The Evolution of Modern Metaphysics: Making Sense of Things (51 page)

5. The Admissibility of Definitions

With senses now available, an obvious account is ready to hand of when a definition is admissible, namely when the
definiendum
and the
definiens
have the same sense. Call this the Simple Account.
⁴³
On the Simple Account the reason why ‘sister’ can be defined as ‘female sibling’ is (simply) that they have the same sense. By application of this definition, and others, the following sentence can be derived from primitive logical laws: ‘If you have no sisters, and that man’s mother is your father’s daughter, then that man is your son.’ On Frege’s conception, this means that the truth in question is analytic, despite its unobviousness – which is just what he wants to say about the truths of arithmetic.

But can we accept the Simple Account? Even if we grant that the notion of sense is as clear as it should be,
⁴⁴
there are grounds for doubting this account. For one thing, it seems not to apply to Frege’s own definitions. It seems not to apply to the definition of numerical identity in terms of the existence of a one-to-one correlation since the
definiendum
in that case involves reference to an object, a number, that might just as well not exist as far as the
definiens
is concerned.
⁴⁵
And it seems not to apply to the definition of numbers as sets since that definition,
qua
stipulation, precisely does not seek only to capture what someone grasps who understands the word ‘number’. True, any discrepancy between the Simple Account and Frege’s own definitions may tell against them rather than against it. To understand Frege’s project is not to endorse it. But the Simple Account seems not to apply to what we
find in standard dictionaries either. These make liberal use of extrasemantic empirical information.
Perhaps
they do this only when they forget that they are dictionaries and start acting the encyclopedia. But something else that we find in standard dictionaries, for which there should surely be provision, are ampliative definitions resting on non-empirical insights that go beyond whatever is grasped in ordinary understanding – for example the definition of a circle as a plane figure bounded by a line every point on which is equidistant from a given point. (And then of course there is the question of how far these insights in turn rest, as they might be thought to in this case, on Kantian intuition.)

All of that said, the Simple Account is very compelling. If the
definiendum
and the
definiens
in a definition do
not
have the same sense, how can the definition be admissible
as a definition
? Is not the very purpose of a definition to convey what must be grasped in order to understand its
definiendum
?

The tension between the intuitive appeal of the Simple Account and the grounds given above for doubting it lies at the basis of what is often called the paradox of analysis.
⁴⁶
I cannot hope in these confines to give a full response to this paradox. As it happens I believe that the Simple Account is wrong, on the grounds that there are many purposes that definitions can fulfil other than to convey the senses of their
definienda
.
⁴⁷
One such purpose, notable from mathematical contexts, is to demonstrate that truths of one kind are mimicked by truths of some other kind, a purpose that can be fulfilled by identifying the subject matter of the former with the subject matter of the latter. This is what Frege does when he identifies numbers with sets, though it remains moot whether that is as much as he himself wants from the identification. (For a non-mathematical application of this technique, see
Ch. 12
, §7.) It is important to appreciate, however, that even when the purpose of a definition
is
to convey the sense of its
definiendum
, it is not obvious that it can fulfil this purpose only if its
definiens
shares that sense. Suppose it is possible to carve a complex sense somewhere other than at its original joints, thereby creating new parts constituting a new sense. (
Necessarily
constituting a new sense? Yes, if senses are individuated finely enough for their identity to be sensitive to any such change of parts.
⁴⁸
) Given
two senses related in this way, the intimacy of the relation makes it an immediate conceptual necessity that they determine the same
Bedeutung
. Suppose next that a complex expression having one of these senses is defined by a complex expression having the other. Then the twin constraints set by the syntactic structure of the
definiendum
, on the one hand, and the intimate relation between its sense and that of the
definiens
, on the other, can ensure that the definition does indeed convey the sense of the
definiendum
. And
this
, arguably, is what we find in Frege’s definition of numerical identity in terms of the existence of a one-to-one correlation.
⁴⁹

I say ‘arguably’. Whether we really do find this depends, of course, on whether such ‘contraconstituent’ carving is possible, and, even if it is, on whether Frege’s definition provides a legitimate case in point.
⁵⁰
There is ample room for doubt. A similar definition that Frege gives of the identity of what he calls ‘courses-of-values’ (
Basic Laws
, §3), of which sets are a type, is demonstrably wrong, for reasons that we shall see in §7(a).
⁵¹
However that may be, Frege’s introduction of senses, which are themselves possible objects of manipulation, reconfiguration, and investigation, has certainly created possibilities for an appealing account of when a definition is admissible, an account that is far more robust, and far more congenial to Frege’s project, than the Simple Account.

But at what price?

6. The Objectivity of Sense. The Domain of Logic

There are several grounds for concern. One is that Fregean senses are in danger of forming a veil between us and the
Bedeutungen
that they determine, somewhat like the veil of perception that Descartes was forced to acknowledge between each of us and the material world (
Ch. 1
, §6).
⁵²
Another ground for concern has to do with Frege’s very characterization of a name’s sense as containing the mode of presentation of its
Bedeutung
. Does this not stand in some tension with his context principle, which, whether understood as a principle about sense or as a principle about
Bedeutung
– Frege formulated it before distinguishing between these – suggests that a name’s
Bedeutung
may have no ‘name-sized’ mode of presentation at all, no mode
of presentation short of whatever is involved in understanding whole sentences that contain the name?
⁵³
If so, so much the worse for senses. For the context principle surely embodies a genuine insight which, depending on how great the tension in question is, Frege’s introduction of senses has either obscured or, worse, violated. These two grounds for concern are related. Fregean senses seem to obtrude. The root problem seems to be that Frege has construed our making senses of things as a tripartite affair in which we are directly related to sense and sense is directly related to things, but we are related to things only indirectly, via this link.

Such concerns are real enough. The notion of sense has to be handled with great care to assuage them. On the other hand, it is not obvious that it cannot be. We can think of the sense of an expression as, in David Bell’s words, ‘the condition which anything must meet in order to be [the expression’s
Bedeutung
]’ (Bell (
1984
), p. 184).
⁵⁴
And we can think of grasping the sense, not as confronting some representation of the
Bedeutung
, but rather as knowing what this condition is, where such knowledge is, in Michael Dummett’s words, ‘manifested in a range of interconnected abilities’ (Dummett (
1991c
), p. 51). There is then no need to regard senses as opaque intermediaries between us and the
Bedeutungen
that we use language to talk about and with which our interests, at least in scientific and mathematical contexts, typically lie. We are nevertheless at liberty to treat senses, along with the
Bedeutungen
that they determine, as thoroughly objective, just as Frege wants us to do.

That said, there is room for doubt about whether Frege himself is always as circumspect and as restrained in his handling of senses as he should be. The objectivity that he accords them is not just thorough; it is Platonic.
⁵⁵
He sees senses as abstract entities whose existence is completely independent of us (see e.g. ‘Thought’). But surely, if senses are to be acknowledged at all, the objectivity accredited to them needs to be less extreme than that. While they may be independent of each of us, and in particular of each individual mind, they are surely not independent of
all
of us, and in particular of the meeting of our minds in communication. The former, less extreme objectivity is objectivity enough. Certainly, it stands opposed to the subjectivity of whatever ideas individuals privately associate with linguistic expressions, which is Frege’s principal requirement (see §4).
⁵⁶

The latter, more extreme objectivity which Frege accords senses – the Platonic variety – is curiously reminiscent of the objectivity that Hegel accorded concepts. For both philosophers, the stuff of thinking stands over against us, no less amenable to scientific investigation than the stuff of nature.
⁵⁷
Both undertake such an investigation. Both call their investigation ‘logic’. The fundamental difference between them is that, for Frege, the stuff of thinking is sharply separated from the stuff of nature (‘Thought’, pp. 336–337/p. 69); for Hegel, it constitutes the stuff of nature, as indeed it needs to in order to achieve its own full being (see §5 of the previous chapter). For Frege, logic is an attempt to make sense of something transcendent. For Hegel, the very distinction between the transcendent and the immanent is to be overcome.
⁵⁸

The fact that the subject matter of logic is transcendent for Frege does not of course mean that he takes it to be irrelevant to the natural world, still less that he takes it to be irrelevant to our thinking about the natural world. Frege expresses very clearly the relation that he sees between these in the following passage from the
Foundations
. (He couches the relation specifically in terms of ‘the laws of number’, but he would say the same about the laws of logic more generally.)

The laws of number [do] not … need to stand up to practical tests if they are to be applicable to the external [i.e. natural] world; for in the external world, in the whole of space and all that therein is, there are no concepts, no properties of concepts, no numbers. The laws of number, therefore, are not really applicable to external things; they are not laws of nature. They are, however, applicable to judgements holding good of things in the external world: they are laws of the laws of nature. They assert not connexions between phenomena, but connexions between judgements; and among judgements are included the laws of nature. (§87)

Elsewhere he clarifies what he means by a ‘law’ here, emphasizing, in radical opposition to Hume (see
Ch. 4
, §4), that he is not talking about regularities in how we, human beings, happen to think. Logic, for Frege, is emphatically
not
‘dependent on the science of
MAN
’ (Hume (
1978a
), Introduction, p. xv). Frege writes:

Logic has much the same relation to truth as physics has to weight or heat…. [It] falls to logic to discern the laws of truth. The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean
prescriptions
…. Laws of nature are general features of what happens in nature…. It is … in this [latter, descriptive] sense that I speak of laws of truth…. [But from] the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. And we may very well speak of laws of thought in this way too. But there is at once a danger here of confusing different things. People may very well interpret the expression ‘law of thought’ by analogy with ‘law of nature’ and then have in mind general features of thinking as a mental occurrence. A law of thought in this sense would be a psychological law…. That would be misunderstanding the task of logic…. I assign to logic the task of discovering the laws of truth, not the [descriptive] laws of
taking
things to be true or of thinking [as a mental occurrence]…. (‘Thought’, pp. 325–326/pp. 58–59, emphasis added; cf.
Basic Laws
, Introduction, pp. 12ff., and ‘Logic’, pp. 246ff./pp. 157ff.)
⁵⁹

The realm of sense, which is set over against the realm of ‘actual’ things in space and time,
⁶⁰
has its own laws, then, and occurrences in our mind had
better
conform to those laws if they are properly to count as thinking, if we are properly to make sense of things. Hence, although Frege is far from endorsing the Hegelian tenet that the rational is actual and the actual rational, he does believe that we cannot properly make sense of what is actual except by doing so rationally. Indeed, he is prompted to ask, rhetorically, ‘What are things independent of reason?’ (§26, translation slightly adapted). And although his question is rhetorical, he adds a comment which shows that he takes rationality to be a prerequisite not only of proper thinking, or of proper sense-making, but of thinking at all, or of making any kind of sense; that is, he takes there to be no doing these things except properly doing them. ‘To answer that,’ he says, in other words to answer his own rhetorical question, or in yet other words to make sense of things beyond the limits of rational sense-making, ‘would be as much as to judge without judging’ (ibid.). Here there is an echo of Hegel’s variation on the Limit Argument (
Ch. 7
, §2), in which there was in turn an echo of Fichte’s variation on the Limit Argument (
Ch. 6
, §3). The conclusion in all three cases is that there is
no limit to that of which (thin, rational) sense can be made: there is nothing which, by its very nature, fails to make (thin, rational) sense.
⁶¹