a line r ' w ill be introduced, when
van hypothesis A is introduced, to the left of the proof but to the
aright o f a ll lines already appearing t o the lef t o f the pr oof
rsequence. This line w ill be continued down the proof until the
ilast member of the sequence before hypothesis A is eliminated
at which stage the line w ill terminate. Because of the condition
a
on hypothesis elimination, lines w i l l never cross. T he lining
b
technique has a similar status to familiar conventions for simplilfying primitiv e bracketing notation. Derivations can als o b e
eclarified by wr iting in the margin t o the left o f a sequence
s
p
i
,
p
n
i
n
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
3
3
element indications as to how that element is obtained. These
techniques will be illustrated in succeeding proofs.
3. ( S x ) ( p v f ( x ) ) D
. p
y
Proof: ( S x ) fPvf(EY(Pvt(Y)))
( x )
S - e l i m i n a t i o n
( Sx )
( p v f rf(ey(pvf(y))) H - i n t r o d u c t i o n
(Sx)/(x) S - i n t r o d u c t i o n
( x ) )
AcY(Pvi(y))) D (Sx)f(x) H - e l i m i n a t i o n
H
pv(Sx)f(x) S - d e r i v a t i o n
(Sx)(pvf(x)) D
i
n
. p V
t
r
4. ~ (Ax)—
( S40)
( S X f(x)-=-.
) /
o
( X )
dProof: r f( x )
H l(Sx)f(x)
S - i n t r o d u c t i o n
u
e
l
i
c
t f(x)D (SX)/(X)
m(SX)/(X)
i n / C O S - d e r i v a t i o n
i
o
a t i o
n
—
nf(x) S - d e r i v a t i o n
r ( Ax)— f(x) A - i n t r o d u c t i o n
(—
S (Sx )0) D (Ax)—
x ~(
) /Ax ) ~ f(x) D ( S 4 0 ) S - d e r i v a t i o n
-( x- )
((Sx)/(x)
S AcYt(Y)) S - e l i m i n a t i o n
x (Ax)— f(x) S - d e r i v a t i o n
) / ~ f(cYl(y)) A - e l i m i n a t i o n
( f(eYl(y)) & A c y f(y)) S - d e r i v a t i o n
x( S x ) f ( x ) & ( A x ) — D . f (E Y / (Y )) & •
)f• ' / ( E )
S-derivation
(
&)C
rt(cYt(y))
(/Y( Y ) )
If(EYt(y))
S-derivation
A/f(EY1(y)) D f(cyt(y))
(
x(SX)f(X)
S-derivation
(Ax )
Y
),) (Ax)— f(x) =-• ( Sx ) 0 )
S-derivation
)
—
— f ( x )
ff
(
(C
xY
)1
(
y
)
)
D
134
R
.
RO UTL E Y
These proofs illustrate several of the techniques used in N D
proofs. I n practice the proofs would be written in a much abbreviated form.
Two derived rules of ND.
I. A quantifier conversion rule:
—(Ax)— B
(Sx)B
The proof directly generalises upon 4. above.
S'x BI
2. Extended A-introduction:
(Ax) B
provided variable y is not free in an hypothesis of the premiss.
Proof: Case 1 S'NB I is B. Then the result is immediate from
A-introduction.
Case 2. S'xI3 i s not B.
(Ay)S'B
S'Y S'xy B II A - e l i m i n a t i o n
S'Y Six B i s B
x
(Ax)B
A
y
introduction;
in this case x is
not free in the
hypothesis
(Ay)S' B D (Ax)B
Now (1) B
(Ay)S' B I ,
provided y is not
free in an hypothesis
of ( I )
(Ax)B
S-derivation
A SIMPLE NATURAL DEDUCTION SYSTEM
1 3 5
Pathological examples whic h illustrate typical violations o f
flagging, ordering and related restrictions i n familiar natural
deduction systems.
1. (Sx )g(x ) D (Ax)g(x)
Attempted proof: (Sx )y (x )
g(cyg(y))
[ (Ax)g(x)
This last step is illegal: no rule permits generalisation on a w f
term.
2. f ( y ) ( A X ) / ( X )
Attempted proof: r f( y )
1(Ax)f(x)
This step is inv alid; f o r generalisation has been made o n a
variable free in the hypothesis.
3. ( Sx ) /( x ) & (Sx)g(x) D (Sx)(f(x) & ex)).
Attempted proof:
But no rule permits S-introduction on different E-terms.
4. (Ax)(Sy)f(x,y) D (Sy)(Ax)f(x,y).
Attempted proof:
(Ax)(Sy)/(x,y)
(Sy)f(x,y)
f(x,eyt(x,y))
(Ax)f(x, EYffx,Y))
(Sy)(Ax)/(x,y)
136
R
.
RO UTL E Y
This step is invalid: the introduction of 'S' here does not accord w i t h t h e S-introduction r ule. (Ax )f(x , tyl(x, y)) is n o t
Sv ( A x ) f ( x , y) I, since x would be bound under the subcy/(x, y)
stitution. This kind of failure can be foreseen: it is preceded by
the binding up of a variable free in an &term, in particular by
A-introduction. Note that:
(Ax)(Sy)/(x,y) D (Ax)f(x,cyl(x,y)) is an c-theorem.
In 1955 Wang ( ) suggested combining use of Hilbert's c-symbol with the method of natural deduction as developed by Quine.
But the system L g whic h Wang proposed is inadequate: f o r
either it permits the derivation of pathological examples lik e 2.
and 4., or, interpreted differently as suggested by the assertion
signs, it sanctions very few theorems.
11
The system H . The vocabulary and formation rules o f H are
like those of ND except that quantifiers 'A' and 'S' are omitted
throughout.
Axiom schemata
HI: A , provided A is a substitution instance of a tautology of
sentential logic.
H2: S x B I Bx ( c y B) ,
z an indiv idual variable or w f
term (weak e-scheme).
Transformation rules
RH1: A , A B ---> B. ( m o d u s ponens)
RH2: B
- - > Bxi(cy —B). (generalisation)
HI c o u l d be extended to include RH2.
(
Peking
and North-Holland Publishing Company- Amsterdam (1963), 3152
316.
A
related idea was introduced by Hailperin: see T. HAILPERIN, ' A
)
Theory
of Restricted Quantification T h e Journal of Symbolic Logic,
S
Volume
22 (1957), 118-126.
e
e
H
a
o
W
A
N
G
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
3
7
Definitions
1. ( Sx ) B = it Bx(EyB).
2. ( A x ) B =1)f Bx(Ey —13).
Theorem 1: Every theorem of H is a theorem of ND.
Proof: T he axioms, rules and definitional equivalences o f H
are derived for ND.
HI
[B
BB B
A
,
provided A i s a substitution instance o f a
tautology of sentential logic: by S-derivation.
H2 S x B I
(Sx)B S - i n t r o d u c t i o n
1Bx(EyB) S - e l i m i n a t i o n
SxzB I D Bx(EyB).
RHI. I f A and A D B are theorems of ND then B is a theorem
of ND. For the completed deductions of A and A D B can be
combined. Then, using S-derivation, B can be obtained as the
conclusion of a completed deduction.
RH2, I f B is a theorem of N D then B is the conclusion of a
completed deduction. T o this deduction the following elements
are added:
(2)
(Ax)B A - i n t r o d u c t i o n , since B has no
hypotheses.
B x ( c y —B) A - e l i m i n a t i o n .
Since ( 2) is the conclusion o f a completed deduction, ( 2) is
also a theorem of N D .
Definition I . r(Sx)B
I Bx(cyB)
(Sx)B D Bx(EyB)
rBx(EyB)
(Sx)B
S-introduction
138
R
.
ROUTLEY
Bx(EyB) D (Sx)B
(Sx)B B x ( E y B )
Definition 2. r(Ax )B
113x(Ey —B)
(AX)B D Bx(Ey—B)
1 Sx)—B
. —B D (SX) B
DB
(
— (Sx)—B
B
r(Ax)B
(Sx) —B D (Ax)B
S-derivation
A-elimination
—(Ax)B D (SX) B
—(Ax)B
(Sx)— B
—Bx(Ey—B)
(Ax)B B x (ey B )
Bx(ey — B) D (Ax)B
(Ax)B B x (Ey — B)
Thus every proof in H can be replaced by a completed deduction in ND.
An E-free formula (sequence) A o f H is a relettered w l f
(theorem, proof) of Church's FIP i f A is obtained from a w f f
(theorem, proof) A ' of F
its
1 corresponding symbol in the vocabulary of H, e.g. b y 'f',
'G'
' bbyy'g', 'V' by 'A' (
3
r e p l a c i n g
Theorem
)e.
a 2:c I f Ahis a relettered wff of Church's F r , then A is a
wff of H.
s
y
m
b
o
Proof:
l By induction over the formation rules of FIP and H once
i correlation
the
n between symbols is set up explicitly.
A
'
Theorem
3: Every relettered theorem o f Church's F IP i s a
b
theorem of H.
y
Proof: T he propositional postulates of F
H
1 1:the rules of FIP are immediate. I t remains to establish relettered
schemata o f F Ir :
P f o quantificational
l l o w
d i r e c t l y
f ( r
o
m
Logic,
3 vol. I , Princeton (1956), 169-173.
)
F
o
r
d
e
t
a
i
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
3
9
(1) ( A X ) A D Sʻ A , z a variable or w f term;
(2) ( A x ) A D B) D . A D (Ax) B, x not free in A ;
and the scheme
(3) ( Sx ) A = ( A x ) — A.
ad (1) S x — A A x ( e y —A).
H2.
Result by contraposition and definition 2.
ad (2) S
Ey—B
N
x not free in A.
H
2
.
—
(*** —
A (A D Bx(Ey — B)) D
in
. ( A.
A
D
D
E y
:BB
1R Ax D (Bx(Ey—B(A
D B)) D . A D Bx(Ey —B), x not free in A.
)— (Ax)(A D B) D ( A x ) B , x not free in A. Definition 2.
A
I(
D (3) 1 . — (Ax)— B — Bx (Ey — B) D e f i n i t i o n 2 .
ad
D
D e f i n i t i o n
) 2. ) ( )S x ) — B
.B
3.
Bx
(
Ey
—
B)
B
x
(
E
y
—
B
) H 2 .
,
—
4. — Bx(EyB) D
(X
.o —
5.
B 1 3x x ( E y — B ) =.Bx(EyB) 3 , 4.
An
6.
(Sx)—
5 ,
Definition 1.
( e y —B ( S x ) B
t
D
7.
(Sx)B
=
—(Ax)—B
2
,
6.
)
rB
Bf
2
eH
xe
Theorem 4:
. Every relettered theorem o f Church's F IP i s a
(theorem of ND. By theorems I and 3.
E
yA deduction theorem for H
A finite sequence o f w f f 13
—
A,, A „
1
(hypotheses
2
) , B, is
1 one
3 „of , A
A, 1 1 (1)
.D
o. .orA „lf i n
(2)
, AB , . . A „ ,
BH
or
fis
)i
f)a o(3)
i s Br
B
and
B
aiar minor
co premiss
p
o
)e
or
kf
in s
,h
(4)
w
h
e
if
iar n
re B j <o i ,
i
ke
m <ifx e i t ,
h
eiri sr r
ie
on
fd
me
rb
, r
e
y d
b
m
140
R
.
RO UTL E Y
or
(5) B
or i
(6)
is B
or i
n
(7)
if se B, is inferred by S-elimination from B
j < i .
'A p
ir nr w h e r e
potheses
A
l
fe ed
i. .
rb r
.Theorem
..Ae
. 5 (Deduction theorem): I f A
y„ ' d
An-1,i—
A
lA
b
S
n
,, , AB 1y.
Proof:- Let B
-1
A
A
i
1
3
„
,
t
h
e
n
l,— B B„,.
2-n , It is shown how to insert a finite number of additional
A
wff
B
, into
, it this sequence so that the resulting sequence is a proof
I/ A„ nD B from hypotheses A
of
„
r
exhaustive:
l,,4
toe
b
,Cases
A
. .lb , 2, T3: ash in Church
e
(
'
rdla,
„a
tions.
f4
l
l
o
w
i
n
g
s.p o ro
u
cCase
)C
pd
y ao
4:
is (Ax)13. Insert before A
o
ca aB,r isst Sx fBe ar n doB, m
sm
i
n
o
r
n
D
u
B
o
f
t i(Ax )BDSx B 1 a n d i t s consequence A „ D (Ax)B D
theorem
a
otco hdr ei f e i c a
i.m
b
n
o
A
-o
tn i
s
nf
A„
l Do
tB
fB, then results by RH1.
D
iCase
n
5.
(r
r B, is (Sx)B and 13
S
s i s fo
1
u
=
A„
x D B, the theorem Sxl3 D(Sx)13, an immediate consequence
e
rm
S
cB I B
,
B
of
H2,
and its consequence A„ DS
ft,s o o
1 r
xsresults
't„
RH1.e
om
3by m
B
D
1
B
h
. is (. Ax A
)t
Case
6:
i B,
)B
D
(
S
) I 3 .
h
3
Iie
n, A „ sD B,et hxe rtheorem
before
f
( AX) ( A
A
t(Ax)(A,
n
,fre wD iw
h eD
B r13,)
e whic h results b y RH2. A
n
rxo RH1.
n
D,he B f
oD
ibby
B
,
e stw
)rin
m
ee
D h .e n
A
f„in
o
h
l
l
o
w
s
h
ory
t D
1 3
(
s
eo e
1ftp
r
4
,a
t ) hrj
e
a
n
d
pA e
ise
<s n
rA
e . jis
oC <
lq
,
H i
o
,u
U
f
,n
Ie
R
o
sn C
e
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
4
1
Case 7: B, is 13x(EyB) and 13, is (Sx)13. Insert B, B , ( fr om
definition 1) and A „ D 13, D D B, before A
apply
R H I.
t
, D
B , .
T h e n
Theorem 6: I f every w f f in the set A
in
I the set CI, C2, C
Co—B.
r, Aa nT he
d proof is an adaption o f Church's *362 (
generalisations
upon individual variables need however be con5
i2 f
A
sidered.
),I . OA n l „ y
a l s
o
,o c A c . u r
s
7: . A .is a theorem of H iff there is a proof of A from
3Theorem
,
null hypotheses in H.
. A „ 1 —
B , I f A is a theorem, then the proof of A provides a proof
Proof:
t A from
of
h null hypotheses. Conversely if there is a proof of A
from
null
e
n hypotheses, then the proof of the deduction theorem
specifies
effective methods f or converting this proof sequence
C
in
a
proof
of A. For each derived rule, used in cases (4)-(7), is
I
there
vindicated
by derivation from rules and theorems o f H .
,
The
result
then
follows
using the definition of 'theorem of H'.
C
2
Theorem 8: Every theorem o f N D is a theorem of H.
,
Proof: I t is shown that a completed deduction with A as last
member in ND can be effectively replaced by a proof of A in H.
The argument is by induction over the length of deductions in
ND. The theorem then follows by generalisation.
Suppose, for induction hypothesis, that a deduction B
B,"k1 in ND of 131
1 3by
111,f
r othe
m set of proofs from hypotheses:
for, H
2
h y p o t h
A
e ,s e s
n
A
i
(12k),
A
I
B
,where the setno f hypotheses A
km
A
2 or improperly)
I(properly
in the set of hypotheses of ( 1 , ,
,
,,I A 2 ,
A n ,
A
o) ( f of r
"( 5 l
i
)
ki )
s
A
ci . n
c
l
u
d
a
e C d
n H
b U
R
e C
r H e
142
R
.
R O U T LEY
each i, and B
1111 ,
B
B On this induction hypothesis i t is shown that a deduction
m
l k in ND to 1 3
ku
sequence
b
.( 1 1e1 l
to B
o
,m k n g
t1hypotheses
,
t h e
c)
o
sa u c c e s s B
r
to r ,
n
a
h
ib
An
I
1
e
te
h,
where
A
,
p
r
e
rIor (ii), Aujust in the case that an hypothesis is eliminated in the
(
o
o
d
e k d
u
e
,fNDkproof
sequence,
I by the set of proofs from hypotheses
cp
t j i
o
ks
e
n
A„ B I
la n d q
uB
o
m i )
a
B
(
e
n
fc
m
( j B
m
cA
e1
I (1
)
ie
B
where
B
k
I 1
d
,1
,
m
+
) ,
fis introduced
P
in the
N D sequence before A
o,o k
+ case that a new hypothesis is introduced in
t I k( iii) , just in the
or
ir s
ithe nN D proof sequence,
,
by the set of proofs from hypotheses:
A
H
k
(
A
ne
_
1
I
ki
I
i
,A
D
ta
)
A
B
l
h
r
,
k
u l „,
k
e
B
1
kr
A
(
a complication
The
can be attributed t o the occurrence o f sub
m
1
a(
„
+
s
proofs.
There
are
three cases to consider:1
-B
ni
)
b
k
d) 1 3 -r
f
(a)
e
•
B
w
a derivation,
1
0sentential
b
A-elimination, S-introduction, A-introducf
3
n
7
(
,ho
„
ytion or S-elimination.
Then case (i) obtains, and ( 1
r
,
e
a
1from
tr1 ) k ( 1k
(
1
.
1
irke
h )s i s
o 4.b t 1a i n e d
,
e
1
j)e
)
: o
)
(
a
n
d
k
iA
n
s
—
n
e/ od r 1
(1
1
ti
k
)
so
_
ttf
1
h
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
4
3
in a proof of B ,
1
of1theorem 6. For example, if 13
,and
1 B
1
) L
(,(Ik
m
f+
iization
ir s o (mA x ) from
C hypotheses and theorem 6. Note that the
lh y pof oa proof
t
aconditions
n
d
B , „ , ,
1
) ) e s eons A-introduction are guaranteed by the N D proof
h
) k
ilsequence.
isii s
s
n
(b)
C b1 3
o
o
(H
tAb in
a N D by elimination of hypothesis A
„a
„ n,
intains,
t k . and (1;) is derived fr om ( 1
kn
e
.for
n (ii) exhausted
c a s proofs
e from hypotheses are discardcase
ka sdTH. hIn e
id
f(ed.
i
i
)
)i b y
t h e
b
A
rod
b u c- t i o n
n
e
d
yo (c) B , , ,
u
te hp ep o
r
e
m
ka
+ 1 ) 1 , introduction. Case (iii) obtains. Since ( l
hypothesis
m
d
lBfrom
c
D
for H, it can be added to the set of proofs from
k i , ishypotheses
fB
a
t
i
hypotheses.
j
o
i
,ir
care
basis.
,o
es also takes
) n, i n(a)
a
p ofr the
o induction
o f
„Case
o
ki d
,nm tIt follows by induction that every stage in the proof sequence
(of A in ND can be effectively replaced for H by a set of proofs
a
o
N
1
from
n
t hypotheses. Therefore the last stage, where A is obtained
D
)
as
the conclusion o f a completed deduction, can be effectively
d
bb h
for H by a proof of A from zero hypotheses. For an
iyreplaced
e
y
hypothesis
is introduced and eliminated in the representing sets
sA p
c
of
proofs
fr
om hypotheses f or H , when and only when an
j- r
hypothesis
is
introduced or eliminated in the ND proof sequena
o
oBut, by the proof o f theorem 7, a proof o f A fr om zero
ece.
ils o
hypotheses
effectively yields a proof of A in H.
e
n
i( f
e
s
m
5
Theorem
9: N D and H are deductively equivalent, i.e. their
d
e
itheorem
sets coincide. By theorem 1 and 7.
)
tn q
g
o
ai uBy eliminating quantifiers as in H, the primitive rule structure
ttof eND could be much reduced. Consider the natural deduction
v
h
ie n
e
on c
d
nu e
e
,n o
d
td f
u
he A
ce i
tnr n
144
R
.
RO UTL E Y
system N D
as
I H and the following rules: Hypothesis introduction and
elimination,
sentential derivation, and
w h i c
h
3'.
p
r
o
v
i
d
e
d
x is not free in an
h a
Bx(cy B ) h y p o t h e s i s of B.
s
B
t
h
6'.
e
Bx(cyB)
sRules 3' and 6' can be combined in one artificial rule scheme.
a
'Deduction', 'completed deduction' and 'theorem' are defined
m
as for ND.
e
pTheorem
r
10: N D
i1 m
iTheorem
, tN iD11 (
vis
5
a a neproof
d o f B fr om hypotheses A
is
a
relettered
(c-free) proof of B from hypotheses A
aI)H: I f
in
F
P.
n
nl,B
a Ar „ e i
,dH
A
2
,
A
„
a
en ,Let
dd B
u be c-free and let the proof o f B from c-free hyProof:
h
e
dtA
c t i Av e n
potheses
e Br
e
ettl y h
fwhere
,e A
B
1
q
u
iplacements:A v 2a
n
l,
n,(i)
=
B &ter m introduced b y S-elimination systemce Replace
on
t each
eB
A
.atically,
n
s. i in an 2order determined by the first appearance of the
dI„sc-term
by the first variable alphabetically
,
t in the proof-sequence,
variable already occurring in the proof sequence
sn
alater than any B
o
ytrf(a).hTerms c x iA and cyB are replaced b y the same variable
iff A is S
m
ite s
n
I
bp
ch(ii) Do- the same
, for remaining &terms introduced by A-elimiY
ore
fnation.r Then a( new sequence
B
.
lo
e
a13
s
ee
so
ais obtained.
)1
q
u
fn
e
n
',
scd ( e
see6HiLBERT and B
BERNAYS, op. cit., 18.
t
e
'
)
h T
q
2
e h
u
,
r i
e
B
e s
n
,
c t
„
e h
'
e
mo
(
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
4
5
With the exception of finitely many, say k , applications o f
(Sy)C,
S-elimination of the form
( 1
i k),
C
the sequence ((3), after supplementation
b y finitely many ini
sertions, satisfies Church's requirements on a proof from hypotheses in PP. First, occurrences of cases (1), (2), (3) and (6)
under a proof from hypotheses in H in the sequence (a) are
guaranteed in the sequence (13) by cases (1), (2) and (7), (3) and
(4), respectively, of a proof from hypotheses in PIP. The substitutions made i n going fr om ( a) t o (13) provide n o obstacle,
because o f the choice of new substituted variables. I t remains
to consider cases ( 4) and (5) under a proof from hypotheses
in H .
ad (4): Suppose 13, in sequence (a) is obtained from B, (where
i < j ) by A-elimination; s o B, and B
i a ( rAxe) B respectively.
o f
t T h e n eB , ' i s s ubs t B , wher e
and
fthe changes
o
r made
m
under Subst. i n B are those, specified i n
S
x above, and where u i s z if z is a variable and an
(i) and ( ii)
B
appropriately chosen new variable i f z is a term. Thus I V is
1 Subst B A l s o 13
1
sequently B
' is
i
before
B
S u b s t
quirements
on a proof from hypotheses in F " . Then B,' results
i'
( A x )
from
B,'
in
tf o l l o the supplemented sequence (13) by proof steps perB ,
mitted
iwns t oin a proof from hypotheses in FP.
i
.
e
(fad (5):
0r Suppose B
.
)o
m
iThe( case
Ais similar to (4) except that further insertions must
B
be
made
in
' i xn
() ((3),
a ) namely a proof of Sx ( S x ) B 1 before Bj'.
i,i s '
S
u
Consequently
'b
d be r si v
ye d A
D
t
A
B
f B- r I o
holds f o r F'P, where a pr oof o f B fr om these hypotheses is
,e
m . l i,
provided by the supplemented sequence (13), and where w
'm
B Ci An
alphabetically later than any in (a), for each
i i ti's" variable
athe
„i
i, —o(1<i e x c e p t variables fr ee i n hypothesis SY C, 1
so
b nn ,.
N
p
y - S
o
e
S - Y
rw
i n C
t r o
id u , c t
m
in
i ot In .
ts
,
e
C
IV
146
R
.
RO UTL E Y
for some L Suppose there are m cases o f generalisation upon
variables free in hypotheses o f the proof sequence, where the
i
t D,
a n d z , i s n o t fr eei n A
h( Az
lA „ .
m
) Then the supplemented sequence gi) guarantees for FlP the
)following
D
proof of B from hyoptheses:
ii
A
sSI,
3 1
o
A
1
fn Now each o f the hypotheses SYk ck I
C
I
,tS
Y,
Y
h
( C I I of (a) can be eliminated in turn. Suppose for induction
S
k
e
A
C
and step the elimination has been completed up to SYi 1
kbasis
fz
B
i
o
.r
C
(r
n
ia
m
F1
))
+It follows by the deduction theorem for 1
:D
A
%
1
l
i +1
„I
S
Y
,, SinceY w
1
i
(s i + 1C
i
A
C
o ,
S
Izt b y Y
generalisation
(
i
,1
h7 i t +
SY
)a s
S
1
+
i
))•
C
D
Y
tSince,
c hfurther,
i
Cw
i1
A*364,
i o i +1
i
,I+ c h e a
C
I
s ( , SY
w.
w
i,n , o D
Ai i
B
1
Af d r . e ewC + 1
(
o
„since
oo
u if,or some j, may contain w
7 c( A c
z ) D,
f,permissible
since
hypotheses
ir )e
f r r ee e n
. +c1 H
o w e ( Av z e r
t(by
the
n
tei T
s
s
hdeduction
e 11theorem. For the more general deduction theorem used,
see S. C. KLEENE, Introduction t o Metamathematics, Ne w York (1952) ,
h
)si Dh
( e A zp )
A
n, , , t,
)*
p. 97.
izDn io
s
(1 r
a
e
s,IB st
StI
n s
o
p
ne, o
i
m YDi
n
a
t
t l
r)eb c d
e
i
B
o
by pc
C
o
nr iu e
l i.
fie sr t
t I
f.e ni r
e D
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
4
7
As w
by
1, relettered *378 and by substitutivity of equivalents,
1
A
d
I
Since,
however,
( Sy
o
,
earlier
stage
in
the
deduction sequence ([3), it follows by *362,
1e
C
„s
A
i
1n
IC
Thus,
by
)o C , ,, , modus ponens
SYi C , I B ,
1t
hS
, a s
ao l rY
e a
and by 1iteration
dc y ,D
A
bc
eC
e
B
I
nu thei. proof fr om hypotheses (b) at the same time supplies
But
,
premises
(Az
d
e( (Az„,)D„„
r
r
A
available
D„,, D
iready
v
S
e
f
are
id
)r D not ynfree in hypotheses A
,
which
guarantees
(6) at the same time guarantees
t,1
ae
,
(
,rt e A
sb p.A.e
y c t i v e
e
A
T
lg
y ,h t ue satisfies
r s a l i the
z requirements of a proof of B from hya
Finally,
i e n,z(E)
tsa
h
e
i
n
c
o
, An
n
1
potheses
n t
p
r r))( o for
f
e
fIrequirements
o o a proof
from hypotheses for F'P.
C
Dr q i ua e
svm ae C
E
, A ,
n
c he s I f A is an E-free theorem of H then A is a reletb
tTheorem
n+ l ,), e12:
ztered
e
•
of F P.
f1 o theorem
, „l
„a
1
r,
Proof:
I.(f A is an E-free theorem of H then, by theorem 7, there
z-F
b
A of A from null hypotheses. Therefore, by theorem 11,
is
tP a proof
,iszan E-free proof o f A i n F P fr om null hypotheses.
y
there
sTherefore
i i there is a proof, necessarily E-free of relettered A in
c
nh so
c )A is a relettered theorem of FIP.
F"';
eo
D
i
1 13: A is an E-free theorem of N D and of N D
Theorem
tc a relettered
is
1 i f f 1 A theorem of PI'.
se
— theorems 12, 10 and 4.
Proof from
po
B
rf
.
ow
(
o,
6
f,
)
si
e,
qi
ut
148
R
.
RO UTL E Y
Theorem 14: N D , N D
1
a nAsd F'P isHconsistent, relettered F " is consistent. ThereProof:
a
fore,rby etheorems 10 and 13, ND, N D , and H are consistent.
c o n s i s t
e n t .
III
Consistency and completeness of ND are now studied directly. Proofs are truncated by proving results for ND,.
From any w f f of ND, ( or of H ) an associated c-free miff is
obtained by deleting ( or eliminating definitionally) a l l quantifiers and replacing every occurrence of the form czA, where
A is a w f f , by an indiv idual variable. From any w f f an associated w f f o f sentential logic (an aws) is obtained by first
forming the associated c-free w f f and then, in the latter, replacing every w f f part f( z
previously
occurring in the wff, in accordance with the requirei
ment
, z that
„ ) two wbf parts
y f( z
by
ia the same sentential variable iff f and g are the same predicate.
,s ze n t e n t i
Theorem 1: Every theorem of ND, (and of H) has a theorem
n
a l
of
sententialglogic as aws.
)v classical
aa rn di
a
(b y l
e
Theorem 2: N D and ND, are simply and absolutely consistent.
in
o
Proof:
I f yND, is simply inconsistent, then for some wff B both
,t
„B and —
) B are theorems of N D
—B*
1 B* and
aaws
r
e are theorems of classical sentential logic. As
this
logic
is
however
r. T eh e pr e f l o r aesimply consistent ND, is simply consistent.
cb
ey d
The
ND,m
now introduced presupposes some nont h semantics
e o rfor e
ontological set theory.
1
t A semantical
h
e basis
i for rND, (or H) is a triple S = < d , e, c >,
where ( i) the indiv idual domain d is a set (possibly null) of
items, (ii) e is a class of relation-in-extension over d, and ( iii) c
is a choice function over d, such that, for each non-null subset
d' of d, c(d') is an item of d', and such that otherwise c(A) =
c(d), and if d = A , c(d) = A . A basis S is usual if e is the
A S I MP L E NA TURA L DE DUCTI O N SYSTEM
1
4
9
class of all relations over d. The null domain is not excluded:
over this domain f(x), (Sx)f(x) and (Ax)f(x) have truth-value F,
since (Aw)(Agw), while — f(x), (Sx)— f(x) and (Ax)—f(x) have
value T.
An interpretation I of N D
is an
1 assignment
a n d
iI which
t s assigns to each sentential constant one
of a
thep truth-values
p l i c a Tt o
i ro F , t o each indiv idual constant a n
element
of
d,
and
to
each
n-place predicate constant an n-place
n s
relation
ovf ,e. Thus
variables are variables having T
o
e sentential
r
andb F as atheir range;
indiv
idual
variables are variables having
s
i
items of d as their range; and n-place predicate variables are
s
variables having n-place relations of e as their range. For each
S
constant
and variable u, I(u) is the assignment-value of u under
I. 'A has truth-value T ' is abbreviated 'TA'. Truth-value assignments are relative to a given interpretation. Now, given I, truthvalues and further assignment-values are defined inductively,
relative to given assignments to free variables, as follows:
(i)
A
w f f consisting of a sentential variable p alone has
value T if f 1(p) is T ;
(ii) A wff f(s
si
i, 1s( 0 , 1( s
<
I(sn));
,n
2
)s, II (( csx A ) c ( f 1 ( x ) : T AI ) ;
(iii)
„c )o >Tn —
(iv)
s iA s — T A ;
ca
(v)
T
(
— TA v TB.
t i n g A B )
in
. f
o
A
w
ff
is valid for I if f it has truth value T under interpretation
d
e
n
I.o for every assignment of values to its free variables; satisfiable
i n d i
for
value T for some assignment of values to its free
fv Ii fif dit has
T
u
variables. A w f f of N D
(n
a
ls
pretation
1 i s vover
a usual
l i d bases. An interpretation I over S is a model
i-for
s a set
u I ' of w ff if f every w f f in I ' has value T under interi f f
,p
b
j I over
e
pretation
S for some simultaneous assignment of values
i
t
slto
c freet variables of w f f of F. A model is denumerable if f the
i
s
a
n
individual
t
e domain o f the semantical basis is denumerable.
v
a
l
i
d
)cr
f
o
r
Ie
m
e
v
e
(p
f
s
r
y
)r
i
n
t
(e
I
e
r
(d
s
i
1
)
,c
a
150
R
.
RO UTL E Y
Theorem 3: Every theorem of N D
I i s v a l i d .
Theorem 4: (Skolem-Lbwenheim): Every consistent set o f w f f
of N D and of N D
numerable
1 ( a n dmodel.
oProof:
f Let A be a consistent set of wff of ND
ND
1 o r)
o f
a n
H
tuting
of w f f of A constant E-terms distinct
'a p nfor
p free
l d i variables
e d
from
any
occurring
in
wff
of A , in such a way that a different
b
o
f
constant
E-term
is
substituted
for the free occurrences of each
ay
p
p
l
different
individual
variable
and
no variables are bound by the
ia
e
d Then, if I ' has a denumerable model, A also does.
substitution.
d
N
Because a
F consists o f closed w f f , since constant E-terms are
id
those that contain no free variables, i t suffices t o prove the
hitheorem for closed consistent sets of wff.
an Let F be a consistent set of closed wff of ND
sg D1, By
N
o rLindenbaum's
o f
alemma
n (
say
J.
actension,
8
a p p l i e d
)do, AI semantical
'
h a sbasis S a
n
specified
thus:
d a l
e I a =x i m
m
is <
io
s d nconstants)
i s, i sof tN De
c-dividual
1
tension
tn e
ht e lover, d
tation
a ceo Imov
a s> l x er
ei,d
A
sentential
i,lin tfnahi nas tdo constant
t rh e p is assigned T i f f J
p
constant
i d u1 ia l
co
1nsassigned
1 si n ditsielfv under
t :ai l ( a{af is
constant
f
is
assigned
an
n-place
relation I ( f ) o f e
;(io
a
n
d
a
n
n
cs xm l) :f oo
ds
closed
terms
tfor
s
u
c
h
t
h
a
t
p
al lcd e
T
ia
l
s. el seA
ip
r
}srtL I e )eol e a Idt1 ii oc na t e
,E s
sreHence
i(ei)n I( -f e x i -p m
2
xcst1 s i f (ii)i TII(f)(si,
)
s
,A s .„ :
(
1
e
(I E dxEA2)
The
of c
li-'= specification
c, s1 „ )
iiwhile
i s i f —T(Sx)A
then, as { I
n
1
tb
e f from
(rs sd
selected
cs1 o n s i s1 t e n
tm
e
it(( x: {) :I esi T A I
a
), s
ra
m
o
.fii (")NSee
o, Awr. CHURCH,
op. cit., " 4 5 2 . Th e s implif ic at ion o f Henk in's
s
o
sn
1
( x of) the
b
2: Skolem-Lbwenheim
proof
theorem used here is suggested by Wang's
ifn usee
f l 1WANG,
l , O . cit., p. 318.
work:
o
)
tT A o}, H.
d
s
T
lfiaE
l ax ( A
=
(oc S w a x s ) n
tn
iE
)
.
A
b
sh
:
„e
-nx
t
h )
eA
e
n )
id;
n
T
A
h
tfa
e
(
E
ax
rn
p
A
)
s,
A S I M P L E N A TU R A L DE DUCTI O N S Y S TE M
1
5
1
(*) A closed w f f A o f I ' has truth-value T f o r 1
The
is by induction over the number n of occurrences of
1 i proof
f f
'c', — ' and '
n
D = 1. Since A is closed A must be of the form f(exg(x)). Then
' i n Tf(exg(x)) -= I
A . I
Ii(t)(ex.g(4)
b
y
( i)
F o ( rj ) ( 1
= H
i n d1 u
it
Otherwise, in the case of certain applied N D
c t i c ox g ( x ) )f )
for sn y=s 0t e
is m
immediate
1
s ,
t from
h (ii).
e T o complete the induction
n
( cto consider:
there
are
three
uses
b a s i s
b
a
x
Case
1.
A
is
of
the
s
i
e form f( s
where
B contains at most x free. Then
is
,
x
Tl(s
,l )
i1
Asi, • •• S
e
)
,Case 2.
i c Ax is B
the ,form
—B. Then
t
b
s
„ TA )= T ,— yB s i n c1 e B is closed
A
i
,
= —TB.=-=
b
( — J
E
Esince
x I is max imal consistent.
B, i b y
e
x
,
B
,
i n id u c
a
B
Ji o n
s
t the
w 3. A is of
) form (BDC). Then
Case
,
A
„
)
h yD C)p
TA = T(B
f
s
b
t v hTC
f
- -o TB
,
i
y
thesis
w
I e s i
,
(
- — B y ii
h
C s
=
consistent
C i
e
b
y
1
i
H
r
i n d u cs i n c e
1
consistent a n d b y
J J )
e
t i o n
(
sentential logic
h
y
p i —s
,
c
mB a x
o
J a
x
i vm
As (
A
B
l Chas
provable
in 1, and therefore
truth-value T for I . Finally
l')
) is a denumerable model ssince the constant terms are deh
,numerable. Since N D is deductively
i
equivalent t o N D
o l
1
theorem
also
follows
for
ND.
n
I dt h e
1
c
s ,
(
e
I
0
I
i
)
i
s
s
a
m
m
a
o
x
d
152
R
.
ROUTLEY
Corollary: T he Skolem-Lbwenheim theorem holds f or F P.
Theorem 5 (Completeness theorem): Every v alid w f f o f N D
is a theorem of N D
1
1
( a n 6dEvery v alid w ff of F P is a theorem of F r .
Theorem
o f Similar to the proof of theorem 5, applying the corolProof:
N
)
lary ofD theorem
4.
( Certain
'
) alternative
.
completeness proofs can also be shortened
in N D
!
Monash University, Aus tralia
R
.
ROUTLEY
u s
i n(°) Fo r a proof, using theorem 5, see A. CHURCH, O . cit.. p.245. Note,
however,
g
that a different definition of validity is adopted.
E (")- Fo r instance the Quine-Dreben proof and R. Smullyan's s implif ication o f this: see W. (DIANE, Met hods o f Logic revised edition, Ne w
t e
York (1959), 253-259, B. DREBEN ' On the completeness of quantification
r
theory',
Proceedings of the National Academy of Science. vol. 38 (1952),
1047-1052.
m
s
(
'
'
)
.