Why Wittgenstein’s Ideal Language Project Failed

I. Introduction

1. The purpose of the following paper is to examine Wittgenstein’s concept of logical analysis; to outline the nature of its central role in the Tractatus. By logical analysis, I am referring to Wittgenstein’s systematic use of reasoning for developing a triadic-aspect method. The first aspect is constructing, through natural language, the elucidations governing the logical structures of a stipulated class of signs. The second aspect is differentiating the philosophical significance, as to the various elements of this theory of signs, in relation to meaning and world. The third aspect is inferring the consequent philosophical implications of this system in terms of the theory of language, knowledge, and ethics—including the question of philosophy’s task as a discipline, as well as the purpose and limitation of the elucidations of the Tractarian system.

My interpretation of this Tractarian method derives from my consideration of two questions. What does this project of reducing sense to the analysis of truth-operations philosophically imply? In judging the success or failure of this project, what is said about the philosophical temperament underlying the motivation for this project? My answer to the first question depends upon the nature of the relationship of the mathematical concepts of limit and completeness, in terms of the logical analysis of the truth-operations that define the sense of general propositions in the Tractarian method. My answer to the second question builds upon this relationship and compares it to explicit Tractarian themes, particularly the notion of saying and showing.

What Wittgenstein’s Tractarian method subsequently commits us to philosophically is what I intend to investigate in the following interpretation of the work. As part of this thesis, I specifically consider the question of the final analyzability of the logical form of x . fx from proposition 5.47. By definition, it is impossible to completely express the logical sum of a mathematical limit. Accounting for the meaning of the limit ultimately relies upon what the limit shows through its application. In logical terms, this involves the valid transformation of variables into constants, of how propositional functions (variables of language) truth-functionally entail propositions (constants of language) and propositions truth-functionally entail propositional functions. Philosophically, the notion of the logical form of a proposition can be expanded to cover a wider range of theoretical issues in the Tractatus. For example, the value of ethical statements cannot be expressly determined within the established limit of the Tractarian elucidations governing the possibility of sense. If nothing can be said about good or evil within the limit of sense, perhaps something can yet be shown in practice. Similarly, since the elucidations of the book are recursive, the conditions of sense concurrently apply to the Tractarian system. I ultimately propose that the logical analyzability of a large class of general propositions cannot be completely determined.

2. I intend to outline my Tractarian interpretation—namely, the status of Wittgenstein’s conception of analysis and the meaning of the elucidations of the book—in accordance with four fundamental philosophical issues. They follow in no particular methodologically dependent order.

Consider that the theory of elementary propositions is a formal component of Wittgenstein’s concept of logical analysis. Unlike Carnap’s logic, an elementary proposition is non-monotonic in Wittgenstein’s theory, defined as a truth-operation that applies directly and only to itself (proposition 5). It is a static system, analogous to some sub-atomic property (the elementary proposition) that chemical bodies are composed of (complex propositions), whereby the components of the sub-atomic property (the truth-possibilities) avoid any means of being internally affected through interaction with any other phenomena. Understanding language becomes a matter of breaking sentences into bits; analysis is then a dissection. Indeed, one could say that the foundation of Wittgenstein’s conception of logical analysis is framed by this physiochemical analogy of the logic of language, as composed of atomic and molecular propositions with a universally shared logical constant (i.e. truth-possibility). Reflecting on the Tractatus, Wittgenstein alludes to this framework in part A of Appendix 4 of Philosophical Grammar when he writes: “…I too thought [like Carnap] that logical analysis had to bring to light what was hidden (as chemical and physical analysis does).”[1] What this philosophically signifies, and whether or not this notion of dissection is an accurate conception, is one of four fundamental issues in my interpretation of Wittgenstein’s conception of logical analysis.

The postulation that there must be elementary propositions is a presupposition of Wittgenstein’s method of constructing the logic of language; for tautologically it is already assumed that a class of signs with this stipulated truth-operational, logical structure exists. The question then becomes: what is the theoretical role that elementary propositions serve? The philosophical purpose of elementary propositions derives from the philosophical temperament that requires their existence; there is an axiological demand for an austere, deterministic account of meaning and truth. We can, therefore, infer that the dual-theoretical purpose of the theory of elementary propositions is intended as a means of preventing an infinite regress in the formation of propositions and their truth-operations, as well as a necessary condition for satisfying the completeness of this truth-operational system in its representation of reality. The consideration of these axiological principles, as well as the nature of the philosophical temperament underlining the Tractarian system, are one of four fundamental issues in my interpretation of Wittgenstein’s conception of logical analysis.

In the Tractatus, the problem of infinite regress depends upon the logical form of propositions. For Wittgenstein, propositions are defined as the aggregate of the truth-operations corresponding to its set of elementary propositions contained therein (4.52). The number of truth-operations is correlated with the degree of the proposition’s complexity kn, as given by the sigma function in 4.42. No matter how complex the logical form of a proposition, complex signs are structured out of simpler signs, whereby the simplest sign can be expressed in terms of elementary propositions. By implication, it follows that we can infer all propositions as results of truth-operations on elementary propositions (5.3). Therefore, since this general propositional form is a variable (4.53) and since variables of propositional calculus are themselves well-formed formulae of the Tractarian system, propositional constants and variables are completely analyzable in terms of their sum of truth-operations. Propositional signs are well-formed recursive definitions because the Tractarian definition governing a ‘proposition’ is derived from examples of itself, as signs containing truth-operations defined by simpler signs containing fewer truth-operations. Analogously, the notion of an elementary proposition has the same theoretical purpose as the concept of ‘oneness’ in the set of all natural numbers in the construction of axiomatic set theory—because all molecular propositions implicitly contain atomic propositions as the base case by which these complex signs are both definable and analyzable. This consequently allows Wittgenstein to introduce the crucial notion of sense (i.e. the truth-operations of a well-formed sign), to the logic of language, in a fully determinant manner, thereby avoiding an infinite regress. But what is here important is the idea that all propositions (elementary and general) are definable in terms of truth-operations. Whether this notion is correct is one of four fundamental issues in my interpretation of Wittgenstein’s conception of logical analysis.

In order to understand the philosophical significance of a non-regressive, sensically infinite logic, we must consider the Tractarian interplay in terms of the antithetical concept of limit: completeness. Sense is defined in the Tractatus as the final declination of a sign’s truth-possibilities. Like set theory, Wittgenstein’s theory of propositions can handle an infinite series of truth-operations. This infinite series of truth-possibilities is the limit to the logical expression of sensible thought. Every propositional sign has one and only one logically complete analysis (proposition 3.25) because there can be but one sum of a sign’s elements (its truth-possibilities) for any given proposition. Therefore, by implication, a Tractarian inconsistency arises if the totality of any proposition’s truth-possibilities cannot, for any reason in its analysis, be completely determined. In conjunction with this criterion, I intend to investigate the relationship of two Tractarian theses. First, I consider the theory of the general form of a proposition, whereby a proposition is defined as consisting of a sum of truth-possibilities. Second, I relate this definition of sense with Wittgenstein’s theory of generality, whereby any general function is defined as consisting of a calculable logical sum of particular elements. Consequently, I wish to consider why the concept of limit, as reflected in general propositions, cannot be completely logically analyzed in terms of a calculable sum of determinable truth-operations. This includes, for example, the analysis of the logical form of the general proposition found in 4.551: x . fx. This also pertains to the notion of mathematical induction of internal relations for a propositional series, as discussed in 4.1252. What this discovery of incompleteness of a limit signifies for Wittgenstein’s theory of sense directly affects Wittgenstein’s conception of logical analysis, and is therefore one of four fundamental issues in my interpretation of Wittgenstein’s concept of logical analysis.

II. Two False Kinds of Tractarian Interpretation

3. By outlining these fundamental issues, I have assumed the Tractatus to be a substantive philosophical work, composed of theories that Wittgenstein himself believed important and definitive to the book. But there are a significant number of philosophers, perhaps a majority in America, directly opposed to any Tractarian reading which supposes Wittgenstein to be engaged in legitimate and substantive theorizing. By degree, these interpretations can be categorized as either nihilist or deflationary.

Those advocating a nihilist reading of the Tractatus conceive of the propositions of the book as primarily consisting of “nonsense sentences.”[2] Such detractors insist that the reader must experience and struggle with the Tractarian propositions as “dissolving from the inside” or attest to their “crumbling in on themselves”.[3] Consequently, they will emphatically disagree with this paper’s positive thesis. For I have hermeneutically granted that the Tractatus is not Finnegans Wake and is indeed engaged in something sensical. I assume that the appropriate interpretation(s) of proposition 6.54 ought to be that which does not render the book either unintelligible or self-defeating.

Those advocating a deflationary reading take the line that although the Tractatus is sensical the book itself offers no traditional philosophical theories. This notion of theoretical quietism is a mainstream if incorrect interpretation of the Tractatus.

By implication, I wish to note that I am particularly opposed to the interpretations of Conanat, Diamond, Floyd, Goldfarb, Kremer, Putnam, Ricketts, Rorty, and any other individual inclined to believe the Tractatus, in some predominant manner, terminates into nonsense or theoretical quietism. While I do consent with these individuals, to the extent that many of the logically positivistic interpretations of the Tractatus are misguided, I do not believe such a radical reaction as decrying the work, as either significantly nonsensical or barren of theoretical substance, is the warranted response to Russell’s, Ayer’s, or Black’s Tractarian interpretations.

My opposition to the nihilist and deflationary reading is an auxiliary thesis of this paper. Since each of these accounts is a ‘negative’ reading, their disproof requires the positive affirmation that Wittgenstein himself intended the Tractatus to contain important theoretical considerations which he himself (and not his shadowy literary interlocutor) took seriously. Therefore, by relying upon “Some Remarks on Logical Form”, Philosophical Grammar, Wittgenstein’s personal notes (procured from Dr. Ian Proops, “The New Wittgenstein: A Critique”, University of Michigan, unpublished), as well as Moore’s Cambridge lecture notes and remembered conversations with Wittgenstein concerning the Tractatus (also procured from Dr. Ian Proops), this textual evidence achieves a dual-aspect purpose. First, it provides arguments in favor of my primary thesis, my conception of Wittgenstein’s logical analysis. Second, I will show in the process of supporting my primary thesis that Wittgenstein truly was engaged in a type of logical analysis that proposed numerous theories; thereby supporting my auxiliary thesis. In the mid-1930’s, far from being dismissive of the sense of the Tractarian propositions, Wittgenstein re-assesses their validity, remarking quite openly to his mistakes and to what he considers to be the proper conclusions. It is these reflections that I take to significantly weaken the arguments of both the nihilist and deflationary reading.

III. The Problem of Logical Form and Its Analysis

4. The purpose of this section and the following section of the paper involves my presentation of evidence for supporting my Tractarian reading. This interpretation comes in two parts. Firstly, I try to capture Wittgenstein’s early 1930’s post-Tractarian interpretation of the Tractatus, by considering the logical mistakes of the book that led Wittgenstein to abandon his Tractarian project (section III). The largest of these logical mistakes is Wittgenstein’s belief that all propositions are truth-functions, and that the general propositional form is correct in its interpretation of all complex propositions as composed of simpler propositions with truth-functions (recall this incorporates fundamental issues 3-4). Secondly, I consider that although original authorial intent of the Tractatus was one oblivious to the knowledge of these logical mistakes that the nature of these logical mistakes are such that they compliment many of the central philosophical elements of the Tractatus, as stated by the elucidations of the work (section IV). This includes, in particular, the thematic relationship of saying and showing, which in some way anticipates the logical mistakes Wittgenstein will discover he commits in his conception of logical analysis (recall this incorporates fundamental issues 1-2).

I wish to begin by considering a paper Wittgenstein published shortly after the release of the Tractatus that went undelivered for the Joint Session of the Aristotelian Society and the Mind Association in 1929. This paper was entitled “Some Remarks on Logical Form”; my interpretation of its primary purpose was to further advance the Tractarian project of outlining the logical form of propositions, particularly emphasizing the rules of syntax necessary to avoid either contradictions or nonsense, as well as to account for the possibility of truth-operational exclusion vis-à-vis the interrelation of the logical product of two propositions that are derived by abstracting the content of ordinary language into Wittgenstein’s Tractarian symbolism. The first aspect of this paper is right out of the Tractatus: the directive to construct a logical system free of contradictions or nonsense. The second aspect is novel, the notion of exclusion, and is Wittgenstein’s first written attempt to deal with the growing realization that in the realm of application atomic propositions do indeed seem to be inferentially related and can truth-operationally affect the form of another proposition—a claim rejected as a possibility in the Tractatus.

Wittgenstein re-affirms in the beginning of the 1929 paper his theory that “every proposition has a content and a form.”[4] In conjunction with the Tractarian statement that “form is the possibility of structure”, (2.033) the notion of propositional form implicitly consists of the necessary principles of logic by which the structure of a proposition is possible. Such thinking is reflected in the 1929 paper, when Wittgenstein states that by “abstracting from the word-content (constants) of a proposition, variables can be substituted to give the pure form in accordance with the formal rules of syntax,”[5] thereby manifesting the underlying propositional form necessary for the possibility of sensical linguistic structures. This includes the domain of ordinary language:

“…where ordinary language disguises logical structure, where it allows the formation of pseudopropositions, where it uses one term in an infinity of different meanings, we must replace [the grammatical deficiencies of ordinary language] by a symbolism which gives a clear picture of the logical structure, excludes pseudopropositions, and uses its terms unambiguously.”[6] (my italicization)

Like the Tractatus, in my interpretation of Wittgenstein, I here take him at his word. While Russell, Carnap, and other logical positivists misinterpreted many aspects of Wittgenstein’s project, I do believe they had substantive justification for believing that Wittgenstein’s temperamental drive for an austere, determinism was directly correlated with a philosophical architecture meant for deciphering the logical form of the ideal language.

My critics would most likely make a dual-aspect counter-argument against the importance of the 1929 paper. Firstly, although published, Wittgenstein never presented the 1929 paper, as he chose another lecture topic for the Joint Aristotelian and Mind Association conference. Secondly, by 1933 Wittgenstein had written to Mind that the 1929 paper was completely worthless. Many philosophers interpret these events as directly related, that the reason Wittgenstein never presented the 1929 paper was because Wittgenstein doubted the work as soon as he wrote the paper. But the outstanding question left for speculation is: why did Wittgenstein believe the paper to be futile, whether in 1929 or 1933?

Nihilists like Conant seem to have no choice but to assume that even though Wittgenstein had us go through the ‘non-sense’ sentences of the Tractatus, for some inexplicable reason Wittgenstein decided to climb back down the ladder and write another ‘non-sense’ paper using ‘non-sense’ things like logic and mathematics that are alright just to ignore because otherwise we would have to actually learn them like a rigorous philosopher. Or perhaps some deflationist could more reasonably and cynically speculate that the paper was originally written as a means of merely appealing to logical positivists in order to facilitate more interest in the recent publication of the Tractatus; that the paper was fulfilling in unwanted responsibility to Russell and others. Then again, Wittgenstein was often a terribly mean, anti-social, and insecure person so maybe he was just enjoying a private joke at the expense of people that believed rigorous, serious thinking could solve real philosophical issues?—After all, Wittgenstein hated authority and liked to read poetry to the Vienna Circle so maybe Wittgenstein was extending his ironic sense of intellect to yet another Oxford paper and yet another Cambridge conference full of robe-wearing, old school intellects.

I believe the nihilist and deflationist reading of the Tractatus have few hermeneutical choices in accounting for Wittgenstein’s continued interest in the logical form of language post-Tractatus. The most sophisticated alternative interpretation that I am aware of concerns the issue of exclusion in the 1929 paper. Some believe that as Wittgenstein came to realize in the Philosophical Grammar that his theory of elementary propositions was a failure that Wittgenstein came to doubt the paper and shelved it. As his ideas evolved over the next four years he would retroactively declare it completely worthless. Since the paper was predominantly on how the complete analysis of a proposition could get entangled with the truth-operations of another proposition, the notion of completeness of a logical product for an atomic proposition could no longer sustain the Tractarian premise of being static. Like Leibniz, Wittgenstein’s logical world gave way to the crushing force of dynamism. But the added force of this interpretation is that it gives a viable explanation for why Wittgenstein wrote in 1933 (the year Wittgenstein was working on the Philosophical Grammar) that the paper was completely worthless. If Wittgenstein had doubts in 1929 about the notion of exclusion, it is quite reasonable to believe that by 1933, when he realized his theory of atomic propositions was wrong (and wrote as such in Philosophical Grammar), then he would consequently see the 1929 paper as worthless.

However, I do not believe any of these interpretations come close to examining the central issue. These interpretations leave out the fact that while Wittgenstein did not present the 1929 paper, he did attend the lecture and did present to the conference. Understanding the gap between the Tractatus and Philosophical Grammar, to really come to grips with Wittgenstein’s intellectual evolution, requires an explanation of the importance of this 1929 lecture. At the 1929 Joint Session of the Aristotelian and Mind Association, Wittgenstein lectured first on the relationship of the mathematical concepts of generality and infinity (i.e. mathematical limit) and second the relationship of these mathematical concepts to logical theory. But why would Wittgenstein the nonsensical, Wittgenstein the theoretical quietist, give a rigorous lecture on issues in the philosophy of mathematics? For the life of me, I have no way of discerning any intelligible reason a nihilist or deflationist reading can give for this lecture. As for the interpretation that Wittgenstein abandoned the 1929 paper because of his rejection of his conception of atomic propositions, I find this answer both correct and insufficient. I believe my interpretation of Wittgenstein’s own view of his project is the only one that can answer precisely the textual and theoretical relationship of the Tractatus, the 1929 paper, and Philosophical Grammar.

5. I have so far assumed that the 1929 paper is a continuation of the Tractarian project and that Philosophical Grammar is its correction. My critics would most likely take the opportunity to try and separate these works in order to prevent the usage of the 1929 paper as textual ground for interpreting elements of the Tractatus. I wish now to defend my assumption that the 1929 paper was a central continuation of the Tractatus, that the work is an addendum to the Tractarian considerations of logical form. By implication, I believe that if the 1929 paper is incorrect, this subsequently affects the very foundations of the Tractarian project well beyond the theory of elementary propositions. The focus concerns the philosophical and mathematical relationship of completeness and limit down to the very elements of method and concept for Wittgenstein’s theory of logical analysis (i.e. this paper’s thesis).

In the 1929 paper, Wittgenstein re-iterates his theory of the general form of a proposition from the Tractatus: “If we try to analyze any given propositions we shall find in general that they are logical sums, products, or other truthfunctions of simpler propositions.”[7] The exception to Wittgenstein’s theory, as mentioned earlier is the elementary proposition:

“We must eventually reach the ultimate connection of the terms, the immediate connection which cannot be broke without destroying the propositional form as such. The propositions which represent this ultimate connexion of terms I call, after B. Russell, atomic propositions. They, then, are the kernels of every proposition, they contain the material, and all the rest is only a development of this material.”[8]

These textual references are interesting because they textually confirm the 1929 paper’s reliance on Wittgenstein’s Tractarian theory of elementary propositions and the doctrine that the sense of non-atomic propositions is definable in terms of a determinable logical product of truth-operations. This textually confirms the central relationship of the 1929 paper with the elucidations of the Tractatus because they share the definition of complete analysis, the criterion for the sense of a proposition, the definition of an elementary proposition, and the same conception of generality. The fundamental difference between the two philosophical works remains the notion of exclusion, insofar as Wittgenstein is struggling with the possibility that truth-operations are inferentially related, that they are monotonic. But this notion of exclusion ‘builds’ off of the Tractarian system, making the 1929 paper akin to an appendix.

6. The question I now desire an explanation is: why does Wittgenstein reject significant themes of the Tractatus and its related work “Some Remarks on Logical Form?” Towards this end, I consider two of the fundamental issues I have with Wittgenstein’s conception of logical analysis. I first challenge the idea that all propositions, elementary and general, are definable in terms of truth-operations. I then reveal the incompleteness of theory of propositions, to show that the underlying problem with Wittgenstein’s theory of sense in relation to the logical form of general propositions.

I wish to begin by considering an excerpt from part A of Appendix 4 of Philosophical Grammar. Reflecting upon the Tractatus Wittgenstein writes:

“The idea of constructing elementary propositions (as e.g. Carnap has tried to do) rests on a false notion of logical analysis. It is not the task of that analysis to discover a theory of elementary propositions, like discovering principles of mechanics. My notion in the Tractatus Logico-Philosophicus was wrong.”[9]

Wittgenstein goes on to say his earlier thinking was victim to two cardinal fallacies:

“…1) because I wasn’t clear about the sense of the words ‘a logical product is hidden in a sentence’ (and such like), 2) because I too thought that logical analysis had to bring to light what was hidden (as chemical and physical analysis does). ”[10]

It is popular knowledge that Wittgenstein abandoned his theory of elementary propositions because of his quite embarrassing discovery of the monotony of inference in the relation of propositions.

But what is not commonly understood is the significance of Wittgenstein’s rejection of the notion of “a logical product hidden in a sentence” results in the rejection of the notion that the complete analysis of the semantics of a sentence can be made explicit through the identical transformation of the logical product of a propositional function with its truth-operationally entailed logical sum, thereby undermining Wittgenstein’s conception of logical analysis. What is at stake from this discovery is not merely the theory of elementary propositions or even the notion that language can be logically dissected into a pyramid of analyzable integral units like physics. The fallout extends to the theoretical feasibility of satisfying the axiological principles underlying the philosophical temperament giving rise to the Tractarian project. What is fundamentally at issue is the fact the Tractarian drive for austerity is interrupted by the logical existence of indetermination.

What is required now is proving this indetermination arising out of Wittgenstein’s theory of generality. In proposition 5.441 of the Tractatus, Wittgenstein writes:

“This vanishing of the apparent logical constants also occurs in the case of ‘~(x) . ~fx’, which says the same as ‘(x) . fx’, and in the case of ‘(x) . fx . x = a’, which says the same as ‘fa’.”[11]

In conjunction with this elucidation is the subsequent Tractarian proposition 5.442:

“If we are given a proposition, then with it we are also given the results of all truth-operations that have it as their base.”[12]

Tractarian proposition 5.441 is not directly concerned with Wittgenstein’s theory of general propositional form but contains an example in the function ‘(x) . fx . x = a’. Both ‘(x) . fx’ and ‘~(x) . ~fx’ are general propositions, whereas ‘fa’ is a constant of the logical product ‘(x) . fx’. What I am interested is the claim that ‘f(x)’ is the same as ‘f(a)’, for this proves to be the critical flaw of Wittgenstein’s analysis of the logical form of the general proposition in both the Tractatus and the 1929 paper.

The claim that ‘(x) . fx . x = a’ is a propositional function with the identity relation ‘x = a’, formally implies that a constant and variable are interchangeably truth-functionally entailed. Since f(x) entails f(a) and f(a) entails f(x), when this syntactic claim of 5.441 is coupled with the semantic claim of 5.442, it deductively follows that the results of all truth-operations of a proposition f(x) must always be identical with all results of all truth-operations of a proposition f(a). That Wittgenstein holds to this position is the theory of general propositional form and is explicitly stated in 5.7 in the text of the Tractatus:

“An elementary proposition really contains all logical operations in itself. For ‘fa’ says the same thing as ‘(x) . fx . x = a’. [13]

Wittgenstein goes on to say in 5.47:

“Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants. One could say that the sole logical constant was what all propositions, by their very nature, had in common with one another. But that is the general propositional form.”[14]

It is important to note that the entire consistency of the book hinges on Tractarian proposition 5.47. This elucidation brings together the necessary elements of Wittgenstein’s conception of logical analysis: the theory of elementary propositions, the theory of sense, the axiological principle of completeness, the theory of generality, and the nature of the logical form of propositions. We further know that 5.47 is the theoretical apex of the book because of the two Tractarian propositions that follow:

“The general propositional form is the essence of the proposition.”[15]

“To give the essence of a proposition means to give the essence of all description, and thus the essence of the world.”[16]

Since Wittgenstein’s conception of logical analysis is akin to a physical system, Wittgenstein requires a unification theory to bring together all the components of his logic of language. This grand unification theory is the general propositional form, where the universe of discourse is able to mirror the universe of matter, where the truth of description corresponds to the truth of the world.

The question of general propositional form is extended in “Some Remarks on Logical Form”, where the logical structure of propositions is once again brought into question. The paper questions how truth-operations might be hidden in an elementary proposition, to try and give examples where the Tractatus gave none. It is in this specific sense that we see the correlation of the Tractatus with the 1929 paper, for the essence of a proposition (5.471) is reducible to the analysis of elementary propositions because the analysis of propositional forms reaches the point where they are not themselves composed of simpler propositional forms but are defined in terms of logical sums of truth-operations (1929 paper). Since truth-possibility of a proposition is a logical constant, the basis of logical argument and function can be constructed out of this semantics. So when Wittgenstein states in the Tractatus that the sole logical constant is that which is responsible for defining the common nature, he is expressing his theory of sense. The 1929 paper takes this theory of sense as the basis of logical form of elementary propositions in order to determine the conditions for its complete analysis (its logical sum).

It is from these considerations that the relationship of limit and completeness dominates the validity of the Tractarian statement that “An elementary proposition really contains all logical operations in itself. For ‘fa’ says the same thing as ‘(x) . fx . x = a’.—as well as the statement of “Some Remarks on Logical Form” that “If we try to analyze any given propositions we shall find in general that they are logical sums, products, or truthfunctions of simpler signs.” Therefore, the reason why Wittgenstein signifies that the 1929 paper is completely worthless is given by Wittgenstein’s explanation in the Philosophical Grammar for the errors in logical analysis of the Tractatus. In “Criticism of my Former View of Generality”, Wittgenstein states:

“My view about general propositions was that (x) . ϕx is a logical sum and that though in terms aren’t enumerated here, they are capable of being enumerated (from the dictionary and the grammar of language)…the explanation of (x) . ϕx as a logical sum and of (x) . ϕx as a logical product is indefensible. It went with an incorrect notion of logical analysis in that I thought that some day the logical product for a particular (x) . ϕx would be found.”[17]

Wittgenstein is criticizing his conception of the general propositional form. But for what reasons is the logical analysis incorrect? I am fortunate to have come across the work of Dr. Ian Proops, who recently flew over to Cambridge University with the aid of a grant from the University of Michigan Rockham Fellowship. With the help of Ms. Kathleen Cann of the Cambridge University Library, they came across G. E. Moore’s personal handwritten notes concerning Wittgenstein’s Cambridge Lectures of 1929-1933, including the lecture given at the 1929 Joint Aristotelian Society and Mind Association Conference. According to Moore’s Cambridge lecture notes, in 1932 Wittgenstein alluded to two fundamental mistakes in the Tractatus. The first mistake involves the theory of elementary propositions, problems that are well understood and accepted today. But the second mistake, I believe to be the more fundamental. Moore recounts:

“The second important logical mistake which [Wittgenstein] thought he had made at the time when he wrote the Tractatus was introduced by him in connexion with the subject of ‘following’ (by which he meant, as usual, deductive following or ‘entailment’—a word which I think he actually used in discussion) from a ‘general’ proposition to a particular instance and from a particular instance to a ‘general’ proposition. Using the notation of Principia Mathematica, [Wittgenstein] asked us to consider the two propositions ‘(x) . fx entails fa’ and ‘fa entails (x) . fx.’ [Wittgenstein] said that there was a temptation, to which he had yielded in the Tractatus, to say that (x) . fx is identical with the logical product ‘fa . fb . fc …’, and (x) . fx identical with the logical sum ‘fa v fb v fc …’; but that this was in both cases a mistake.”[18]

And why did Wittgenstein believe he got the notion of entailment incorrect? What was wrong with Wittgenstein’s identification in 5.47 that fx = fa? The ability to calculate a complete analysis of the truth-functions of general propositions, as constituted out of elementary propositions, proved to be an inaccurate conception of logical analysis (both in the Tractatus and the 1929 paper). As Moore continues:

“[Wittgenstein] went on to say that one great mistake he made in the Tractatus was that of supposing that in the case of all classes ‘defined by grammar’, general propositions were identical either with the logical products or logical sums (meaning by this logical products or sums of propositions which are values of fx) as, according to him, they really are in the class of the class ‘primary colours’. [Wittgenstein] said that, when he wrote the Tractatus, he had supposed that all such general propositions were ‘truth-functions’; but he said now that in supposing this he was committing a fallacy, which is common in the case of Mathematics e.g. the fallacy of supposing that 1 + 1 + 1 … is a sum, where it is only a limit, and that dx/dy is a quotient, whereas it is only a limit.”[19]

The reason that ‘(x) . fx . x = a’ in 5.47 is false, is because (x) . fx is not a logical sum of calculable truth-operations made out of a calculable sum of elementary propositions (x) . fx that compose this general proposition. Therefore, fx ≠ fa because fx is a general proposition that does not truth-functionally entail the particular instance fa. Wittgenstein’s fallacy is to confuse (x) . fx as a summation of components, when it is actually expressing a limit of a sum and not a sum itself. What is so damaging about this fallacy is it leaves most of propositional calculus outside of Wittgenstein’s conception of logical analysis. Perhaps the most fundamental component necessary for the unification of the Tractarian project, the theory of general propositional form, turns out to be incorrect. Of course, since Wittgenstein’s conception of the logic of language is one built out of a conception of physics, the discovery that propositional functions built out of the variables of general propositions does not truth-functionally entail the constants of propositions leaves the primary Tractarian elucidation governing the interrelationship of formal theory and real-world application as sensically incommensurable logical forms.

It would be analogous to taking the Euler-Lagrange theorem away from non-linear mathematics, the possibility of dynamic relations in mechanical physics would collapse. So in a sense Conant and Diamond are right, if for all the wrong reasons. Therefore, I wish to consider this in the following section, to bring out the philosophical implications of this inconsistency in the Tractarian project and how, in many ways, it ironically compliments the original authorial intent of the book.

7. I believe I have now given sound justification regarding why Wittgenstein gave his 1929 lecture. According to my interpretation, the 1929 paper is completely worthless, insofar as the logical form of propositions, namely general propositions, turn out to be mistaken because their logical sum of truth-operations cannot be determined, as Wittgenstein’s Tractarian method claimed. For this very reason, Wittgenstein abandoned the 1929 paper in favor of an alternative lecture at the Joint Aristotelian Society and Mind Association Conference. This lecture dealt with the problem of the completeness of mathematical analysis as an impossibility in the light of the infinity of a function. This is the very topic Wittgenstein mentioned in the Philosophical Grammar for his abandonment of the Tractarian conception of logical analysis. Therefore, I believe Wittgenstein realized his mistake in the Tractatus by 1929; that by 1933 he made this public to Mind.

With the abandonment of a singular system for accounting for the logical form of language, Wittgenstein would later turn to the logical multiplicity of language-games, setting the stage for Philosophical Investigations, and its abandonment of the Tractarian crystallization of logical analysis (PI §108). This provides a reasonable narrative for the majority of Wittgenstein’s life, though it requires an in-depth consideration of the seven textual uses of lebensformen in Philosophical Investigations and certain other notions that I cannot here pursue. But I would like to believe, at the very least, I have offered an internally consistent interpretation of Wittgenstein’s philosophy.

8. As an addendum to this research, I just want to offer an interesting side-note I discovered in the process of thinking through what it means for Wittgenstein to be wrong about the propositional general form. It would subsequently follow that 3.333 is false. Wittgenstein failed to solve Russell’s referential paradox because Wittgenstein’s solution confuses the entailment suffers the same fallacy as 5.47 because Wittgenstein writes: “if instead of ‘F(Fu)’ we write ‘(ϕ):F(ϕ) = Fu.’…that disposes of Russell’s paradox.”[20] Of course, this is obviously false, and does not offer any solution whatsoever. This is the same basic mathematical confusion made earlier, the substitution of a constant for a variable where there is no ground for this in the abstract function. But it would be easy to write a paper dedicated to this topic alone, so I only mention it in passing.

IV. Wittgenstein’s Ladder: Ethics and the Fall of Logical Analysis

9. I take Wittgenstein’s conception of logical analysis to be that which is most at stake in one’s interpretation of the Tractatus. Apropos, Wittgenstein’s Tractarian analysis allows him to justify the limits and possibilities of human understanding as reflecting what he conceives the logic of language as philosophically capable of achieving. For example, the notion that thought agrees with reality is a consequence of the theory of pictorial form (2.17); for how a picture depicts reality first necessitates that the form of the picture contain something in common with that which it depicts (2.16). This is an important assumption of Wittgenstein’s logical analysis, framing the applicability of the Tractarian propositions—particularly since Wittgenstein believes the limits of language are the limits of the world (5.6). What elementary propositions exist depend upon the application of the logic to the world (5.557) because their existence is not a priori (5.5571). But if Wittgenstein is correct, the means by which philosophy is capable of analyzing the relationship of language to the world is itself limited by logic—for philosophy cannot say what cannot be shown through an a posteriori connection. Therefore, it follows that the limits of philosophy would itself, by implication, here be disclosed.

I believe a critical component of understanding the Tractarian propositions requires for us first to understand what the idea of ‘complete analysis’ philosophically commits us to in the application of the logical form of elementary propositions to their presupposed agreement with their underlying pictorial form of reality. Therefore, the logical relationship between completeness and limitation I explored in section III in order to give a general account of the incompleteness of the general propositional form, in order to begin revealing the limitation of the Tractarian concept of logical analysis as a means of depicting the complete picture of reality through the tabulation and dissection of truth-operations. What I take to be the meaning of 6.54 and the philosophical role of elucidations manifests from an understanding of the incompleteness of this theory of sense, to show its correspondence with the nature of the elucidations themselves. From this understanding, we can begin to recognize how Wittgenstein is genuinely making as fundamental break with traditional philosophy as Heidegger (albeit extraordinarily different). This is because, I believe, Wittgenstein knew all along that the Tractarian method was an incomplete mirror-image of what we know about the world—that what he defined as the “mystical” (e.g. the ethical) was in truth his way of showing what lies beyond the limit of the logic of language, of what he could validly say within the confines of the Tractarian system before climbing out of that framework. For it was the nature of ethics that first indicated to Wittgenstein the incompleteness of logical analysis as a means of solving or even affecting our understanding of the solutions to the problems of life.

10. In reading the Tractatus, I have stated, as an auxiliary thesis, my insistence on the fundamental premise that Wittgenstein is genuinely committed to tackling fundamental issues in the philosophy of logic, language, metaphysics, and ethics. Accordingly, I interpret the Tractatus to be a philosophical work predominantly concerned with answering four questions. What relation must one sign have to another to be capable of accurately symbolizing sense? What is necessary, to the logical form of our language, for the possibility of embodying our reality? What is the consequence to traditional philosophy of limiting analysis of reality to the logical form of propositions depicting natural science? How does this relationship of Wittgenstein’s theory of sense and its symbolism, epistemologically dualistic theory of pictorial form and its metaphysical realism, and meta-philosophical conception of philosophy’s traditional subject matter as senseless, affect both the possible completeness of this project, as well as the internal consistency of the Tractarian system vis-à-vis the elucidations? How Wittgenstein answers these questions ought to determine how we judge the book.

I judge the Tractatus to be a philosophical work concerned with how our language embodies our reality, of understanding the nature of this agreement by understanding the underlying logical framework that is necessary for the possibility of this agreement. As Wittgenstein inquires: how is logic a mirror-image of the world? (6.13) For if logic is the limit not of thinking but how thinking can possibly be expressed (Tractatus introduction), then the construction of logic can itself only be sensibly thought in conjunction with the aide of elucidations using natural language. To employ formal logical rules in the elucidations would invite questions as to the meaning of the logical constants, requiring further logical rules. This would create an infinite regress of logical rules over logical rules, a meta-logic of meta-logics, just as surely as an infinite regress in the logical form of propositions that Wittgenstein avoided with the aide of his theory of elementary propositions. Sine qua non, logic cannot itself analyze its own foundation; any logical construction cannot sensibly express its essential properties. There must be an alternative philosophical means of accounting for their theoretical purpose. To draw an analogy, I believe Wittgenstein’s philosophical analysis of logic to be engaged in a Husserlian-styled ‘transcendental reduction’. Logic must assume its essential properties as given without further ado; they can only be examined from a different frame of reference than a logical investigation.

What points to the nature of Wittgenstein’s ‘transcendental’ reduction? All along I believe Wittgenstein realized the pluralism of ethics would undermine any rigid conception of logical analysis; that when we consider these four questions we must understand that Wittgenstein knew all along that which was mystical was that which was beyond the scope of logical analysis. Logical positivism was not philosophically equipped to deal with certain kinds of theoretical issues. The inability to reduce good and evil to the absolutist confines of a stark ordering of truth-operations of propositions, representing the picture of the world, could not be achieved within the limits of the Tractarian conception of logical analysis. But what even Wittgenstein didn’t realize, when writing the Tractatus, was that not only were ethical, religious, or logical constants the mystical a posteriori realms we could not directly speak of in an epistemological manner. The very project of a deterministic epistemological system, built out of the interrelationship of concept with its object, that seductive siren, could not be achieved with logical analysis. What we had to climb out of was not a consistent, working system that we had to leave behind but an evaporating specter that promised the world but only delivered illusion. The elucidations may have been self-ironic, they may have indeed folded inward and collapsed, but this was only because the logical framework imploded for objective, measurable reasons of logical and mathematical theory that Wittgenstein never anticipated. The difference is in the journey. And in philosophy, we all know the journey is what really matters.

11. Wittgenstein realized that if the logic of our language can only expressed by the language of our logic, then the primitive signs of logical grammar require an understanding of their application in order to know what these signs mean. Propositions that express these applications are the elucidations necessary for showing the meaning of the logic. Therefore, the limits to what thinking can possibly be expressed are implicitly determined by indicating the possible applications of sign relations through the special language (the propositions) of the Tractatus. But even with the help of this special language our logical analysis must often settle for the power of merely indicating how it itself functions. By implication, to elucidate, in this manner is to leave in reality something less than completely analyzed. By Tractarian definition, anything less than completely analyzed is somehow mystical, for there is something left undetermined beyond the limits of what thinking we can express and thereby understand its sense.

In order for someone to understand the method of logic, logic can only show through its use, namely the sign-rules of its logical grammar, what it is capable of performing. But these sing-rules, these propositions that Wittgenstein employs are tautologies, senseless proofs of how signs are inferentially related as syntactic connectives. They can say nothing as to the philosophical meaning of logical constants, the inputted values, or even the operations themselves. One must ‘transcend’ logic through a ‘reductive’ method, if one is to be philosophically capable of analyzing the nature and structure of logic, to know both why and how the logic of language can possibly mirror our knowledge of the world. This requires a frame of reference of one looking in to the book—it is impossible to accomplish this task from ‘inside’ the thinking of the book itself.

Therefore, what is nonsensical about the Tractarian propositions is not that they are literally unintelligible like the nihilist reading would have us believe. On the contrary, when taken together as a project, the elucidations self-negate the very philosophical purpose that were intended to prove—the analytic objective that all truth can be commensurably expressed through language, and that this language is capable of determining how all truth is depicted in reality by the use of a singular system of logical analysis that completes our knowledge. Since the propositions of the book concern primitive signs of the logical grammar, whereby the Tractarian propositions explain their applications, the Tractarian propositions are themselves tautologiesthey cannot possibly be false because they are only rules for defining how signs are used. Their truth is stipulated so they are not real propositions because their values are assigned. In this manner, to know the Tractatus as consisting of elucidations is to know the book as both intelligible and nonsensical. This line of reasoning, I believe, to be the genuine philosophical role of the Tractarian propositions, and the force behind both the introduction and proposition 6.54. What Wittgenstein is breaking us free of is a philosophical predilection, an attitude or a way of seeing philosophical method in terms of the perfectionist’s search for complete determination—Wittgenstein is establishing limits to the possibilities of transcendental research, to show the philosopher all those issues that we must give up.

12. I wish to conclude by briefly mentioning Wittgenstein’s philosophical temperament because I feel the legacy of the Tractatus is linked to William James’ description of the two kinds of philosophers. In many ways the Tractatus reminds me of a young Peirce, someone convinced that traditional philosophy is on the wrong track and that a more nuanced framework is required, by one that must first work through rigorous symbolization to establish the groundwork (not to mention they were both very inclined towards diagrammatical reasoning).

The role of philosophical temperament is the axiological foundation grounding the Tractarian project. The drive for determination is following the need for theoretical perfection, the attainment of the holy grail of intellectualism: a singular systematic foundation for the explication of all truth. By climbing out of this thinking, by seeing the limit of what the elucidations point (i.e. the limit of a structured theory of sense to reflect all possible experience), a new way of seeing philosophy awaits over the horizon. From this predilection, the light of language is refracted by the great prism of life, whereby the experience of some observable frequencies shall be expressively incommensurable with the determination of others, but by encountering this pluralism does not imply any part of the spectrum as being erroneous, irrelevant, or inconsistent.

What is lost is the ideal pursuit for a singular, formalistic system of unlimited application, one capable of the declination of all sense in a commensurable, transparent manner. What is lost is Wittgenstein’s conception of logical analysis: the elementary propositions, the philosophical temperament (the ideal of completeness in a singular construct), and method of analyzing language as a kind of Carnapian formalism (i.e. logic as a physical analysis). From here the road to Philosophical Investigations does not look so irreconcilable.

V. Conclusion

13. That which cannot be said cannot be known, therefore it cannot be determined—but that does not imply that what cannot be said cannot be experienced. The elucidations reveal the absurdity of the logical positivist project, the incompleteness of what this philosophical temperament, defined by a drive for demonstrable perfection, can account for in terms of the human condition.

Concurrently, the notion of logical analysis falls apart, crumbling from the weight of its inconsistencies, its inability to accurately represent the nature of the world through its theory of sense. But Wittgenstein was already aware of this incompleteness of logical analysis in terms of ethics, and desired for us to move beyond it by moving beyond the problem. Traditional philosophical systems, as built out of a logic that grounded an epistemology, generate artificial philosophical problems. These problems have existed for hundreds of years because we have no way of determining their solution. This is because the completion of science, as delineated by the possibilities of logical analysis, leaves completely untouched the problems of life, any real way of accessing their subject matter. But this cannot mean that we simply throw life and its vibrancy away; that we merely forget our existential or transcendental questions. Wittgenstein is not a logical positivist, suggesting that the reduction of the logic of language to issues of the domain of natural science, the linguistic depiction of world eliminates the need for life, culture, or practice.

On the contrary, I believe Wittgenstein’s reduction of philosophy to science expressed the limit of philosophy as a discipline. The ladder that the Tractarian method invites us to climb is the road back towards accepting the life-world’s complexity, to acknowledge that the forms of life may not be reducible to the austerity of a singular formalistic method, but that the pluralism of this experiential network of choice and change must be cut out from under this project forever.

The upshot is letting life ‘speak for itself’ by experiencing what the world has to show us.

[1] Ludwig Wittgenstein, Philosophical Grammar (Los Angeles: University of California Press, 1978): 210.

[2] Ian Proops, “New Wittgenstein: A Critique”, University of Michigan, 2005 (unpublished): 1.

[3] Ibid.

[4] Ludwig Wittgenstein, Wittgenstein Collected Papers “Some Remarks on Logical Form” (Oxford: Oxford Press, 1998): 29.

[5] Ibid.

[6] Ibid.

[7] Ibid.

[8] Ibid.

[9] Wittgenstein, Philosophical Grammar: 210.

[10] Ibid.

[11] Ludwig Wittgenstein. Tractatus Logico-Philosophicus (London: Routledge Classics, 2001): 54.

[12] Ibid.

[13] Ibid. 56

[14] Ibid.

[15] Ibid.

[16] Ibid.

[17] Proops, “Wittgenstein” A New Critique: 9.

[18] Ibid.

[19] Ibid.

[20] Ibid. 20


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