On an alleged contradiction lurking in Frege's Begriffsschrift.

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285
Notre
Dame Journal of Formal Logic
Volume XIV, Number 2, April 1973
NDJFAM
ON
AN ALLEGED CONTRADICTION
LURKING
IN
FREGE'S
BEGRIFFSSCHRIFT
TERRELL
WARD BYNUM
Jean
van Heijenoort, in his introduction [l] to Bauer-Mengelberg's
translation of Frege's
Begriffsschrift
[2], claims to see a contradiction
lurking in the logical system of that work:*
Frege
allows
a functional letter to occur in a quantifier. . . . The result is
that
the difference between function and argument is blurred ... in the
derivation of formula (77) he substitutes 5 [a quantificationally bound
function
letter] for α [a quantificationally bound individual variable]
in/(α),
at
least as an intermediate step. If we also observe that in the derivation of
formula (91) he substitutes S for / [a "free" function letter], we see that he
is on the brink of a paradox. He
will
fall
into the
abyss
when (1891) he in-
troduces the course-of-values of a function as something "complete in
itself,"
which may be taken as an argument.1
Van Heijenoort is mistaken in supposing that any paradox can arise
from the derivations he cites in the
Begriffsschrift.
In that early work,
Frege is pioneering the development of quantificational logic.
While
he
does not yet have all the machinery or the terminology to precisely spell
out
the distinction between what he would later call
"first-level"
and
"second-level" functions, he never confuses the two. And because his
functions occur in
"levels,''
Frege's functional calculus (including that in
the
Begriffsschrift
of 1879) is free of the kind of paradox which, beginning
in
1891,2 does afflict his set theory. Frege himself points this out in a
letter to Russell in June of 1902 [4] when responding to Russell's letter [5]
about the discovery of a paradox.
*I
am indebted to Peter Geach for helpful discussions and some of the points
raised in this paper.
1. See [l],p. 3.
2. In that year, in [3] Frege
first
introduced the notion of the "course-of-values"
{Wertverlauf}
of a function.
Received
August
13, 1971

286 TERRELL WARD BYNUM
Consider now
the
"suspicious" derivations that van Heijenoort men-
tions
:
In
the
first
one,
while
proving formula (77), Frege cites the
following
principle (68):
68
I 1 1 f(c)
I
b
((->!r./(α))S6).
Rather
than
this principle,
he
actually needs—but
has
not yet developed the
machinery
to
express—an analogous second-order principle (call
it 680
involving
quantification over functions.
In the
later notation
of the
Grundgesetze
[6]
(ignoring the,
for
this purpose, irrelevant switch from
<=>
to
<=>)
it
would look
like
this:
68'
I 1
Mβf(β)
I
b
((-^-Mβϊ(β))=b)
.
The
appropriate substitution table (placed horizontally
for
convenience)
would
then
run
as
follows:
f
MβT(β)
b f
8
1—I T(y) γ F
Uθη-Γ(α)
~A*γ,yβ)
L/(*,
α)
δ
/
Γ(α)
U/(δ,α)
These substitutions
in
formula (68'),
and
the
detachment
of
the
defini-
tionally true equivalence
(76)
in
the
Begrίffsschrift,
yields
formula
(77)
with
flawless
correctness.
Similarly,
the
second "suspicious" derivation—that
of
formula (91)
—
requires
a
second-order principle
(a
confinement rule)
involving
quantifi-
cation
over functions. Such
a
principle would be: Given
a
formula with the
form
I
1
Mβf(β)
A
we can derive
a
formula with the form
i
r^L—MβHβ)
A
if
Ά' is an
expression
in
which
/
does not occur and
if/
stands only
in
the
argument places
of
Mβf(β).

ON
AN
ALLEGED CONTRADICTION
287
The "substitution"
of S
for/
in
the
derivation
of
formula
(91)
referred
to
by van
Heijenoort
is
actually
the
two-step procedure
of
applying such
a
second-order confinement
and
then substitution
8 for f. In a
footnote
to
that derivation, Frege himself calls attention
to his
use
of a
confinement
rule,
but
van
Heijennort mistakenly interprets
the
footnote
as a
mere
acknowledgement
of
quantifying over functions.3
Thus,
at
the
time
he
wrote
the
Begriffsschrift, Frege
was not yet
able
to express some needed second-order principles
for the
derivations
which
van
Heijenoort mentions. Nevertheless, there
is no
contradiction
lurking
in
them,
and
machinery that Frege would later develop
can
easily
clear
up any
difficulties.
REFERENCES
[1]
van
Heijenoort, Jean, "Editor's introduction,"
in G.
Frege, Begriffsschrift,
(trans.
S.
Bauer-Mengelberg),
in J. van
Heijenoort,
ed.,
From Frege
to
God
el,
Harvard University Press, Cambridge, Massachusetts (1967).
[2] Frege,
G.,
Begriffsschrift, Nebert, Halle (1879).
[3] Frege,
G.,
Funktion
und
Begriff Pohle, Jena (1891).
[4] Frege,
G.,
"Letter
to
Russell,"
in van
Heijenoort,
ed., op. cit., pp.
127-128.
[5] Russell,
B.,
"Letter
to
Frege,"
ibid.,
pp.
123-124.
[6] Frege,
G.,
Grundgesetze
der
Arithmetik, Band
I,
Pohle, Jena (1893).
State University of New York at Albany
Albany, New York
3.
See van
Heijenoort's addition
to
Frege's footnote
to the
derivation
of
formula
(91);
[1],
P 66.