3 Frege, Analyticity and Elucidation

Frege’s project is often characteriz ed as the attempt to show that arithmetic –

pace Kant – is a system of analytic a priori truths. Frege suggests as much
himself
in his Grundlagen of 1884. But it is not how he described it in his major work,
the Grundgesetze der Arithmetik (1893, 1903), where instead he talks simply of
‘reducing’ arithmetic to logic. By the time of the Grundgesetze, the introduction
of his distinction between Sinn (sense) and Bedeutung (reference or meaning)
had complicated matters, and, so it seems, Frege had come to have doubts about
the idea of analyticity.2 However, if we do characteriz e Frege’s project in a
Kantian context, and bring the Grundlagen and Grundgesetze together, then the
following can be of fered as Frege’s criterion for analyticity:

(ANF ) A truth is analytic if its proof depends only on general logical laws
and definitions. (Cf. GL,§3.)

This looks quite dif ferent from Kant’s of ficial criterion. So has Frege simply
changed the terms of the debate? If so, then Kant and Frege cannot be
considered to be of fering dif ferent answers to the same question (‘Are
arithmetical judgements analytic or synthetic?’), since they interpret the
question in dif ferent
ways. Indeed, the conflict between their positions appears to disintegrate, since
Frege would agree with Kant that arithmetical propositions are not analytic in
the sense captured in (ANK ). Frege is well aware of the methodological issue
here. In explaining his own distinction between analytic and synthetic truths in
the Grundlagen, he comments: ‘By this I do not, of course, wish to introduce
new senses, but only to capture what earlier writers, in particular Kant, have
meant’ (§3, fn. E; FR, 92). He returns to the issue in the concluding part of the
Grundlagen. The key passage is §88:

Kant obviously underestimated the value of analytic judgements – no doubt as a

result of defining the concept too nar rowly, although the broader concept used
here does appear to have been in his mind. On the basis of his definition, the
division into analytic and synthetic judgements is not exhaustive. He is thinking
of the case of the universal af firmative judgement. Here one can speak of a
subject concept and ask – according to the definition – whether the predicate
concept is contained in it. But what if the subject is an individual object? What
if the question concerns an existential judgement? Here there can be no talk at
all of a subject concept in Kant’s sense. Kant seems to think of a concept as
defined by a conjunction of marks; but this is one of the least fruitful ways of
forming concepts. Looking back over the definitions given above, there is
scarcely one of this kind to be found. The same holds too of the really fruitful
definitions in mathematics, for example, of the continuity of a function. We do
not have here a series of conjunctions of marks, but rather a more intimate, I
would say more organic, connection of defining elements. The distinction can
be clarified by means of a geometrical analogy. If the concepts (or their
extensions) are represented by areas on a plane, then the concept defined by a
conjunction of marks cor responds to the area that is common to all the areas
representing the marks; it is enclosed by sections of their boundaries. With such
a definition it is thus a matter – in terms of the analogy – of using the lines
already given to demarcate an area in a new way. But nothing essentially new
comes out of this. The more fruitful definitions of concepts draw boundary lines
that were not there at all. What can be infer red from them cannot be seen from
the start; what was put into the box is not simply being taken out again. These
inferences extend our knowledge, and should therefore be taken as synthetic,
according to Kant; yet they can be proved purely logically and are thus analytic.
They are, in fact, contained in the definitions, but like a plant in a seed, not like
a beam in a house. Often several definitions are needed for the proof of a
proposition, which is not therefore contained in any single one and yet does
follow purely logically from all of them together. (GL,§88; FR, 122.)3

In a footnote to the first sentence of this section, Frege refers to a passage in the
Critique in which Kant alludes to a dif ferent criterion, which might be
formulated
as follows:

(ANL ) A true judgement of the form ‘A is B’isanalytic if f its negation ‘A is

not
B’ is self-contradictory. (Cf. Kant, CPR,B14;A150–1/B189–91.)

This of fers a broader criterion, which can be readily extended to judgements

not
obviously of subject-predicate form, and which is closer to Frege’s own
criterion.
But Frege is right that Kant does think primarily of universal af firmative
judgements, and that Kant’s of ficial criterion is too nar row for application in
the case of mathematics. There is therefore some justification in Frege’s earlier
claim too that what he is doing is not introducing an entirely new meaning, but
merely making explicit what Kant really had in mind – in the sense that, had
Kant considered it properly, he would have endorsed Frege’s reformulated
criterion.
However, whether or not one agrees with Frege’s claim, the crucial point
for present purposes is that Frege recogniz es the need to address the
methodological issue. It is not enough for him simply to stipulate what he
means by ‘analytic’, and then proceed to prove that arithmetical judgements are
analytic in his sense. In calling them ‘analytic’, he is using a term with an
established history, and some comment is necessary on the relationship between
his use of the term and previous uses. Furthermore, Frege realiz es that he
cannot depart too radically from earlier uses, on pain of being misunderstood
and of making no significant contribution to the traditional debate. In motivating
his own project, Frege himself has to do history of philosophy. He may do so in
a crude form, and may rely (as he does in the Grundlagen) on second-hand
sources, but he cannot avoid saying something about previous philosophers, and
this requires interpreting them. If he of fered incor rect interpretations of Kant,
say, then he would be confusing rather than clarifying the debate, and hence not
showing the value of his own project.
But would it not have been better to have dispensed with the concept of
analyticity altogether? As noted above, this is just what Frege does from the
Grundgesetze onwards. After the Grundlagen, he never again characteriz es his
project by talking of arithmetical truths as ‘analytic’. Perhaps he came to the
view that he was not in fact using the term in the same sense as Kant – or at
least, that there were problems in claiming that he was. However, if we
characteriz e his project simply as showing that arithmetic is reducible to logic,
then the problem only re-emerges in a dif ferent form. For what is now required
is explanation of the relevant conceptions of reducibility and logic. 4 According
to Kant, arithmetic is not reducible to Aristotelian logic, and on this Kant is
quite right, as Frege would have been the first to agree. But what Frege did, of
course, was to broaden the scope of formal logic, by means of which the
reducibility of arithmetic to logic became a genuinely feasible project. But this
too is something that needs to be spelled out, as Frege both recognized and
undertook. He was at least as much concerned with expounding and justifying
his ‘Begrif fsschrift’ – his ‘concept-script’, as Frege called his logical system –
as he was in using it to demonstrate logicism. And he was well aware that this
also required comparing his system with those of others, as he does, for
example, in his two papers on Boole’s logic (BLC and BLF ), and criticizing the
vari- ous accounts of arithmetic and logical conceptions of other
mathematicians and philosophers, as he does in the first three parts of the
Grundlagen and at various points in the two volumes of the Grundgesetze. Once
again, in other words, Frege recognized the need to do a certain amount of
history of logic and philosophy to motivate his project.
This is not to say that Frege saw himself as doing history of logic and
philosophy as such. Nor is it to say that he did it well. The caustic tone he
increasingly adopted in his criticisms of other writers reflects a lack of the
charity and sensitivity that is essential in the proper interpretation of other
points of view.5 But Frege did realiz e that clarification of the most basic
concepts of a system or theory was a necessary complement to its formal
construction, and that this clarification had a dif ferent methodological status.
This comes out most explicitly in his late work, ‘Logic in Mathematics’, written
in 1914 (and only published posthumously). What he draws attention to here is
the role of what he calls ‘elucidation’ (‘Erläuterung’), which makes clear the
sense of an expression by example – by using it in context. 6 Since not
everything can be defined, we must rely on something else – elucidation – to
explain the meaning of the most basic terms of all. Since there is always the risk
of misunderstanding in attempts at elucidation, Frege writes, ‘we have to be
able to count on a meeting of minds, on others’ guessing what we have in mind’
(LM, 224; FR, 313). But, Frege goes on, ‘all this precedes the construction of a
system and does not belong within a system’ (ibid.).
Clearly, Frege sees elucidation as a non-scientific or pre-theoretical
activity. But, one might suggest, the most ef fective method of elucidation will
involve history of philosophy and science. We may still have to count on a
meeting of minds, but drawing on, making explicit, and to some extent refining,
what is our common conceptual heritage may be the best way of achieving this.
Frege relegates elucidation to mere propaedeutics, but if our concern is
genuinely with the foundations of a system, then elucidation is just as important
a part of the foundationalist enterprise as the formal construction itself.
Significantly, Frege himself uses the term ‘logical analysis’ (‘logische
Zerlegung’) in describing the work of elucidation (LM, 225–8; FR, 314–8),
suggesting an admission of its fundamental role. So although Frege’s own
emphasis is much more on systematic construction, even Frege, who is often
regarded as the archetypical denigrator of historical considerations, in fact
makes room for them.
To make explicit that the elucidation required here has an historical
dimension, let us call it ‘historical elucidation’. On Frege’s account,
while historical elucidation serves a preliminary pedagogic purpose, it plays no
further role once the concepts are fixed. To use Wittgenstein’s famous metaphor
of the Tractatus, elucidation is merely a ladder to be kicked away once the
conceptual framework is in place. But fixing concepts is itself an essential part
of the foundational enterprise, and if fixing concepts requires historical
understanding, then historical considerations enter at a much deeper level than
Frege was willing to admit. Indeed, if the aim of a foundational enterprise is to
reveal and articulate the underlying conceptual framework (and the ontological
commitments, and so on) of one of our actual practices, then does this not
presuppose understanding of that practice as it has been manifested up to now?
And how can we assess whether a prof fered set of concepts is foundationally
adequate without reference to our use of such concepts – and related concepts –
in the past?
By ‘elucidation’ Frege meant explaining the sense of the most basic terms
by using them in context. But clarity about their senses is achieved not just by
giving examples illustrating those senses but also by saying how those terms are
not to be understood.7 To find good examples, we need to draw on our past and
present practice. But since that practice will also involve the use of related
terms, and those same terms used in slightly dif ferent senses, and indeed, those
same terms used in confusing ways, we also need to distinguish ‘cor rect’ from
‘incor rect’ uses. So critique of past and present practice will also be required.
Properly understood, in other words, elucidation requires critical awareness of
our practice to date, and hence possesses an historical dimension. It is this that I
want to reflect by talking of ‘historical elucidation’.8