This preview shows pages 1–2. Sign up to view the full content.
A Proper Inductive Proof of the Interpolation Theorem for
P
Branden Fitelson
02/14/07
Theorem
. Let
A
and
B
be formulas of
P
, such that (1) they share at least one propositional symbol in com
mon, and (2)
±
P
A
⊃
B
. For any two such formulas of
P
, there exists a formula
C
(called the
P
interpolant
of the formulas
A
and
B
) such that (3)
±
P
A
⊃
C
, (4)
±
P
C
⊃
B
, and (5)
C
contains only propositional
symbols that occur in both
A
and
B
(
i.e.
, only propositional symbols shared by
A
and
B
).
Setup for an inductive proof
. Let
S(A)
be the set of propositional symbols occurring in
A
,
S(B)
be the set
of propositional symbols occurring in
B
, and
q
be some propositional symbol that is shared by
A
and
B
(
i.e.
,
q
∈
S(A)
∩
S(B)
). We will focus on the set
X
=
S(A)

S(B)
of propositional symbols that occur in
A
but not in
B
. We will prove the interpolation theorem by strong mathematical induction on the cardinality
of
X
. That is, we will prove that the following property of natural numbers holds for all
n
≥
0:
S
(n)
: The interpolation theorem (above) holds when
X
=
S(A)

S(B)
=
n
.
Proof.
As always, a proper inductive proof involves a Basis Step and an Inductive step.
Basis Step
. Prove
S
(
0
)
. That is, we must prove the interpolation theorem for the case in which there
are
zero
propositional symbols occurring in
A
that do not also occur in
B
(
X
=
0). In this case, the set
of propositional symbols in
A
is a subset of those in
B
[
S(A)
⊆
S(B)
]. Let
C
=
A
. Then, obviously, (3)
±
P
A
⊃
C
, since
±
P
A
⊃
A
. And, since the assumption of the theorem is that
±
P
A
⊃
B
, we also know that
±
P
C
⊃
B
. All we need to show is that (5)
C
contains only propositional symbols that occur in both
A
and
B
.
But, this follows from the fact that
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '07
 FITELSON

Click to edit the document details