Interpolation for Church’s pure functional calculus of second order under secondary interpretations

Abstract

A proof is given of Craig’s Interpolation Theorem for Church’s Pure Functional Calculus of second-order, which is provably complete w.r.t. the class of all secondary interpretations (a subclass of Henkin’s general interpretations). The search for the proof allows for a deeper analysis of interpolation within the framework of second-order formalisms.

Full text

Logique A n a l y se 129-130 (1990), 69-74
INTERPOLATION FOR CHURCH'S PURE FUNCTIONAL CALCULUS
OF SECOND ORDER UNDER SECONDARY INTERPRETATIONS
CriStina CORREDOR-LANAS
Abstract
A proof is given of Craig's Interpolation Theorem for Church's Pure
Functional Calculus of 2nd-order, which is provably complete w. r.
t. the class o f all secondary interpretations (a subclass of Henkin's
general interpretations). The search for the proof allows for a deeper
analysis o f interpolation within the framework o f 2nd-order forma-
lisms.
1. Introduction
The result stated above seems not to be in harmony with other well-estab-
lished facts. For, on the one hand, the class of all 2nd-order standard sys-
tems verifies interpolation, but no axiomatization exists which characterizes
this class. O n the other hand, Henkin proved in [1950] that it is possible
to define a wider class of 2nd-order formalisms (called general systems)
which satisfy certain closure conditions, in such a way as to be characterized
by a calculus that extends the usual classical 1st-order one. Nevertheless a
counterexample can be found for definability and a fortiori for interpolation.
In his [19561 Church defined some 2nd-order calculi and proved them to
be complete w. r. t. the class of all secondary systems. In fact, he relativizes
the notion of validity in Henkin's general sense to that of truth in a par-
ticular class of normal systems, these latter defined as those in which all
axioms of F2
21
' ( t h e
P u r e
F u n c
t i o n
a l
C a
l c u
l u s
o
f
2 n
d -
o r
d e
r)
a
r
e
v
a
l
i
d
a
n
d
all the rules of F 2
4
p r e s e r v e
v a l i d i
t y .
It
s t r a i
g h t f o
r w a r d
l y
f o
l l
o w
s
t
h
a
t
e
v
e
r
y
secondarily valid w ff in the calculus is a theorem of it.
As Church points out, it is not possible to extend F2
2
P b y a d d i n g r u l e s
a n d
axioms, in such a way that the theorems come to coincide with the wffs
which have the value true for all systems of values of their free variables,
under all standard interpretations. This follows from Gôdel's incompleteness
theorems. In particular this incompleteness result applies to any 2nd-order
system containing 2nd-order Peano arithmetic, provided that at least one

70 C. CORREDOR-LANAS
predicate constant has been added to the vocabulary. Then it is possible to
define a categorial finite set of axioms for the set of positive integers. When
general systems are defined (by means of making sure that to each way of
compounding formulas in the calculus is associated an operation defined on
the domains of the systems, w. r. t. which these must be closed), it is also
the presence of a predicate constant what leads to the failure o f Beth's
Definability Theorem, henceforth of Craig's Interpolation Lemma (since
both results are equivalent within the framework of regular logics).
This last remark makes Church's calculus F 2
4 p a r t i c u l a r l y
i n t e r e s t i n g .
F o r
it is defined as including among its primitive symbols all the individual,
propositional, and functional variables, but no (individual or functional)
constants. Then the calculus turns out to be closed under definability and
the strategy for showing the failure of interpolation is no longer available.
This fact strongly suggests that interpolation might hold for this 2nd-order
formalism, complete under the class of all secondary interpretations.
2. The Calculus F2
2
P
From now on the notations and definitions are taken to be those to be found
in Church [1956]. In particular, the Simple Functional Calculus of 2nd-order
has, in addition to notations of the Functional Calculus of 1st-order, quan-
tifiers with propositional or functional variables as operators variables. The
Pure Functional Calculus of 2nd- order includes (as it has just been said)
all the individual, propositional and functional variables, but no individual
or functional constants. The primitive vocabulary includes individual vari-
ables a„ a
2
,..., b
1
, b
2
,...,
p r o
pos
i t i o
n a l
v a
r i
a b
l e
s
p
,
q
,
p
r
e
d
i
c
a
t
e
v
a
r
i
-
ables P, Pa,..., (instead of original Church's functional ones, for reasons that
will be explained below) and sentence letters A, B, S
B
P
A ( o r A [ P / 1 3 ] )
stands for the result of substituting P for all occurrences of B in A. More-
over the following two definitions are introduced: f = df (S)S, t = df (3s)s,
where s is any variable. The syntactical and semantical notions are also
those to be found in Church [1956]; we only specify here what is fundamen-
tal for the proof.
Thus the rules of inference and axiom schemata for F2
2
P a r e t h e f o l l o w i n g .
Rules o f Inference
RI M o d u s Ponens: From A and A D B to infer B.
R2. Generalization: From A, if a is any variable, to infer (a)A.

INTE RPOLATION FO R CHURCH' S PURE FUN CTION A L CAL CU LU S 7 1
R3. Alphabetic Change of Bound Individual Variables: From A, if a is an
individual variable not free in C, and b is an individual variable which
does not occur in C, i f B results from A by substituting C[atb] for a
particular occurrence of C in A, to infer B.
R4. Substitution for Individual Variables: [This rule requires an explana-
tion. In Church's original work, this rule permits substitution of in-
dividual variables which are not bound; since the calculus is func-
tional, it is necessary to carefully explicit the cases in which such a
substitution is possible. Nevertheless, we have assumed a vocabulary
that only contains predicate variables and no functional ones, by per-
forming the usual transformation from functions into their correspon-
ding diagrams. Furthermore, it will be seen that the set of valid wffs
in the calculus coincides with the set of closed secondarily valid wffs.
Thus it is easy to see that under these conditions rule R4 reduces to
R31.
Axiom Schemata
al. pD.ciDp
a2. sD(pDci)D.sDpD.sDig
a3. D D.11 D p
a4. (x ) ( p D F(x)) D . p D (x)F(x)
a4
0
.
(
p
)
(
A
D
B
)
D
.
A
D
(
p
)
B
(
p
i
s
n
o
t
f
r
e
e
i
n
A
)
a4
n
.
(
P
n
)
(
A
D
B
)
D
.
A
D
(
P
'
)
B
(
P
n
i
s
n
o
t
f
r
e
e
i
n
A
)
a5. (x )F ( x ) D F(y )
a5
0
.
(
p
)
A
D
S
B
A
a5
n
.
(
P
)
A
D
S
B
P
(
x
l
-
x
n
)
A
(
x
i
x
j
i
f
i
j
)
.
As for the semantic notions, just remember that a system is an ordered
pair (F, I) where F = F
o
, F , ,
F 2 , . . .
; F o
i s
a
non-
e m p t y
s e
t
( t
he
d o m
a i n
o
f
individuals) and F, g P( F
o
i
) ; a n d I
i s a n
i n t e r p r e t
a t i o n
on
F ,
d e f i
n e d
i
n
t h
e
usual manner in 2nd-order. In particular 1(s) E F
o f o r a n y
i n d i v i d u a l
s y m b o l ;
and I(Ps,...s
r
) = I
( P ) ( 1
( s
1
) . . . I ( s
n
) ) ,
w h
e r
e
l
(
P
)
E
F
n
;
m
o r
e
o
v
e r
1
(
A
)
E
{
t
,
f
}
for any closed wff A of F 2
4
. A w f f
A i n
t h e
c a l c u l u
s
i s
s a i
d
t o
b
e
v a l
i d
w.r.t. (F, if I(A) = I ) is said to be normal if all the axioms of F 2
4
are valid w.r.t. it, and every rule of F2
2
P p r e s e r v e s
v a l i d i t y
w . r . t .
i t .
A
w f f
A is said to be secondarily valid i f it is valid w.r.t. every normal system.
It can now be stated the completeness result referred to above.

72 C. CORREDOR-LANAS
Theorem (Church [1956])
Any wff A of F2
2 i s
s e c o n d
a r i l y
v a l i
d
i f
f
i
t
i
s
a
t h
e o
r e
m
o
f
t
h
e
c a
l c
u l
u s
.
3. Interpolation lemma for F2
4
' u n d e r
s e c o n d a r y
i n t e r p r e t
a t i o n s
There are two possible strategies for proving interpolation: a model-theoreti-
cal one and a proof-theoretical one. The latter can be characterized as
consisting of two steps. Firs tly, to specify an interpolant for the axiom
schemata (e.g. for A A in a Gentzen calculus), and secondly, to find an
interpolant for the conclusion of each rule, assuming it exists for the premi-
se(s). The only difficulty arises within classical 1st-order when the introduc-
tion rule for the universal quantor is considered. For the critical condition
establishes that the proper parameter (eigenvariable) cannot occur in the
conclusion in
A(a) / r e , (x)A(x)
Yet it is possible to overcome this difficulty by means of defining a notion
of deducibility apparently more restrictive than, but provably equivalent to,
the usual classical one (cf. Smullyan [1968]). Thus, a deduction of A from
is to be a finite tree of wffs, starting in A, finishing in axioms or mem-
bers o f f , and such that each node which is not a final node is derived from
one of its predecessors by application of one of the rules. Furthermore (and
here the crucial point lies), whenever [Ni] is applied, the subtree starting
in the conclusion must be such that the proper parameter occurs in the final
nodes only in axioms.
The underlying idea is that the eigenvariables be seen as they were impli-
citly general (i.e. universally quantifiable). This is feasible by making sure
that they are introduced only via the logical axioms. And this is precisely
what happens in the case of F n. For none of the rules authorizes a replace-
ment o f individual or predicate variables with constants — they do not
appear in the vocabulary. Only such substitutions are available as those
involving alphabetic changes of individual or predicate variables. Moreover
these can be restricted to the case of closed wffs without loss of generality,
as said before.
Consequently, any not bound variable apparently playing the role of a
proper parameter can be generalized (i. e. universally quantified), provided
that conditions stated in rules R3 (and R4) be satisfied. Namely, that no
other free variable would turn out to be bound, or conversely that no van-

INTE RPOLATION FOR C HURC H'S PURE FUNCTI O NA L CALCULU S 7 3
able originally bound be freed by accomplishing a substitution. Generaliza-
tion over all free variables of a valid wff A of F 2
4 y i e l d s a c l o s e d
w f f ,
which constitutes a provable valid sentence:
Theorem (Church [1956])
A wff A of F 2
4 i s
v a l i d
i n
a l l
s t a n
d a r d
s y
s t e
m s
[ a l t
e r n
a t i
v e l
y ,
i
n
a
l
l
s
e
c
o
n
-
dary systems] i f its universal closure is valid in all standard systems [in all
secondary systems resp.].
This leads the path to a direct proof of the Interpolation Lemma, given
the following definition. I f A, B are two valid wffs of F2
2
P a n d A D B
i s
secondarily valid, let w(A) n w(B) denote the set of their common varia-
bles, and let L , ,
)
,
1 3 ) b e
t h e
s u b l a
n g u a g
e
i
n
w
( A
)
n
w
(
B
)
( i
. e
.
,
t
h
e
s
e
t
of wffs of the calculus whose variables belong to that set). A w ff C is said
to be an interpolant for A D B i f (i) C — w(A) ve(B) and (ii) A D C, C
B are both theorems of F 2
4
.
Now the announced result can be proven.
Theorem Interpolation holds for F2
21
'.
Proof Let A D B be any secondarily valid wff of F 2
4
. B y h y p o t h e s i s
( s i n c e
F2
2-
1
'
i
s
c
o
m
p
l
e
t
e
w
.
r
.
t
.
t
h
e
c
l
a
s
s
o
f
a
l
l
s
e
c
o
n
d
a
r
y
i
n
t
e
r
p
r
e
t
a
t
i
o
n
s
)
,
A
D
B
is a theorem of the calculus. Let a „ , P
1
, P „ , b e
t h e f r e e
i n d i v i d u a l
and predicate variables resp. which occur in B but not in A. Replace each
of them with a new variable, b, instead of a„ Q
J i n s t e a d o f 1
3
j
, w h i c h
do
not
occur in B, and let B' be the result of effecting this replacement. Take then
the universal closure of B', ( b n ) ( Q 1)- d e n o t e d by B". I t is
immediate to see that the following holds:
I) B''ELA) r),( B
)
;
2) A D B " is a theorem of F2
2
P ( b y r u l e
R 2 t h e
c l o s u r e
o f
A
D
B
i s
s o
,
and by rule R4 [R3] it holds also for A D B" ;
3) B " D B is a theorem of FI
2
P ( b y a x i o m
a 5
n
) ;
4) A D B " , B" D B are valid wffs in F 2
4 ( b y C h u r c h
preceding
t h e o -
rem).
Thus B" is the looked-for interpolant.
University of Valladolid.

74 C. CORREDOR-LANAS
REFERENCES
[1] CHURCH, A . [1956]: Introduction to Mathematical Logic. Princeton
Univers. Press.
121 HENK1N, L. 119501: Completeness in the theory of types; Journal of
Symbolic L ogic 15 (1950), pp. 81-91.
131 S MUL L Y A N, R. M . [1968]: First-order Logic. Berlin: Springe r.