CONTENTS

1    THE PROBLEM

2    A CONJECTURAL SOLUTION

3    TESTING THE CONJECTURE

4     CONCLUSIONS - A THEORY OF THEORY

2   A CONJECTURAL SOLUTION

My conjecture is that theorisation involves a process which can be called Fregean Analysis. The thesis of the “Primacy of Analysis” then explains why we need theory as an indispensable part of science. This section consists of an exposition of the thesis, a defence thereof against the counter-arguments put forward by Dummett, and a deeper account of what is involved in this analysis, under the heading of “articulation”. It is worth noting at this point that this essay stands as an example of its own subject matter. It strives to develop a better understanding of understanding, and a theory of theory.

2.1       The Thesis of the Primacy of Analysis

The part of Frege’s thought to which I am appealing and which I call the thesis of the Primacy of Analysis (PA), is that discussed by Sluga under the heading of “The priority of judgment over concepts”. The passages in which Frege expresses this doctrine most clearly are the following.

 “For in Aristotle, as in Boole, the logically primitive activity is the formation of concepts by abstraction, and judgement and inference enter in through an immediate or indirect comparison of concepts via their extensions. ... As opposed to this, I start out from judgements and their contents and not from concepts. ... I allow the formation of concepts to proceed only from judgements. ... And so instead of putting a judgement together out of an individual as subject and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of a previous judgement.”

“What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word ‘true’, and then immediately go on to introduce a thought as that to which the question ‘Is it true?’ is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgement; I come by the parts of a thought by analysing the thought.”

The idea of the primacy of analysis can also be found underlying the notion of incomplete expressions. Sentences cannot be regarded as being made up of elements all of which are complete. The incompleteness of the predicate comes from the fact that it is not an elemental building block out of which sentences are formed; the incompleteness is a sign of the irreducible sentence-ness which remains with predicates. The sentence comes first and the predicate is derived from it. At the more fundamental level of sense, the thought comes first and, as Frege says, its parts are then found by analysis. If a name-sense is extracted from the thought, what remains of the sentence is the predicate-sense.

The sentence has the essential power of being able to say something, of being able to express a thought. No mere collection of names can do this. If a sentence is analysed into predicate and name, the expressive power stays with the incomplete piece, the predicate.

Different aspects of the PA thesis can be displayed by expressing it successively at the three levels of reference, sense and language.

Reference (PA-R):  What we are primarily given by experience are a range of truths. We must then carry out Fregean Analysis to discover the referents (at different levels of the Fregean hierarchy) which these truths are truths about. This idea reappears at the beginning of the Tractatus: “The world is the totality of facts, not of things” – albeit expressed in terms of Wittgenstein’s logical ontology, not Frege’s.

Sense (PA-S):  The sense we are primarily given by experience is a thought. We must then carry out Fregean Analysis to discover the components of the thought, in order that we can be said properly to grasp the thought we have been given.

Language (PA-L):  The linguistic items which correspond directly to our primary experience are unanalysed sentences, and our job is then to find the right vocabulary to convert these into properly articulated linguistic items.

The doctrine is at its weakest at the level of language, and it is here, we shall see, that it bears the brunt of Dummett’s counter-arguments. The nearest we get to the unanalysed truth-bearer may be the sentence “Yes” (and its negation, “No”). Here we could be thought to be expressing a truth, but without being able to say anything about what it is true about. (Of course in the actual use of “Yes” and “No” in language, the sentences being expressed are determined by the context, for example, by the question to which they are given as answers.) It is better though to abandon the PA thesis at the level of language, saying that sentences are intrinsically composite, and only to be deployed once one has produced an analysis of the underlying thought, however conjectural and provisional it may be.

This strand of thought is brought together with the problem of understanding, posed in the first section, in the central conjecture of this essay.

CC      Fregean Analysis is what converts knowledge into understanding.

To build a realist theory of theory, we combine this with the thesis of Fregean realism: Fregean analysis does not create the items it uncovers; they were objectively there in the phenomena, awaiting discovery by us. If the analysis has been successful the resulting structure is an objective feature of the thought, which was there before we began the analysis but which was then unknown to us.

The motivation behind the conjecture is that the picture of first being presented with a brute fact and then having to hack out some way of articulating what the fact is, feels intuitively similar to what we do when struggling with the problems of understanding, be it in science or in philosophy. This similarity suggests that by exploring the notion of the Primacy of Analysis in a Fregean context we might obtain an understanding of understanding. This then might in turn explain why in science we need theory as well as experiment.

Dummett has subjected the idea of the primacy of analysis to a penetrating critique. This involves splitting the Fregean notion of analysis into two processes which Dummett calls “analysis” and “decomposition”, and then making a parallel distinction between “simple” and “composite” predicates. Dummett contends that he “did not really mean to correct Frege, but only to emphasize something which he glosses over, but which is not merely consonant with his views but important for the avoidance of a misunderstanding of them” (IFP p 292). We can not hope to build an understanding of understanding upon the PA thesis without taking into account Dummett’s work on the subject.

In connection with the PA thesis, Dummett points out that Frege appeared to hold  two pairs of contradictory theses.

“Frege held the following theses:

A1.  A thought my be analysed in distinct ways.

A2.  A thought is not built up out of its component concepts; rather, the constituents of the thought are arrived at by analysis of it.

But he also held the following two:

B1.   The sense of the parts of a sentence are parts of the though expressed by the whole.

B2.   A thought is built up out of its constituents, which correspond, by and large to the parts of the sentence expressing it.

On the face of it thesis A1 and A2 are in conflict with B1 and B2; A2 and B2, in particular appear to be flatly contradictory. It thus seems that there is a radical inconsistency at the heart of his philosophy. The difficulty is not to be circumvented by denying that he held any of these theses; there are numerous expressions of all four of them.” (IFP, p261)

Thesis A2 is effectively PA-S, the primacy of analysis expressed in terms of sense. Thesis B2, states the opposite, the primacy of synthesis. As Dummett points out, in the same “Notes for Ludwig Darmstaedter” which provided one of the statements of A2 quoted above, there is also the following statement of B2.

“We can regard the sentence as a mapping of a thought: corresponding to the whole-part relation of a thought and its parts we have, by and large, the same relation for the sentence and its parts.”

The problem is not merely that Frege may have contradicted himself, but rather that the contradiction appears to be organic to his philosophy. There are powerful motivations from within the body of his thought for both A2 and B2. If we are to build upon the intuitions underlying A2, we must also take into account those underlying B2.

One of the most striking features of language, and our usage of it, is our ability to take elements from a common vocabulary of expressions, and construct entirely new sentences. Even when never heard before, these sentences are intelligible to all those who share an understanding of the vocabulary (at least in principle – complex sentences, such as may be found in works of philosophy, often baffle even when every individual word is understood). According to Frege this is because the users of the vocabulary grasp the senses of the expressions, and then express thoughts by showing how to combine these senses by the way they combine the words to form sentences. In this picture the sentence fragments and thought fragments are primary, and the sentences and thoughts secondary creations.

Dummett resolves the contradiction by making a distinction, which he claims to be implicit in Frege’s work, though not brought out explicitly, between:

            analysis of a sentence into its constituents,  and

            decomposition of a sentence, revealing components.

For a full explanation of this distinction, see IFP, chapter 15. What follows here is enough of an account to show how what Dummett is doing differs from the ideas concerning the PA thesis developed below.

The process of analysis simply reverses the original synthesis of a sentence from its constituent expressions. Since each sentence is made out of a unique set of expressions, analysis has a unique end point. As an example let us take a sentence:

(1)                                            "x [k(x,a) Ž h(x)]

Successive analyses of this sentence, reversing the manner in which it was built up, yield the following constituents:

• the quantifier “"x [F(x)]”,
• the sentential operator  “pŽq”,
• the relational expression “k(x,h)”,
• the predicate “h(x)”, and
• the name “a”.

(where the following notation for argument places is used: F for predicates, p and q for sentences, and x and h for names).

The process of decomposition by contrast works by extracting instances of constituents from sentences, leaving behind complex predicates which are not constituents of the original sentences, that is, they were not used in building up the sentences in the first place. While analysis is a set of steps, prising the constituents apart, with a unique end-point, decomposition takes place in a single step, removing the constituent instances. The result is not unique; another choice of removals would yield as a component a different complex predicate. As an example let us take the sentence:

(2)                                            k(a,a) Ž h(a)

The list of its ultimate constituents is the same as (1), with the omission of the universal quantifier. However, by removing the first and third instances of the name “a”, we arrive at the complex predicate

(3)                                            k(x,a) Ž h(x)

This is not one of the constituents of (2). It is however one of the patterns to be found within it, and it is this sort of pattern which Dummett calls a component. We need to recognise it as as being in some sense contained in (2) if we are to recognise the validity of the entailment: (1) entails (2). This validity depends on having the same pattern (3) in both (1) and (2). In (1) it is actually an intermediate constituent, but in (2) it is only a component.

Once this distinction has been made, the apparent contradiction between the A and the B theses can be resolved. The A theses refer to decomposition, while the B theses to analysis, in Dummett’s terminology. Within this terminology the “Primacy of Analysis” must be renamed the “Primacy of Decomposition”, and now it refers only to the complex predicates like (3). They can be understood only in terms of taking a pre-existing sentence and removing parts, to expose an incomplete expression implicitly contained within the sentence, even though it was not used in the original construction of the sentence from atomic expression.

Dummett goes on to argue (IFP chapter 16, and earlier in FPL chapter 2) that there is therefore a strong distinction to be made between simple and complex predicates. (For convenience we use the word “predicate” here to cover all predicates, relational expressions and functional expressions, for want of a general word which does all these things.) Frege’s central doctrine that predicates are incomplete (unselbständig) applies strictly only to complex predicates. Dummett’s argument (IFP pp 292-293) can be paraphrased as follows. Consider the sentence

(4)                                            "x$y [(x<y) Ł p(y)]

If the domain of quantisation is the natural numbers, “<” is “less than” and “p(x)” is “x is prime”, then (4) asserts the infinity of the primes. The first step in its analysis is to split it into the universal quantifier and the complex predicate:

(5)                                            $y [(x<y) Ł p(y)].

This predicate is a component, though not a constituent, all sentences of the form

(6,n)                                         $y [(n<y) Ł p(y)]

where “n” is the name of some natural number. To grasp the sense of the predicate (5) is to grasp the common feature of how the truth-value of each sentence (6,n) is determined. This grasp in turn derives from an understanding of the constituents of (6,n). Since (5) is not among these constituents, this explanation of the sense of (5) is not circular.

On the other hand if we take the simple predicate “p(x)” (assuming for sake of argument that it is indeed simple, and not to be revealed as complex by means of a definition), we might try to explain its sense in terms of a grasp of sentences of the form “p(n)”. But when in turn we try to explain this sense in terms of its constituents, we find that “p(x)” turns up there, and the explanation is circular.

Concerning simple predicates, Dummett concludes (FPL p32)

“Simple predicates are selbständig in the way that complex ones are not: they are merely words or strings of words which can quite straight forwardly be written down. In one sense, of course, they are incomplete – they do not constitute a sentence, a ‘complete utterance’: but in that sense, proper names are equally incomplete. It is true that, in order to give an account of the rules govering the formation of atomic sentences, we must explain the ‘valencies’ belonging to different words – which expressions can, and which cannot, be juxtaposed, and when we have a whole sentence and when only a fragment on one. It is also true, as we have seen, that, in stating these rules, it is to the simple predicates and relational expressions that we must assign slots into which singular terms have to be fitted, rather than ascibing to the singular terms slots into which the predicates and relational expressions have to be fitted. But this does not make the simple predicates incomplete in the sense that Frege intended when he spoke of incomplete expressions. We might say that, in the case of simple predicates, the slots are external to them, whereas in the case of complex predicates, they are internal. That is, we can know what linguistic entity, considered just as a sequence of phonemes or of printed letters, a simple predicate is, without knowing anything about the slot it carries with it: the slot consists merely in the predicate’s being subject to a certain rule about how it can be put together with a term to form a sentence. But the complex predicate cannot be so much as recognized unless we know what slots it carries: they are integral to its very being.”

Before returning to the distinction between analysis and decomposition, let us consider this assertion about the completeness of simple predicates. Although the way simple predicates are formed by removing names from sentences may not have the full explanatory power it has when complex predicates are thus formed, surely the presence of slots, to indicate the “valency” of the expression is surely sufficient for incompleteness. The final argument that a predicate can be expressed as a row of letters or phonemes, without reference to its slots, seems implausible. How can I begin to understand “is prime” unless I realise that it is a predicate, with a name-slot in front. The presence of the letters “is” indicates to the English speaker that there needs to be a name in front to complete a sentence, but this is really a convention for indicating the presence of a slot, using an English word, rather than the Greek letter (which is a much more powerful way of locating the slots). The idea that we do not need the slots seems to be harking back to the bad old pre-Fregean days of singular and general terms, to be joined by a copula, a muddle and a mystification which Frege’s notion of predicates as incomplete expressions swept away utterly.

As to the circularity in the explanation of the sense once we arrive at the simple predicates, this chicken-egg cycle has to be broken somehow. While it is true that once we have “p(x)” in our vocabulary, we can use it to build up new sentences, the question remains: how did it get there in the first place. Rather than simply being presented with the predicate and its sense directly, I wish to argue that we start out with an inarticulate perception of something special about and held in common between the natural numbers: 2, 3, 5, 7, 11, 13, 17, and so on. From the instances of these truths, we then extract the sense of “p(x)”. (This is, unless “p(x)” is just a shorthand for a complex predicate – in which case we do get the predicate sense itself directly, by synthesis and if decompostion from other elements in our vocabulary.) It is how we might extract the simple predicates from inarticulate thoughts that I now want to discuss.

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