It is common knowledge that one cannot understand Wittgenstein's Tractatus without being acquainted with Frege's logical theory. Yet there is little agreement about exactly how the two doctrines are related to each other. The aim of the present paper is twofold. Firstly, I want to trace Wittgenstein's Grundgedanke to a deep-seated flaw in Frege's theory of Functions. Secondly, I will briefly discuss a modification of the Tractarian doctrine which would suggest itself if the flaw was to be rectified.

The problem in Frege's theory is best brought out with an example. Consider the following two methods of computing a number from an arbitrary couple of given numbers. One method consists in multiplying the members of the couple and then adding the second member to the result. The second consists in taking the successor of the first member and multiplying the result by the second member. Symbolically, the two calculation methods can be represented as

+(.(x,y),y) and .(Succ(x),y)

respectively. For any two numbers as arguments, each of the two methods yields a unique number, the number which one ends up with if one applies the method of those arguments. In this sense each of the method realizes a binary mapping from numbers to numbers. As it happens, the two methods realize one and the same mapping, call it M, which takes 1 and 1 to 2, 1 and 2 to 4, 2 and 1 to 3, 2 and 2 to 6, etc. Computational methods are thus not to be confused with the mappings they realize. A computational method, or procedure, is a structured complex containing mappings. The first of our two methods, for example, contains the multiplication mapping and the addition mapping, while the second contains the multiplication mapping and the successor mapping. The mapping M realized by such a method, on the other hand, is simply the association which the method sets up between arguments and the corresponding values. In +(.(x,y),y) and .(Succ(x),y) we have two different methods for calculating the value of M for any couple of arguments. Yet another, more direct, method consists simply in applying the mapping M itself:

M(x,y).

This trivial method is easily confused with the mapping, but the two should be kept apart: a mapping is one thing and the calculation method consisting in applying it to an unspecified couple of arguments is another. Let us call trivial computational methods li M(x,y) primitive.

It is tempting to liken a computational method to a computer program. In my view, however, a computer does not really compute. It does not, for example, apply arithmetical mappings to numbers. A computer is basically a symbol-shuffling machine, one which transforms strings of symbols into other strings of symbols. One can program a computer in such a way that a numeral corresponding to a number will be transformed into a numeral corresponding to the value of a mapping at that number. But this does not mean that the computer itself applies the mapping to the number. A

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computer works on the numeral, paying no heed to the number the numeral stands for.

Physiological processes taking place in the brain of a mathematician may be to a large extent analogous to the electronic processes taking place in the circuitry of a computer. But, unlike the internal states of the machine, the mathematician's mental states are intentionally directed to numbers, mappings, and computations involving such. The electronic states of a computer lack intentionality. Any connection between the realm of mathematical objects and the printouts disgorged by the machine has to be mediated by a human operatorr.

What I have to say, however, depends nowise on my view that computers cannot think. Those who insist that computers really calculate - in the sense of applying functions to numbers, quantifying over numbers, etc. - will readily agree that calculation methods (or, as they call them, programs) are not to be confused with the mappings they effect. A mapping is a self-contained entity lacking any internal structure. There is nothing to a mapping over and above the correlation it sets up between arguments and the corresponding values. A calculation method, on the other hand, is a structured complex containing mappings, numbers, and gaps waiting to be filled with numbers or other entities.

When the argument gaps in a calculation method are filled with definite numbers we get a gapless calculation. Filling the gaps in +(.(x,y),y) and .(Succ(x),y) with 2 and 4, for example, produces

+(.(2,4),4) and .(Succ(2),4).

These are two different calculations, two structures consisting of numbers and mappings. Hence they are not to be confused with the single number, 12, they both yield. Another way of closing a gap in a calculation method is abstraction. By abstracting on the first argument of +(.(x,y),y) we obtain 8x.+(.(x,y),y); a two-argument computation of a number has thus been transformed into a one-argument computation of a unary mapping.

I have argued elsewhere⁽¹⁾ that one of the main flaws in Frege's logical theory, indeed the source of virtually all that is obscure and counter-intuitive about it, is the fact that in his notion of Function⁽²⁾ the concept of mapping and that of a calculation method are hopelessly conflated. Much of what Frege says about Functions presupposes that they are structured entities. They are like calculation methods in having gaps fillable with objects, each gap occurring once ore more than once. It is the presence of these gaps which makes a Function what Frege called unsaturated, something that requires completion with appropriate arguments. Frege's Functional names are perfect diagrams of calculation methods: (> .0)+0, for example faithfully depicts the calculation schema +(.(x,y),y) and (Succ > ).0  no less faithfully depicts the calculation schema .(Succ(x,y),y). And Frege makes it abundantly clear that, on his view, a Function is structurally analogous to its name.

But Frege also repeatedly says or implies that Functions which take the same arguments to the same

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values are the same Function. On this extensionality principle, Functions cannot be simply identified with calculations: the calculations +(.(x,y),y) and .(Succ(x),y), for instance, are distinct despite the fact that they yield the same values for the same arguments.

But Fregean Functions cannot be identified with mappings either. A Function must, according to Frege, be an unsaturated entity, for otherwise it would not logically adhere to its argument(s). A mapping is not an entity of this sort it is a self-contained, gapless, object.

Still, the extensionality principle and the unsaturatedness thesis are not irreconcilable. If one identifies Fregean Functions with what I have called primitive calculation methods - i.e. calculation methods which, like Succ(x), +(x,y), and M(x,y), consist simply in applying a mapping to an appropriate number of arguments - then they will be unsaturated in Frege's sense. But since there is only one such Function for any given mapping, the extensionality principle will also be satisfied. For example, there will be only one Function realizing the mapping M: the primitive calculation M(x,y).

To restrict Functions to primitive calculations is equivalent to saying that a Function cannot contain another Function as a component part. Evidence can be adduced that Frege indeed thought of Functions, at least occasionally, long these lines. But he failed to draw what seems an inevitable consequence of this point of view. If a Function cannot contain other Functions, a notation in which a Function name is allowed to contain another Function name is inadequate and misleading.

It seems to me that the main aim of Wittgensteins's Tractatus is to draw and spell out this consequence for Frege. Wittgenstein accepted the Fregean notion of Function and said explicitly that the Functions must not be confused with mappings, or, as he called them, operations. To set the two notions apart he unfortunately used a sloppy formulation which makes little sense as it stands. 'A Function (he wrote) cannot be its own argument, whereas an operation can take one of its own results as its base.'⁽³⁾ Read literally, the aphorism entirely fails to draw any contrast. But it seems natural to read it sympathetically as follows. When a Function is combined with an appropriate system of arguments, one obtains a calculation, a complex in which the Function and the arguments are preserved as constituent parts. This complex cannot serve as another argument for the same Function. But when an operation (i.e. mapping) is applied to an argument, one gets simply the value of that operation at that argument, and the operation may well be applicable to that value. The hypothesis that this is what Wittgenstein meant seems confirmed by the immediately following text, where Wittgenstein emphasizes that, as distinct from a Function, an operation can counteract another operation and thus cancel it out, or counteract itself and in this sense 'vanish'.

If this interpretation is right, Wittgensteins's Grundgedanke - the thesis that logical symbols are not names - can be explained as a direct consequence of the Frege's Extensionality Principle. To see this, consider the proposition

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(1) Peter is black or Fred is not tall.

In Frege's conceptual notation the proposition receives a symbolization which has the same 'mathematical multiplicity' as, and is indeed isomorphic to, the two-argument calculation method

Ú (p, ~ (q))

ie. the method which consists in applying the disjunction mapping to the couple consisting of the first argument and the result of applying the negation mapping to the second. This calculation contains the Fregean negation Function ~ (q) and the Fregean disjunction function Ú (p,q) as constituent parts. But the compound calculation Ú (p, ~ (q)) as a whole cannot count as a Fregean Function because, as we have seen, Functions cannot occur in Functions. If it counted as a Function, so would the equivalent, yet completely different calculation

~ ( & ( ~ p),q))

and the extensionality principle would be violated. Hence Wittgenstein's Grundgedanke: the logical connectives which occur in Frege's symbolization of (1) cannot refer to Functions. And since there is nothing else they could conceivably refer to, they do not refer at all: they are mere punctuation marks.

Given this conclusion, how can a sentence like (1) be analyzed?

It would hardly do to postulate a relation M(x,y) and claim that (1) signifies the result of saturating it with Peter and Fred: M(Peter, Fred). It is not only that we would need inconceivable many distinct relations of this sort. The relations themselves are hardly acceptable, mixing as they do logic with matters of contingent fact. If logic and fact are to be kept apart the analysis of (1) must look rather like this:

(2) Q(Peter is black, Fred is tall).

It must construe the statement as the result of saturating a Function with whatever the atomic sentences like 'Peter is black' and 'Fred is tall' stand for.

But if so, Frege's view that those sentences stand for truth-values cannot be right. For imagine that each of the sentences 'Peter is black', 'Fred is tall', 'Mary is white', and 'Ann is short' stood the truth-value T. Then (1) and the sentence

(3) Mary is white and Ann is not short.

would receive one and the same analysis. Both sentences would express the gapless calculation Q(T,T), where Q is a mapping which takes couples whose second element but not the first is T to F, and all other couples to T. The difference between (1) and (3) would thus be obliterated. If the difference is to be preserved the sentences must be construed as signifying calculations in which the

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role of the saturating arguments is played by the distinctive contents, call them A and B, of the atomic sentences 'Peter is black' and 'Fred is tall', not by their truth-values.

Thus the Function involved in the Tractarian analysis of (1), must work directly on the contents of A and B and effect a purely logical operation.

The Tractarian account of propositional content is undoubtedly Wittgensteins's most important contribution to twentieth-century philosophy. The insight that the assertive power of a proposition depends on what truth-values it takes in possible alternatives to the actual state of affairs is strongly suggested by Frege's definitions of the propositional connectives. But the philosophical point was first spelled out by Wittgenstein in the Tractatus.

Let P₁, P₂, ..., P_m be distinct atomic states of affairs. If they are logically independent, there are 2^m possibilities as to which of them are true and which false these are Wittgenstein's truth-possibilities with respect to P₁, P₂, ...., P_m. The assertive content of a proposition can then be represented as a class of truth-possibilities, namely those truth-possibilities under which the proposition comes out true. Wittgenstein calls those truth-possibilities the truth-grounds of the proposition.

Assuming that the truth-possibilities with respect to any m-tuple are ordered in the usual way, any particular class of truth-possibilities with respect to P₁, P₂, ..., P_m can be represented by an ordered 2m-tuple of T's and F's. Where 8 is such an 2^m-tuple, let (8) be the m-ary mapping which takes any atomic states of affairs P₁, P₂, ..., P_m to the class of truth-possibilities (with respect to P₁, P₂, ...P_m) represented by 8. Mappings of this sort can be called truth-mappings and primitive calculations which realize them truth-functions. Consider, for example, the couple whose first constituent is the state consisting in Peter's being black, and second the one consisting in Fred's being tall. The value of the truth function (TTFT)(P₁,P₂) at this couple is the class of all the corresponding truth-possibilities except the one whereby Peter is not black and Fred is tall.

A proposition, according to Wittgenstein, arises by saturating a truth-function with specific atomic states of affairs. The proposition expressed by (1), for example, arises by saturating (TTFT)(P₁,P₂) with the atomic state A of Peter's being black and the atomic fact B of Fred's being tall. In other words, (1) signifies the primitive calculation

(TTFT)(A,B).

On this analysis, nothing corresponding to the symbols v or ~ enters into the meaning of the sentence. Indeed the sentence

(4) It is not the case that Peter is black and Fred is tall.

receives exactly the same analysis as (1). Hence the formulas (Bp Ú ~ Tf) and ~ ( ~ Bp & Tf), which correspond to (1) and (4) in the Frege/Russell notation are, according to Wittgenstein, totally misleading. They wrongly suggest that the sentences have something to do with the distinct non-primitive calculations

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Ú (p, ~ q)) and ~ ( & ( ~ (p),q)).

The notation is thus entirely inadequate.

It's hard to see how this conclusion can be avoided by anyone who takes seriously Frege's thesis that Functions obey the extensionality principle. Yet the conclusion is not easy to live with. It is a powerful source of incoherence and ambiguity in the Tractatus itself. For example, it flies in the face of Wittgensteins's asseveration that ordinary language is in perfect logical order as it stands. The original ordinary-language sentences (1) and (4) are equiform with the formulas (Bp Ú ~ Tf) and ~ ( ~ Bp & Tf). Hence they must be deemed misleading by the same token. Ordinary language thus also turns out to be an inadequate notation. Sentences built up by means of logical particles from 'Peter is black' and 'Fred is tall' come in infinitely many syntactic forms and yet according to Wittgenstein there are no more than sixteen logical constructions, namely

(TTTT)(A,B),(TTTF)(A,B),..., and (FFFF)(A,B),

for them to express. The sentences contain words like 'or' and 'not' and yet nothing corresponding to these words plays any role, according to Wittgenstein, in what the sentences signify. The 'mathematical multiplicity' of syntactic forms thus seems sharply at odds with the multiplicity of meanings.

But it is not only the logical connectives '~', 'Ú', and '&' that make the formulas (Bp Ú ~ Tf) and ~ ( ~ Bp & Tf) inadequate notations. The compounds Bp and Tf which occur in them are also inappropriate. Wittgenstein agrees with Frege that a property like being black is a one-argument Function. Hence the compounds Bp signifies the calculation arising from filling the Function's argument place with Peter. On the extensionality principle, however, this calculation cannot be part of the calculation expressed by sentence (1). For even if the negation and disjunction Functions were eliminated in favour of Wittgenstein's (TTFT)(P₁,P₂), sentence (1) would still be construed as expressive of the non-primitive calculation (TTFT) (B(p),T(f)), a calculation which contains Functions as component parts. As we have seen, such calculations cannot be tolerated if the extensionality principle is to prevail.

But if the argument places of (TTFT)(P₁,P₂) are not filled with calculations, what are they filled with? According to Wittgenstein the saturating arguments are atomic states of affairs, the state consisting in Peter's being black and the state consisting in Fred's being tall.

Wittgensteins notion of atomic state of affairs, however, is fraught with unsurmountable difficulties. A state of affairs, we are told, is something which consists of several items, it is a configuration of objects. Peter's being black, for example, consists of Peter and the property of being black. But if the two items are not joined together as a Function with its argument in the calculation consisting in an application of the former to the latter, what is it that holds them together? What combines Peter and blackness into a configuration? It cannot be the instantiation relation, for a state of affairs need not

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obtain. If Peter fails to be black there is still such a state as Peter's being black, yet Peter does not instantiate blackness. So it cannot be instantiation that provides the bond.

There are further problems, however. Consider the state of affairs consisting in Peter being taller than Fred. According to Wittgenstein, this state of affairs consists of three simple objects, Peter, Fred, and the taller-than relation. Now, consider the state of affairs consisting in Fred's being shorter than Peter. This state of affairs should consists of Peter, Fred, and the shorter-than relation. But surely nobody would deny that Peter's being taller than Fred is the very same state of affairs as Fred's being shorter than Peter. If the state obtains, we have just one fact of comparative height, not two. Bt one and the same entity clearly cannot be correctly analyzed as a composite of one triple of simple items and at the same time as a composite of another triple of simple items.

The idea that a state of affairs is a complex made up of parts is thus quite indefensible. Wittgenstein soon realized this himself. A repudiation of the idea can be found already in Philosophische Grammatik. But Wittgenstein never offered anything to replace the Tractarian state of affairs.

Yet the Tractatus itself contains, in rudimentary form, all that is needed for an adequate account. I have already stressed the import of Wittgenstein's insight that a propositional content is best explicated by reference to possible alternatives to the way things in fact are. Now the content of a proposition reporting an atomic state of affairs should clearly be nothing other than the state of affairs itself. Hence, just like any other propositional content, an atomic state of affairs should be representable as a class of truth-possibilities.

On this approach, of course, we cannot define a truth possibility as Wittgenstein did, namely as a distribution of truth-values through atomic states of affairs. This would be moving in circle. But there is no need to proceed in this circular way. Any m-ary state of affairs consists in some m-ary attribute being satisfied by an m-tuple of particulars, and is thus characterizable by the m+1-tuple consisting of the attribute and the particulars. To the state consisting in Peter's being black, for example, there corresponds the ordered couple <blackness,Peter>. We can redefine Wittgenstein's truth-possibilities as distributions of truth-values through such m+1-tuples. Let us agree, moreover, that a distribution will always be through all such tuples. All-embracing truth-possibilities of this sort have come to be known as possible worlds. A state of affairs, whether atomic or not, can then be identified with a class of possible worlds, namely those in which the state of affairs obtains. Peter's being black, for instance, will be the class of possible worlds which assign truth to the couple <blackness,Peter>. This class does not consist of Peter and blackness. For one thing, classes do not have parts, but elements. Besides, the elements of the class at issue are possible worlds, not individuals or properties.

Now, let A b the class of possible worlds in which Peter is black and B the class of those in which Fred is tall. Then the Tractarian analysis of (1) can be represented as

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(1') (TTFT)(A,B),

where (TTFT) is mapping whose arguments are classes of possible worlds. Its value is also a class of possible worlds, namely the union of the first argument and the complement of the second.

Structurally, sentence (1) is as unlike construction (1') as it can possibly be. Not a single word occurring in the sentence has a counterpart in the construction. No wonder that Wittgenstein concluded (after having credited ordinary language with being in 'perfect logical order') that

[l]anguage disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes.⁽⁴⁾

Wittgenstein does not say what those purposes are. But as far as (1) and (4) are concerned, an answer suggests itself. Although they both allude to the same class of possible worlds, they portray two different ways of arriving at that class, two different constructions thereof. These two unlike constructions or calculation of the single affirmative content are entirely obliterated in Wittgenstein's analysis (1').

If my line of argument is correct, Wittgenstein arrived at this impoverished analysis by unswervingly following the consequences of Frege's extensionality principle for Functions. In the rest of this paper I shall briefly consider how Wittgensteins's theory could be improved if the principle were to be repudiated.

Let us then set clearly apart what Frege conflated. A mapping is simply a correspondence between argument and values, hence two mappings effecting one and the same correspondence are really one and the same mapping. But the values of one and the same mapping may be obtainable from its arguments by more than one calculating method. Thus two calculating methods may be distinct even if they yield the same correspondence between arguments and values. Setting the two notions apart obviates the need to restrict oneself to primitive calculations. Let us then legitimize calculations like +(.(x,y),y) and .(Succ(x),y) which contain other calculations as component parts, and see how the theoretical landscape changes as a result.

Consider the atomic sentence

(5) Peter is black.

Clearly each possible world W determines a unique class as the class of those individuals which would be black if W was realized. Call it the extension of blackness relative to W. Blackness itself can be identified with the mapping, call it B, which takes any possible world to the corresponding extension. Thus the extension of blackness in W can be arrived at by applying B to W, i.e., by means of the (primitive) calculation B(W). Now Peter is or is not black in W according as ho does or does not belong to the extension of blackness in W. Hence, where X is Peter, the truth-value of 'Peter is black' in W is constructed by the (no longer primitive) calculation [B(W)](X). The assertive content of 'Peter is black', i.e., the class of possible worlds in which it comes true, can then be constructed by abstracting on the possible-world parameter W:

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(5') 8w.[B(w)](X).

By constructing (5) as expressive of (5'), we dissent from Wittgensteins's thesis that a sentence stands for a class of possible worlds. Instead, we espouse the thesis that it stands for a definite calculation, or construction, which yields such a class. We can assent, on the other hand, to another of Wittgensteins's theses, namely, that a sentence is a kind of picture. But again, a sentence is not a picture of a state of affairs as Wittgenstein thought. A state of affairs is simply a class of possible worlds; as such it has not Gestalt and cannot therefore be pictured. A sentence is rather a picture of a construction of a state of affairs. A construction like (5') is a compound made up of Peter and blackness. These constituents are represented in the sentence by the words 'Peter' and 'black' and the way the words are combined into (5) represents the way Peter and blackness combine into (5'). This is what makes the sentence a picture of the construction.

My claim that states of affairs cannot be pictured will probably be challenged. Is, say, a black silhouette of Peter not a picture of the state of affairs consisting in Peter's being black?, it will be asked. The answer, however, is No. Imagine that, apart from being black, Pater has a ski-jump nose. A good silhouette will show this. But how can we then decide whether the drawing pictures the state of affairs of Peter's being black or rather the state of affairs of Peter's being ski-jump nosed? The answer is obvious. The drawing is a picture of Peter, not of any particular state of affairs involving Peter. It is a picture of Peter because Peter is a definite arrangement of bodily parts and pictorial counterparts of those parts are arranged analogously in the drawing. States of affairs do not have parts and cannot be therefore pictured. How cold one possibly picture the state of affairs consisting in Peter's being a tax dodger?

Let us now look at the sentence

(6) Peter is not black.

which negates sentence (5). The truth-value of (5) in world W can be constructed by [8w.[B(w)](W), hence the truth-value of (6) in W can be constructed by ~ ([8w.(B(w)](X)](W), where ~ is the familiar negation mapping taking truth to falsehood and falsehood to truth. The assertive content of (6) is thus constructible by

(6') 8w. ~ ([8w.[B(w)](X)](w)).

(6) is a picture of the construction (6'), the words 'Peter', 'tall', and 'not' deputizing for the constituents Peter, blackness, and ~ of the construction.

The ordinary language sentence is, of course, only a rough picture of the construction. no separate part of it represents the bound possible-world variable w, for example. But ordinary language is notorious for suppressing bound variables. It is for a perspicuous conceptual notation to make them explicit.

If we look at a sentence as an expression of a construction it is easy to answer the question what it is about. We can simply say that it is about the constituents of the construction it expresses. (6), for example, is bout Peter, blackness and negation.

This, of course, is in direct opposition to Wittgenstein's vies.

[I]f there were an object called '~' [he wrote], it would follow that '~~ p' said something

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different from what 'p' said, because one sentence would then be about ~ and the other not.⁽⁵⁾

But if this argument were sound one could similarly argue that there is no such object as the taller-than relation, for then it would follow that 'Peter is taller than Fred' said something different from what 'Fred is shorter than Peter' said, because one of the sentences would then be about the taller the taller-then relation and the other not. But of course the sentence tells us something about the taller than relation, namely that its application to Peter and Fred in the actual world yields truth. Or, what amount to the same thing, it tells us something about a construction involving Peter, Fred an the taller-than relation, namely, that it yields a class of possible worlds containing the actual one. The sentence 'Fred is shorter than Peter' tells us something analogous about a different construction, one involving Peter, Fred, and the shorter-than relation. Similarly, a sentence of the form '~~ p' tells us something about the negation mapping that 'p' does not. It tells us, that a double application of the mapping to p in the actual world yields truth. Unless p is trivial, this is a contingent property of that mapping.

The notation of calculation or construction, which lingered in an impoverished form in the philosophies of Frege and Wittgenstein, has since disappeared from philosophical debate altogether. With it disappeared the notion of a linguistic expression as a diagram. The contemporary logical semantics entertains only linguistic expressions, on the one hand and structured objects like individual, sets, mappings, etc., on the other; it does thus has nothing in its ontology for linguistic expressions to be diagrammatic of. It creates an unexplainable discrepancy between the infinite variety of syntactic forms and the structural paucity of the entities that those forms are supposed to represent.

If tones did not combine with other tones into sound patterns like themes, movements, and symphonies, there would be little point in combining dots with other dots into visual patterns on the musical staff. By the same token, if numbers, mappings, and operation did not combine into logical constructions, there would be little point in combining numerals, functors, and operators into typographically composite terms and formulas.

Yet the obvious fact that logical constructions are what logic and mathematics are all about seems to be the closest guarded secret of contemporary semantic theory. Little wonder then that the theory is hard put to explain, for example, what it means to translate a sentence from one language into another. For it obviously means is to produce a sentence which expresses the same construction as does the original. And little wonder that the theory is hard put to give a satisfactory account of belief, knowledge and their ilk. For they are attitudes subjects take to constructions.

NOTES

1. The Foundations of Frege's Logic, Berlin et New York, De Gruyter, 1988

2. I shall write the word 'Function' with a capital 'F' to remind the reader that it is Fregean Functions that are under discussion rather than the functions of modern mathematics (ie. mappings).

3. Tractatus, 5.251

4. Tractatus, 4.002

5. Tractatus, 5.44

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