Copyright © 1997, Allan Randall

  Logic, Idealism and Materialism in Early and Late Wittgenstein

Allan F. Randall
Toronto, Ontario, Canada,



Wittgenstein's philosophies, from both the Tractatus and the Philosophical Investigations, are explained and developed. Wittgenstein uses a primitive version of recursion theory to develop his attempt at a purely logical metaphysics in the Tractatus. However, due to his implicit materialist assumptions, he could not make the system completely logical, and built in a mystical division of possible worlds into the true and the false. This incoherence eventually lead him to reject logic as a method for doing metaphysics, and indeed to reject metaphysics entirely. I argue that his move from the Tractatus to the Investigations was valid, but only given his materialist assumptions. If he had been willing to drop this unnecessary baggage, recursion would have played a very different role in his system, since he would then have had no need to separate static objects from processes, which he saw as purely mental. F.H. Bradley developed such a nonmaterialist metaphysics in the nineteenth century, but was crippled by a mentalism that Wittgenstein was free of. The anti-mentalism and anti-materialism that Wittgenstein considered as given were not so obvious to his predecessor, Russell, who revolted against Bradley's idealism in much the same way Wittgenstein ended up revolting against Russell's logical atomism. In my view, none of these positions was the right approach, which would require nonmentalism and nonmaterialism.  But for some reason, these things (which seem to go together quite naturally to me) have been widely considered to be incompatible. Bradley was appropriately a non-materialist, but suffered from mentalism. Russell and the early Wittgenstein were appropriately nonmentalists, but suffered from materialism. The later Wittgenstein was, I would argue, still an ardent materialist and anti-mentalist, in spite of the fact that he had long since realized the contradictions to which materialism leads; he just had not recognized that it was his materialist assumptions that had lead him there, since these assumptions were so firmly engrained in his thinking as to be invisible. Hence, he could publicly claim to have rejected metaphysics, while continuing to argue philosophically from a strongly materialist, and hence deeply metaphysical, position.

I. Introduction

Wittgenstein, in the Philosophical Investigations,[1] is concerned with the meaning of "meaning". His working definition of the "meaning" of a word or statement in a language is its use in the community that speaks the language [I.43][2]. Philosophy should not, he insists, abstract out the logical form of a statement, and present that as its meaning. Any logical analysis will leave out part of the actual situation in which the statement is used [I.89]. So we must leave language as we find it, unabstracted [I.124-6].

Wittgenstein offers a nonmetaphysical account of meaning in the Investigations, as an alternative to the traditional metaphysical approaches, in particular his own earlier metaphysical treatment, found in the Tractatus Logico-Philosophicus[3]. In the Tractatus , the early Wittgenstein treats the propositions of language in just the way the later Wittgenstein warns against: a proposition has a particular logical form, which is either true of the world or not [T.1-2.0]. The world is just the division of all possible logical forms into the true (positive facts) and the false (negative facts) [T.2.06]. What makes this world a world of true facts, and some other imaginary world a world of false facts, is not something one can really even say, as it is outside of logic [T2.022-23]. This just is the world in which I find myself: the "real world" is simply "my world" [T.5.62-63].

The early Wittgenstein thought the logic-based metaphysics of the Tractatus was the last word on all traditional philosophical problems. So why did he turn completely away from this view in later years? I will argue that Wittgenstein was driven to reject his early work for much the same reasons that drove nineteenth century philosophers into idealism, the idealism rejected by the analytic tradition within which Wittgenstein operated. I will examine the logic-based metaphysics of the Tractatus, discuss its main problems, and compare the nonmetaphysical solution of the Investigations with the earlier solution of nineteenth century idealism, in particular the work of F.H. Bradley.[4,5] I will suggest that Wittgenstein's logic-based metaphysics runs into problems, not because it is too centred on logic, but precisely because it does not take logic seriously enough. By dividing logical space into positive and negative facts, the early Wittgenstein stops short of a fully logic-based metaphysics, and allows mysticism to creep in. If the Tractatus had taken its own logical analysis to its logical conclusion, the result would have been a more sophisticated version of nineteenth century idealism, one based on formal logic rather than mentalism. Yet Bradley and the other idealists also stopped short of such a view, largely because they shunned formal logic and saw metaphysics and logic in mental, not formal, terms.

However, the later Wittgenstein shows no particular idealist tendencies, working firmly within the anti-idealist tradition of Russell and Frege. Yet without some kind of idealism, his only alternative, taken up in the Investigations, is to attempt an account of meaning that simply does not require metaphysics: not to disprove metaphysics, but to ignore it.

II. The Tractatus

A World of Facts

The striking thing about the Tractatus is that it seems at times to be constructing a completely logical metaphysics, but ultimately leaves truth outside of logic. Tractarian metaphysics is logic-based because it considers that the world is just the existence of facts having certain logical forms (and likewise the nonexistence of certain other logical forms) [T.1-2]. These logical forms, if they are to have existence, must be independent of each other [T.1.21]. The existence of logical form A cannot depend on whether some other form B exists or not, else A and B would be aspects of some larger form, C, which is actually the thing that exists (or not) in the world. Wittgenstein calls these independent logical forms "atomic facts". An atomic fact that is true (has existence in the world) is a "positive fact", or just a "fact". The world consists of all existing atomic facts (i.e. all positive facts) [T.1-1.1].

This leaves many other atomic facts which are logically just as valid, but which do not exist as true in the world. These are "negative facts". Although the negative facts do not exist, there is a sense in which they are as much part of the world as the positive facts. The world is simply the division of all atomic facts into the positive and the negative. The set of all positive facts is the real world we live in, the world of our experience. The set of all negative facts yields all possible imaginary worlds that do not exist in reality [T.2.022-23].

Wittgenstein seems sometimes to allow a fact to be either positive or negative, and at other times, he assumes by default that a fact is positive. I will use "fact" in this essay to refer to either positive or negative facts; a fact will not be taken to be true unless explicitly stated as such.[6]

It is also possible, in Wittgenstein's system to have nonatomic facts. These need not be independent of each other; they are built up from more fundamental atomic facts. A nonatomic fact can always be reduced, however, to a set of independently existing (or nonexisting) atomic facts. So it is the atomic facts that have metaphysical significance. Below is a rough overview of the relationship between the different kinds of facts we have encountered so far, and the world.

Although this may seem thoroughly logic-based, there is something quite nonlogical about such a metaphysics. The difference between a positive fact and a negative fact is its existence. This is a thoroughly nonlogical property, and is hence literally indescribable. It is this nonlogical distinction that will lead the early Wittgenstein to an ultimately mystical metaphysics.


So just what is an independent logical form, or atomic fact? Wittgenstein describes it as a relational structure [T.2.03], for instance aRb (a is related to b). This could also be written in a more functional notation as R(a,b) or as an ordered set, or list, as (a b), taking R as a ubiquitous and assumed relation. One could certainly imagine building up complex structures from networks of such simple relations, although it is not immediately obvious that such a formalism is really adequate to describe the world. However, we will accept for now that such a structure of related objects forms an atomic fact, which can either exist or not.


So what are the a and b in aRb? These noncomposite components are what Wittgenstein calls "objects". They are the things which exist in the world. It may not be obvious why these objects should be the things of the world, rather than the atomic facts, or relational structures, but in a way it does not matter. From a purely logical perspective, the objects do not exist in themselves. After all, in aRb, what is a, if considered apart from the whole structure? It surely cannot be the physical ink on paper (or electrons on screen) that form a certain pattern that we recognize as the letter a. We are using an abstract notation, after all, and the letter a simply stands for some object. So, insofar as we can capture the object in logical notation, the object must itself be defined in terms of the other objects it is related to. Together they all form a structure, which is an atomic fact.

So what is the object? It seems from the above logical perspective that the "object" is a fiction which can be done away with, and it is the atomic facts which are the true things of the world. Indeed, it is not entirely clear why Wittgenstein needed to speak of objects at all. He seems to be unsatisfied thinking of relational structures as "things", since the things that make up the substance of the world must, to his mind, be noncomposite [T.2.021]. If, instead, the substance of the world lay in composite relational structures, then one relational structure that contained another could only have any sense at all (i.e. be able to be determined as true or false in the first place) if its constituent relational structures were already true. So aRbRc can only have sense if aRb and bRc are true. But then each of these can only have sense if a, b and c are true. So trying to define the things in the world as composite will take us into a regress that ends up placing the truth in noncomposite objects, after all. So without the foundation of noncomposite objects, we could not draw any relational picture of the world in the first place, whether it was true or false [T.2.021-2.0212].

In my view, this argument works only if we assume that relational structures are adequate to describe the world to begin with. If relational descriptions are inadequate to define the objects of the world (as I believe they are) then the argument has no force. Furthermore, since the argument seems to imply noncomposite objects, which from a logical point of view seem to be mere fictions, Wittgenstein's regress argument could even be taken as an argument for the inadequacy of relational descriptions. From the purely logical point of view I will develop later in this paper, the objects really are fictions, but then so is the "existence" or "nonexistence" of the relational structures.

Since objects, apart from the relational structure they partake in, seem to be mere fictions, Wittgenstein defines them as that which can potentially combine with other objects in relational structures. So a may be related to b and not to c, in the real world, but it could have been related to c instead. From a logical perspective, it is related to c in some imaginary worlds, in the space of negative facts. So, if we stick to logic, the object a cannot be considered without pulling in all possible logical forms, since it could partake of them all in possibility, even if it does not in reality.

Here we see that Wittgenstein's notion of a noncomposite object does go beyond logic, since from a purely logical perspective, the a in atomic fact aRb cannot be the same object as that in atomic fact aRc, even if one is true and the other false and merely possible. a in itself is logically void of content. It is only when we add a nonlogical "material existence" property to it that we can imagine the a's in these independent logical forms to be the same a (the same material world object). So Wittgenstein's metaphysics is fundamentally materialist, relying as it does on the material substance of noncomposite objects. From a logical perspective, it is nonsensical to say that the a in an imaginary possible world is in a nonexisting relationship with c, while in the real world the object is related to b instead. From a logical perspective, the two a's simply are not the same object, since all we have are two different relational structures.

Imagine, for instance, that a cup is on top of a saucer (ignoring for now that the cup and saucer are actually noncomposite).[7] Does it make sense to suggest that this actual cup could have been under the saucer instead (is under the saucer in some other possible world)? That may be fine as a rough way of talking, but it seems to me thoroughly nonsensical if one is trying to be rigorous about what one says. The cup's being under the saucer makes it a different cup in a different, although similar, world. It cannot literally be this cup in this world, which is on top of the saucer. Of course, I am assuming a completely logical point of view, which if we restrict ourselves to relational structures would require that the relational structure be the metaphysical "thing", rather than the noncomposite object. Wittgenstein might counter all this with his regress argument. However, that argument applies to relational structures, and I think there are good reasons to reject his assumption that relational structures are adequate in the first place. These reasons will be taken up later in the paper.

In spite of his fundamental faith in materialism, however, Wittgenstein seems to flirt with a purely logical metaphysics throughout the Tractatus. One can almost imagine dropping the references to material substance, dropping his objects as anything substantial, and still being left with a coherent, purely logic-based idealism. This is a fundamental tension throughout his work. He seems to want to adopt a logical metaphysics, but is continually driven back from it by what he sees as the manifest reality of his particular world.


Even if Wittgenstein had dropped material substance and adopted some kind of logical idealism, it would not have been the mentalist idealism of the nineteenth century, which saw reality as fundamentally mental in nature.[8] A Tractarian fact is not necessarily a mental object, although we do picture facts to ourselves, representing them in our heads. Such a "picture" also has a logical form, which shares some similarity of structure with the existing (or nonexisting) logical form that it refers to (the atomic fact). So a picture is a kind of fact (either positive or negative); it is a fact that represents another fact. In other words, it is the expression of a thought [T.3]. Pictures are important kinds of facts to humans, naturally, since they are the means by which we know our world. But their mental character is not fundamental to the things they represent, which are purely formal logical constructions (except for their existence or nonexistence, of course, but that is not mental either).

From a purely logical point of view, then, a picture represents an atomic fact, by virtue of similarity in logical form. The question then arises as to whether the truth or falsity of the atomic fact (its existence or nonexistence) is part of the mental picture we form of it. Wittgenstein says that a picture has a "sense", which is the possibility of existence (or nonexistence) of the atomic fact it represents [T.2.22]. If the atomic fact really does exist, the picture's sense agrees with reality and the picture is true, otherwise it is false. The existence is not part of the picture, which is just a fact instantiated in a mind. Its being a picture establishes it as a fact, but does not determine whether it is a positive or negative fact, which is determined by its existence or nonexistence, something outside of logic [T.6.4].


What is shown, displayed before us, by the picture is the logical form of the fact it represents. This is its sense, and includes its possibility of existing, but not its actually existing or not existing. Yet, when we think of a picture in our minds, or express the picture in speech, we are often trying to say more than just what is shown; we are trying to say that the atomic fact does indeed exist, not just that it may or may not exist but here it is in any case [T.4.022]. Such expressions of pictures are called "propositions" [3.1-13]. In a proposition, we actually do say what we cannot show: that the picture's sense is true. Propositions that state the existence of atomic facts are "elementary propositions". An elementary proposition says that an atomic fact exists, but it can only say this by showing us the logical form and then flatly stating that it exists. Yet a simple statement of existence is quite empty logically. It would seem, again, that Wittgenstein is getting mystical. Logical form is all that can ever be shown, yet there are material objects in the world that partake in some forms that "exist"and some that do not, which we somehow manage to say even though all we can actually show are logical forms.

From my own logical idealist point of view, the distinction between showing and saying will, of course, prove illusory, since the showing of the picture is all that can be done within logic. Saying and showing become equivalent. An idealist Wittgenstein would probably not have made the distinction. But the unquestioningly materialist Wittgenstein insists that when we express a picture as an elementary proposition, we are attempting to say what cannot be shown--that the atomic fact exists, not just what it is.

Since atomic facts are independent, elementary propositions are too. One cannot have two contradictory elementary propositions [T.4.22]. As we will see, however, it is possible to combine and transform elementary propositions into nonelementary ones, which need not be independent of each other. These are "truth-functions" of the elementary propositions. However, such truth functions do not have the metaphysical significance to Wittgenstein that the elementary propositions do, since all such higher level propositions can, if analyzed, be reduced to the constituent elementary propositions they were built from.

Tautology, Possibility and Contradiction

In section 4, Wittgenstein discusses what can and cannot be expected of a logical notation [T.4.2]. Here we get a clearer idea of what atomic facts really are, as Wittgenstein is forced to grapple with the issue of how we represent them logically in propositions. As we saw earlier, an atomic fact is a relational structure of "objects", which individually can only be defined in terms of their potential to combine into such structures. Elementary propositions, since they represent atomic facts, are likewise combinations of "names", combined into relational structures, where each name refers to an object. But just as an object cannot exist apart from the whole relational structure, a name does not by itself refer. It only refers by virtue of how it is used, so as to form such relational structures."a" in itself is just a notational device and cannot on its own have meaning.

"a=a", then, cannot possibly be an elementary proposition [T.4.24]. There is no relational structure here, nothing with a sense that it could be true or false. We call such a proposition a "tautology", since it is true for all truth assignments of elementary propositions. If, on the other hand, a proposition were false for all truth assignments of elementary propositions, then it would be a "contradiction". Neither of these extreme kinds of propositions, according to Wittgenstein, really have sense[T.4.46]. Since a tautology is always true, it literally says nothing about the world. A contradiction likewise says nothing about the world, since it cannot possibly be true. Tautologies and contradictions show through their logical form that they say nothing [T.4.461]. Yet in between certainty (tautology) and impossibility (contradiction) lies the possible: the propositions with sense (they can possibly be true or false, but which they are cannot be shown logically).


Propositions are either elementary or can all be built out of elementary propositions through a series of "operations". Each application of an operation transforms one or more propositions into a new, higher-level proposition [T.5.2], always starting with elementary propositions at the beginning. This process of repeatedly applying an operator could go on indefinitely, producing an infinite stream of new propositions, such as in a mathematical series, or the counting of the integers.

An operator can combine one or more lower-level (closer to elementary) propositions into a new high-level proposition. A higher-level nonelementary proposition is a nonatomic fact, and differs from an atomic fact in that the "objects" that are related in the structure are not true noncomposite objects; each is a sort of "pseudo-object" that can actually be further analyzed into a lower-level relational structure. A nonelementary proposition can always be recursively reduced to its constituent elementary propositions (just as it was recursively built up from them in the first place).

Although the atomic relational structures have truth-values in Wittgenstein's system, there is no inherent reason why an operator-based series of transformations on such a structure should produce nonelementary relational structures with truth-values. However, since Wittgenstein is interested in nonelementary propositions, which attempt to actually say something true or false about the world, he is interested in those operators which do produce nonelementary propositions with a truth-value. These include the standard logical connectives, such as AND, OR and NOT. These truth-values are, of course, ultimately defined in terms of the truth-values of the elementary propositions (the "truth-grounds"). However, as we will see more clearly later, these truth-values have nothing to do with the series of operator transformations that produce the truth-functions. A series of operator transformations could literally run on to infinity, and produce no truth-value at all. But each relational structure it produces at each step has a truth value dependent on the initial truth-grounds.

At first, Wittgenstein talks about operators in quite general terms, allowing us to make up any that we want, including but not limited to the standard logical connectives. Later, Wittgenstein simplifies things by reducing all possible operators into a single operator which alone is sufficient to do the job [T.5.5]. I will therefore call this the "universal operator". The universal operator provides the familiar truth-functions of logic, formed from elementary propositions. This gives us nonelementary propositions, like ((a & b) v (c -> d))). This higher-level proposition's truth or falsity is completely determined by the truth or falsity of the component elementary propositions: a, b, c and d. It is produced by a repeated application of the universal operator on elementary propositions. For instance, in the above nonelementary proposition, (c -> d) can be reduced to (~c v d), which can be reduced to ~(c & ~d). The whole expression can be reduced, then, to ((a & b) v ~(c & ~d)), and finally to ~(~(a & b) & c & ~d).

It has been proved, in fact, that any expression of standard propositional logic can be given solely in terms of this kind of NOT AND, or NAND, operation. In computer science, it has been similarly proven that any computer program can be wired up as a combination of NAND gates. In Church's lambda-calculus, it is widely believed that all possible relational structures that can be computed can be computed through the successive application of a single rule: sometimes called the "rule of substitution", or the "abstraction operator".[9] This has also been shown to be equivalent to both the NAND operator as well as "Turing Machines", an historically important model of computing. In fact, all these apparently different notational systems have the same generality in that they seem to be able to express just about anything. Although it cannot be proved that there is nothing left out, it is widely believed to be complete.

Following Wittgenstein's use of the letter "N", if we abbreviate ~(x & y) as N(x,y), then the above proposition is N(N(a,b),c,N(d)). In fact, assuming that any other operator can be expressed in terms of "N", we can take "N" to be our sole operator, and make do with just the bracket notation. So we could just write ( (a b) c (d) ).

Although I have chosen to define N() above in terms of NAND, Wittgenstein actually defines his universal operator, not in terms of NAND, but in terms of NOR (NOT OR).[10] Either operator will do equally well as a universal operator, and the choice between the two is actually arbitrary, so long as we stick to a purely logical perspective. The two functions are inverses of each other (the Ts in one truth table become Fs in the other, and vice-versa). But the distinction between the two symbols T and F as one of "truth/falsity" is an arbitrary external interpretation of a logical system, not something inherent to the system itself (for how could there be any real distinction between the arbitrary symbols T and F, purely within the formal system?). Wittgenstein is quite insistent that this distinction be made, and all such arbitrary external meaning be recognized as nonlogical in character. If he were truly seeking a logical metaphysics, he would insist on extending this requirement to meaning in general, since all meaning would be somehow a case of logical meaning. But Wittgenstein is assuming a materialist metaphysics, and so will end up insisting that reference to the world is fundamentally nonlogical, depending on a nonlogical distinction between T and F (the division of atomic facts into the true and false).

Note that a negation sign in front of a sign for a single atomic proposition has no meaning in itself. The negation sign only has meaning as it is used within a system of building more propositions out of existing propositions, in combination with other operators that yield an expressibility equivalent to that of the N() operator. This means that the negation operator needs at least the AND (or OR) operator plus the notion of repeated application (or "recursivity") to have meaning. Likewise, the other logical connectives have meaning only by virtue of the rules by which they are used, within such an N-equivalent system. So ( (a b) c (d) ) only has meaning insofar as the brackets are interpreted as a rule for manipulating the elementary propositions in a certain way, according to certain truth-functional rules (the NAND operator). So NAND and NOR are in a sense more meaningful than the others logical connectives, since one need only add to it the notion of recursivity (repeated application of the operator) to get a fully expressive system. However, without recursivity, even the NAND and NOR operators are as fully meaningless as the other logical operators.

An interesting thing about ( (a b) c (d) ) is that it can also be viewed as a relational structure, just like each of the original elementary propositions of which it is composed. Here, the relation is N, set up between elementary propositions. We could even choose to use the more relational-looking "xNy" instead of the more functional-looking "N(x,y)". The purpose, according to Wittgenstein, in repeatedly applying the N operator, is to transform elementary propositions (relational structures) into nonelementary propositions containing the elementary propositions as names for pseudo-objects in a higher-level relational structure. So a truth-function is a relational structure, like an atomic fact, but its "objects" are not noncomposite (each one can be expanded into a further relational structure, perhaps elementary, perhaps not).

For Wittgenstein, there is a huge difference between the higher levels and the atomic level. Although the high-level relations are instances of NAND, the atomic relations have no operational meaning at all. Wittgenstein does not state what a relation is in atomic facts. One might think that at an atomic level, the universal operator would be the only logical choice, since it is the one operator (i.e., relation) that is fully expressive. Certainly, one would think that if logical connectives need rules of use to give them meaning, relations in atomic facts ought to require the same thing. And if our language is to be metaphysically significant, surely the most universal operator (or set of operators) we can find would be the appropriate choice for describing metaphysical objects. But Wittgenstein does not believe that elementary relations need rules or process to give them meaning. Like Frege, he believes that the truth-grounds of a proposition must be simple, undefined things (that which exists in the world).

The atomic fact, represented by an elementary proposition, can, in itself, be true or false, existing in the world or not. But a nonelementary relational structure requires a procedure to go through to determine truth or falsity, since each of its pseudo-objects has its own truth value, instead of the proposition as a whole having a truth value. In the nonelementary proposition above, ( (a b) c (d) ), there is a process or procedure that must be gone through to compute the final truth value. This process in itself is necessarily finite, and thus not enough to account for the equivalence of N with general models of computing. But if self-reference is allowed, as in Wittgenstein's system, the calculation of the truth value might theoretically go on forever, and the full notion of a recursively computable function is attained. In fact, this is the reason that Wittgenstein's N() operator has the universality of the Turing Machine, the lambda-calculus, and the NAND gate. The N() operator, as it is used in the finite calculation of the truth value of a particular truth-functional proposition, is not a general recursive model of computing (it is not Turing-equivalent). Only when used in the potentially infinite process of building truth-functional propositions out of other propositions does N() become Turing-equivalent. Self-reference is required for full expressiveness. Wittgenstein tells us that the truth-function itself (the relational structure) that results from the recursive building up process cannot take itself as an argument [T.5.251], so self-reference is not allowed at that level. This is because the truth-function itself is simply a relational structure, and there is nothing in a relational structure per se that could possibly refer to itself. This is an important distinction between a static relational structure and one subject to a process or algorithm. So it is crucial to Wittgenstein's system that he does allow (in the repeated application of N on the relational structures) that the operation can take one of its results as the base for the next step [5.251]. So relational structures, as truth-functions, are static. When coupled with one or more operators, however, they can represent dynamic, recursive computational processes. Operations, in producing new truth-functions, can theoretically loop on and on forever. So, for some truth-assignment of its atomic components, the computation required to determine the truth-falsity of an operational series (if we imagine for now that it could be said to have one) also loops forever. But, of course, if it loops forever, it cannot produce any final relational structure and truth value as a result. It just keeps on producing more and more structures and truth values ad infinitum. An operational series thus has no truth value. An infinite sequence of operations is more like a mathematical series than a proposition. It does not necessarily have any final result at all. While it may consist of an infinite series of truth-value computations, there is no "truth" to the series as a whole.

 Fig. 1: Relational vs. Procedural Metaphysics

To Wittgenstein, this means that the operator, applied to a proposition, defines a difference between the relational structures, and thus defines a relation that is "internal" to the nonelementary proposition--a relation between two of its internal, lower-level relational structures (an inherent similarity/difference in their logical form) [5.2-23]. Although the pseudo-objects in a nonelementary proposition can have internalized relations, an elementary proposition's atomic objects are noncomposite, and thus are related only to the other independent material objects via the external relations of the relational structure. For Wittgenstein, real material objects are related only externally to other material objects. This is opposed to the Leibnizian tradition, which holds that all relations are ultimately internal to the object. For Wittgenstein, all relations can ultimately be reduced to external relations between objects, and no relation is truly, metaphysically internal to an object, which is fundamentally noncomposite. Internal relations are mental abstractions we make about the metaphysically existing objects and their external relations. We form these abstraction through a recursive process, but nature itself contains no such recursive computation (this is, to a computationalist such as myself, a fundamental error on Wittgenstein's part).

An infinite mathematical series, for instance, has an internal relation that unites all its terms, which is the operator. Recall that it is a similarity in form that allows a picture to refer to some other fact. If this "similarity" is just identity, then the picture represents an atomic fact directly, and its reference to the world is "built-in", so to speak--no abstract internal relation is required. But for higher-level abstractions, our nonatomic pictures have internal relations that allow us to refer to the atomic facts through similarity in form. A relational structure cannot be said in itself to be similar to some other nonidentical relational structure. It thus cannot reflect it or refer to it except through a process or procedure of applying an operator. The operator provides abstraction or generalization. If full information about the complete elementary propositions that we started from is not retained in the repeated application of the operator, then the result is an abstract generalization that leaves much of the actual state of affairs out of the picture. This is necessary if we are to refer usefully to a world that is too complex to represent completely in our heads, identically to the original. But in spite of the arbitrary, abstract nature of the higher-level picture, it still manages to really refer to the original atomic facts because it can, at least in principle, be reduced to a ground of elementary propositions, which immediately refer to the atomic facts through the identity, not mere similarity, of their form.

To sum up, given truth-values for its atoms, we can compute a truth value for a proposition, but not for an infinite series of operations. Such an operational truth value would have to be computed for every step in an infinite series of operations. So the series has no truth/falsity, and cannot be what exists in the world, according to Wittgenstein. It is in the repeated application of a universal operator that computation and process, or abstraction, really enters Wittgenstein's system. Although each individual truth-function also needs a process to compute its truth value, this is a finite process with no self-reference, that always produces a definite result, completely determined by the simple truth-values of the elementary propositions. Truth-functions are not an adequate model of process or computation. But the repeated generation of new relational structures, and the computation of the series of truth values, does have this general feature of abstraction or process, which for Wittgenstein (but not for me) is a mental process, not a fundamental metaphysical principle.

In my view, Wittgenstein is making a mistake in separating off process from existence. Modern computer science has taught us that any attempt to mathematically model the world requires process. Specifically, it requires a language that is Turing-equivalent (or N-equivalent or lambda-equivalent, etc.). This means that its propositions or representations are equivalent to some expression in Turing-machines or NAND gates or the lambda-calculus or the N-operator, all of which are formally equivalent. Any representation, such as Wittgenstein's elementary propositions, that does not have this feature of Turing-equivalency is, in my view, metaphysically inadequate. Wittgenstein cannot in the long run get away with calling computation mental and insisting that the world lacks this feature. My own view is much like that of Leibniz, who saw reality as the computational process of unfolding an atomic "monad" that has all its relations internal to itself.[11] But for Wittgenstein, the N operator (computation/abstraction) is what we use to picture the world, rather than being fundamental to the world itself, which consists of static objects externally related to each other.

In elementary propositions, the atoms are not propositions (i.e., pseudo-names) but genuine names for noncomposite objects. These objects, as the substance of the world, are not true or false, they simply are. Wittgenstein believes that the metaphysical objects of the world cannot be "possibly true" or "possibly false" (what would it mean to have a substantial object that did not exist?). However, some of the atomic relational structures that they partake in may be false. The relational structure is a static thing. No process or procedure is required to determine is truth/falsity, since it just is true or false, as an atomic fact, in a way that is outside of logic.

While I agree with Wittgenstein that the ultimate metaphysical objects should just be, without being "possibly true" or "possibly false", I disagree that they need to be noncomposite. I see the recursive series itself as a world object, rather than just as a mental abstraction. Each world object is thus a recursive function, or computer program, unfolding perhaps to infinity. These recursively defined objects, like Wittgenstein's static objects, are not inherently true or false, they just are. Unfortunately, Wittgenstein does not even consider this possible way of looking at things. He seems, like Berkeley [14], to take for granted that process or computation is something mental.

The following is a summary of the three levels of representation we have discussed: atomic facts, truth-functions and operational series (the T's and F's below represent arbitrary truth assignments for the elementary propositions, T for true and F for false):

Each term in the recursive operational series (III) is itself a finite truth-value computation (II). The things that are thus manipulated are all ultimately atomic facts (I), or more precisely elementary propositions which refer to atomic facts. Note that the operational series in (III) above is a simple chain of propositions all built from a single elementary proposition. However, this was done only to give a simple example. As in fig. 1, the operational series can actually pull in various elementary and lower-level propositions in just about any combination desired. The important thing to note here is that the nonelementary proposition, as a procedural thing, is an abstraction that generalizes about the already metaphysically existing things. So while it depends wholly on the truth values of the elementary propositions, it has no actual existence in the world itself, independent of the existence of its atomic components. Although it is a relational structure, like the elementary propositions, its pseudo-objects (represented by pseudo-names) are noncomposite. It thus only has any meaning in the context of the internal relation that relates it to the structures it was built up from.

Process and Existence: Wherein lies reality?

The universal operator is widely believed to be powerful enough to perform any conceivable computation (although this has never been proven). This hypothesis is known as the Church-Turing Thesis, which comes in various versions. The two most important philosophically are:[12]
(1) The cognitive version: any thinkable thought can be described as an elementary proposition, plus a truth assignment, plus the universal operator.

(2) The ontological version: any existing thing can be described as an elementary proposition, plus a truth assignment, plus the universal operator.

Our central metaphysical question here concerns the reality of Wittgenstein's atomic facts, represented in elementary propositions. I will consider two possible metaphysical interpretations of Wittgenstein's elementary propositions:
(i) A proposition refers to reality only in combination with a truth assignment and an operator.

(ii) The operator is just the means by which humans abstract and form new higher level propositions about that which is represented in the elementary propositions, which in themselves are independent of the operator, and represent atomic facts that exist in themselves.

The ontological version of the Church-Turing thesis would be equivalent to interpretation (i). Wittgenstein seems sometimes to lean in that direction when discussing logic alone. A piece of a logical notation has meaning only through its use in the whole logical system of which it is part. Noncomposite names do not refer inherently on their own [T.3.3]. Yet, to speak solely from this logical perspective would require pulling in the whole logical system, and admitting all possible worlds, both imaginary and real. To say that the reality of "p" is "p + N + TTFT" is to say simply that the reality is "N", since one cannot define N without defining the domain of propositions and truth values to which it can be applied [T.5.524]. So "p + N + TTFT" is already part and parcel of "N". Thus our answer to every question about the reality of any proposition would be the same: "the reality is the universal operator." Furthermore, the notion of truth and falsity drops out of the picture when we take this view, since the N() operator is formally equivalent to formulations that do not use truth values (such as Turing machines or the lambda-calculus), and the operational series itself has no truth value. The reality of anything in this view would thus lie in its place within the whole of logic, with no role for some things being "true" and others being "false". All the worlds consisting of negative facts, which Wittgenstein labels "imaginary", would be as really true as any other other, including our own world.

This would be a kind of "all is one" idealism that Wittgenstein's background makes him loathe to consider. Looking around, he sees one particular world. This "manifest" truth leads him away from (i) and towards (ii). Since one particular world must be true as opposed to all the others, which are false, there must be a material, substantial reality to some kind of structure that is, from a logical perspective, incomplete. It is not that Wittgenstein failed to realize that his separation between process and existence left him with logically incomplete objects. Rather, this incompleteness was specifically required by his assumption that only one world is true. For Wittgenstein, one must be able to point to something and say "that exists in the world", without pulling in all logical possibilities. Objects, for instance, are obviously logically incomplete--apparently mere fictions. To ask oneself what they are, logically, will necessarily lead one to consider all possible logical forms they could be involved in. So Wittgenstein builds existence/nonexistence into his objects, in the form of truth values for the atomic facts they take part in. This divides atomic facts into the true and the false, in a way not determined by logic, but only by the world: by the application of logic in the world [5.557].

Note that interpretation (i) accepts the Ontological Church-Turing Thesis, while (ii) accepts only (perhaps) the Cognitive version. Wittgenstein's separation of objects and process/abstraction is a traditional empiricist distinction (see, for example, Hume[13] and Berkeley[14]). In the realist-empiricist tradition from which he came, Wittgenstein places reality firmly in static objects, not in a process, which he sees as an abstraction performed for convenience by humans, and not a metaphysical absolute.

Relations: Bradley, Russell and Wittgenstein

Even if we accept that some logical forms (atomic facts) exist and some do not, we would likely not take Wittgenstein's materialism too seriously if we did not feel that we could at least coherently conceive of the atomic fact itself, as a logical form. If an atomic fact were nothing more than a noncomposite object, we would likely decide that Wittgenstein was speaking gibberish. His system seems plausible because we can imagine (or think we can imagine) what a relational structure is. So even if we cannot imagine its existence, which we perhaps can accept as a minimal amount of mysticism, at least we can imagine everything else about the relational structure other than its existence (i.e. the logical component).

Yet idealists have maintained that relational structures as things in themselves actually cannot be imagined. Indeed, their reasons for believing this were very much like Wittgenstein's reasons for believing that, within logic, one cannot separate out a proposition from the logical system it is embedded in. The nineteenth-century idealists, however, had no commitment to materialism, and no problem saying that reality was nothing more than pure logic (although for them, logic had a fundamentally nonformal, mental character).

In the early days of this century, the great analytic philosopher Bertrand Russell debated this issue with the great idealist F.H. Bradley, in the journal Mind. Russell, whose philosophy was a reaction to Bradley much as Wittgenstein's was a reaction to Russell, believed that a relational structure could exist independently, as a metaphysical object. Bradley thought this was nonsensical. Relational structures, he thought, were no more real than the individual notational name "a". As objects, they were completely dependent on the entire notational system used, with all its quirky charm.[4][15] For Bradley, as for Leibniz, the internal relations, defined in terms of process, are the full reality of the thing, and truly internal to the metaphysical object. Wittgenstein, inheriting Russell's legacy, not Bradley's, sees these internal relations as abstraction or generalization from reality, as mentally relating metaphysically existing structures to each other, not as fundamentally part of the things themselves. For Wittgenstein, what has metaphysical existence/nonexistence is a static system of external relations. But Bradley maintained that static relational structures are not "real". When we look at a relational diagram on a piece of paper, it makes sense because the symbols on the paper really represent abstractions, processes, in our minds. We apply meaning from the outside, via some process (an operator such as N, perhaps?).

Figure 2 illustrates the "fission to unreality" that Bradley talked about. We start with the relational structure in fig. 2(a): "John =loves=> Mary". But of course the words "John", "loves" and "Mary" are not really part of the relational structure, but are just there as reminders to us. They are simply notational devices that in themselves mean nothing. So far, this agrees with Wittgenstein, who says that a name, like "John" or "a", has meaning only through its use in a logical system, that places it within a relational structure [T.5.45]. So the relational structure, if it exists in the world "for real" apart from our own conception of it, might just as well be expressed with different, "unmeaningful"  names, as in fig. 2(b): "a =b=> c". Already, the meaning seems to be slipping away (if it was there at all in the first place). Yet the relational structure itself still seems somehow intact.

 Fig. 2: Are relations unreal? What is the difference between a and g?

But, of course, "a", "b" and "c" are also just arbitrary labels. In fig. 2(b), there is really no reason to say that the relation "b" has anything about it that makes it "relational", as opposed to being just another object. So really, we just have three connected objects of different types, drawn in fig. 2(c) as three objects of different and "unmeaningful" shapes. But what is holding these objects together? If they are just three completely independent objects, then there is no relation anymore. One problem here seems to be that there is nothing in our notation to properly distinguish a relation from an object. So we need to better define what the relation is. Either we have no structure at all, or we need to say something like "Well, our original diagram was underspecified, so when you consider the relation as an object, you need to fill in more relations, as in fig. 2(d)." But then, we can do the same thing and consider these as objects, as in fig. 2(e). So we end up with a regress that gives us, in the limit, an infinite number of completely unconnected objects, as in fig. 2(f). Of course, there is no point in continuing to draw different shapes, since the objects are now completely unconnected. In fact, if all we have is an infinite set of unconnected objects, why can we not say that there is really just one featureless object, as in fig. 2(g)?[16]

According to Bradley, relations exist only insofar as the objects within the relational structure get manipulated, used, or processed so as to form such relations. The static relations themselves fall apart unless viewed within some larger process. Without process, there is no meaning or reference, and hence no reality, since the complex system of "real" objects might as well be a single object. The Bradley regress is familiar to computer programmers who attempt to represent the world as a static system of related objects. One soon realizes in programming such a system, that there is really no justification for calling the "relations" relations rather than mere objects, unless the relation is used by some kind of procedure that describes how to manipulate the connected objects in a certain way. So a process is needed, or the relational structure will melt away to nothing.

Wittgenstein sees mental abstraction in much the same way, but does not believe reality inheres in this abstraction. Logical abstraction/process are mental for Wittgenstein, as for Bradley. But Wittgenstein believes, unlike Bradley, that there must be a nonabstract truth grounding for our mental abstractions, which exists in a real, particular world. The reason Bradley felt he could do away with this truth grounds is that he believed that all possible worlds exist, and therefore reality is simply all of logical possibility, which is necessarily procedural. For Wittgenstein, there is a fundamental separation between logical process/abstraction (mental), and existence (static and nonmental). This was necessary for him, because he assumed that only his world of experience was objectively real, and the other possible worlds were objectively false, or merely imaginary.

I tend to agree with Bradley that existence and process cannot be separated. Bradley's regress argument, however, does not quite work as it stands. Such an argument can be applied to any notational system, since any logical system needs some fundamental entity or entities that are left undefined. Bradley perhaps feels that the defender of static relations is forced into viewing relations as themselves related to objects, since a relation on its own has no meaning and, since static relations are supposedly adequate to describe the world, the only way to define the relation is with more relations. So Bradley's regress cannot be accepted unless we agree that the relations are inadequately defined to begin with, and since any notational system will contain some such "inadequacy", Bradley's regress is insufficient as it stands. Today, post-Turing, the regress can be invoked without this problem, since today we have good reason to believe that static relations really are inadequate, since they do not provide Turing-equivalency. Of course, Bradley was ultimately a mentalist who did not see formal logic as adequate in the first place, so he escapes the criticism through a bit of mysticism of his own.

In summary, Wittgenstein needs his static truth-grounds to accommodate his materialist split of logic into the real and the imaginary. Unfortunately, this leaves him with a metaphysical language that lacks Turing-equivalency and hence is insufficient to describe the world in the first place.

Bradleystein: A Hybrid Philosophy

A universal operator, such as the abstraction operator in the lambda-calculus or Wittgenstein's operator, provides the process of abstraction spoken of by both Bradley and Wittgenstein. In fact Wittgenstein seems to echo Bradley in claiming that it is the use of the notational signs and other jits and jots on paper that give them meaning. Likewise, a nonelementary proposition has meaning only when considered within the entire logical notation in which it is expressed. This expression is a repeated application of an operator, transforming one relational structure into another, into another, et cetera. So, if we stop here, considering nonelementary propositions only, Wittgenstein seems on the verge of rediscovering Bradley-style idealism. The truth is the whole. The real thing is its use in the entire system of which it is a part.

While Wittgenstein comes close to this, he stops short of it, and the reason lies in his (in my opinion) mystical belief that some logical forms exist and some do not. Such "forms" are thus not purely formal in nature at all. If Wittgenstein restricted himself to logic, the notion of existence/nonexistence would disappear, and all he would be left with would be all possible logical forms (atomic facts): all possible worlds would be equally true. This "truth" becomes uninformative and tautological, of course, since there is no longer any notion of absolute falsity to oppose truth. But to Bradley, ultimate truth is tautological. Truth as opposed to falsity applies only from the perspective of a particular world. Absolute reality now lies in the universal operator, and Wittgenstein's metaphysics has become a demystified, analytic version of Bradley's idealism.

According to Bradley, what we normally think of as the real world is simple one person's perspective on the whole of what is logically possible: one possible world out of many, each of which can be considered as a relational structure, but only if not separated from the overall procedural system in which all possible worlds are embedded. This is similar to what we get if we modify Wittgenstein's metaphysics into a purely logical one, but without Bradley's mentalism--call it "Bradleystein". Substance (existence/nonexistence) would be erased from his system, leaving no distinction between elementary and nonelementary propositions. An elementary proposition would be defined only in terms of its operational expansion into a series, via an internal relation or operator. An elementary proposition would have meaning only as combined with an operator and truth assignments--in other words, as generalized into nonelementary propositions. It would only be through this process/abstraction that the elementary proposition would refer to something that exists. For Wittgenstein, the nonelementary proposition accomplishes reference through formal (mental) abstraction, but the elementary proposition achieves reference without abstraction. Yet one cannot say, within logic, what makes an atomic fact true or false, which is determined not by logic, but by the use of logic in the real world.

The problem here for Wittgenstein is that, by his own admission, this real world existence cannot be described, or shown in logic. So he must resort to mysticism, and simply declare it as a brute fact without explanation [T.6]. In Bradleystein's hybrid system, the world is simply the set of all atomic facts (there is no absolute distinction between positive and negative facts), with all possible truth-assignments, coupled with the universal operator. This reduces truth-values to nothing, yielding all possible worlds (and many structures we probably would not apply the term "world" to at all), defined as relational structures, all of which are on equal footing, but none of which exists apart from the universal operator. The "reality" of the now procedural atomic facts becomes a formal, but nonmental, process. I tend to prefer the term "process" over "abstraction". The word "abstraction" (and even sometimes "formal") has mental connotations in some circles. Although I personally do not insist on distinguishing between "formal" and "abstract", I think "abstract" is more likely to refer to a process that we mentally use, or could use, to refer to some other more complex, difficult-to-represent, "formal" object in our world of experience. Current usage of these words is not, however, consistent enough to insist on making this distinction or not.

So any individual relational structure cannot really be conceived of without admitting all the rest that are part and parcel of the formal system that we are using to create the structures, one out of the other. Since we are sticking purely to logic, this whole logical system is reality, not any one world, which is just one person's perspective on reality. What I normally think of as "my real world" is simply "all of logic, seen from my own perspective, as one aspect of it".

Bradleystein's system can be contrasted with that of both Bradley and Wittgenstein on their own by calling it a "formalist" metaphysics. Reality is nothing more than formal systems, which are independent of the arbitrary notation in which they are expressed, so long as the notation is Turing-equivalent. Wittgenstein is not a formalist because he believes formal systems are mere mental abstractions of the truly real objects, which are static. But Bradley is not really a formalist either, because he believes a formal system gets its reality from something mental. But a Bradleystein-style computational metaphysics removes the materialism from Wittgenstein and the mentalism from Bradley, leaving us only with the formal core, which is essentially the same for both. Reality is Turing-computability.

So far, Wittgenstein's treatment of relational structures and operators seems to follow the strict material-realist tradition of Russell and Frege, insisting that "my real world" be real in an absolute way, independent of the other imaginary worlds (or negative facts). Yet, in grappling with this issue, Wittgenstein seems in later sections of the Tractatus very close to an idealist solution much like Bradley's. Since the only way I can determine what the elementary propositions are (or which atomic facts so represented are the true ones) is by using logic in the real world of my experience, and since this is a completely nonlogical matter, it follows that "the real world" is simply "my world" [T.5.63]. This seems very close to saying, a la Bradleystein, that all possible worlds exist within the unity of formal logic, and I just happen to view them from the particular perspective of the one I happen to be in. If the existence/nonexistence distinction were erased from the Tractatus, the apparent brute fact of my own world of experience would no longer be something mystical outside of logic, but simply the accident of my being me in this possible world, as opposed to some other conscious being in some other possible world. Again, Wittgenstein seems close to actually saying this [T.5.6], but in the end he is too firmly mired in the anti-idealism of Frege and Russell to take this further and give the universal operator the metaphysical significance that an idealist Bradleystein would have.

To Wittgenstein, such a move would turn the world into nothing but formal logic, and deny that "my real world" is objectively real, which is an unwritten assumption throughout the Tractatus. It would also mean that Wittgenstein would have to dump the idea of "objects" and "elementary propositions", and place the reality of the world in the universal operator used to turn one relational structure into another, rather than in the relational structures themselves (or in the elementary versions from which they are built).

As a result of his implicit rejection of idealism, Wittgenstein is forced not to take some of his own claims about logic seriously enough. Take, for instance, his elegant treatment of the logical versus purely notational component of our real-world logical systems [T.4]. If a notational device has meaning only through its use in a logical system, then the distinction between tautologies, propositions with sense, and contradictions disappears. Wittgenstein warns against giving a logical form meaning apart from its use, yet his notion of "sense" seems to do just that. I personally can see nothing about any logical form that, within the logical system, means that the logical form has the possibility of existence. What is there about a proposition, from a logical perspective, that says that a relational structure is true as opposed to false (a thoroughly undefinable distinction)? I see nothing that could possibly perform this function, nor do I see anything that could make a proposition inherently contradictory. Wittgenstein agrees that, within logic, everything is just as it is: we can make no mistakes [T.5.473]. All propositions of logic are tautologies [T.6.1]. Contradictions and falsities only appear in the application of logic to the real world [T.5.55]. Yet Wittgenstein turns around and insists that we must admit that we do say something about an external world beyond logic in our propositions, but he cannot define what this is. Logically, there is no such thing as a "proposition with a sense" or a "contradiction". All there are, logically speaking, tautologies. To Wittgenstein, these say nothing about the world. To Bradley, they say all that can be said.

Both Wittgenstein and Bradleystein agree that to refer usefully to a complex world we must use abstractions and leave out part of reality. But Wittgenstein believes these are abstractions of (operations on) something nonformal, static and thus particular and nonabstract, which is required for something to be real in the world. For Bradleystein, however, mental abstraction is just one kind of formal process. I am being noncommittal here as to whether we can use the word "abstract" for a particular thing. But even if we decide not to call a particular thing "abstract", it is still not a static material thing, but a procedural, mathematical, "ideal" entity that any materialist would almost certainly want to call "abstract" in any case.

The notion of truth/falsity is not a fundamental metaphysical concept to an idealist. The procedural forms have no truth-grounds. The truth just is; it has no falsity opposed to it. An operational series, defined by an internal relation, is a metaphysically existing entity for Bradley, even though it has no definite truth value. When one speaks purely metaphysically, and not from one's own arbitrary vantage point, the notion of a truth value disappears as illusory. So to accept a purely logical view, Wittgenstein would have to admit the reality of the imaginary (negative) worlds. And this, which Bradley did willingly, he does not do, at least not explicitly. Wittgenstein's position could perhaps be interpreted as it stands in idealist terms, since he sees the existence/nonexistence of relational structures to be a matter of the world one finds oneself in, and in this sense he seems not to directly declare the false worlds as absolutely false. However, this is not explicit and does not inform the development of his metaphysics enough for him to make the idealist move of locating reality in process, rather than static relational structures. For a Bradlian idealist, in the tradition of Leibniz, all metaphysical relations are internal relations, and procedural in nature. However, while Bradley realized this, he was nowhere near coming up with a solid analytic treatment of the kind Wittgenstein did. He saw process in purely mental terms, since it involved abstraction. Reality was "All Experience" to Bradley.

We will see in the next section on the Philosophical Investigations that it is Wittgenstein's reluctance to take his own metaphysics to its logical idealist conclusion--that the truth is the whole of logic--that ultimately brings down the Tractatus. The later Wittgenstein realizes, through a series of paradoxes much like the Bradley regress, that his logical forms (the static relational structures) cannot possibly be said to be true or false of the world, as they will always be subject to reinterpretation under some alternative abstraction. Wittgenstein finally comes head to head with the problem of how it is that elementary propositions can directly refer, through identity, to the real world without some kind of abstraction, which must ultimately be arbitrary. Once again, Wittgenstein has an opportunity to embrace idealism. He could have decided then and there that the reality must lie in the whole (the universal operator), and not in its finite parts (the particular relational structures). But, still insisting on objective material reality for his personal world, and moving away from any idealist tendencies implicit in the Tractatus, Wittgenstein cannot find a way out. In the end, he resorts to dropping metaphysics altogether, declaring it a nonsubject, and placing the meaning of propositions as, indeed, mere parts in a whole, but not the all-encompassing logical whole of idealism. A proposition will now get its meaning only through its use ,via the application of rules, within the more limited whole [T.6.45] constituted by a real-world community of language speakers.

III. The Investigations

The Paradox

The later Wittgenstein's main argument against a Tractarian-style metaphysics can be divided into two parts: (1) a sceptical part, which shows that any definition of meaning that abstracts a statement out of the overall (real-world) context in which it is used can be reduced to paradox, and (2) a solution, which defines "meaning" in terms of the entire context of a statement's usage.[17]

I will use Kripke's[18] example of addition to introduce Wittgenstein's main sceptical argument.[19] It is similar in many ways to Bradley's regress. Bradley said that a relational structure cannot refer to reality because its relations depend on the meaning we give it from outside, by using them in some logical manipulation.[20] Some kind of external interpretation of a relational structure is required for it to have any meaning at all. And from the point of view of the relational structure, our choice of interpretation is arbitrary, so the structure could be made to mean anything we want by an appropriate choice of interpretation. This is essentially the paradox Wittgenstein rediscovered, convincing him that the Tractatus was ill-founded (although his arguments in the Investigations are aimed more generally at all metaphysical approaches to meaning, not just the Tractatus).

Kripke's example asks us to perform the following addition: 68+57. We do, and we get 125. When asked to justify this answer, we reply that "+" is defined by the following algorithm, at which point we proceed to describe in detail the method for adding two numbers. But suppose, Kripke suggests, that we have in the past only added numbers less than 57. Then there is no way to justify our claim that "+" means addition (which we will call "plus"). It is perfectly consistent with all our past behaviour to assume that, all along, we meant "quus" when we used "+". "Quus" is a function that behaves just like "plus" when both numbers are less than 57, but yields the value 5 in all other cases. Thus, we now have no justification for answering 125 instead of 5!

But, we protest, I know that I meant "plus". True, "quus" is equally consistent with my past use of the term "+", but I meant "plus", not "quus", and surely my own memory of what I meant should not be questioned.

It is precisely this sense of "meaning" as an internal understanding or mental process that Wittgenstein is against. Supposedly, we had an internally felt idea of addition in all those past cases. This idea implicitly contained all possible additions within it. So even though our use may have been consistent with both "plus" and "quus", this internal idea had the meaning "plus" and not quus--it was an algorithm or procedure of some kind that described a rule for how to add. This rule produces 125 for "68+57", not 5. And it was this rule we used in the past, not quus.

That sounds air-tight and quite intuitive. But Wittgenstein seems to reduce many such examples of supposed meaning to paradox. In this case, the rule we give for adding cannot be said to actually be the act of adding; it is an abstraction and so leaves part of the actual situation out of the picture. There will always be alternative interpretations to such rules. For instance, our description of the addition rule probably included a reference to "counting". But, as Kripke points out, how can we justify our meaning of "count" as the traditional 1,2,3,...,56,57,58,...? Perhaps what was really meant by "count" was to "quount", which is just like traditional counting except when a group being quounted is formed by combining two smaller groups, one of which has 57 or more elements, in which case the result of quounting is 5.

Many other ways can be suggested to define addition, but if they involve abstraction or interpretation, they will leave out something of the real-world context and will be susceptible to the following paradox: an abstract interpretation will itself have more than one interpretation, and each of these will in turn have more than one interpretation, et cetera (see also I.86).

 Fig. 3: Why are we justified in choosing one interpretation as the meaning?

So we cannot claim that the meaning of "+" is "plus" and not "quus" on the basis of an abstract interpretation of the rule. Wittgenstein puts it like this: "no course of action could be determined by a rule, because every course of action can be made out to accord with the rule" [I.201]. The problem, as shown in figure 3, is that any choice of an interpretation will always be an arbitrary choice amongst equally possible interpretations. "+" can be interpreted as "plus", "quus", or in innumerable other ways. If we try to define "+" in terms of one interpretation only, say "plus", we find ourselves in exactly the same boat: "plus" in turn has multiple possible interpretations, call them "plus1", "plus2", et cetera. If we decide that what we really meant was "plus4", then that will again have multiple interpretations. Meaning has somehow slipped through our fingers. True, some of these possibilities may seem too odd and off-key to really be taken seriously, but surely if they are so obviously wrong, we ought to be able to say just what it is about them that excludes them from consideration.

The Solution

Wittgenstein's way out of the paradox is to find "a way of grasping a rule which is not an interpretation" [I.201]. An interpretation is analogous to the use of an operator on an elementary proposition in the Tractatus. Wittgenstein now realizes that, if he separates the elementary proposition from the operator (taking reality as "p + existence", instead of "p + N + TTFT"), then one's choice of operator is arbitrary, and p cannot be said to have meaning on its own. This realization was, of course, already there in the Tractatus! But there, Wittgenstein resorted to mysticism as a solution. Wittgenstein is really just coming to terms here with a problem he already saw in his earlier work.

The idealist solution, as we have seen, would be to admit all possible interpretations, but place reality in the interpretations themselves, not the referent of a "fictional" elementary proposition. This solution is to see the meaning truly as the use within the entire logical system. The truth is the whole (an old idealist catch-phrase). But, Wittgenstein is still not considering idealism as an alternative. The real world of his experience still has an absolute status for him above logic. Yet Wittgenstein's solution is similar to idealism in many ways. The meaning is the use within a whole, but not an all-encompassing logical whole that would admit all possible worlds. Wittgenstein's more limited whole [T.6.45] is, not surprisingly, his own world of experience, what he sees as the "real world".

So by "interpretation", Wittgenstein means the abstraction of a rule out of its real-world context. Now, instead of pointing to the way we use a name within some universal logical system, he is actually pointing to the jits and jots on paper, and their use within our culture and society, as the meaning. This context includes a whole community of language speakers in which one is raised, along with the rest of one's environment, one's brain, genetic makeup, et cetera. Within this whole environment (making up my "form of life"), I play a game with the other members of my society. In this game, there are rules (such as how to add). The right way to follow a rule is determined by the customs and conventions of the community, embedded in a particular shared form of life. My meaning of "plus" is not a logical or a mental process, although it involves my mental processes in combination with the rest of the game and game-players. The entire situation determines the "meaning" of plus.

So "meaning" is necessarily a characteristic of the entire context: "world + me". There is no "meaning" that is completely internal, contained entirely within my mind: "to think one is obeying a rule is not to obey a rule. Hence it is not possible to obey a rule 'privately'" [I.202].

But how can that be? Why can't I simply imagine a rule inside my head, and follow it? This is what I claimed to be doing when I tried to defend "plus" against "quus", and it is what I am doing when I argue that the reality referred to lies in the universal operator--in whatever can be thought of. But Wittgenstein does not consider such all-encompassing idealism as an alternative. He assumes (as in the Tractatus) that the meaning of a proposition refers to something beyond what can be thought of--to a material world. So he is forced to conclude that when we imagine the "plus" interpretation, there cannot possibly be any meaning there, since our purely logical imagining cannot on its own refer to an external world beyond logic.

So has Wittgenstein resorted to saying that I cannot even have internal ideas or thoughts with any sense to them at all? Is he perhaps even questioning my very ability to think? I do not think Wittgenstein is being that radical, although he does seem to be rejecting "sense" in the Tractarian sense of the word, as involving existence. But he does not deny that you can have a completely internal mental idea of a function "plus" that always yields "125" for "68+57" and not "5". In generating the multiple possible interpretations in figure 3, the paradox arose, not because each possibility was incoherent or somehow nonsensical, but only because there was no way to choose between them. Wittgenstein's "meaning", then, requires that it be a unique meaning, and purely private "meanings" are not unique. Without the external context of a community that by custom has used "+" in a particular way, your internal idea cannot be said to be the meaning of "+". What we are talking about here is one idea ("+") representing, referring to, or meaning, another idea ("plus"). Wittgenstein is allowing you to have whatever ideas your imagination can muster. He is simply not allowing that you can make one idea "mean" another by an internal act of will. He is not (I think) suggesting that you need be sceptical about your having had the idea "plus", nor that it yields 125 and not 5. Neither does he deny that you can imagine running through the rule in your head and getting the answer 125. All these are immediate facts of your experience, and were Wittgenstein to deny them, he would be on shaky ground indeed.

I think what Wittgenstein is doing is much more mundane than this. He is simply saying that without the custom of your community as a standard by which your answer is judged, you are completely free to imagine either the "plus" or the "quus" processes as the meaning of "+". Just by imagining that you are following the "plus" interpretation in your mind does not mean that you are really following a rule. The phrase "to follow a rule", for the later Wittgenstein, implies that there is only one possibility open to follow. That possibility must be justifiable. If you are free to choose any old interpretation, then imagining following the rule is not actually following a rule.

The footnote to I.38 is illuminating. Wittgenstein suggests that one could decide, in one's imagination, that "bububu" means "If it doesn't rain, I shall go for a walk." One certainly could decide to imagine this. But "meaning" for Wittgenstein is more than an idea's making sense on its own terms (as for the idealist). Meaning is supposedly a correspondence between this idea and something in the material world. Traditionally, material realists see this as a metaphysical issue: to determine the meaning, you must show how an idea in the mind can pluck out some object in the world as its referent. This is the position Wittgenstein is most concerned with refuting: the idea that a word or idea can somehow carry its meaning with it [I.1,97]. For lack of a better term, let us call this "the interlocutor's view", since it is the view often taken by Wittgenstein's imaginary dialogue partner (who has much of the early Wittgenstein in him, but also includes other metaphysical materialists). Wittgenstein sees this as an "occult" view of meaning [I.38]. If the meaning is a correspondence between an idea in my head and a thing-in-the-world, then the meaning must therefore be a characteristic of this entire situation (me + world), not something within me that somehow on its own terms is able to successfully point to the correct object in the world.

"The grammar of 'to mean'", says Wittgenstein, "is not like that of the expression 'to imagine' and the like" [I.38 footnote]. I am free to imagine "bububu" and what it means, but "imagining meaning" is not by itself "meaning". It can only really be meaning if it is embedded in a form of life, within a community of speakers that have a common use for "bububu" in their language game. We are free to use abstractions to describe this entire situation, of course--Wittgenstein is not denying us even that. But as soon as we describe the entire situation with such an abstract interpretation, we are open to Wittgenstein's paradox. What Wittgenstein forbids is the declaring of one particular abstraction as the meaning of an idea. The entire situation must in the end be left as it is. Abstraction can be a tool to understand it within a certain domain, but no single abstraction of it can be given special status. To do so is to delve into a kind of metaphysics that Wittgenstein is anxious to avoid in the Investigations.

Logic and Meaning: Several Possible Solutions

What I have been calling "abstraction"--what is exemplified in the interpretations of "+" as "plus" and "quus"--is discussed by Wittgenstein in many different guises: as the mechanistic process of a machine [I.193], as an algebraic formula [I.186-7], as a lookup table [I.86], et cetera. In the Tractatus, it is a universal operator repeatedly applied to elementary propositions. In [I.89-102], Wittgenstein discusses it in terms of logic, obviously harking back to the Tractatus, although this is not explicitly stated. It is this logical view that Wittgenstein still seems to find most compelling. Logic, it seems, is the principle of abstraction that underlies all these different formulations. Logic seems to be what is left after one translates between all these different languages. In the Tractatus, Wittgenstein even defines meaning in just these terms: what is common between expressions in different languages [T.3.344]. This is very much like my own definition of meaning, alluded to earlier, in terms of computability: meaning is what is retained in translation between Turing-equivalent languages. Could it be that logic thus encapsulates what it is for an idea to mean something--the essence of a rule in a language game?
The Early Wittgenstein's Solution
In order to get a firmer grasp on this issue, let us first look at logic as viewed by our interlocutor, whom we shall consider as roughly equivalent to the early Wittgenstein. The interlocutor sees logical formulations either as the referents of our language or somehow as intermediaries between the ideas in our minds and the objects in the world. For instance, I have an idea of "plus" in my mind, to which I attach the symbol "+". In the future, when I see "68+57", I call up the "plus" idea, which can be described as a logical form, and apply it to 68 and 57. Now perhaps there is also a rulebook (or custom, or whatever) in the real world that this idea corresponds to. If there is such an object, then my idea "plus" can in turn refer to this object in the world (perhaps by virtue of having the same structure, as in the Tractatus).

Thus, the symbol "+" refers to this rule in an external rulebook, through the intermediary of a logical form. But there is no necessity that it do so in order to have meaning. "+" could be used completely privately by me to refer to my internal idea of "plus" even if there were no rulebook in the world, and even if nobody else had ever performed this function before. Thus, the interlocutor is allowing a "private language", known and understood only by me. Thoughts have meaning on their own terms. This is the Tractarian position. A picture of an atomic fact can have a sense (that it exists), whether or not the atomic fact really does exist or not. The meaning of a proposition is carried with it; its truth or falsity is not.

Figure 4 sketches the relationship between my thoughts, logic and the external world, within the interlocutor's system. Here I have three ideas in my mind: "p1", "p2" and "p3". Thoughts, in Wittgenstein's view, are associated with possibilities [I.97]. I can imagine any logical possibility I want. Each imagined possibility (such as p2) can be interpreted in multiple ways. The interlocutor presumably would claim that each of these possible ways of interpreting p2 (such as t2 and t6) is a meaning (possibly private, possibly public). If the idea happens to refer to something in the world, then there is an appropriate logical form (t2) that has a correspondence with this object (o2). So the logical form t2 is in some sense an intermediary between the private idea and the real-world object [I.92]. Logical forms are what the world and my thoughts can have in common [I.97], and thereby they provide the connecting link between the two.

 Fig. 4: The role of logic in the interlocutor's system.

Unfortunately for the interlocutor, there will always be more than one logical ordering to any possibility one can think of. Not only that, but there will always be more than one that is in accord with the world (as with plus and quus). So the interlocutor must admit that both t2 and t6 could correspond to o2. How can we know which possibility is the "correct" one? In figure 4, there is no justification (if the reasoning in figure 3 is correct in general, then there can be no justification) for choosing t2, and not t6, to refer to the world. The choice is completely arbitrary. But what if, the interlocutor might reply, t2 somehow "agrees with" the external world and t6 does not? But as we saw in figure 3, and in the Tractatus, there will always be an alternative logical ordering (perhaps t6, perhaps some other ordering) that will refer p2 to the "wrong" world object.

In the Tractatus, p2 is an elementary proposition, which refers to o2, an atomic fact. Yet, as we saw earlier, what makes p2 elementary is beyond logic. Separated from the universal operator, p2 could be subject to any operator (i.e., interpretation). So this whole problem that the later Wittgenstein sees with the interlocutor's system is already there in the Tractatus. However, the early Wittgenstein's rather vacuous solution no longer satisfies the later Wittgenstein.

The Idealist Solution
The idealist solution to this kind of paradox--the kind of solution adopted by Bradley--assumes that logical form is all there is to the external world in the first place. Once we assume this, there is no longer a problem in choosing between t2 and t6, for they stand on equal ground. There is no longer a problem in choosing between operators, since the reality (the thing referred to) is in the operator (so long as the operator is universal, or Turing-equivalent). This, as Wittgenstein noted in the Tractatus, inevitably drags in the entire logical system, but that is no problem for the formal idealist, who sees ultimate reality as logic and allows all possible worlds equal objective status. Meaning no longer need correspond to a single "correct" and "true" world; the existence of two or more meanings (and hence worlds) no longer bothers us.

It is tempting to object that this places the reality in the meaning act, rather than in the actual thing referred to by the proposition or thought that performs the meaning. Traditionally, this is indeed a mistake the idealists made: equating thought with reality. But this is not necessary. A more objective idealism equates reality with "what can be thought of", not "thought itself". This does not make reality dependent on thought. Reality can be what is thinkable, and thus definable using thought, without being identical with it. In this respect, Bradley could do well to listen to Wittgenstein. Although less mentalist and more objective than many idealists, Bradley still thought that the fundamental metaphysical things were somehow mental. In the early Wittgenstein's system, what is mental (thoughts and pictures) can be defined logically, but this does not mean that logic depends on the mental. Mental pictures are a subset of logical form, not all of logical form.

In figure 5, the objective idealist has no problem deciding which of t2 and t6 are "correct", since they are both correct. The external world is literally nothing but logic. One idea can be said to equally refer to two contradictory things. In the formalist idealism I am advocating (the Bradleystein hybrid), it is possible for an idea (such as p2) to refer to more than one logical form. Using the distinction between formal and abstract that I suggested earlier, p2 is an "abstract" idea because it refers to more than one other particular logical form. As such, it is a mental object, involving abstraction. We have arbitrarily decided to represent the (possibly extremely complex) forms t2 and t6 by the simpler, easier to imagine, form p2, which is also one of the t's, but one that is simple enough to contain in our heads. This is abstract reference, and is justified purely privately. We have decided, in our own minds, to use one form to refer to others. Wittgenstein's Tractarian view of this as higher-order abstract propositions (defined in terms of N) is perfectly applicable here. What is not Tractarian is the assumption that the things referred to are themselves just other logical forms of the same kind as p2 (just possibly too complex to be contained in a human mind).

In the formalist idealism I have argued for, the p's can thus be "abstract", leaving it unspecified whether they refer to say t2 or t6. But, and this is very important, they can also concretely refer to only a single logical form, like p1 does. Such a form does not require the arbitrary mental justification of using one form to refer to more than one other form. Although we certainly can build recursive structures, like p2, that through an act of will refer to other recursive structures like t2 and t6, this is not required for reference, since we are taking the objects that are referred to as being themselves also recursive structures, not something fundamentally different in kind. That which is referred to is the same sort of thing as that which does the referring. Therefore, incomplete abstract reference is not required for reference to take place, but is simply a practical requirement given the resource limitations we are faced with. So we can also build formal structures that are complete and nonabstract (such as when we add two numbers together). Here, the formal recursive structure we wish to refer to is, unlike most world objects, simple enough by human standards to build completely within our heads. We can entirely build it in our minds, and so do not need to make other, simpler forms that somehow mirror its structure. We have, almost, what Wittgenstein wanted: a meaning for "meaning" that does not require an interpretation! At least, it requires no interpretation beyond the universal notion of Turing-equivalency (some sense of the adequacy of such formalisms must still be assumed). When a recursive structure is simple enough to refer to in this complete and nonabstract way, we feel a kind of certainty about our knowledge of it that we do not feel when we abstractly and thus incompletely refer to structures that are too complex to fully deal with. These more complex structures we thus feel are somehow tinged with "contingency" and uncertainty. But this contingency and uncertainty are products only of our inability to fully grasp the structures we are talking about--not, as material realists would have it, because the objects themselves are somehow inherently, mystically, nonlogical and ungraspable in their very essence.

This is an even more extreme case of allowing private meaning than that of the interlocutor. But that is precisely why it works. Here, all meaning is in some fundamental sense private--since what can privately be thought of is equated with external reality. The paradox of meaning is dealt with by the radical move of destroying any distinction between possible ideas and the external world.

 Fig. 5: The role of logic in an idealist system.

I think the idealist solution really does work to solve the paradoxes, but for many it is a solution that is simply too uncommonsensical to be accepted. They see problems with idealism in explaining our world of experience as the ordered, stable entity that we all know. If all possibilities are equally real, they argue, how can our world be as ordered and stable as it is, with objects persisting through time and obeying set rules of physics, etc. Bradley certainly felt he had a rough solution to such problems, using his coherence theory of truth (and I would argue that quantum mechanics provides us today with an even better answer, although there is not room here to defend this position). However, there are many others (even amongst those who understand quantum theory) who see the problems as intractable . My own sympathies lie with a truly objective, nonmentalist idealism (the Bradleystein hybrid constructed by removing mystical "existence" from the Tractatus, or by removing mystical "experience" from Bradley's system). Such an idealism might solve the metaphysical paradoxes without the mentalism that was largely responsible for the early Analytic rejection of idealism in the first place.

The Later Wittgenstein's Solution
The later Wittgenstein's solution can be seen, in one light, as a compromise between idealism and material realism, extremes between which the Tractatus seems implicitly torn. The idealist attempts to solve the paradox inherent in the interlocutor's system, but at the cost (some would say) of the reality of our common-sense world. So instead of erasing the external world, Wittgenstein keeps it around, but does not allow ideas to "mean" logical forms (at least not in general--an idea can certainly refer to a logical form if that logical form is embedded in a language game played by the community).

 Fig. 6: The role of logic in the later Wittgenstein's system.

If logical forms are indeed what the world and ideas can have in common, then meaning does not arise from this commonality in Wittgenstein's system. There are many possible orderings to an idea. Each of these could itself be an idea. So t1 could be instantiated in my mind as idea p1. Logic describes what is possible, and I am allowed to think of any of these possibilities. But for the later Wittgenstein "meaning" has to do with more than unconstrained possibility. Meaning constrains my thoughts of possibility, according to my form of life, in the world of my experience. So the "private meaning" arrows in figures 3 and 4 are no longer drawn as "meaning arrows" at all--call them "possibility arrows" instead. The meaning arrow that gives p2 its reference to o2, I have simply drawn in figure 6 straight from the "mind bubble" to the world. Understand that this diagram is not an illustration of how meaning works in the Investigations--it is simply meant to show where logic and meaning stand in relationship to each other. As can be seen in figure 6, Wittgenstein has separated the two. A logical form can still be "had in common" between a thought and something in the world, but this cannot be the meaning of the thought.

An idealist such as Bradley would see the solution in the Investigations as naive, since Wittgenstein takes the existence of his personal world as an absolute. Wittgenstein believes he is being rigorous in not attempting to answer meaningless questions about what "lies beyond" and restricting himself to all he can really know: his personal world of experience. But Bradley would claim that Wittgenstein has already admitted the other possible worlds in his use of logic, which is implicit in his very thinking in the first place. Indeed, by claiming that his personal world is somehow the real world, he is once again, as in the Tractatus, claiming that the world is ultimately mystical. Of course, it is just this mysticism that Wittgenstein hopes he is avoiding, but Bradley would claim Wittgenstein has already lost the argument the moment he uses logic in his own mind. But Wittgenstein's solution seems not to address such issues at all. Partly, this is because the idealist alternative was never explicitly recognized by Wittgenstein, even though it lurks in the background. But partly, it is also because the later Wittgenstein has become convinced by his own paradoxes that there is no solution to the metaphysical questions, and they are thus meaningless nonquestions. The later Wittgenstein has, in frustration, literally given up on metaphysics.

Against Private Meaning

One of the greatest difficulties in reading the Investigations is understanding Wittgenstein's stance against private meaning in such a way that it does not contradict the obvious fact that we do have thoughts and sensations that are private, which we need not share with others. I do not believe that Wittgenstein is necessarily talking only about thoughts that are in principle noncommunicable (although these would certainly not have meaning in his system). His denial of meaning to private thoughts, I believe, includes thoughts that are perfectly coherent internally, and could in principle be communicated, but nonetheless have no reference, because they are not embedded in a form of life. Recall that Wittgenstein, unlike the idealist, views meaning as distinct from imagination. A private thought that did not refer by virtue of a public language game would simply have no meaning. Does this mean it would be noncommunicable in principle? It would indeed remain noncommunicable until it was embedded in a public language game, and thus a form of life. Then it would have meaning, but only with respect to that form of life.

It is worth asking whether there is any sense in which the private possibility lines in figure 6 could become lines of meaning. However, by the later Wittgenstein's notion of meaning, the line from p2 to t2 cannot be said to "refer" or "have meaning" unless t6 (and all others) are eliminated as choices. This requires that t2 be a commonly shared custom in the language-speaking community, and hence it cannot be viewed as a purely formal structure. Everything that the game-players actually do in the real world to bring t2 into their play must be considered: such as the exact nature of the squiggles they make on paper, et cetera.

Of course, if we allow the idealist's claim that the real world is simply an aspect of logic, with no independent existence, then Wittgenstein's insistence that we consider the "entire situation" becomes equivalent to considering "logic alone", and Wittgenstein's claims start to sound very much like Bradley's. But Wittgenstein shows even fewer such idealist tendencies in the Investigations than in the Tractatus.

I would also add that Wittgenstein is not launching a full Berkeley-style general attack on abstract ideas,[14]or on logic in general (although he shares Berkeley's distinction between abstraction/process and objects). Nowhere in the Investigations does he claim that abstract ideas are invalid or incoherent, although the role they play is very different from their role in the interlocutor's system or the idealist's system. For the interlocutor, abstractions are an intermediary between the idea and the world. For the idealist, the abstractions are in some sense the world. For the later Wittgenstein, abstractions, while perfectly valid, do not contain within them any meaning. We can use them to talk about thought, the external world, and the logic of possible thoughts, but they do not by themselves refer to reality, which is a mistaken notion according to Wittgenstein, a piece of unnecessary metaphysical baggage that is not needed to understand what language is, how we can use it, or how we obtain it.

In Defence of Private Meaning

A primary difficulty with Wittgenstein's concept of meaning is found in the argument against private language. Why can an individual not have a purely private thought, complete with multiple possible interpretations, that nonetheless has a particular meaning due to its being embedded in the individual's own private form of life? Wittgenstein uses the example of a man recording the letter "S" in his diary whenever he feels a certain sensation [I.258]. Couldn't this man be said to be playing a game with himself? The players here consist of the individual, plus all his past selves whom he remembers recording "S" at appropriate times in the past. Could he and his past selves not form a "community of mind" that agrees by custom on the meaning of "S"? As in Wittgenstein's other examples, the entire context of the language game would have to be considered as part of the meaning. This game is played out largely in the individual's head, but also includes the surrounding environment.

But this would make Wittgenstein's notion of meaning just a special case of the private notion of meaning, and he would once again be faced with either his paradoxes of meaning, or the possibility of idealism. So Wittgenstein insists that such a person cannot mean anything by "S" because he has no external criterion for correctness. He can make "S" mean whatever he wants, and so there is no "right" or "wrong" to his usage. While I think that Wittgenstein's formulation of meaning is useful, since in verbal communication we usually do play a language game with others, I think he goes too far in not allowing individuals to play the game with themselves. There may be no absolute reason compelling me to use "S" correctly, but nonetheless, I have made up my own game, and have decided to stick to it. Surely, there is meaning here. To me, Wittgenstein has constructed an account of one of the more common ways that words can refer. But his elimination of private meaning is ultimately arbitrary. If a community can decide on a standard, why can't I do the same within my own mind? True, no force of will can actually create an "occult" connection between my private idea and its referent. But, then, the customs of society do not create such a connection, either.

Wittgenstein also objects to private meaning by pointing out that one's left hand cannot give one's right hand some money [I.268]. Likewise, it makes no sense for the "agreement" required for meaning to consist solely in agreement with oneself, for that is not agreement at all. Still, is there really any problem in giving one's right hand money with one's left, if one plays a game with oneself and imagines a concept of "hand-ownership". This is the sort of thing the individual with the S-diary is doing. Perhaps it is not too un-Wittgensteinian to say that this is not a common use of "meaning", but there is no reason to disallow it altogether.

It is possible that, if pushed on the matter, Wittgenstein might have allowed this kind of meaning to a certain extent. However, given that the Investigations was largely motivated by Wittgenstein's rejection of the Tractatus, one suspects that Wittgenstein would remain intent on the strictly public nature of meaning, even if pushed, since to allow private meaning at all would be to automatically drag in the rest of logic as part of the external world, since now we can refer to anything that is logically conceivable. If I am allowed to make up my own rules, independently of society's rules, then I can choose whatever operators I want to interpret elementary propositions. So Wittgenstein's notion of public meaning must be absolute, or else it will force him into idealism, via all the logical arguments he has already expounded in the Tractatus. To the extent that private meanings are possible for Wittgenstein, they are only possible meanings, and have meaning only with respect to a form of life.

But even from the idealist perspective, could we perhaps allow that Wittgenstein's public definition of meaning applies most of the time to most of our propositions? I think this is perhaps valid to some extent. The public language game captures a large and important part of how we use language in the everyday world. Even so, it seems to me that in the everyday world, we also often play games with ourselves. Perhaps it would be possible to place much of the private game in terms of a hypothetical public game, but if it does not actually get used publicly, it remains a private game. Surely, Heisenberg's quantum theory[21] was meaningful, even while it was still just an idea in his own mind, before he had published or communicated it to anyone. Yet, it contained many new terms and mathematical constructions, with no public agreement as to their meaning. The later Wittgenstein would allow these terms meaning only to the extent that the new theory could be embedded in a public language game. Anything else is vacuous tautology. But the fact remains, that if Heisenberg dies before he has a chance to publish, the theory never does get so embedded, and the whole structure, the whole system of new terms and rules by which they are used, is subject to reinterpretation. So, for the later Wittgenstein, there should be no meaning in it, if his paradoxes are taken as having any force at all. To Bradley, of course, the interpretation given by Heisenberg privately is the meaning. But this idealist conception of purely private meaning only works if we allow meaning to be completely a matter of logic, in which case we deny that propositions make reference to something external to logic, and are forced to admit the reality of imaginary worlds.

Wittgenstein never properly addresses the issue of imaginary worlds in either his early or later works. He simply assumes they must be "false". He thus has failed to convince me that private terms cannot refer. It is not at all clear to me that the private thought p2, in figure 6, cannot "mean" t2, if that is how I decide for myself that I am going to use it. I agree that the next step from t2 to o2 is unjustified, but that is because as an idealist, I do not recognize an external world separate from logic in the first place. It is this that is meaningless, not private reference.

I think perhaps we use "meaning" in the private sense a good deal more than Wittgenstein would like to admit. I do not think he has really shown that meaning is solely a matter of "playing the game" or "usage according to custom" or a "form of life". Certainly it is possible to define meaning in this way, and thus private thoughts of the imagination simply do not refer because we have defined reference so as not to include them. But the notion that meaning is a purely logical thing, inherent in the thought itself, still has a powerful pull for me. In my formalist idealism, incomplete abstract reference is a matter of an embedding in a form of life, for all the reasons that Wittgenstein cites. But such incomplete reference is only a make-shift way of talking about things too complex to fully talk about. The things talked about are still in principle formalizable in a complete, nonabstract manner. I see no evidence that the world of my experience could not in principle be understood as a huge, gigantically complicated recursive structure. So, since throughout my life, recursive structures are the only things that I have been able to refer to with a feeling of certainty, it seems a reasonable working hypothesis to try and understand the world in such purely logical terms. Logic I can understand on its own terms within my mind, but what is this external world of language speakers? It seems like it must be a material entity, like the world of the Tractatus, given that Wittgenstein's development of it depends on certain materialist assumptions. But even this is never explicitly addressed in the Investigations. Wittgenstein says really nothing at all to explicate in rational terms what this fundamental entity is. He simply accepts it as a given, in a radically anti-rationalist, empiricist manner. To my rationalist tendencies, this extreme empiricism is most unscientific, and even superstitious. Philosophy and science should attempt to discover what our everyday world really is, not accept it as a given. But Wittgenstein, in a fundamentally anti-scientific move, has rejected this entire enterprise.

In summary, while Wittgenstein's interlocutor does not seem to have a workable theory, I do not see a reason to prefer Wittgenstein's conception of meaning over that of the idealist. However, Wittgenstein does not discuss the idealist conception of meaning, and does not seem to be arguing against it. He is largely silent on metaphysics in the Investigations. Of the everyday world of the language game, he neither claims that it is the one real world, ordered in a particular way, nor that it is simply a logical form. Rather, he just takes it as it comes, and does not address its absolute metaphysical status. Indeed, he tries very hard to avoid this.

IV. Conclusion

To my mind, the Tractatus came close to a proper solution: a nonmentalist, objective idealism, in which everything, including mind and the so-called external world, is built out of formal logic. Remove certain key passages that pay homage to material realism, and the Tractatus almost is an argument for nonmentalist idealism. But Wittgenstein, compelled by his background and training, ultimately cannot consider such a thing. As a result, the Investigations takes Wittgenstein in exactly the wrong direction, yet the only conceivable direction he could take, given his anti-idealist assumptions.

Perhaps the Investigations is best interpreted in this light. Wittgenstein has given up on metaphysics, for good or ill, and now is simply examining what we can sensibly speak about in our everyday life. So requiring public agreement, while metaphysically untenable, is perhaps a reasonable stance to take from the later Wittgenstein's more pragmatic perspective. The everyday publicly shared world is what our normal discourse is already about (most of the time). Given that we are already talking about it, Wittgenstein asks us to try the best we can to clarify what we can and cannot say about it. Within this context, his meaning of "meaning" has, I think, much utility.

However, the real problem is not so much the pragmatic utility of his idea of meaning, but the fact that in an odd kind of way, he really has not given up on metaphysics at all in the Investigations. The whole enterprise is motivated and directed by the metaphysical failure of the Tractatus. As a result, Wittgenstein is compelled to couch his "common use" definition of meaning only in such terms that do not raise metaphysical issues. This causes him to be extremist on certain aspects of his theory, such as the public nature of meaning, where there is really no justification for such extremism from the pragmatic, nonmetaphysical perspective he is claiming to take. Wittgenstein is not, as he thinks he is, avoiding metaphysics. Rather, in trying to avoid it, he is pushed into untenable positions by his absolute, unquestioned faith in the materiality of his personal world. A more metaphysical position one could not ask for.

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