Hauptli’s Introduction to the Tractatus

     Copyright © 2014 Bruce W. Hauptli

I. Introduction:

As I have indicated, Wittgenstein began working in philosophy because of an interest in symbolic logic.  One of his primary concerns was with providing an account of meaning (a “theory of meaning”) which would explain the necessity of logical truths.^[1]  As David Pears notes in his The False Prison: A Study of the Development of Wittgenstein’s Philosophy:

...the Tractatus, as its full title implies, is a philosophical treatise on logic.  It was the foundations of logic that presented the problem, because, though we all know how to establish ordinary contingent truths, it is not clear how the familiar necessary truths on which we rely in everyday arguments are established.  It is no good saying that we prove them, because anyone who has looked into Euclid knows that you cannot prove anything from nothing.  Axioms and rules of inference are always needed, but then their status too can be questioned.^[2] 

It should be remembered that upon the advice of Gottlob Frege, Wittgenstein went to Cambridge to discuss logic with Bertrand Russell.  Frege and Russell were codifying the new (“symbolic”) logic—Russell and Whitehead’s Principia Mathematica (which axiomatized the new logic) was published in 1910. 

     Wittgenstein treats logical formulae as tautologies which do not depend upon anything that happens in the world.  As David Pears notes, tautologies

are not hostages to contingency.  Factual sentences make claims and they get a grip on the world, which then verifies them or falsifies them.  Tautologies make no claim and they ride loosely on the world, being neither supported nor let down by any contingency.  They levitate because they say nothing.  Logical formulae are radically independent when their necessary truth is explained in this way.  Each of them can be validated directly without any help from the others.  There is, therefore, no need to string them together in a calculus, giving some of them the role of premises and proving others as conclusions. 

  If this is what logic is like, it is very unlike anything to be found in factual discourse.  It is not a system of connected truths, like science: it is not even a medley of independent truths, like the ordinary record of what goes on around us.  In the Tractatus Wittgenstein spends a lot of time on these differences between the formulae of logic and factual sentences, but people read this part of the book rather rapidly, because they are already converted.  They ought to pause and ask themselves how he saw the point which strikes them as so obvious.  He saw it as a deep difference.  It is not only that logic does not cover the same ground as factual discourse: it does not cover its own ground in the same way—or, rather, it does not cover any ground.  Its formulae do not express knowledge of any subject.  They merely reveal connections between different forms of sentences, and so between different forms of facts.  But these forms do not belong to another world, to be explored after the world of facts, as it were, on a separate expedition.... 

  The system of the Tractatus is built on an idea that is the exact opposite of Russell’s idea: the forms revealed by logic are embedded in the one and only world of facts and, therefore, in the language that we use to describe it.  If Russell’s view was Platonic, this view is approximately Aristotelian.  Logic is immanent in factual discourse from the very beginning, and it emerges when we take factual sentences and combine them in various truth-functional ways—that is, in such ways that the truth or falsehood of the combinations will depend entirely on the truth or falsehood of what went into them.^[3] 

     Consider Tractatus 5.5303:

roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.^[4] 

What this passage claims is clear-cut, but what we need to do is to try and understand what it says about the world.  What is it true of?  Here we get down to the core of the Tractatus.  As I have indicated, Wittgenstein initially wanted to title the book “The Proposition.”  The following three passages summarize what he takes the point of his book to be:

in his Notebooks, 1914-1916 he maintains that:

in philosophy there are no deductions; it is purely descriptive.  The word “philosophy,” ought always to designate something over or under, but not beside, the natural sciences.  Philosophy gives no pictures of reality, and can neither confirm nor confute scientific investigations.  It consists of logic and metaphysics, the former its basis.  Epistemology is the philosophy of psychology.  Distrust of grammar is the first requisite for philosophizing.  Philosophy is the doctrine of the logical form of scientific propositions (not primitive propositions only).  A correct explanation of the logical propositions must give them a unique position as against all other propositions.^[5] 

in a letter to his friend Ludwig von Ficker, he says:

the book’s point is an ethical one.  I once meant to include in the preface a sentence which is not in fact there now, but which I will write for you here, because it will perhaps be a key to the work for you.  What I meant to write, then was this: My work consists of two parts: the one presented here plus all that I have not written.  And it is precisely this second part that is the important one.  My book draws limits to the sphere of the ethical from the inside as it were, and I am convinced that this is the ONLY way of drawing these limits.^[6] 

finally, in his “Preface” to the Tractatus Wittgenstein says that:

the book deals with the problems of philosophy, and shows, I believe, that the reason why these problems are posed is that the logic of our language is misunderstood.  The whole sense of the book might be summed up in the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence. 

  Thus the aim of the book is to set a limit to thought, or rather—not to thought, but to the expression of thoughts; for in order to be able to set a limit to thought, we should have to find both sides of the limit thinkable (i.e. we should have to be able to think what cannot be thought). 

  It will therefore only be in language that the limit can be set, and what lies on the other side of the limit will simply be nonsense.^[7] 

To properly understand the meaning of the “logical” propositions, we need, of course, to understand Wittgenstein’s view of the meaning of the ordinary empirical propositions.  In the Tractatus, he holds that our factual language (better, the propositional portion of it)^[8] pictures the world.  According to the theory which he offers, the world is composed of simples which are named by certain words (the meaning of a name is the simple in question).  Wittgenstein holds that if there were not such simples and names, there would be no way for our language to be meaningful.  For sentences to have sense, they can not depend exclusively on the sense of other sentences—ultimately there must be elementary propositions which get their sense not from other sentences, but rather directly from the world.^[9]  Wittgenstein never indicated what he took the simples to be—unlike Russell and other Logical Positivists and Logical Atomists, who contended that they were uncovered empirically.^[10] 

     The simples are combined together into simple facts which are pictured by propositions.  Other propositions picture possible combinations of simples (or facts).  Thus the world consists of simples which are arranged in a “grid” of possible combinations, and some of these possible combinations actually obtain. 

     But, you will note, we have not yet indicated what the “true” sentence in Tractatus 5.5303 says or pictures!  If the proposition “The cat sat on the mat” is an elementary one, it pictures a cat sitting upon a mat.  What facts do tautologies picture?  According to Wittgenstein, they show rather than say something.  They show how the world of facts is “structured”—they exhibit (or describe) the grammar, form, or logic, of our language (and of our world). 

     Suppose, for purposes of illustration, that there are four (and only four) elementary propositions about the world.  Then the following represents the sixteen possible states-of-affairs:

Remember, each of p, q, r, and s is a proposition, and the central characteristic of propositions is that they are sentences which are capable of being true or false.  The “truth-table” above represents the only possible combinations of the truth and falsity of these four elementary propositions.  Because they are each elementary propositions, Wittgenstein holds, the truth and falsity of each is independent of the others, and, thus, there are sixteen “possible worlds” (or better, at least to my metaphysical taste, possible ways in which the world comprised by these four elementary propositions could be).  Now this statement (that there are sixteen possibilities) is true of the “world” of these four propositions, but it does not picture any fact in that world! 

     To muddy the waters still further, we need to note that Wittgenstein believes that his statements about the meanings of logical and empirical propositions themselves occupy a peculiar status.  He contends that one thing a picture can not do is to portray its pictorial capacity.  While one could draw a picture of an artist drawing a picture of a cat sitting on a mat, and while that picture could be used to portray how the portrayed artist represents cats sitting on mats in the pictured pictures, the picture of the artist picturing the feline on the mat can not, itself, picture its own picturing capacity.  Thus, Wittgenstein says:

6.124 The propositions of logic describe the scaffolding of the world, or rather they represent it.  They have no ‘subject matter’.  They presuppose that names have meaning and elementary propositions sense; and that is their connection with the world.  It is clear that something about the world must be indicated by the fact that certain combinations of symbols—whose essence involves the possession of a determinate character—are tautologies.  This contains the decisive point. 

In the above example, that there are sixteen possible combinations of the four propositions is part of the logical scaffolding of that world (as is the proposition “p or not-p”).  Moreover, according to Wittgenstein (and here, of course, it is important that I don’t say “says Wittgenstein”), this can be shown but not said—and he is most interested in showing this.  As he says in notes written in 1930:

it is all one to me whether or not the typical western scientist understands or appreciates my work, since he will not in any case understand the spirit in which I write.  Our civilization is characterized by the word ‘progress’.  Progress is its form rather than making progress being one of its features.  Typically it constructs.  It is occupied with building an ever more complicated structure.  And even clarity is sought only as a means to this end, not as an end in itself.  For me on the contrary clarity, perspicuity are valuable in themselves. 

  I am not interested in constructing a building, so much as in having a perspicuous view of the foundations of possible buildings. 

  So I am not aiming at the same target as the scientists and my way of thinking is different from theirs.^[11] 

Of course, Wittgenstein did, at around the time he wrote this, construct a building (he was the co-architect for his sister’s house in Vienna).  This remark indicates something about his concern—he is not primarily interested in uttering factual sentences, or in determining whether uttered factual sentences are, in fact, true.  Instead, he is interested in the foundations of factual sentences.  Thus his concern with the “logic” or “grammar” of our language (and, of course, of the world). 

     Many of us find logic and mathematics to be a very “dry” topic in the first place, and we find it difficult to understand how there can be a “history” of field, or how there could be “intellectual developments” or “progress” in the field.  This will make it very difficult for one to understand Wittgenstein’s motivations however.  The following thinkers, however, represent a historically oriented list of major figures who affected the history of thinking about logic: Aristotle (in his Organon—consisting of: Categoriae, De Interpretatione, Analytica Prioria, Analytica Posterioria, Topica, and De Sophisticis Elenchis [384-322]), Euclid (“The Elements” [~323]), Boole (The Mathematical Analysis of Logic [1847], DeMorgan (Formal Logic [1847]), Russell and Whitehead Principia Mathematica [1910], Carnap (Logische Syntax der Sprache [1934]), and Wittgenstein’s Tractatus [1922].^[12]  The history of the field can be divided into two main periods: Aristotelian (Syllogistic) Logic and Modern (Symbolic) Logic.  A brief overview of the differences between these two periods is supplied by Raymond Bradley and Norman Swartz:

the details of traditional syllogistic analysis need not concern us here.  The main point to note is that the validity of many arguments can be determined simply by analyzing them to the level made possible within that tradition and checking to see whether the form which, on analysis, that argument is found to have, is one of the certifiably valid ones.  We do not have to analyze the terms themselves or even understand what concepts they express in order to show that certain arguments are valid....Note that we do not say that any argument whose form is not certifiably valid is an argument which can be certified as invalid.  Plainly, that would be a mistake.  As we have already seen, in passing from the logic of unanalyzed propositions to the logic of unanalyzed concepts, an argument whose form at one level of analysis is not certifiably valid may turn out, at a deeper level of analysis, to be valid nonetheless.  Having a certifiably valid form is a sufficient condition of the validity of an argument but it is not a necessary condition.^[13] 

The analytical and notational resources of modern predicate logic are much richer than those of traditional syllogistic.  Accordingly, many more arguments yield to its treatment.  Like traditional syllogistic, it recognizes that so-called quantifier-words, like “all” and “some”, express concepts which feature in the internal structure of a proposition in such a way as to determine that proposition’s logical connections with other propositions independently of what other concepts feature in that proposition, and independently, too, of the analysis of those concepts.  But unlike traditional syllogistic it utilizes a symbolism which blends with that of propositional logic to provide a much more versatile means of exhibiting the internal structure of propositions.^[14] 

While much of this may be “Greek” to the uninitiated, the important point to note here is that from Aristotle until the middle of the Nineteenth Century, syllogistic logic and logic were, in effect, synonymous.  In the last 150 years, however, an importantly new formal technique of logical analysis has been developed.  The development of this new logic has not been without its problems and paradoxes however. 

     Russell and Whitehead’s axiomatization of the modern logic was both ground-breaking and problematic.  As David Pears notes, Wittgenstein had a number of problems with it:

his main criticism of Russell’s system is that it is not self-contained.  When the need arises, Russell helps himself to additional axioms which, Wittgenstein argues, are, at best, only contingently true....Therefore, after the point at which Russell adds contingent truths to his axioms, the theorems or formulae that he proves will all belong to science rather than logic.  But that violates the fundamental distinction between logical formulae and factual sentences. 

  Another criticism that he makes of Russell is the one that occurs to everybody.  Even if none of the axioms were contingent, they would all remain unproven, and so the task of proving the theorems from them was not worth undertaking.  However strong your rope, you cannot hang anything on it unless it is attached at the other end, and then what you can hang on it will depend on the strength of that attachment.^[15] 

     To better understand Wittgenstein’s theory, however, we must turn to the work itself. 

II. Overview of the Tractatus:

The Tractatus Logico-Philosophicus consists of a series of numbered paragraphs.  One way to begin to get a handle on what he is saying is to list the seven “integer” propositions, as they should provide us with an outline of the book. 

1 The world is all that is the case.  [p. 7, less than one page long]

2 What is the case—a fact—is the existence of states of affairs.  [p. 7, 12 pps. long]

3 A logical picture of facts is a thought.  [p. 19, 16 pps. long]

4 A thought is a proposition with a sense.  [p. 35, 38 pps. long]

5 A proposition is a truth-function of elementary propositions.  [p. 73, 46 pps. long]

6 The general form of a truth function is [       ].  This is the general form of a proposition.  [p. 119, 32 pps. long]

7 What we cannot speak about we must pass over in silence.  [p. 151, one proposition only]

Note that since Wittgenstein claims that all the propositions of logic are “equal” (that is, since he rejects the axiomatic approach to mathematics and logic wherein certain propositions are held to be more fundamental than others), one should not think that proposition 1.1 is “more advanced” or “less basic than” proposition 1! 

     David Pears maintains that Wittgenstein’s “metaphysic” in the Tractatus

…starts from the existence of factual language.  We evidently do succeed in using this language to describe the world, how is it done?  His answer is that we succeed only because there is a fixed grid of possible combinations of objects to which the structure of our language conforms.  The grid must exist and connections must be made with it if language is going to work.  But it clearly does work and so the metaphysical conclusions follow.^[16] 

In the Tractatus Wittgenstein offers only a metaphysic of experience deduced from the existence of factual language, but not expressible in it.  What we need to know is why this deduction of his metaphysic prevents us from expressing it as a very general factual truth.  Why is it impossible for factual language to express the fundamental condition of its own existence?  This is a difficult question.^[17] 

In short, there will be two things we will have to get clear on as we read the book:

(a) what his metaphysical theses are, and

(b) why they can not be said (in factual language).