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equals 1 and hence
h
(
e,n
+1)+
h
(
f,n
+1)+(2
q
2)
h
(0
,n
+1)+(2
2
q
2
q
)
g
(0
,n
+1)=1
.
(22)
−
−
From this and (21) (using some 0
<e,f<n
+ 1), from (19) with
c
=
e
=0and
from the inductive hypothesis we have
e
+
ˇʽ
ˇʽ
h
(0
,n
+1)=1
+
f
+
ˇʽ
ˇʽ
2
q
)
ʾ
ˇ
+(2
q
2) + (2
2
q
−
−
so
((
n
+1+2
ˇʽ
)+(2
q
2)
ˇʽ
+(2
2
q
2
q
)
ʾʽ
))
h
(0
,n
+1)=
ˇʽ
−
−
ˇʽ
n
+1+
ʽ
.
Hence by
and since 2
q
ˇ
+(2
2
q
2
q
)
ʾ
= 1, it follows that
h
(0
,n
+1)=
−
(21) and (19),
c
2
+
ʾʽ
n
+1+
ʽ
c
+
ˇʽ
n
+1+
ʽ
h
(
c, n
+1)=
and
g
(
c, n
+1)=
(
c
∈{
0
,
1
,...,n
}
)
.
From (22) (with
e
=
n
+1
,f
=0)wehave
h
(
n
+1
,n
+1)=
n
+1+
ˇʽ
n
+1+
ʽ
.
Finally considering some
h
1
=
d
1
,from
⊛
⊝
h,d
⊞
n
+1
n
+2
⊠
=1
w
ʳ
[
k,k,h,d
]
ʲ
k
(
a
j
)
∧
ʳ
[
k,k,h
1
,d
1
]
(
a
1
,a
j
)
j
=1
j
=2
we have
2
g
(
n
+1
,n
+1)+2
q
h
(0
,n
+1)+(2
2
q
2
q
−
−
2)
g
(0
,n
+1)=1
which yields
n
+1
2
+
ʾʽ
n
+1+
ʽ
g
(
n
+1
,n
+1)=
completing the proof.
References
[1] Goodman, N.: A Query on Confirmation. Journal of Philosophy 43, 383-385 (1946)
[2] Quine, W.V.O.: Two Dogmas of Empiricism. The Philosophical Review 60, 20-43
(1951)
[3] Carnap, R.: A Basic System of Inductive Logic. In: Carnap, R., Jeffrey, R.C. (eds.)
Studies in Inductive Logic and Probability, vol. I, pp. 33-165. University of Cali-
fornia Press (1971)
[4] Carnap, R.: A Basic System of Inductive Logic, Part 2. In: Jeffrey, R.C. (ed.) Studies
in Inductive Logic and Probability, vol. II, pp. 7-155. University of California Press
(1980)
