The validity of this argument can be accounted for on Frege’s argument-function model, but not on the subject-predicate model. On the latter model, we can account for the validity of the move from 2 to 3, but not from 1 to 2. From 2 to 3, the same predicate, ‘killed Cato’ is applied to different subjects. But from 1 to 2, different predicates are applied, ‘killing himself and ‘killed Cato’. On Frege’s model, however, the same two-place relation-term ‘... killed ...’ is applied in each step. Cf
I. Rumfit’s ‘Frege’s Theory of Predication’, Philosophical Review (1993).
quantifier by disjunction. Together with other ways of listing characteristics, namely by use of negation and the conditional, one would have at hand the elements for representing the most complex of concepts. The criticism is rather that the Kantian (or Boolean) method lays down elements of the definition in ‘no special order’. Many concepts have structure involving quantifiers and other operators of differing scope. Yet no language before Concept Script was sufficiently equipped to represent this special ordering. Ambiguity of scope can mean failure to distinguish between a true and false s e n t e n c e . A l s o unless differences in scope are made clear, the validity of reasoning involving a definitional transform, or any other sentence, will not be represented, precluding it thereby from use in proof. This problem became pressing in mathematics of the nineteenth century. On the one hand, nowhere are sentences involving multiple generality more common than in mathematical discourse. On the other hand, mathematicians demanded greater rigour in proofs, but lacked sufficient techniques. Frege’s Concept Script provided the means. Consider the sentence, ‘F is everywhere continuous at <2 = (Vx) (Vy) (y > 0 —> -i (Vz) (z > 0 —> -i (Vw)
(|w| < z —> |f(x + w)-f(x)| < y)))’.*’ Because the scope differences of the quantifiers are clear, so too are the truth-conditions of the sentence, and thus also its contribution to the validity of the proof. Clear too is the irrelevance of the subject-predicate model to the logical syntax of this sentence.
More importantly for our purposes, Frege identified this more fruitful method of defining as warranting the view that, pace Kant, analytic truths could
For example ‘Every boy loves some girl’ is ambiguous as it stands. It says either that ‘Every boy loves the same girl’, or that ‘Every boy loves some girl or other’. Depending on what one
intended to say, it could be regarded either as (Vx) [B(x) —> (3y) G(y) a xLy], where [B(4)-^(3y)
G(y) A ^ y ] is a first level concept, taken as an argument for (Vx); or as (3y) (Gy a ( Vx) ( Bx- > xLy)), where the existential quantifier takes the remaining part as argument.
Frege extols his own method of forming concepts, emphasizing the use o f the quantifier, negation and conditional as central to it. ‘The content is to be rendered more exactly than is done by verbal language. There is only an imperfect correspondence between the way the words are concatenated and the structure of the concepts.’ See his ‘Booles Rechnende Logik und die Begriffschrift’ (1880-81), pp. 12-13, PW. For a useful survey of the kind of concept-formation identified here as ‘less finitful’ see H. Sluga’s ‘Frege Against the Booleans’ (1987), esp. p. 83, p. 84, p. 87, p. 89. However, H. Sluga’s paper falls short of making clear the kind of contrast Frege is drawing between ‘more fruitful’ and ‘less fruitful definitions’.
extend our knowledge. Referring back to the definitions proved in earlier sections of Grundlagen, Frege, in the following three pivotal extracts, draws a contrast between Kantian analtyic propositions and his own, cognitively more powerful, variety.
‘What we find in these is not a simple list of characteristics; every element in the definition is intimately, I might almost say organically, connected with the others.’^
Here again, in his reference to ‘simple list’, contrasted with his reference to
elements organically connected, Frege is advertising the more fruitful method of definition: namely as residing in greater representation of structure, particularly of scope differences, whether of quantifiers or other truth-functional operators. Frege continues
‘A geometrical illustration will make the distinction clear to intuition. If we represent the concepts (or their extensions) by figures or areas in a plane, then the concept defined by a simple list of characteristics corresponds to the area common to all the areas representing the defining characteristics; it is enclosed by segments of their boundary lines. With a definition like this ... we ... use the lines already given in a new way for demarcating an area [Fn: Similarly, if the characteristics are joined by “or”.] Nothing essentially new, however, emerges in the process.’^*
The section from which this pivotal passage is extracted makes clear, I think, that Frege has in mind here the containment view of analytic propositions, and the method of definition that had accompanied, or indeed informed, it. Hereafter, however, a swift switch occurs from discussion of the containment variety to the merits of the Fregean type. Frege indicates the latter below with reference to inference and proof.
‘But the more fruitful type of definition is a matter of drawing boundary lines that were not previously given at all. What we shall be able to infer from it, cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from it extend our knowledge, and ought therefore, on Kant’s view, to be regarded as
^ Op. cit. pp. 1(X)-01.
21
synthetic; and yet they can be proved by purely logical means, and are thus analytic
Unfortunately, these three (pivotal) passages, replete as they are with metaphor, do not, as they stand, make clear the contrast Frege seeks to draw here between, on the one hand, analytic propositions that, given fruitful definitions, do extend knowledge, and on the other hand, analytic propositions that, given the less promising variety, do not. In particular, one will still wish to ask what precisely is meant here by ‘extension of knowledge’; as well as ask what it is about quantifier- variable structure—a central characteristic of the ‘more fruitful method of defining’— that makes extension o f knowledge possible. For instance, does the remark ‘something new emerging’ pertain, as many Frege scholars have maintained, to a process of arriving at something conceptually new— hereafter the ‘inflated sense’ of extension of knowledge? And what precisely would that mean? Or does Frege have in mind a more deflated sense of ‘extension of knowledge?
Consider, first, what the latter possibility might mean. Analytic propositions do not extend our knowledge, Kant claimed, because they fail to comply with the condition:
Proposition, S, extends our knowledge if S's predicated material is not part o f the other constituents o f S.
Frege reconstructs the notion of analyticity in terms of logical proof,^^ involving the requisite kind of premises. In that case, he could claim that some analytic propositions comply with the above Kantian condition in the following deflated or minimalist sense:
Proposition S is an extension o f our knowledge if it is inferred from a set o f premises, included among which are 'fruitful definitions \ and such that while S is not a member o f that set, the elements o f S are elements^'^ o f the propositions o f that set.
22
Op. cit.
Nothing radical is intended here by my use o f ‘reconstructs’. Frege of course takes himself to be
explicating what he takes Kant to have had (albeit dimly) in mind. See Grundlagen, §3, fn. 1; §88,
fn. 1. Precisely what we are to make of that claim is controversial. We take up the issue in chapter six when we consider Frege’s alleged epistemology.
^ I use ‘element’ here neutrally with respect to whether Frege had a merelogical view of the structure of propositions or judgeable content.
Clearly this construal of Frege’s use of ‘extension of knowledge’ meets Kant’s condition. The epithet ‘extension of knowledge’ would apply to analytic propositions because, given ‘fruitful definitions’, the source of the predicated material would be shown to stem from other propositions alone— and because these propositions would be derivable exclusively from logical laws. But notice that no presumption is made here regarding the grasp of concepts not grasped prior to the deductive proof, or prior to any analysis of the content of our ordinary arithmetical language. If we could substitute the terms of, e.g. the sentence ‘7 + 5 = 12’, for definitions in purely logical terms and could prove the proposition by means only of these definitions and logical laws, then that proposition would be analytic. More, it would represent an extension of knowledge—but in the above
minimalist sense. No presumption would be made here about the ‘conceptually new’ because none would be required.
But just for that reason might this minimalist notion of extending knowledge be criticised. For prima facie, Frege in the pivotal extracts wishes to understand the pivotal connection—i.e. between fruitful definitions and extension of knowledge—in terms of the idea that analytic propositions can be informative, that they can yield the conceptually new: the inflated sense of the pivotal connection. As just seen, he says
(1) ‘[W]e ... use the lines already given in a new way for demarcating an area [Fn: Similarly, if the characteristics are joined by “or”.] Nothing essentially new, however, emerges in the process. (2) But the more fruitful type of definition is a matter of drawing boundary lines that were not previously given at all. (3) What we shall be able to infer from it, cannot be inspected in advance; (4) here, we are not simply taking out of the box again what we have just put into it.’
(Because the purport of this passage is not, to my mind, immediately clear, and for ease of exposition, I have divided the passage into four parts.) First, we have a contrast in sentences (l)-(2) between ‘nothing essentially new’ emerging and ‘what was not previously given at all’. But in what, precisely, does this contrast consist? It is not altogether clear that Frege does have ‘new concept’ in mind.
however one might understand the term ‘new’. Here ‘boundary lines’ refers to the extension of concepts, given in this case by definitions. By use of disjunction or conjunction, new concepts can be formed: we combine the boundary lines in new ways, yet without altering the existing lines of any single concept. If we use only these old methods, then the boundary lines will not reflect with sufficient clarity the structure of the concepts identified as the more fruitful: e.g. the ancestral relation, continuity, concept of limit, and number. The reason is that given above. Without Frege’s quantifier-variable technique, the boundary lines will not represent scope differences, and other important structure. The true boundary lines will not have been ‘previously given at all’. Accordingly, what is ‘essentially new’ need mean no more than that of rendering of the logical structure hidden by old methods of demarcating boundary lines. Hence, if we read the sentences of (2) in this way, then, modulo further consideration, my minimalist
construal of ‘extension of knowledge’ need not be unfaithful to the spirit, nor indeed to the letter, of the above pivotal extracts.
It may appear, however, that my interpretation of sentences (l)-(2) fails to cohere with lines (3) and (4). In (3) Frege says ‘[w]hat we shall be able to infer from [the fruitful definition] cannot be inspected in advance’. And in (4), he says ‘... we are not simply taking out of the box again what we have just put into it’. Surely, it will be said, these two lines speak of the inflated sense of ‘extension of knowledge’? But let us look more closely. In lines (3)-(4), Frege continues his contrast with the containment view of analytic proposition begun in lines (l)-(2). Given analytic propositions of the containment variety, we can inspect in advance what can be inferred from it, at least to a certain extent.^ If we know what is contained in the subject-concept f i . . . / n , then we know from our understanding of it what can be deduced from it. (If the subject term is A, then we can infer that ‘A is / i ’, ‘A is /2’, ‘A is f \ \ and so forth.) Line (4) gives the reason: we are merely
taking out of the box—the subject term—what we have just put into it. But this is not possible with Frege’s variety of analytic propositions. Firstly, there is no box
out of which to take what one has just put into it—Frege’s propositions are not of that form. Secondly, we cannot inspect in advance what can be inferred from a Fregean proposition alone; because to make such an inspection we require not one proposition, but several. Again, my minimalist reading of ‘extension of knowledge’ is not incompatible with the pivotal extracts.
But what, if anything, is the connection— alluded to in line (3)—between our alleged lack of foresight regarding what is inferable from a more fruitful definition, and the advantage of Frege’s method of defining identified above? And why might the connection matter? The advantage of Frege’s method identified above was that it could represent more concepts more thoroughly than could the Boolean method. On the inflated sense of extending knowledge, the connection in question lies with the grasp of new concepts, and that is why it matters. On the minimalist reading, by contrast, no such connection is there, nor need it be. And it is not there. To see this, we should recognise that lines (3) and (4) are apt to mislead. Taking line (3), suppose we have two analytic propositions, (A) and (B), both converted from a definitional identity, where (A) involves only Boolean methods, and (B) involves Frege’s more fruitful approach. Let them be used in separate proofs. Now proposition (A) might be of enormous complexity, prohibiting us from seeing what is derivable by its means. Indeed, the proof involving (B), consisting of nested generality, might be of considerably less complexity than the proof in which (A), the Boolean counterpart, occurs. We may well be able to see immediately, or even after some contemplation, the inferential possibilities availed by the definition, given other premises. Of course, given a suitably astute cognitive agent, this might be true of the most taxing definition, be the definition Fregean or Boolean. After all, Frege himself says
‘No doubt these propositions are in a way contained covertly in the whole set
taken together, but this does not absolve us from the labour of actually extracting them and setting them out in their own right’.“
My point is that, taken as a general remark about what is inferable given a fruitful definition, line (3) fails to capture what is crucial to the notion of fruitful
definition. The remark appears to rest on an observation about our psychological limitations.
The passage just cited returns us to line (4), in particular, to its potential to mislead. For the passage appears to conflict with the claim, suitably broadened, that ‘ ... we are not simply taking out of the box again what we have just put into it.’ For any analytic propositions that extends knowledge precisely does depend on taking something out of the box—now the set of propositions—whose contents we supplied on constructing the proof system. Unless of course the qualifier ‘simply’ carries some special, but as yet unspecified, weight. So if we do not read (3) and (4) simply in terms of the contrast set out above,^^ they would appear to be both misleading and false. Moreover, they will appear to be of little importance to the pivotal connection—between a fruitful-definition and extending knowledge. Again, fruitful definitions enable us to prove truths otherwise unprovable. This is because Frege’s method permits a more comprehensive representation of the structure of certain concepts than does the Boolean method. This by itself secures an extension of knowledge, given our minimalist reading of that notion. Pending the viability of alternative readings, it appears that that is all that is essential to the pivotal extracts. The rest, I suggest, is misleading rhetoric.^® As we shall now see, our construal of ‘extension of knowledge’ would take us a step forward in rebutting an objection, outlined above and considered below, to our strong hermeneutic claim. First let us further consider the inflated construal of the
pivotal extract.
§4.3. The Value of Deduction and Analysis: An Inflated Interpretation. M. Dummett has suggested that the pivotal extracts are a response to a two pronged challenge.^’ The first challenge is to explain how deductive reasoning from purely
^ Grundlagen §17, p. 23. My emphasis. See p. 122.
^ Of course its purpose might be to persuade a sceptical audience of the merits of analytic
propositions.
This is explicit in, for example, his ‘The Value of Analytic Propositions’, and his Frege:
logical notions can yield epistemically valuable truths: viz. extend knowledge in the robust sense. The second challenge is taken to be implicit in Frege’s overall response to the general question of how deduction can yield new truths, and concerns the validity of deductive reasoning. The validity of an argument is recognised by the common structures between the premises and conclusion. To account for the validity of deductive reasoning, we need to say something to the effect that knowledge of the premises, in particular knowledge of their logical form, could carry with it knowledge^® of the conclusion. But while its validity, if any, can be accounted for along these lines, it would not explain how the unexpected might arise in the course of the deductive proof.^^
The problem of the justification of deduction has of course a counterpart at the level of conceptual analysis—the paradox of analysis— and in his recent book, Michael Beaney claims that Frege is concerned to meet the problems that analysis presents. The paradox of analysis can be formulated as a challenge to say how ‘fruitful definitions’, which Beaney interprets (in part) as the analysis of concepts already grasped,^^ can be both correct and of ‘epistemic value’: viz. extend our knowledge in the inflated sense.^^ The analysans will be a correct analysis of the analysandum just in case they have the same content, but will be informative just
central passage in this way: ‘Extension of Knowledge, and “Fruitful Concepts”: Fregean Themes in
the Foundations of Mathematics’ Noûs (1995b), p. 432, as well as Dummett’s interpretation of the
solution. See below.
This is not to say that knowledge is closed under entailment—that if I know P, then I know the