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FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT
CALIXTO BADESA AND JOAN BERTRAN-SAN MILL ´
AN
Abstract.
It is well known that the formal system developed by Frege
in Begriffsschrift is based upon the distinction between function and
argument – as opposed to the traditional distinction between subject
and predicate. Almost all of the modern commentaries on Frege’s work
suggest a semantic interpretation of this distinction, and identify it with
the ontological structure of function and object, upon which Grundgesetze
is based. Those commentaries agree that the system proposed by Frege
in Begriffsschrift has some gaps, but it is taken as an essentially correct
formal system for second-order logic: the first one in the history of logic.
However, there is strong textual evidence that such an interpretation
should be rejected. This evidence shows that the nature of the distinction
between function and argument is stated by Frege in a significantly
different way: it applies only to expressions and not to entities. The
formal system based on this distinction is tremendously flexible and is
suitable for making explicit the logical structure of contents as well as of
deductive chains.
We put forward a new reconstruction of the function-argument scheme
and the quantification theory in Begriffsschrift. After that, we discuss
the usual semantic interpretation of Begriffsschrift and show its inconsis-
tencies with a rigorous reading of the text.
1. Introduction
Gottlob Frege’s foundational work, Begriffsschrift, eine der arithmetischen
nachgebildete Forlmelsprache des reinen Denkens [
Frege, 1879
], has been
studied intensely over the last decades
1
. Throughout this period a unitary
interpretation has been established and generally shared by scholars – with
the exception of certain particular cases. One specific element of this in-
terpretation is a particular understanding of the distinction – developed in
Begriffsschrift – between function and argument. A careful reading of the
Earlier versions of this paper were presented at the APA/ASL 2014 Spring Meeting in
San Diego, at the ECAP8 2014 in Bucharest and at the meeting of the Philosophy of Logic
group, Department of Philosophy, University of Barcelona. We are grateful to Ignasi Jan´e
and Jos´e Mart´ınez for their careful reading of the paper and their helpful remarks and
suggestions. Thanks to Ansten Klev and Danielle Macbeth for comments, and to Laura
Cort´es for linguistic advice. This work was partially supported by the Spanish MEC under
the research project ‘Reference, self-reference and empirical data’ (FFI2011-25626).
1
From now on and throughout the whole paper, for the sake of clarity, we will use
‘Begriffsschrift’ to refer to the book published by Frege in 1879 and ‘concept-script’ to
refer to the formal system developed in it.
1
2 C. BADESA AND J. BERTRAN-SAN MILL´
AN
sources shows not only that this usual understanding of the distinction has
some inaccuracies, but that it is fundamentally incorrect.
When Frege presents the notion of function in Grundgesetze der Arithmetik
[
Frege, 1893
] – from now on, Grundgesetze –, he acknowledges that his
understanding of this notion has been modified from 1879 to 1893:
“My Begriffsschrift (Halle, a.S. 1879) no longer corresponds entirely
to my present standpoint; it is therefore to be consulted as an
elucidation of what is presented here [Grundgesetze] only with
caution.” [Frege, 1893, §11, p. 5, footnote 1]
We will argue that the modifications this new understanding involve imply
an essentially different notion of function from that of 1879. This is precisely
what has been systematically minimised – if noted at all – in contemporary
readings of Begriffsschrift. Our aim is to faithfully depict Frege’s account of
the distinction between function and argument that he developed in 1879.
We will propose a new reconstruction of the function-argument scheme
and the quantification theory developed in Begriffsschrift. After the present
introduction, in the second section, we will, on the one hand, state how an
analysis in terms of function and argument is applied, and what characterises
it; and, on the other, lay out the notion of generality. We then leave the
expository part of the paper and begin the historical discussion. We will
clarify, in the third section, the linguistic nature of the function-argument
distinction and the relation between this structure and the elements of the
language of the concept-script. In the fourth section, we will discuss the
nature of the notion of generality and raise specific considerations concerning
the syntactical guidelines Frege himself provides to handle it. Finally, in the
fifth section, we will show that the reading of Begriffsschrift’s quantification
done by most modern scholars is untenable.
2. Function-argument and generality: Exposition
2.1.
Basic notions of the concept-script.
Before considering the distinc-
tion between function and argument, we lay down some terminology. Frege
divides the symbols of the concept-script into those that have a fixed meaning
and those that express generality over different objects [
Frege, 1879
,
§
1, p.
111]. The former group includes the logical symbols; the latter consists of
what Frege calls ‘letters’ and is the group that we will consider first.
Letters express generality and have no determinate meaning. For explana-
tory reasons we differentiate between two sorts of letters: ‘
f
’,‘
g
’,‘
h
’,
. . .
are
function letters; and ‘
a
’,‘
b
’,‘
c
’,
. . .
– which do not receive a specific name from
Frege – will be referred to as argument letters. Our terminology is, in any
case, in agreement with Frege’s use.
The logical symbols of the concept-script are the content and the judgement
strokes, the conditional and the negation strokes, the equality symbol and
the generality symbol. The content stroke indicates that the content
of certain combination of symbols is taken as a whole, and the judgement
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 3
stroke expresses the act of assertion by which a content is affirmed. Frege
does not accurately explain what is the content. For our purposes, it will
suffice to say that the content of a statement is what it means – an assertible
content –, and that the content of a term is its denotation.
The conditional and the negation concern contents. If
A
and
B
are
assertible contents, then the negation of Ais denoted by:
A,
and the conditional “if B, then A” is rendered by:
A
B,
which links an antecedent Band a consequent A.
The equality symbol differs from the conditional stroke and the negation
stroke in that it relates names, and not contents [
Frege, 1879
,
§
8, p. 124].
For instance, ‘2
·
3
≡
5 + 1’ expresses that ‘2
·
3’ and ‘5 + 1’ have the same
content, i.e. denote the same object
2
. We will later discuss in detail the
generality symbol and its meaning in the concept-script.
2.2.
Function-argument analysis in Begriffsschrift.
In Begriffsschrift,
the distinction between function and argument is explained with the help of
simple expressions. From a technical point of view, a simple expression has
neither quantification nor propositional structure: it is an atomic statement or
an individual term. In his examples, Frege borrows expressions from natural
language. We will also use concept-script expressions and arithmetical
examples.
Frege introduces the function-argument scheme as a particular decom-
position of expressions. According to his exposition, an expression can be
decomposed into two components: a variable part, which is the compo-
nent that can be replaced (the argument) and a fixed part (the function)
[Frege, 1879, §9, p. 126].
The main characteristic of this distinction is the fact that it does not
obey any pre-established guideline: any component can be the argument.
It is taken for granted that the division has some reasonable restrictions,
which are mainly dictated by common sense
3
. A prominent feature of this
distinction is the absence of a rule for drawing the division. In Frege’s words:
2
As a consequence of Frege’s rendering of the equality of content, this relation introduces
an inconsistency in the interpretation of the letters of the concept-script. According to his
exposition, the same symbol ‘
a
’ would be interpreted as a content – outside an equality
statement – and, at the same time, – in an equality statement such as ‘
a≡b
’ – as a
symbol. We will not consider the difficulties this inconsistency could produce. For a detailed
discussion of this matter, see ‘Frege’s Begriffsschrift Theory of Identity’ [
Mendelsohn, 1982
]
and ‘What Frege’s Theory of Identity is Not’ [May, 2012].
3
Each component must be a significative unit; for instance, articles or prepositions
cannot be taken as a function or as an argument.
4 C. BADESA AND J. BERTRAN-SAN MILL´
AN
“This distinction has nothing to do with the conceptual content,
but only with our way of viewing it.” [Frege, 1879, §9, p. 126]
This means, on the one hand, that the division between function and
argument does not reflect the semantical structure of the expression where
it is applied and, on the other, that there may be different ways in which
an expression can be decomposed; each decomposition depends only on our
particular interests.
For instance, the statement ‘3 is an odd number’ can be decomposed in at
least two different ways:
(1) The function ‘is an odd number’ and the argument ‘3’.
(2) The function ‘3’ and the argument ‘is an odd number’.
Frege provides a somewhat rigorous definition of the distinction between
function and argument:
“If, in an expression (whose content need not be assertible), a
simple or a complex symbol occurs in one or more places and we
imagine it as replaceable by another (but the same one each time)
at all or some of these places, then we call the part of the expression
that shows itself invariant a function and the replaceable part its
argument4.” [Frege, 1879, §9, p. 127]
The function-argument scheme can thus even be applied to non-assertible
expressions. Therefore, any complex term – as well as any simple formula –
can be divided into function and argument.
The letters of the concept-script are not suitable for informally explaining
the meaning of the logical symbols of this formal system, because they
express generality. This is the reason why, in addition to them, Frege uses
capital Greek letters in order to represent particular contents without further
determination: ‘
A
’,‘
B
’,‘
Γ
’,‘
∆
’,
. . .
and ‘
Φ
’,‘
X
’,‘
Ψ
’,. . .. These symbols do not
belong to the language of the concept-script and neither do they express
generality. Frege uses them mainly in chapter I of Begriffsschrift to assist
his explanations of the components of the formal system – as we have done,
for instance, for the conditional.
According to the use of these symbols, ‘
Φ
(
A
)’ represents a particular
expression that has already been divided into two parts: under an initial
analysis, ‘
Φ
’ is the function of the argument ‘
A
’. Moreover, Frege provides a
reading for ‘
Φ
(
A
)’, namely, ‘
A
has the property
Φ
’ [
Frege, 1879
,
§
10, p. 129].
This reading comes from the basic sentence represented by ‘
Φ
(
A
)’, which
expresses that an object has certain property – either simple or complex.
Hence,
A
takes the place of the object and
Φ
represents the property. However,
since ‘
Φ
(
A
)’ is a generic scheme for any expression decomposed into function
and argument, this reading becomes merely suggestive. Therefore, from a
purely logical point of view, it does not impose any pre-established way to
4
We have removed the expressions enclosed in brackets Terrell W. Bynum adds in his
translation. We will do the same in what follows. All remaining brackets have been added
by the authors.
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 5
handle the decomposition. In fact, as Frege suggests, the initial analysis of
‘
Φ
(
A
)’ – according to which ‘
Φ
’ is the function of the argument ‘
A
’ – can be
revised:
“Since the symbol Φoccurs at a place in the expression
Φ(A)
and since we can think of it as replaced by other symbols
Ψ
,
X
– through which other functions of the argument
A
are ex-
pressed
5
–we can consider
Φ
(
A
)as a function of the argument
Φ
.”
[Frege, 1879, §10, p. 129, emphasis added]
In other words, the analysis in terms of function and argument is not
absolute, and can be changed afterwards: once a division has been made,
what was taken to be the replaceable part might become the fixed compo-
nent and both symbols can thus be taken as either function or argument.
Accordingly, nothing prevents us from considering ‘
Φ
’ as the argument of
‘
Φ
(
A
)’. Consequently, in the expression ‘
Φ
(
A
)’ we can regard, on the one
hand, ‘
Φ
’ as the function having ‘
A
’, ‘
B
’, ‘
Γ
’,
. . .
as arguments and, on the
other, ‘A’ as the function having ‘Φ’, ‘X’, ‘Ψ’,. . . as arguments.
As we have said, expressions such as ‘
Φ
(
A
)’ are useful for the sake of
explanation, but they do not belong to the language of the concept-script.
In order to express properly in the language a generic expression divided
into function and argument, the concept-script provides ‘f(a)’.
Frege adds an important modification to this scheme: the possibility of
a function having more than one argument [
Frege, 1879
,
§
9, p. 128]. This
provides – for the first time in the history of logic – an appropriate way to
analyse relational statements.
The considerable flexibility with which the function-argument structure
can be applied to simple cases is not suitable for all expressions. Frege
warns of the inadequacy of applying the distinction between function and
argument when taking the gramatical structure as a guide. This inadequacy
is especially manifest in the decomposition of quantified statements. In order
to demonstrate this, Frege uses the following examples [
Frege, 1879
,
§
9, p.
127]:
(3) The number 20 can be represented as the sum of four squares.
(4)
Every positive integer can be represented as the sum of four squares.
These two statements exhibit a clear difference: they have a radically distinct
subject, in spite of sharing the same predicate. According to Frege, there is
a relevant element underlying this linguistic divergence:
“What is asserted of the number 20 cannot be asserted in the
same sense of “every positive integer”; though of course, in some
circumstances it may be asserted of every positive integer. The
expression “every positive integer” by itself, unlike “the number
5
We do not follow here Bynum’s translation. He renders this phrase “(...) replaced by
other symbols [such as] Ψ,X– which then express other functions of the argument A”.
6 C. BADESA AND J. BERTRAN-SAN MILL´
AN
20”, yields no independent idea; it acquires a sense only in the
context of a sentence.” [Frege, 1879, §9, p. 128]
Frege seems to recognise the difference between these two statements in the
fact that ‘the number 20’ denotes a particular object, while ‘every positive
integer’ does not have a specific denotation, but refers generally to a variety
of entities, namely, the positive integers.
The difference between these two statements is generalised with the help
of the distinction between being determinate and being indeterminate:
“For us, the different ways in which the same conceptual content
can be considered as a function of this or that argument have
no importance so long as function and argument are completely
determinate. But if the argument becomes indeterminate (...),
then the distinction between function and argument acquires a
substantive significance. It can also happen that, conversely, the
argument is determinate, but the function is indeterminate. In
both cases, through the opposition of the determinate and the
indeterminate or the more and the less determinate, the whole
splits up into function and argument according to its own content,
and not just according to our way of looking at it.” [
Frege, 1879
,
§9, p. 128]
Even though Frege does not specify what it means to be determinate,
it can be said that an indeterminate component is either an argument
or a function that expresses generality and has no fixed denotation. The
meaning of indeterminate expressions cannot be expressed in the concept-
script without letters. In contemporary terms, this is equivalent to the
presence of either variables – in terms or in open formulas – or quantified
fragments of an expression, such as ‘every positive integer’, which do not
express an independent idea6.
Every quantified statement contains indeterminate components. As Frege
states, the distinction between function and argument has to be drawn
according to the content in those expressions that contain indeterminate
components. The reason for this variation is the need to avoid incorrect
analyses, which are suggested by the gramatical structure of many expressions.
6
We can find at least three different uses of the pair ‘bestimmt-unbestimmt’ in Begriffs-
schrift, which is commonly translated by ‘determinate-indeterminate’. The distinction
between the uses relies on their application. We have already explained the first use.
According to the second use, in the case of an expression such as ‘
Φ
(
A
)’, the symbol ‘
Φ
’
is indeterminate because it represents a particular expression – which is, under an initial
analysis, taken as the function of ‘Φ(A)’ –, but without specifying which one.
Finally, a property is indeterminate if it is denoted by an expression such that, when
the latter is taken as the function of an statement, it is not the case that the result of
combining this function with any argument denotes an assertible content. There are cases
of properties for which the expression of the circumstance that an object has them is
not always a judgement. Frege’s own example is the property of being a heap of beans
[Frege, 1879, §27, p. 177].
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 7
Hence, Frege deals with the decomposition of quantified statements in a
specific way in order to impose a correct logical analysis of the content that
the concept-script manipulates. Now, it is not Frege’s aim in Begriffsschrift
to show how logically analyse content – although it certainly helps to do
so. Rather, the aim of Begriffsschrift is to offer, once the content has been
appropriately analysed, an adequate way of expressing it; a way which in
turn is free of the ambiguities of natural language. Thus, at this stage of
explanation, Frege does not specify how this particular analysis must be
performed: he only warns that the general decomposition is not adequate
for those expressions that contain indeterminate components.
After this exposition, we can state that ‘the number 20’ is a determinate
expression and ‘every positive integer’ an indeterminate one. We know that
‘the number 20’ can be the argument of the function ‘can be represented as
the sum of four squares’. Hence, in (3), the most natural argument coincides
with the subject. In this sense, the subject-predicate analysis is innocuous
with respect to the general decomposition, since the latter can be applied
without guidelines – just depending on the point of view adopted. However,
as we have already pointed out, it is not appropriate to replicate this same
decomposition with (4), for a previous formal analysis has to be carried out.
This reveals the fact that, since the predicate ‘can be represented as the
sum of four squares’ is taken to be simple – although actually it should be
analysed –, (3) is an atomic expression while (4) is a complex quantified
statement. Therefore, (4) must be analysed taking its content into account.
This leads to a symbolisation such as that of universal affirmative statements
made nowadays.
2.3.
Generality.
The generality symbol of the concept-script is related to
a letter and, in Frege’s words, “delimits the scope of the generality signified
by this letter” [
Frege, 1879
,
§
11, p. 131]. This symbol is composed of two
different elements. First, a concavity is placed in the middle of a content
stroke – not necessarily attached to a judgement stroke – in such a way
that its position indicates the scope of the the generality expressed. Second,
the German font points out which letter is bound by the concavity. When
a concavity occurs, ‘
F
’ will be used as function letter and ‘
a
’,‘
b
,‘
d
’,‘
e
’ as
argument letters. We will conventionally refer to the generality symbol as
‘quantifier’ and thus mention ‘quantified statements’7.
It is clear from the very presentation of this notion in Begriffsschrift how
it is deeply linked with the function-argument scheme. According to Frege’s
proposal, a symbol in the expression of a judgement – either a function letter
or an argument letter – is taken to be the argument and is replaced with
a German letter [
Frege, 1879
,
§
11, p. 130]. In fact, he explicitly says that
the function is what remains when a symbol in an assertible expression is
7
This convention, however, should not induce to conclude that statements that do not
contain any concavity are not quantified in our contemporary sense. Our account of Frege’s
use of italic letters makes this explicit.
8 C. BADESA AND J. BERTRAN-SAN MILL´
AN
regarded as the argument and a German letter replaces it. Thus, there is
no particular link between the predicate of a quantified statement and the
component taken to be the function.
We have said that the following statement:
Φ(A)
can be analysed in a natural way taking ‘
A
’ to be the argument and ‘
Φ
’ to
be the function. Hence, if the symbol ‘
A
’ is seen as the variable part, then it
can be replaced with a corresponding German letter and thus be quantified.
The result:
aΦ(a),
stands for the judgement that
Φ
(
A
) is a fact – that is,
Φ
(
A
) is true – whatever
argument ‘
A
’ may be put in place of ‘
a
’. Frege also mentions that ‘
Φ
’ can be
the argument as well, and it too can be thus replaced with the German ‘
F
’
in order to get:
FF(A).
We will consider this matter in detail in section 4.1.
The concavity and German letters are not the only means to render
generality in the concept-script; italic letters express generality as well, but
their scope encompasses the totality of the judgement where they occur. In
Frege’s words, “An italic letter is always to have as its scope the content of
the whole judgement, and this need not be signified by a concavity in the
content stroke” [Frege, 1879, §11, pp. 131–132]. In fact:
a > b
a+b>b+b
stands for the same judgement as:
a b a>b
a+b>b+b.
3. The nature of the distinction function-argument
We have so far considered Frege’s exposition in Begriffsschrift of the
distinction between function and argument, as well as his notion of generality.
We will now depart from this expository approach in order to contrast our
results with the relevant aspects of the traditional reading of Begriffsschrift
8
.
8
Throughout this section, for the sake of simplicity we consider only unary functions in
order to set our position.
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 9
3.1.
Functions in Begriffsschrift and mathematical functions.
Prob-
ably because of the groundbreaking character of Frege’s 1879 work, many
modern historical commentaries tend to identify the notion of function in
Begriffsschrift with a modern one – which has strong similarities with the
notion of mathematical function: an assignment of a unique value to every
appropriate argument
9
. The notion of function in Begriffsschrift is relative
in the sense that a component can be either a function or an argument of a
single expression, whereas the mathematical notion of function is absolute,
that is, function and argument are not interchangeable in the same sense.
The divergence between the way in which Frege introduces functions in
Begriffsschrift and modern accounts is often considered as mere inaccuracy.
Frege indicates that the notion of function developed in Begriffsschrift
takes as a guide the notion of mathematical function, but at the same time
he points out that there are great differences between the two notions:
“[S]ince we can think of it [
Φ
in
Φ
(
A
)] as replaced by other sym-
bols
Ψ
,
X
– through which other functions of the argument
A
are
expressed
10
– we can consider
Φ
(
A
) as a function of the argument
Φ
. This shows quite clearly that the concept of function in [math-
ematical] analysis, which I have in general followed, is far more
restricted than the one developed here.” [
Frege, 1879
,
§
10, p. 129]
While the reading of an expression such as ‘
Φ
(
A
)’ – divided into function
and argument – is, in Frege’s words, ‘
A
has the property
Φ
’, he would never
read the mathematical expression ‘
f
(
x
)’ as ‘
x
has the property
f
’
11
. And,
contrary to what happens with
Φ
(
A
),
f
could never be the mathematical
argument of
f
(
x
). In this sense, the Begriffsschrift function cannot be seen
as a generalisation of the function in mathematical analysis. In other words,
the divergence between these two notions is not a matter of an extension of
the domain of possible arguments and values.
What Frege takes from analysis is the fact that in every arithmetical
expression of a function – such as, for instance, ‘2
x2
+ 3’ – there is always a
component that takes values, ‘
x
’, while the remaining component is fixed.
Accordingly, function and argument in Begriffsschrift are only loosely related
to their arithmetical counterparts12.
9
Gordon Baker and Peter Hacker firmly and repeatedly defend this possibility in
‘Functions in Begriffsschrift’ [
Baker; Hacker, 2003
]. More recently, in Gottlob Frege: A
Guide for the Perplexed [
Kanterian, 2012
, pp. 128–129], Edward Kanterian contemplates
the need for a Begriffsschrift function to yield a value – either an object in the universe or
a truth-value – once combined with a suitable argument.
10See footnote 5.
11
This is exactly the aspect Frege emphasises when he compares these two concepts of
function in ‘Booles rechnende Logik und die Begriffsschrift’ [Frege, 1880, p. 26].
12
Our reconstruction of the two notions of function essentially coincides with the account
by Carl Theodor Micha¨elis in his review of Begriffsschrift [Micha¨elis, 1880]:
“The form of the function symbols is the same as the usual one of
mathematics. It differs in sense from the mathematical one since it
10 C. BADESA AND J. BERTRAN-SAN MILL´
AN
The sense in which the function-argument scheme is more general than
that of mathematical analysis is exemplified by Frege in ‘Anwendungen der
Begriffsschrift’:
“According to the more general conception of function [allge-
meinerer Funktionsbegriff ] that I took as a basis, we can regard
u+ 1 = v
as a function of
u
and
v
and can therefore view it as a particular
case of f(u, v).” [Frege, 1879b, pp. 204–205]
In analysis, ‘
u
+ 1 =
v
’ would never be seen as a function of two variables.
However, it is perfectly natural for Frege to take this expression – according
to Begriffsschrift – as a function of ‘u’ and ‘v’, that is, as ‘f(u, v)’.
There is another difference we want to consider regarding the two notions
alluded here. In mathematical analysis, every function yields a unique number
for every number. As a mathematician, Frege should know this and yet, for
some reason, he decides not to incorporate this fundamental characteristic
into his account of function in Begriffsschrift: in this work, a function is not
an assignment of a value to an argument. Only later does Frege generalise
the mathematical notion of function in such a way that it assigns objects
– and not merely numbers – to objects, as he explicitly affirms in Function
und Begriff [Frege, 1891, pp. 144–146]13.
3.2.
Function-argument vs. concept-object.
One of the first things we
have highlighted about the distinction between function and argument is
the fact that it is applied only to expressions – once they have been pre-
sented in an adequate way. We defend the claim that there is absolutely
no hint in Frege’s exposition which suggests that the field of applicability
of this distinction is more than linguistic in nature
14
. According to our
signifies, not the whole of the dependent expression of magnitude, but,
unlike the argument, only the invariant part of the expression. Also, the
logical function symbol allows interchange of argument and function.”
[Micha¨elis, 1880, p. 215]
This is specially relevant as Micha¨elis was a contemporary of Frege.
13
On some occasions, Frege uses the mathematical practice of taking
f
(
x
) as a function
of
x
in order to exemplify his function-argument distinction in Begriffsschrift ; for instance,
by saying that “we can consider
Φ
(
A
) as a function of the argument
Φ
” [
Frege, 1879
,
§
10,
p. 129, emphasis added]. However, these examples are not proof that the Begriffsschrift
functions yield values. First, they can be read coherently, according to our reconstruction,
with the other of cases in which Frege expresses himself in different terms concerning
the same distinction. Second, if there is a sense in which the combination of function
and argument provides a value in Begriffsschrift, it is just the trivial result of taking the
resulting expression of this combination as a value. It would be extremely difficult to clarify
the rest of Frege’s exposition of the function-argument dichotomy assuming the requirement
for every Begriffsschrift’s function to yield a non-trivial value for each argument.
14
Nevertheless, there has notoriously been controversy regarding the linguistic nature of
the function-argument distinction in Begriffsschrift. Michael Dummett, on the one hand,
and Baker and Hacker, on the other, are prominent contenders of the resulting polemic.
See Frege: Logical Excavations [
Baker; Hacker, 1984
, pp. 104–144] and the succession of
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 11
position, there is no denotation in pure concept-script and hence there is no
supra-linguistic dichotomy with which the function-argument scheme can
be related
15
. Furthermore, the analogy between the function-argument di-
chotomy – developed in Begriffsschrift – and the distinction between concept
and object – deployed in Frege’s later works – is problematic. Although the
two structures are clearly related, there is absolutely no reason to identify
them16.
In the previous section we have provided evidence against a particular
absolute sense commonly associated with Begriffsschrift’s functions. We will
now complement our discussion there by claiming that there is no absolute
sense that can be attributed to the notion of function in Begriffsschrift
according to which the function of an expression can be related with a
concept or a property.
First of all, there is no direct correspondence in Begriffsschrift between
a decomposition in terms of function and argument and the ontological
structure expressed by a statement. To put it in another way, the former
is not determined by the denotation of the components of an analysed
expression.
We have already argued that the analysis in terms of function and argument
is not absolute: what has been initially seen to be the function in a given
expression can be taken to be the argument afterwards. This could never be
the case if functions had an ontological counterpart. A direct consequence
of this is that the function-argument scheme cannot reflect the semantic
structure of atomic judgements. The expression ‘Socrates is mortal’ can be
analysed in such a way that either ‘Socrates’ or ‘is mortal’ be the argument.
In contrast, neither Socrates can be the concept of the atomic content
reviews and replies its publication generated: ‘An Unsuccessful Dig’ [
Dummett, 1984
],
‘Dummett’s Dig: Looking-Glass Archeology’ [
Baker; Hacker, 1987
], ‘Reply to ‘Dummett’s
Dig’, by Baker and Hacker’ [
Dummett, 1988
], and ‘The Last Ditch’ [
Baker; Hacker, 1989
].
Dummett argues that the distinction between function and argument in Begriffsschrift
is not tied to any supra-linguistic structure. The opposite thesis, according to which
the distinction between function and argument should be applied to the meaning of
expressions, has been more recently defended by Baker in “Function’ in Begriffsschrift:
Dissolving the Problem’ [
Baker, 2001
], by Baker and Hacker in ‘Functions in Begriffsschrift’
[
Baker; Hacker, 2003
] and by Michael Beaney in ‘Frege’s use of function-argument analysis
and his introduction of truth-values as objects’ [Beaney, 2007].
15
We refer with ‘pure concept-script’ to the isolated use of this formal system, such as
that of chapter II of Begriffsschrift. In pure concept-script, the only non-logical symbols of
the language are letters. See section 3.3 for a treatment of this notion and a discussion
about the differences between pure and applied concept-script.
16
A careful historical analysis of the sources shows that the general approach of defending
an analogy between Frege’s mature position concerning the notions of function and object
and his account of function and argument in Begriffsschrift – an analogy defended, for
instance, for the sake of an expected uniformity in Frege’s thought – is out of place.
See the comments on this matter by Hans Sluga in Gottlob Frege. The Arguments of
the Philosophers [
Sluga, 1980
, p. 139], Wolfgang Kienzler in Begriff und Gegenstand
[
Kienzler, 2009
, p. 57] and, especially, Richard G. Heck and Robert May in ‘The Function
is Unsaturated’ [Heck; May, 2013, pp. 826–827].
12 C. BADESA AND J. BERTRAN-SAN MILL´
AN
expressed by ‘Socrates is mortal’, nor can the property of being mortal be
its object: that would not only be impossible, but also completely senseless.
This shows that the analogy between a property or a concept and a function
is in general impossible to sustain. In fact, when Frege tries to explain
the distinction between concept and object or individual – as he does, just
after the publication of Begriffsschrift, in ‘Booles rechnende Logik und die
Begriffsschrift’ [
Frege, 1880
, pp. 16–18] –, he does not turn to the function-
argument dichotomy17.
On the second place, an absolute notion of function will not appear in
a substantial way in Frege’s writings until 1891, in Function und Begriff
[
Frege, 1891
]. The fact that Frege takes in this text the mathematical
notion of function as a basis for a precise characterisation of the notion of
concept has a singular relevance for our present discussion. It should not be
minimised that until 1891 the author does not discover a means to rigorously
associate functions and concepts. The completion of this association requires,
in addition to taking truth-values to be objects, considering concepts and
properties as functions – in the mathematical sense – from objects in the
universe to objects in the universe. This is exactly what Frege claims to have
accomplished in Grundgesetze:
“Moreover, the nature of functions, in contrast to objects, is char-
acterised more precisely [in Grundgesetze] than in Begriffsschrift.
Further, from this the distinction between functions of first and
second level results. As elaborated in my lecture Function und Be-
griff, concepts and relations are functions as I extend the reference
of the term, and so we also must distinguish concepts of first and
second level and relations of equal and unequal level.” [
Frege, 1893
,
p. x]
If Frege had already pressed the analogy between the notion of function
and that of property or concept in Begriffsschrift – as it is so often claimed
–, then he would not speak of the extension of the reference of the term, but
just of a matter of detail, and he certainly would not say that “Begriffsschrift
(...) no longer corresponds entirely to my present standpoint” [
Frege, 1893
,
§
11, p. 5, footnote 1] concerning the notion of function in Grundgesetze. In
the light of the previous text, this possibility is hard to accept18.
17
The fact that, in the papers elaborated straight after the publication of Begriffsschrift,
Frege does not allude to the distinction between function and argument when he wants to
render the concept-object structure is a further indication of the mistaken nature of Baker
and Hacker’s claim in ‘Functions in Begriffsschrift ’ that “he tied his concept of a function
to concept formation” [Baker; Hacker, 2003, p. 283].
18
Notice that the most prominent constituent of the Fregean notion of concept, namely,
insaturation, is completely absent in Begriffsschrift. It appears for the first time in a letter
to the philosopher Anton Marty (1847-1914) dated from 29th August 1882 [
Frege, 1976
, p.
101].
The differences between Frege’s perspective in Begriffsschrift and Grundgesetze – espe-
cially concerning the divergence between the function-argument scheme and the concept-
object scheme – are seldom noticed and almost never properly considered.
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 13
Most of the arguments employed to defend the close analogy between the
schemes function-argument and concept-object take the notion of conceptual
content as essential. For instance, authors such as Baker, Kanterian, Mark
Textor or Joan Weiner claim that there is a correspondence between the
linguistic structure of expressions – articulated by the distinction between
function and argument – and the semantic structure of their content
19
.
Accordingly, Weiner argues that “the value of [the function] is a planet
for the Earth as argument is, according to the views of Begriffsschrift, the
conceptual content of the sentence ‘The Earth is a planet’.” [
Weiner, 2004
,
p. 77].
However, the formulas of the concept-script cannot be linked to any
particular semantic structure, because they acquire such a variety of readings
that they cannot be assigned properly a definite meaning. We will see that
the letters of the concept-script can have instances of very different kind.
Consider Proposition (52) of Begriffsschrift [Frege, 1879, §20, p. 161]:
f(d)
f(c)
(c≡d).
(5)
This proposition is a basic law of the concept-script and has different readings.
We will express them in a contemporary formal language.
First, ‘
c
’ and ‘
d
’ can be seen as individual variables; this is the usual
interpretation when the concept-script is reconstructed as a system of first-
order logic. In that case, the letter ‘
f
’ would be a formula variable and
‘
f
(
c
)’ and ‘
f
(
d
)’ formulas where ‘
c
’ and ‘
d
’ occur as free variables. This first
reading is:
x=y→(φ(x)→φ(y)).(6)
A particular case of this reading consists of taking ‘
f
’ as a predicate
variable:
x=y→(Xx →Xy).(6’)
Nevertheless, Heck and May identify in ‘The Composition of Thoughts’ [
Heck; May, 2011
,
pp. 129–134] the reasons for the deep divergence in this matter that appears between
Begriffsschrift and Grundgesetze. They also defend this view throughout ‘The Function is
Unsaturated’ [
Heck; May, 2013
]. However, there is some tension in their analysis, as their
fluctuating position regarding Frege’s observance of the distinction between use and mention
in Begriffsschrift shows. As opposed to Heck and May’s account, our reconstruction of the
notion of function in Begriffsschrift defends clearly that no absolute sense can be attributed
to it and, at the same time, explains how this notion is related to the mathematical
notion of function. Moreover, in the following sections we clarify the link between the
function-argument scheme and the content expressed by an statement and suggest how
this distinction constitutes Begriffsschrift’s notion of generality. All these elements are
absent in Heck and May’s account.
19
See “Function’ in Begriffsschrift: Dissolving the Problem’ [
Baker, 2001
, pp. 537–538],
Frege: A Guide for the Perplexed [
Kanterian, 2012
, p. 139], Frege on Sense and Reference
[Textor, 2011, pp. 76–77] and Frege Explained [Weiner, 2004, pp. 76–77], respectively.
14 C. BADESA AND J. BERTRAN-SAN MILL´
AN
Second, ‘
c
’ and ‘
d
’ can also be propositionally interpreted. Hence, ‘
f
’
cannot be a second-order variable at all. In fact, it has no equivalent in the
formal language: it should be seen as an expression in which a sub-formula
is replaced with a logically equivalent formula. Accordingly,
(5)
can also be
read as:
If φand ψhave the same content, then if Φ(φ) then Φ(ψ).
This formula could be specified as:
(φ↔ψ)→(Φ(φ)→Φ(ψ)).(7)
Frege’s equality symbol ‘
≡
’ has two different readings in
(6)
and
(7)
: either
as an equality between terms or as a biconditional. We will see below an
example taken from Begriffsschrift which shows that the reading of
(5)
as
(7) agrees with Frege’s practice.
The processes of substitution of the concept-script demonstrate the variety
of kinds of instances a single letter can have. There is a uniformity in
the substitutions performed in contemporary logic: individual variables
are replaced with terms; propositional variables are replaced with complex
formulas; and so on. However, Frege’s practice in Begriffsschrift does not show
the same uniformity; most substitutions involve – according to contemporary
standards – a change in the interpretation of the original formula. Even
though, from Begriffsschrift’s point of view, they just imply a restriction on
the possible readings a single formula can have.
We will exemplify this last claim with a substitution performed by Frege
in some derivations. The formula:
d
c
(c≡d).
(8)
is obtained from the substitution of ‘
c
’ for ‘
f
(
c
)’ and ‘
d
’ for ‘
f
(
d
)’ in
(5)
.
These replacements can be schematised just by saying that Frege substitutes
‘α’ for ‘f(α)’ in (5)20.
20
In order to ease our explanation, we will use, from now on, small Greek letters – such
as ‘
α
’, ‘
β
’, ‘
γ
’ – as place-holders for the argument in a complex function. This is especially
useful in the explanation of the substitutions. Frege uses capital Greek letters – such as
‘
A
’, ‘
B
’, ‘
Γ
’ – for the same purpose. However, this introduces an ambiguity in the use
of these symbols, since they are, on the one hand, taken as representatives of particular
contents and, on the other, used to indicate how the argument and the function of a given
expression must fit together. Frege does not distinguish between these two contexts.
The substitution of ‘
α
’ for ‘
f
(
α
)’ in
(8)
occurs in the derivation of Proposition (89)
[
Frege, 1879
,
§
28, p. 181]; it is used by Frege only in those contexts where the argument
letters are propositionally interpreted. According to this propositional interpretation, ‘
f
(
α
)’
is a formula in which ‘
α
’ occurs: in particular, it can be ‘
α
’. A modern translation of
(8)
would be:
(φ↔ψ)→(φ→ψ).
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 15
While
(5)
can be read as
(6)
or
(7)
, the formula
(8)
only allows a proposi-
tional interpretation. Now, through the transition from
(5)
to
(8)
, we see
that it is difficult to determine in general the kind of instances ‘
f
’ can have
in
(5)
: this letter is eliminated in the substitution that yields
(8)
– which is
considered by Frege as an acceptable replacement – and hence can hardly be
taken as a letter that stands for properties.
We can draw two conclusions from this last remark. First, a single letter
of the concept-script must not have instances of a single and fixed kind. On
the contrary, it can be interpreted in a wide variety of ways
21
. Among the
letters of the concept-script there is no distinction between those expressing
generality over predicate symbols – symbols that denote properties – and
the rest. Moreover, there is no substantive and invariable role a letter will
play in all particular analyses in terms of function and argument. A function
letter can be the argument of a proposition and an argument letter can be
taken to be the function in some contexts. This does not preclude the fact
that an initial decomposition of an expression such as ‘
f
(
a
)’ will take ‘
f
’ to
be the function and ‘
a
’ to be the argument. For this reason we have used
the denominations ‘function’ and ‘argument’ to refer to these two sorts of
letters.
Second, the variety of possible readings that a single letter of the concept-
script acquires definitely shows the enormous generality such a letter ex-
presses, which Frege not only does not mean to avoid, but indeed is seen
by him as a tremendous advantage. This generality is a direct proof of the
plasticity the letters of the concept-script exhibit and also a strong argu-
ment against the thesis that the expressions of the concept-script properly
have a definite meaning – whose structure is reflected in their linguistic
decomposition.
3.3.
Application of the concept-script.
The usual discussion concerning
the nature of functions in Begriffsschrift is almost exclusively based on Frege’s
explanation of this notion in chapter I of this book. However, this introductory
chapter does not contain the concept-script proper. Frege uses examples of
natural language for explanatory purposes, but those examples are often taken
by advocates of a supra-linguistic interpretation of the function-argument
21This conclusion might be reinforced with the following formula:
(f(c)≡f(d))
(c≡d).
(9)
It does not appear in Begriffsschrift, but it can easily be shown to be a theorem of the
concept-script. While readings that are analogous to
(6)
and
(7)
can be extracted from
(9)
, there is another possible reading. Additionally, the letter ‘
f
’ can be interpreted as a
function variable. In that case, ‘
c
’ and ‘
d
’ are taken as individual variables. Hence, the
formula (9) would also be read as:
x=y→fx =fy.
16 C. BADESA AND J. BERTRAN-SAN MILL´
AN
structure as the only representatives – and as the unique object – of Frege’s
analysis. If they were indeed the case, the semantical interpretation of
the distinction between function and argument those particular examples
suggest would be reproducible in pure concept-script; but it is not. Even
if commentators such as Baker claim that Frege’s exposition in
§§
9–10 of
Begriffsschrift must be applied to judgements expressed in the language of the
concept-script, their arguments do not actually apply to pure concept-script,
but to a certain complementation of its language and that of a particular
subject matter22.
The language of the concept-script should not be seen as a formal language
in the contemporary sense, but as a device for the accurate expression of
the logical relations that link statements from a given subject matter –
paradigmatically, arithmetic. From this perspective, this formal system –
leaving aside provisionally what Frege does in chapter III of Begriffsschrift
(see section 4.2) – can be seen as a tool for developing a discipline such
as arithmetic, so that it provides neither a formalisation nor a reduction.
The concept-script, on the one hand, allows to express all logical relations
between arithmetical statements; and, on the other, makes manifest the
fact that any mathematical proof can be formulated as a series of strictly
regulated steps, regimented with a small set of inference rules. As a result,
the concept-script eliminates all trace of natural language from arithmetic.
The product of this complementation was known as logistics23.
In logistics, the symbols of a particular discipline are mixed with the
symbols of a formal language – and, in this case, with the concept-script
symbols. The result is a regimented language to which the formal resources
of the concept-script can naturally be applied. In particular, all expressions
can be analysed in terms of function and argument. Now, since the meaning
of the statements of this scientific discourse is structured – according to
its semantics – in terms of concepts, properties and objects, it is thus not
surprising to find certain analogy between a function-argument decomposition
and the semantical structure of a statement in the given subject matter.
If we consider expressions of pure concept-script, it should be acknowledged
that their only non-logical symbols are letters and therefore that they lack
interpretation. Such expressions do not appear in formalised arithmetic, but
they are components of a sort of superstructure or metalanguage. Recall
our formula
(5)
. Since it is not expressed in a logistic language, there is
no direct link between its symbols and any precise semantic content; the
formula is meant to be complemented with the language of a discipline. Only
the symbols of this applied language – and, consequently, its formulas – can
rigorously express the specific meaning with which the discipline is concerned.
In this sense, the relation between the formulas of pure concept-script and a
22
See “Function’ in Begriffsschrift: Dissolving the Problem’ [
Baker, 2001
, pp. 528–529].
23See, for instance, [Lewis, 1918, p. 3].
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 17
specific meaning is merely indirect. An example will help us to shed light on
our position. Consider the following instances of (5):
2+1>0
3>0
(3 ≡2 + 1)
32>1
3>1
(3 >1≡32>1)
As this example shows, the plasticity exhibited by the formulas of the
concept-script emerges as a tremendous advantage when this formal system
is used as part of logistics. Accordingly, the possibility to naturally and
fruitfully applying the formal resources of the concept-script to a particular
discipline is due to the fact that the expressions of pure concept-script do
not have any particular meaning and can be read in different ways.
4. Remarks on Frege’s exposition of generality
4.1.
Reconstruction of Frege’s account of generality.
Most modern
commentators defend the notion that the quantifiers of the concept-script
are interpreted over a general domain. Now, Frege’s explicit formulation of
the notion of generality is the following:
“In the expression of a judgment we can always regard the combi-
nation of symbols to the right of as a function of one of the
symbols occurring in it. If we replace this argument by a German
letter and introduce in the content stroke a concavity containing the
same German letter, as in
aΦ(a),
then this stands for the judgment that the function is a fact whatever
we may take as its argument. Since a letter which is used as a
function symbol, like
Φ
in
Φ
(
A
), can itself be considered as the
argument of a function, it can be replaced by a German letter in
the manner just specified.” [Frege, 1879, §11, p. 130]
After this explanation of the meaning of the quantifiers of the concept-
script, Frege specifies certain conditions for the instances of a quantified
letter. These conditions help to understand how the generality the ‘whatever’
expresses – that we interpret syntactically, as we defend in section 3.2 above
– should be handled:
“The meaning of a German letter is subject only to the obvious
restrictions that [1] the assertibility (
§
2) of a combination of symbols
following the content stroke must remain intact, and [2] if the
German letter appears as a function symbol, this circumstance
must be taken into account.” [Frege, 1879, §11, p. 130]
Departing from the traditional interpretation of this fundamental element
of the concept-script, we claim that, following what Frege establishes in these
18 C. BADESA AND J. BERTRAN-SAN MILL´
AN
two quotations, his account of the quantification in Begriffsschrift can be
systematised as follows:
(∗)
If ‘
f
(
a
)’ is any assertible expression where the letter ‘
a
’ is taken to
be the argument:
af(a)
stands for the judgement that
f
(
A
) is a fact whatever argument
‘
A
’ may be put in place of ‘
a
’ preserving the assertibility of the
expression.
This way of reconstructing quantification can be naturally applied to any
kind of quantified expression. In particular, Frege introduces a new function
letter – ‘
F
’ –, which is used only in chapter III of Begriffsschrift. As Frege
mentions, function letters such as ‘
F
’ can be taken to be arguments as well
and hence be replaced with a corresponding German letter
24
. In this sense,
a quantification over function letters should be seen as a particular case of
(∗). Accordingly:
(∗∗)
If ‘
f
(
F
)’ is any assertible expression where the letter ‘
F
’ is taken to
be the argument:
Ff(F)
stands for the judgement that
f
(
B
) is a fact whatever argument
‘
B
’ may be put in place of ‘
F
’ preserving the assertibility of the
expression.
The transition from (
∗
) to (
∗∗
) is just a matter of application. Let us see
an example of this kind of application. The following formula is applied by
Frege:
f(c)
b
(af(a)) ≡b
(10)
as follows:
f(F)
b
(Ff(F)) ≡b.
(100)
We have labelled Proposition (68) of Begriffsschrift as
(10)
. It should be
noted that, although Frege proves Proposition (68) as in
(10)
, he then uses it
– for instance, in the derivation of Proposition (77) [
Frege, 1879
,
§
27, p. 174]
– as in
(100)
, which is a particular case of
(10)
. Observe that the following
expression:
af(a)
is any quantified formula and, accordingly, that ‘
a
’ represents any argument.
In particular, ‘
a
’ can be a function symbol as well – as it is made explicit in
24
The first use of the quantification over function letters in Begriffsschrift can be found
in Proposition (76) [Frege, 1879, §26, p. 173].
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 19
the transition from
(10)
to
(100)25
. We will discuss the relationship between
the quantification of argument letters and that of function letters in section
4.2.
As we have said above, the distinction between function and argument is
not absolute: a function symbol ‘
Φ
’ can be the argument of an expression
such as ‘
Φ
(
A
)’, independently of the fact that – in other contexts – it would be
considered its function. Now, since the quantification of ‘
Φ
’ in an expression
such as ‘
Φ
(
A
)’ consists in a quantification of a function letter, it can be
expressed in the language of the concept-script as follows:
FF(a).
This stands for the judgement that
Φ
(
a
) is a fact whatever argument ‘
Φ
’ may
be put in place of ‘F’ preserving the assertibility of the expression26.
Our reconstruction in (
∗
) can be used to explain Frege’s restrictions to
the meaning of a quantified letter. First, Frege introduces a general criterion
for limiting the possible instances of a quantified letter: “the assertibility of
a combination of symbols following the content stroke must remain intact”
[
Frege, 1879
,
§
11, p. 130]. Accordingly, the acceptable instances are exactly
those expressions that, once added to the functional component, yield an
assertible expression. This criterion brings into question the possibility of
there being unlimited quantification in the concept-script, say, over absolutely
any meaningful expression.
The second restriction, as Frege puts it, establishes that “if the German
letter appears as a function symbol, this circumstance must be taken into
account” [
Frege, 1879
,
§
11, p. 130]. From a syntactical perspective, this case
of quantification does not differ from quantification over argument letters:
they both concern arguments. But there are two relevant circumstances
that only affect function letters: their arity and certain specific conditions of
assertibility applied to the judgements in which they occur.
The basic and minimal expression of the concept-script is a function letter
with one – or several – argument letters enclosed in parentheses; for instance,
‘
f
(
a
)’ or ‘
g
(
a, b
)’. Regardless of whether the function letter is the function
or the argument of a given – simple or complex – formula, it always occurs
with some symbols enclosed in parentheses. The number of these symbols
determines the arity of the function letter. Frege points out that every
instance of a quantified function letter must share the arity of this letter.
Thus, for example, a binary function letter is not an acceptable instance of a
unary letter. Moreover, as we will see in an example in section 4.2, in some
contexts the function component of an expression has to adapt itself to the
25
We have indicated only some of the substitutions that affect
(10)
. Specifically, Frege
substitutes a complex instance for ‘f’.
26
An expression such as ‘
f
(
a
)’ consists of two elements: either of them can be taken to be
quantified, and thus be the argument. The case (
∗
) generically represents the quantification
of an argument, whatever nature it has. Hence, this last form of quantification is only a
particular case of (∗).
20 C. BADESA AND J. BERTRAN-SAN MILL´
AN
circumstance that a quantified function letter occurs with the number of
symbols corresponding to its arity. This determines further delimitations for
preserving the assertibility of the whole expression.
An example will help us to clarify our exposition concerning this last
limitation. Some possible instances of the judgement:
FF(a)
are the following:
f(a) or 2 + a > 3,
or even:
a
a,
where the connective is an instance of the quantified letter ‘
F
’
27
. All these
examples preserve the assertibility of the whole expression. Moreover, even
the last instance can be schematised with Φ(A).
4.2.
Quantification and assertibility.
Our reconstruction of the notion
of generality can be supplemented by focusing on two particular issues:
the specificity of the quantification over function letters and the notion of
assertibility.
First, there is no hierarchy of quantification in our reconstruction: (
∗∗
)
is a particular case of (
∗
). The fact that we have made the case (
∗∗
)
explicit is simply a convenient way of providing a specific account of function
letters. This specificity should be clear in our discussion above. However,
the particularities of the quantification over function letters by no means
modify the unitary account Frege offers
28
. In fact, an example will show that
the author has no substantive difference in mind between quantification over
function letters and that over argument letters. In the derivations contained
in chapters II and III of Begriffsschrift, Frege gives expressions which are to be
substituted for letters in a given formula. Since letters express generality, we
can view this process of substitution as an instantiation. Hence, in the process
of derivation of a proposition and in the application of the substitutions, we
find a clear and reliable testimony to the way Frege handles quantification.
In the derivation of Proposition (93) [
Frege, 1879
,
§
28, pp. 182–183], Frege
proposes certain modifications to one of its premisses – Proposition (60). For
27But ‘a’ is not an acceptable instance of ‘F’.
28
The fact that (
∗∗
) is a particular case of (
∗
) could have induced Susan Russinoff – who
partly follows what George Boolos suggests in ‘Reading the Begriffsschrift’ [
Boolos, 1985
,
p. 339] – to state in ‘On the Brink of a Paradox?’ [
Russinoff, 1987
, p. 128] that the
quantification over function letters is a particular case of the quantification over argument
letters. However, according to her account, function letters take values over concepts and
relations, while argument letters take values over every entity in the universe. We will
discuss such a pre-established domain for quantified letters in section 5.3 below.
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 21
the sake of simplicity, we will consider these changes only in the antecedent
of (60):
af(a)
g(a)
h(a).
Frege applies this formula in the following way:
Ff(F)
g(F)
h(F),
which results from replacing ‘a’ with ‘F’.
The quantification of ‘
F
’ in this last formula has to be seen as an example
of our general reconstruction: it consists of a specification of the argument.
From a syntactical perspective, the reading of ‘
a
’ as ‘
F
’ involves the need
to consider two circumstances: the arity of the function letter and the
adaptation of the function to the new quantified argument. In the end,
function letters have a specific place in the expressions of the concept-script.
Accordingly, in the same derivation Frege makes certain substitutions in
the function, that is, replaces some of the letters that occur in the fixed
component with expressions that fit with the new argument. For instance,
Frege substitutes:
aα(a)
F(x, a)
for ‘h(α)’. Hence, the result of this change is:
Ff(F)
g(F)
aF(a)
F(x, a).
In this substitution, we can appreciate both the preservation of the arity
of ‘
F
’ in ‘
F(a)
’ and the adaptation of the function – through an instance of
‘
h
’ – to the assertibility of the whole expression, i.e., to the circumstance
that the argument is a function letter.
Second, we want to shed some light on the notion of assertibility. We
mention this notion through the foregoing exposition without properly ex-
plaining how it is to be understood in Begriffsschrift. Frege does not discuss
this matter further; even though it is a complex issue. In the concept-script
there seem to only be syntactical rules for the evaluation of the assertibility
of an expression; those rules establish, for instance, the correct formation of
a complex formula or the acceptable instances of substitution of a letter. An
22 C. BADESA AND J. BERTRAN-SAN MILL´
AN
example of a non-assertible expression in the concept-script can be:
a(f(a)≡b)
b
af(a),
for it produces a clash between the scope of two different quantifiers of the
same German letter29.
However, the notion of assertibility that is relevant in Begriffsschrift is
more than syntactic. In chapter II of this work, the concept-script is handled
in an abstract and isolated way. However, as we discuss in section 3.2, the
concept-script is also meant to be used as the basis for logistics.
In light of this use, it is thus not surprising that there is no definition of
atomic formula in Begriffsschrift: all atomic expressions – and the proper
symbols necessary for expressing it – are provided by a certain subject matter.
The language of the discipline this subject matter belongs to offers a set of
symbols and syntactic rules for the formation of atomic expressions, which
are supplemented by the formal resources of the concept-script. These make
possible to form complex expressions or to render quantification rigorously.
In consequence, the semantical and syntactical elements of the concept-script
that remain to be clarified are resolved by the application of this formal
system to some subject matter.
A particular example of this application occurs in chapter III of Begriffs-
schrift. In this chapter, Frege, on the one hand, introduces new letters, such
as ‘
x
’, ‘
y
’, ‘
z
’ – which are only interpreted as letters that stand for objects
–, or ‘
F
’ – which is interpreted as a letter that stands for properties –; and,
on the other, he uses the letter ‘
f
’ – which in chapter II has been used as
a generic unary function letter – as a binary letter that stands for proce-
dures. The new symbols are the minimum requirement in order to justify
how objects inherit properties in sequences generated by a procedure. The
content of chapter III is, according to Frege – and even from a contemporary
point of view –, pure thought and can be naturally associated with what
we now call “logic”. However, this chapter should be distinguished from
chapter II, since in the latter we find an example of what we have called
“pure concept-script”. First, Frege introduces in chapter III symbols that
have a fixed interpretation and, second, these symbols acquire a proper
meaning, though an abstract one. Hence, although ‘
x
’ and ‘
f
’ are letters
and still express generality, they do not acquire such a variety of readings as
an ordinary letter of the concept-script and it is possible to generate atomic
formulas – such as ‘
f
(
x, y
)’ – with these new letters. The formulas obtained
29
Frege does not rule out the possibility of having nested quantifications in the concept-
script, but he explicitly indicates that the nested quantifiers must bound different letters
[
Frege, 1879
,
§
11, p. 131]. Notice that the occurrence of nested quantifiers of the same vari-
able – as in
∀x
(
φ
(
x
)
→ ∀xψ
(
x
)) – does not establish a formula as incorrect in contemporary
logic.
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 23
using the specific symbols of chapter III can be interpreted exactly in the
same way as the formulas of contemporary formal languages. In this sense,
it can be said that, in chapter III of Begriffsschrift, the concept-script is
complemented with (applied) logic and therefore is used as the basis for
logistics.
The scientific discourses to which the concept-script is adapted determine
a specific domain of entities. The proper symbols of this subject matter
denote entities in that context, or relations and properties applied to the
basic entities. The concept-script does not alter the interpretation of these
symbols, but allows that its formulas be interpreted according to the scien-
tific discourse. Hence, the meaning of the adopted terms and the field of
application of the concepts of the subject matter indicates how the instances
of the quantified letters must be handled in the calculus. In this way, the
syntactic conditions of the concept-script are supplemented with semantic
ones; only the meaningful expressions of the scientific discourse taken into
consideration can be acceptable instances of the letters. Any such discourse
allows the intended interpretation of the letters of the concept-script to be
specified.
In this context, quantification is handled in the same way as in Begriffs-
schrift – that is, in a purely syntactical way – in spite of the fact that it
acquires a meaning it had not before. There is no modification in the use of
the quantifiers in the calculus, they are rather interpreted in such a way that
semantical restrictions are added to the usual syntactic ones, which we discuss
in section 4.1 above. Nevertheless, the possibility of reading the quantifiers
in the usual manner in arithmetic – as Frege does, for instance, in several
examples in ‘Booles rechnende Logik und die Begriffsschrift’ [
Frege, 1880
, pp.
21–27] – does not mean that they are semantically interpreted in the pure
concept-script. Quantification of a letter in logistics thus involves considering
those expressions that may be put in place of the letter. The difference
between quantification in logistics and in pure concept-script is that the
instances in the latter are only syntactically determined, while in the former
they are also semantically delimited.
5. The nature of the quantification of Begriffsschrift
The final issue we want to address is the possibility of offering an interpre-
tation to the quantifiers of the concept-script that is similar to contemporary
ones. This issue is of particular relevance, since there is a common ten-
dency to regard the concept-script of Begriffsschrift as a formal system of
second-order logic.
5.1.
Interpretation of the quantifiers.
In accordance with the traditional
interpretation of Begriffsschrift – shared, for instance, by Peter Sullivan
30
–,
it really is quite surprising that Frege had developed as early as 1879 almost
30See ‘Frege’s Logic’ [Sullivan, 2005, p. 662].
24 C. BADESA AND J. BERTRAN-SAN MILL´
AN
all those elements that a formal system must have according to contemporary
standards of rigour. Up to now, we have tried to show that most of the
similarities that lead to such a conclusion to be reached are merely apparent
31
.
In what follows, we defend that the usual reading of Begriffsschrift is biased
because of its submissiveness to a contemporary formal perspective in one
particular aspect: the presence of a semantics – in the contemporary sense –
for the concept-script. Specifically, scholars such as Beaney claim that Frege
offers a semantic interpretation of the quantifiers32.
Frege does not draw a neat distinction between syntax and semantics. Even
if all components of the language are handled exclusively from a syntactical
perspective, Frege offers explanations with semantic elements. After all,
the author is introducing a completely novel device and, through these
semantic remarks, Frege is probably trying to make all the new symbolism
he introduces both accessible and understandable to the reader. The three
instances of formula
(5)
we have considered provide evidence that these
remarks are not systematic. In spite of his particular exposition, Frege deals
with quantification in a purely syntactical way. Our reconstruction in the
last section makes this plain: the universal quantifier is a logical symbol
whose use in the concept-script is fixed exclusively by certain inference
rules. In the presence of a quantifier, the only elements to be considered are
those expressions that can be put in place of the quantified letter. We have
discussed two syntactical rules that delimit these instances; but there is no
trace in Begriffsschrift of an account of a general interpretation of quantified
letters.
Therefore, in each context of application of the formal resources of the
concept-script we should consider a specific range of expressions that could
be substituted for a quantified letter. This context of application is closely
related to the possibility of using the concept-script as logistics: the comple-
mentation of its language with the proper symbols of a given discipline affects
the assertibility of the expressions and, hence, the generality of the letters of
the concept-script. There are very general contexts of application as well,
such as chapter III of Begriffsschrift, where there are just a few syntactic
conditions that delimit the generality that letters express and, additionally,
certain letters – such as ‘
x
’ – that have a unique interpretation. We have
discussed these assertibility-related aspects in section 4.2.
Now, the range of expressions that can be taken as instances of a quantified
letter is almost completely dependent on this context of application. This
31
Nevertheless, we will further state elsewhere our position against an interpretation of
the concept-script as a second-order formal system. Specifically, all proofs and substitutions
in Begriffsschrift can be reconstructed coherently with a faithful interpretation of the
distinction between function and argument, and the particular theory of quantification
developed in this book. One of the main benefits of such a perspective is that there is
no need to save Frege from his so-called mistakes: each proof can be explained naturally
without constraining it to the tools available in a second-order calculus.
32See The Frege Reader [Beaney, 1997, pp. 378–379].
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 25
should be clear in our discussion concerning formula
(5)
; without a specific
formula, the possible instances of a quantified letter – either a function or
an argument letter – cannot be fully specified. Furthermore, the fact that
the quantified letter is either a function or an argument letter provides no
decisive information about the kind of expressions that can be substituted
for it. Unlike the variables in a contemporary formal system, the substitution
range of the letters in the concept-script is not determined beforehand.
5.2.
The possibility to quantify over functions.
We can discern in
the traditional interpretation of Begriffsschrift three fundamental claims:
argument letters must be considered individual variables; the domain of
these variables is the universal class – that is, the collection of all objects
in the universe; and the functions of Begriffsschrift – which are identified
with properties – can be quantified. The first and the third claims entail
the thesis that the concept-script of Begriffsschrift is a formal system of
second-order logic; while the second is part of the common interpretation of
Frege’s works.
In a substantial part of chapter II of Begriffsschrift only the propositional
fragment of the concept-script is used
33
. One conclusion of our discussion in
section 3.2 is that, regardless of this propositional fragment, in an isolated
formula the argument letters of the concept-script cannot be uniformly
interpreted and, in particular, they should not be taken to be individual
variables. This is our account of the first mentioned claim. In what follows,
we will confront the two remaining claims in detail.
In the previous section we deal with Frege’s account of generality. Ac-
cording to that account, any symbol in a judgement can be taken to be
the argument and then be replaced with a quantified letter. In particular,
function letters can be quantified. Many modern commentators – such as
Sullivan, Bynum or Beaney
34
– have taken this as an evidence that functions
are quantified in Begriffsschrift and hence, that the concept-script contains –
or should contain – a higher-order quantification.
Let us be clear concerning the account we want to oppose. When the
possibility to quantify over functions is defended, it is usually accompanied
by an association between Fregean functions and properties, that is, by
an attribution of an absolute sense to Begriffsschrift’s functions. This
view is probably influenced by Frege’s position on the matter expressed
in Grundgesetze. From this point of view, a function letter – seen as the
representative of a function in an absolute sense – expresses generality over
properties or relations and is therefore interpreted as a predicate variable
35
.
33
Specifically, in the presentation or derivation of Propositions (1) to (51) [
Frege, 1879
,
§§14–19, pp. 137–161].
34
See ‘Frege’s Logic’ [
Sullivan, 2005
, p. 667], ‘On an Alleged Contradiction lurking in
Frege’s Begriffsschrift’ [
Bynum, 1973
, p. 286] and The Frege Reader [
Beaney, 1997
, p. 76,
footnote 52], respectively.
35
T. W. Bynum makes manifest the association between function letters and functions in
‘On an Alleged Contradiction lurking in Frege’s Begriffsschrift’ [
Bynum, 1973
]. Moreover,
26 C. BADESA AND J. BERTRAN-SAN MILL´
AN
Hence, what is really claimed is that in Begriffsschrift there is a quantification
over properties, that is, some sort of second-order quantification. However,
as we discuss here, this interpretation is not tenable.
Certainly, there are many contexts of application in which the instances
of function letters are predicates or conceptual expressions. We are not
defending that this is not the case; but that this is not the only kind of contexts
in which a function letter can appear. Therefore, we are ultimately arguing
against a homogeneous interpretation of function letters in Begriffsschrift and,
in particular, against the claim that – according to such an interpretation –
only predicates can be substituted for function letters.
We have defended, on the one hand, that a function letter has no domain
of quantification, but a range of expressions that can be substituted for it;
and, on the other, that this range cannot be fully determined beforehand.
Even if under particular circumstances – for instance, in a proof – it is
possible to state what are the expressions that can take the place of a
function letter, those expressions do not necessarily refer to a property. In
consequence, function letters do not have a fully specified and general domain
of interpretation consisting of properties or relations.
The reading of Frege’s account of generality we are considering, according
to which functions – in an absolute sense – can be quantified, has led to
semantical interpretations of the quantifiers. The most substantive of them
can be called “type-neutral” interpretation
36
. According to it, a quantified
expression such as:
aΦ(a),
is ambiguous and, in particular, can be interpreted as expressing that
Φ
(
F
)
is the case whatever function – in an absolute sense –
F
(
α
) is taken as
in an editorial note in his translation of Begriffsschrift, he renders Frege’s phrase “
F
(
y
),
F
(
a
) (...) are to be considered different functions of the argument
F
” as “treating
F
(
y
) as
a function of the function F” [Frege, 1972, §27, p. 175, footnote 2, emphasis added].
36
See, for example, Peter Sullivan’s account of some applications of Proposition (58) in
‘Frege’s Logic’:
“(...) Frege appeals to (direct consequences of) his quantifier axiom
[Proposition (58)],
∀afa→fc
, in justifying second-order inferences,
substituting (as we would say) second-level predicates for first-level
predicate variables. It would be surely be a misdiagnosis of this to
hold that Frege cited a first-order axiom when he needed a second-order
one. Rather, his citing the axiom in these cases shows that he did not
understand it as first order, but instead as the type-neutral principle
that if a function holds of every argument it holds of any. It is indeed
overwhelmingly the natural view that there are type-neutral logical
principles, e.g. that there is a single principle of Barbara, exemplified
by properties of any level, to the effect that if wherever one property
applies a second does, and wherever the second applies a third does,
then wherever the first applies so does the third.” [
Sullivan, 2005
, pp.
672–673, our emphasis]
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 27
its argument
37
. Apparently, this type-neutral reading of Begriffsschrift’s
quantification could also explain the substitutions performed in the example
we have considered in section 4.238.
We argue in section 3.2 that there is no absolute sense that can be attrib-
uted to the notion of Begriffsschrift’s function. Accordingly, no hierarchy
of levels of functions can be defined in Begriffsschrift. In this work, Frege
considers properties and objects as ontological categories instead; for instance,
throughout chapter III he repeatedly says that certain objects have certain
properties. However, there is no evidence in Begriffsschrift that he intended
properties to have other properties – which would resemble the possibility
for a function (in the absolute sense) to be the argument of another function.
According to our reconstruction, the expression ‘
Φ
(
F
)’ can be analysed in
such a way that ‘
F
’ is the argument and ‘
Φ
(
α
)’ the function – or, at least,
part of it. Now, if ‘
F
(
β
)’ is a function in an absolute sense, say, if it stands
for properties, then an absolute sense can be attributed to ‘
Φ
(
α
)’ as well. A
natural question would then be: which one? Certainly, ‘
Φ
(
α
)’ cannot stand
for a property, at least in the same sense as ‘
F
(
β
)’. In order to make sense of
this claim, ‘
Φ
(
α
)’ cannot stand for a property of the same kind as ‘
F
(
β
)’, that
is, there must be a distinction between properties of different levels. However,
in Begriffsschrift there can be found no such distinction; there simply are
no types in this work
39
. Moreover, it is extremely implausible that Frege
intended to attribute an absolute sense to the notion of function in the 1879
work and, at the same time, ignored or even did not consider this question.
Therefore, the answer to this question is, in one way or another, unacceptable.
The ambiguity of a quantified expression – in the sense proposed by the
type-neutral interpretation – is thus hardly sustainable. In particular, the
reading of the quantifier as involving a quantification over functions – in
37
In order to render a function in an absolute sense in this discussion, we use ‘
F
(
α
)’,
where ‘
α
’ marks the incompleteness of the function. This use disagrees with our purpose
of introducing small Greek letters, but we think that it is clear enough in this particular
context.
38
Taking the substitutions performed in this example – and in all other examples of
similar nature – at face value, and recognising that all steps made in these substitutions are
valid have persuaded us to rule out an alternative interpretation, according to which the –
to the contemporary eyes – oddities of these substitutions are Frege’s inaccuracies. See, for
instance, T. W. Bynum’s account of the substitutions that affect Proposition (60) in the
derivation of (93) in his translation of Begriffsschrift [
Frege, 1972
,
§
28, p. 183, footnote 5].
39
Departing from a comparative analysis of the presentation of quantification in Grund-
gesetze and in Begriffsschrift and, additionally, a remark on the particular character of the
notion of function in Begriffsschrift, Heck and May come to the same conclusion in ‘The
Function is Unsaturated’:
“[T]here isn’t really a distinction between function and object in Be-
griffsschrift, and accordingly (...) there is no distinction between lev-
els, although there is a distinction between function and argument.”
[Heck; May, 2013, p. 827].
However, they do not definitely endorse there the claim that a function and an argument
are exclusively particular components of an expression, which we take to be essential.
28 C. BADESA AND J. BERTRAN-SAN MILL´
AN
an absolute sense – is incompatible with Frege’s account of generality in
Begriffsschrift.
When Frege alludes to the quantification over function letters and substi-
tutes function letters for argument letters, he is not indicating the ambiguity
of the quantifier ‘
a
’, but considering one particular application of its
general definition. To say that Begriffsschrift’s quantification involves exclu-
sively arguments is completely independent of the nature of the denotation
that any suitable argument of certain expression can have. In fact, there
is no need to consider a type-neutral interpretation of the quantifiers once
we overcome the possibility to attribute an absolute sense to the notion of
function. As Frege shows several times, we can take the letter ‘
F
’ – which
stands for properties in chapter III of Begriffsschrift – to be the argument of
a suitable function and hence quantify it. In this particular context, there
can be no sense of a function – relevant to Begriffsschrift – that can be
associated with ‘F’.
5.3.
Universality in Begriffsschrift.
It is common to defend that the
quantifiers of the concept-script are not restricted, and that the letters express
generality over an absolutely general domain
40
. We will finally discuss this
particular claim and defend the idea that there is no general and unrestricted
domain of interpretation in Begriffsschrift. The presence of a specific range
of expressions suitable for taking the place of a quantified letter goes directly
against this idea. In fact, it is possible that the substitution range of the
same letter does not contain the same expressions in two different settings.
By ‘setting’ we understand a collection of formulas that appear together for
some reason – paradigmatically, because they are premisses or assumptions
in a proof. The interpretation of the symbols must be homogeneous in order
to be consistent with their occurrence in all formulas in a setting. Even if
the notion of context of application is more general than that of setting, they
are closely related; they both impose conditions that delimit the assertibility
of the expressions of the concept-script and, consequently, the generality
its letters express. Thus, a formula in a single context of application – for
instance, that of arithmetic – can be part of different settings.
40
Bynum’s account regarding the domain of argument letters in Begriffsschrift exempli-
fies such a standard reading:
“Arithmetic, according to Frege, is the scientific discipline concerned
with numbers; therefore the range of arguments of its functions is the
numbers. Maintaining a strict analogy with this view of arithmetic, one
can say that, if logic is that scientific discipline concerned with anything
at all, then the range of arguments of its functions should be anything
at all.” [Bynum, 1972, p. 62]
Bynum later affirms that this range is the entire universe, but “not a limited “universe of
discourse”, but the universe” [Bynum, 1972, p. 64, footnote 44].
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 29
Consider Proposition (53) [Frege, 1879, §21, p. 162]:
f(d)
(c≡d)
f(c).
(11)
We will exemplify two different settings with the guidance of some proofs in
Begriffsschrift. The first setting can be found in the derivation of Proposition
(92) [
Frege, 1879
,
§
28, p. 182], where Proposition (53) – which we have
labelled as
(11)
– is used as one of the premises. In this derivation, Frege
does not use
(11)
as it stands, but performs certain substitutions in it. In
particular, ‘d’ is replaced with ‘x’, from which we arrive at:
f(x)
(c≡x)
f(c).
(12)
The letter ‘
x
’ – as is suggested by its arithmetical use – is interpreted
throughout chapter III of Begriffsschrift as a letter that stands for objects.
This means that ‘
c
’ has to be read as a letter that stands for objects as well.
Now, in the setting of the derivation of Proposition (92),
(11)
is interpreted
as (12).
As we have discussed, the letters ‘
c
’ and ‘
d
’ can be propositionally in-
terpreted. Proposition (57), which is very similar to
(11)
, is one of the
premises of the derivation of Proposition (68) – which we have labelled as
(10)
– [
Frege, 1879
,
§
22, p. 166]. The second setting is composed of formula
(10)41:
f(c)
d
(af(a)) ≡d
(10)
and formula (8):
d
c
(c≡d).
(8)
Leaving aside the fruitfulness of such a setting, since ‘
c
’ and ‘
d
’ must
be uniformly interpreted both in
(8)
and in
(10)
, the adequate instances
of ‘
c
’ are only formulas. Therefore, neither ‘
g
’ nor ‘2’, to mention just two
examples of different kinds, belong to the range of ‘
c
’. This is plainly coherent
with reading the formula
(11)
in the way indicated by
(7)
. Without such a
reading, it would not be possible to interpret the transition from (5) to (8).
According to modern standards of rigour, these kinds of substitutions entail
41
We have substituted ‘
d
’ for ‘
b
’ in
(10)
for the sake of the example. Clearly, this change
is only notational and does not affect the interpretation of (10) at all.
30 C. BADESA AND J. BERTRAN-SAN MILL´
AN
an inconvenient lack of distinction. They are, however, a clear advantage for
Frege, for a formula like
(11)
can be used in settings that have very different
logical interpretations.
No instance of ‘
c
’ here is an acceptable instance of ‘
c
’ in the previous setting:
a formula cannot be substituted for ‘
c
’ without affecting the assertibility
of
(12)
. This circumstance is plainly incompatible with the presence of an
absolutely unrestricted domain for ‘
c
’. In fact, as our two examples show, it
is even possible that the ranges of a letter in two different settings may be
disjoint.
6. Concluding remarks
In spite of what is usually asserted by modern scholars, the formal system
of the Begriffsschrift is extraordinarily singular. First, the concept-script
does not properly have either a language or syntactic rules for the definition
of such an important element as the notion of atomic formula. This can be
seen as an evidence that the concept-script is devised as a tool for developing
a discipline such as arithmetic, so that it expresses the logical relations
that link the statements of the discipline and also provides the laws of
reasoning needed to rigorously conduct proofs. In fact, Frege did not aim
at a formalisation or at a reduction. He confirms this in ‘
¨
Uber den Zweck
der Begriffsschrift’ [
Frege, 1882
] considering the nature of the propositions
in Begriffsschrift:
“These formulas
42
are actually only empty schemata; and in their
application, one must think of whole formulas in the places of
A
and
B
– perhaps extended equations, congruences, projections.
Then the matter appears completely different.” [
Frege, 1882
, p.
97]
In fact, when the formal elements of the concept-script are applied to
express, for instance, arithmetical sentences, the reason for the lack of a
proper syntax is patently clear. Atomic formulas are given by the arithmetical
language and are organically put in relation with the logical symbols of the
concept-script, thereby giving rise to complex formulas. The distinction
between function and argument in these formulas stands out as a fundamental
tool in this complementation, as it provides a natural and useful way to
articulate the acquired symbols and apply to them the formal resources of
the concept-script; for instance, quantification.
Second, the concept-script does not have semantics in the contemporary
sense of the term. Its quantification theory, based on the division between
function and argument, is only syntactically handled. This formal system
provides the inference rules that govern the introduction and the elimination
of quantifiers in the calculus and some syntactical limitations for the correct
application of these rules. There is no semantic interpretation for the letters,
42
Frege refers to some examples of combinations of conditional strokes and negation
strokes, where he uses Aand Bto mark the place of relevant expressions.
FUNCTION AND ARGUMENT IN BEGRIFFSSCHRIFT 31
whose generality appears to be of a complete different kind from that of
modern variables.
Defending the claim that concept-script of Begriffsschrift is a second-order
formal system entails contradicting key elements in Frege’s exposition. Such a
thesis demands, on the one hand, the presence a specific ontological structure
that determines both the decomposition in terms of function and argument,
and the semantics of a formal system; and, on the other, a complete and
anachronistic limitation to the flexibility that characterise the letters of the
concept-script. We have argued that there is strong textual evidence in
Begriffsschrift against the presence of these elements.
We have studied Frege’s exposition of the distinction between function and
argument, and the notion of generality in Begriffsschrift in detail, in order to
compare our findings with the traditional reading of this book. However, a
comprehensive evaluation of the more general thesis that the concept-script is
a second-order formal system would require a complete reconstruction. That
would far exceed our aim in this paper, and deserves a proper consideration
elsewhere.
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