Lecture Supplement on Wittgenste

Lecture Supplement on Wittgenstein’s Tractatus Logico Philosophicus[1] [1921]

We can’t limit ourselves to the atomic things if we want to understand the world—we must understand how they are related. The simples in the world are essential, but it’s the relations between them which makes the world what it is. This is the reason why the world is the totality of facts—not things.

In this light, note the difference between the “relation” of p and ~p and the “relation” of p and q (put each relation in a truth-table, and note that it is the nature of simples that all their “relations” are “in them,” but there is a difference between the logical (scaffolding) and the empirical (possible T/F) relations.

1.12 For the totality of facts determines what is the case, and also whatever is not the case.

-That is, we do not need to introduce such things as “false facts,” or “the false,” so that false propositions have something which they can refer to! Cf., 3.42.

1.21 Each item can be the case or not the case while everything else remains the same.

The claim here is that the sense of each name and proposition must be a definite sense—they can not rely upon other signs [names] and propositions for their sense. As we shall see (2.0211-2.0212), if they did, then there would be no way sense could be given to language. The example I offered earlier of the four elementary propositions and the sixteen possibilities (although it deals with propositions rather than names, gives an example here—the truth or falsity of each proposition is independent of the others.

Objects [things, “the substance of the world”], and states of affairs:

In his Wittgenstein: A Religious Point of View? Norman Malcolm’s characterization of the early Wittgenstein’s views involves pointing out the centrality of the “picture theory” of meaning for propositions. It involves a distinction between simple and complex objects, a view of “names,” and a distinction between elementary (or atomic) and complex propositions. Names mean simple objects:

what is the composition of an elementary proposition? It is composed of simple signs called ‘names’. The ‘simplicity’ of a name consists in its meaning a simple object.[2]

A simple object is not a word, nor any other kind of sign. It cannot itself occur in a sentence. But a sentence can contain a sign that ‘takes the place of’, ‘deputizes for’, ‘acts for’, a simple object. The sign, called a ‘name’, will have all of the powers that the object has for which it deputizes, but these powers belong to the name in the medium of language, not in the medium of reality.[3]

In an elementary sentence (proposition), one name deputizes for one simple object, another for another and so on. The names are arranged, linked together, in such a way that the proposition, as a whole, is a picture of a possible state of affairs in the world. It depicts the simple objects as related to one another in the same way as the names are related to one another in the proposition.[4]

States of affairs (Sachverhalte) are atomic facts—they have no facts as components. Facts (Tatsachen) are facts which have facts as components.

In his Pulling Up the Ladder, Richard Brockhaus notes that while many presumed that Wittgenstein meant objects to be sense data, he did not give any examples. Brockhaus contends that the important point regarding “simples” is that a decision regarding what the simples were “…could not be made on logical grounds.”[5] Indeed I think it is best to conceive of Wittgenstein discussion on these points as only being concerned with “the logical prerequisites for meaningful use of language.” Brockhaus goes on to cite Norman Malcolm:

I asked Wittgenstein whether…he had ever decided upon anything as an example of a simple object. His reply was that at the time his thought had been that he was a logician, and that it was not his business, as a logician, to try to decide whether this thing or that was a simple or a complex thing, that being a purely empirical matter.”[6]

For him, as for Leibniz, the existence of (complex) facts and propositions clearly presupposed the existence of objects (simples) and names. This, perhaps, is why such a view is often referred to as “logical atomism.” Whereas Bertrand Russell and most of the logical positivists (perhaps to a lesser extent Rudolf Carnap) were concerned to identify the objects and names, I believe Wittgenstein was only interested in the “logical structure,” and willingly left the question of what the simples, and names were to empirical science.

2.012 In logic nothing is accidental; if a thing can occur in a state of affairs, the possibility of the state of affairs must be written into the thing itself.

That is, it is a function of the sort of object an object is that it can be a constituent of a certain state of affairs.

-Note, though I am not certain what to make of this, that in 2.0121 Wittgenstein moves from speaking in the first person plural (“just as we...”) to speaking in the first person singular (“if I can imagine...”). In what “person” does he write from this point on? What do you (does one[?]) make of this?

2.0122 Things are independent in so far as they can occur in all possible situations, but this form of independence is a form of connection with states of affairs, a form of dependence. (It is impossible for words to appear in two different roles: by themselves, and in propositions.)

Words (especially names) and objects are independent, but this independence is tied to all possibilities and, hence to connections).

2.013 Each thing is, as it were, in a space of possible states of affairs. This space I can imagine empty, but I cannot imagine the thing without the space.

2.0141 The possibility of its occurring in states of affairs is the form of an object.

2.0201 Every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely.

2.021 Objects make up the substance of the world. That is why they cannot be composite.

2.0211 If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true.

2.0212 In that case we could not sketch out any picture of the world (true or false).

2.022 It is obvious that an imagined world, however different it may be from the real one, must have something—a form—in common with it.

In 2.02-2.023 we have Wittgenstein’s argument for the existence of a “grid” of elementary possibilities.[7]

2.0231 The substance of the world can only determine a form, and not any material properties. For it is only by means of propositions that material properties are represented—only by the configuration of objects that they are produced.

-Space, time, and color are “forms” of objects and, thus, specify how they can “come together” to form spatio-temporal-colored facts.

2.0271 Objects are what is unalterable and subsistent: their configuration is what is changing and unstable.

Objects, then, are independent of what exists (facts—all possible facts, which are composed of them). They are unchanging and unalterable. Atomic facts are contingent composites of them and are independent of one another (1.21). Moreover, there is no “metaphysical glue” holding objects together in states of affairs.

Although the relevance of this can become clear only later, note that Wittgenstein’s “picture theory” holds that propositions (language) and thought picture the world and not vice-versa. But words are merely dead signs without a “projective relation” to the world (3.12) and this will require a self with intentions (and, hence, a will). The occurrence of the ‘we’ here, then, marks an additional metaphysical element—the metaphysical self (or selves). But more on this later! Note that here he uses the first person plural (cf., 2.0121).

2.13 In a picture objects have the elements of the picture corresponding to them.

2.131 In a picture the elements of the picture are the representatives of objects.

2.15 The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way.

Let us call this connection of its elements the structure of the picture, and let us call the possibility of this structure the pictorial form of the picture.

2.151 Pictorial form is the possibility that things are related to one another in the same was as the elements of the picture.

-2.1511 That is how a picture is attached to reality; it reaches right out to it.

-2.15121 Only the end-points of the graduating lines actually touch the object that is to be measured.

-2.161 There must be something identical in a picture and what it depicts, to enable the one to be a picture of the other at all.

-2.17 What a picture must have in common with reality, in order to be able to depict it—correctly or incorrectly—in the way it does, is its pictorial form.

Here we have an early, and as yet unclarified occurrence of his saying/showing distinction. Imagine the following on the black board: (1) a drawing of a cat sitting on a mat, (2) a picture of (1), (3) a picture of a person drawing a picture of a cat sitting on a mat while the person is looking at a cat sitting on a mat, etc. This “progression” shows that the one thing which a picture can not depict is its pictorial capability (though this can be portrayed in another picture (which, of course) may not depict its pictorial capability).

2.18 What any picture, of whatever form, must have in common with reality, in order to be able to depict it—correctly or incorrectly—in any way at all, is its logical form, i.e. the form of reality.

2.222 The agreement or disagreement of its sense with reality constitutes its truth or falsity.

2.223 In order to tell whether a picture is true or false we must compare it with reality.

-2.224 It is impossible to tell from the picture alone whether it is true or false.

So far we have discussed objects, atomic facts, facts; names, atomic propositions, propositions; and logical form. Here he introduces “thoughts”—are they a “new” metaphysical “category?” In his Wittgenstein: A Religious Point of View?, Norman Malcolm notes that:

in an exchange of letters between Russell and Wittgenstein in August 1919, Russell asked: ‘What are the constituents of a thought?’ Wittgenstein replied: ‘I don’t know what the constituents of a thought are, but I know that it must have such constituents which correspond to the words of language’….To Russell’s further question, ‘Does a Gedanke [thought] consist of words?’, Wittgenstein replied: ‘No! But of psychical constituents that have the same sort of relation to reality as words. What those constituents are I don’t know’….Notice that Wittgenstein said, without any qualification, that thoughts are composed of ‘psychical constituents’, i.e. mental elements. The straightforward interpretation of his remarks is that all thoughts are composed of mental elements. No thought consists of words, spoken or written. Of course the Tractatus holds that a thought can be expressed in physical signs….But a thought does not have to be expressed in a physical sentence….

A thought is a structure with a sense. A meaningful sentence is also a structure with a sense. The view of the Tractatus would seem to be that when a thought is expressed in a sentence, what happens is that the sense of the thought is thought into the sentence.[8]

If a particular proposition is true there are three structures which, in a sense, are equivalent. There is a configuration of simple objects which constitutes a state of affairs. There is a configuration of mental elements which depicts that state of affairs. There is a configuration of signs, which also depicts that state of affairs. These are three parallel structures in three different domains of reality, thought and language. Two of these structures are pictures of the other one.[9]

3.04 If a thought were correct a priori, it would be a thought whose possibility ensured its truth.

3.1 In a proposition a thought finds an expression that can be perceived by the senses.

In his Blue Book, Wittgenstein characterizes the view he will offer here as follows: “it seems that there are certain definite mental processes bound up with the working of language, processes through which alone language can function. I mean the processes of understanding and meaning.

3.12 I call the sign with which we express a thought a propositional sign.—And a proposition is a propositional sign in its projective relation to the world.

Brockhaus notes that: “the propositional sign is a fact; its being so is a necessary condition for its picturing another (possible) fact. But its being such a fact is not also sufficient; we also need a “method of projection” to map grapheme[11] on element and form on form. But what precisely is this “method of projection”? We know its purpose; to connect a given Name with an Object, and to map the form of the proposition-fact onto the form of its sense [the other (possible) fact].”[12]

He also maintains that we can learn something important when we ask: ““On Wittgenstein’s account of picturing, why isn’t the world a picture of language?” The answer is that the picturing relation requires not only the homology of logical form between picture and pictured...but the “method of projection,” the intending act which maps sign onto signified. This cannot come from some constituent of the world, but only from the metaphysical subject.”[13]

Thus, this passage is important in the same way that 2.1 is: it also indicates the important role of the metaphysical self for the early Wittgenstein. The passage should be read with p. 32 of Wittgenstein’s “Blue Book” as a contrasting and comparative passage.

3.1431 The essence of a propositional sign is very clearly seen if we imagine one composed of spatial objects (such as tables, chairs, and books) instead of written signs.

Then the spatial arrangement of these things will express the sense of the proposition.

3.2 In a proposition a thought can be expressed in such a way that the elements of the propositional sign correspond to the objects of thought.

naming is as it were an act of “pure intending.” This intending being a sort of willing, it requires a willing subject; the knowing subject...cannot possibly perform this vital act.[14]

Frege saw names and propositions as quite similar, but...for Wittgenstein they mirror the radical difference between facts and Objects. Propositions picture facts, which requires that they be “articulated,” have parts. Names, on the other hand, are “simple signs.” They have no logical parts, and cannot be analyzed.[15]

Note how in Wittgenstein’s semantics the work of Frege’s sense-reference distinction is neatly split between Name and proposition. Every proposition has a sense, but not necessarily a reference, while every Name is guaranteed reference, but has no sense, no content.”[16]

What is common to every symbol that can be used as a Name for a given Object is that the user of that Name intends it to be the name of that Object, where “intending” can be taken as the primitive relation that turns signs into symbols....

Names are connected to Objects via a primitive, unanalyzable intentional relation.”[17]

3.23 The requirement that simple signs be possible is the requirement that sense be determinate.

-This is a statement of what may be called his “principle of the definiteness of sense.”

--This “determinateness” is of central importance to him as the notions of both simplicity and analysis are at the core of the Tractatus.

3.3 Only propositions have sense; only in the nexus of a proposition does a name have a meaning.

In short, names have a guaranteed reference, but no sense (“meaning”) [cf., 3.203], while a proposition has a guaranteed sense (meaning), but no guaranteed reference [cf., 3.23].

3.31 I call any part of a proposition that characterizes its sense an expression (or a symbol)....

-3.313 Thus an expression is presented by means of a variable whose values are the propositions that contain the expression.

3.317 To stipulate values for a propositional variable is to give the propositions whose common characteristic the variable is.

The stipulation will therefore be concerned only with symbols, not with their meaning.

And the only thing essential to the stipulation is that it is merely a description of symbols and states nothing about what is signified.

3.318 Like Frege and Russell I construe a proposition as a function of the expressions contained in it.

-Simple propositions are function of names, complex propositions are [truth]-functions of simple propositions.

3.324 Using words in several different senses is the most fundamental of confusions (the whole of philosophy is full of them).

3.325 In order to avoid such errors we must make use of a sign-language that excludes them by not using the same sign for different symbols and by not using in a superficially similar way signs that have different modes of signification: that is to say, a sign-language that is governed by logical grammar—by logical syntax.

(The conceptual notation of Frege and Russell is such a language, though, it is true, it fails to exclude all mistakes).

-Here, of course, we have the “ideal language philosophers’” credo! The problem with Frege and Russell is that their axiomatization allows for logical paradoxes!

3.332 No proposition can make a statement about itself, because a propositional sign cannot be contained in itself (that is the whole of the ‘theory of types’).

Note that this sentence seems to do exactly what it says can’t be done! How can this be the case? Why is Wittgenstein asserting this? In his Pulling Up The Ladder, Richard Brockhaus maintains that: “Russell’s Theory of Logical Types, introduced to deal with certain mathematical paradoxes, constructs a hierarchy of languages, each of which talks about the syntax and semantics of the language at the next lowest level. Examination of the theory of Types introduces Wittgenstein’s fundamental distinction between what a proposition can say and what can only be shown by a proposition. This distinction allows us to introduce Wittgenstein’s theory of logic, contrasting as it does with the classical view of logic as a theory of deductive hierarchies. Wittgenstein advances the then-startling claim that logical propositions say nothing, make no claims about the world. Thus in a sense both the realistic and psychologistic logicians are mistaken, since they differ only on the issue of what facts correlate with logical propositions. But although logical propositions say noting, as “tautologies” they show the structure of the world.”[18]

3.4 A proposition determines a place in logical space. The existence of this logical space is guaranteed by the mere existence of the constituents—by the existence of the proposition with a sense.

3.42 A proposition can determine only one place in logical space; nevertheless the whole of logical space must already be given by it....The logical scaffolding surrounding a picture determines logical space.

Ordinary language disguises thought—outward vs. inward form, and analysis: [the 4.0s]

4.002 ...Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes.

The tacit conventions on which the understanding of everyday language depends are enormously complicated.

But, if this is the case, then the activity of “analysis” begins to look very difficult, if not hopeless.[19] This suggests what some call “The Paradox of Analysis”—in his Metaepistemology and Skepticism, Richard Fumerton offers the following characterization:

how can a philosopher have so much difficulty finding the correct analysis of something X when by hypothesis to even formulate and understand the question “What is X?” one must already know what an X is? How can so many philosophers end up providing such radically different answers to a question like “What is causation?” if they all began with the same thing (a property, thought, or state of affairs) before their consciousness?[20]

4.003 (cf., 4.11-4.112) Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only establish that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language.

(They belong to the same class as the question whether the good is more or less identical than the beautiful.)

And it is not surprising that the deepest problems are in fact not problems at all.

-4.0031 All philosophy is a ‘critique of language’....It was Russell who performed the service of showing that the apparent logical form of a proposition need not be its real one.

4.0312 The possibility of propositions is based on the principle that objects have signs as their representatives.

My fundamental idea is that the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts.

4.0621 But it is important that the signs ‘p’ and ‘~p’ can say the same thing. For it shows that nothing in reality corresponds to the sign ‘~’....

...The propositions ‘p’ and ‘~p’ have opposite sense, but there corresponds to them one and the same reality.

4.064 Every proposition must already have a sense: it cannot be given a sense by affirmation. Indeed its sense is just what is affirmed. And the same applies to negation, etc.

Propositions vs. “philosophical” and “logical” propositions: [the 4.1s]

4.1 Propositions represent the existence and non-existence of states of affairs.

....Philosophy does not result in ‘philosophical propositions’, but rather in the clarification of propositions.

-Brockhaus wonders why analysis (or philosophical clarification) is called for if illogical propositions are impossible, and if ordinary language is already in perfect logical order [5.5563]?[21] This, of course, is a version of the above mentioned “paradox of analysis.”

4.1121 Psychology is no more closely related to philosophy than any other natural science.

4.115 It will signify what cannot be said, by presenting clearly what can be said.

4.12 Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality....

4.1272 The variable name ‘x’ is the proper sign for the “pseudo concept” object.

4.1273 ....In order to express the general term of a series of forms, we must use a variable, because the concept ‘term of that series of forms’ is a formal concept. (This is what Frege and Russell overlooked....).

4.1274 To ask whether a formal concept exists is nonsensical. For no proposition can be the answer to such a question....

4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.

4.22 An elementary proposition consists of names. It is a nexus, a concatenation, of names.

-4.24 Names are indicated by ‘x’, ‘y’, ‘z’; elementary propositions are indicated by functions ‘fx’, ‘Æ(x,y)’, etc. (or by ‘p’, ‘q’, ‘r’).

-4.243 ....Expressions like ‘a=a’, and those derived from them, are neither elementary propositions nor is there any other way in which they have sense.

4.27 For n states of affairs, there are K_n...possibilities of existence and non-existence.

4.3 Truth-possibilities of elementary propositions mean possibilities of existence and non-existence of states of affairs.

4.4 A proposition is an expression of agreement and disagreement with truth-possibilities of elementary propositions.

4.431 The expression of agreement and disagreement with the truth-possibilities of elementary propositions expresses the truth conditions of a proposition....

-Wittgenstein critiques Frege’s claim that ‘the true’ and ‘the false’ are objects.

4.441 Wittgenstein contends that just as there is nothing corresponding to the brackets in logical propositions, there are no ‘logical objects’.

-4.46 Among the possible groups of truth-conditions there are two extreme cases.

In one of these cases the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.

In the second case the proposition is false for all the truth possibilities; the truth-conditions are contradictory.

--4.4611 Tautologies and contradictions are not, however, nonsensical. They are part of the symbolism....

--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.[22]

4.5 It now seems possible to give the most general propositional form: that is, to give a description of the propositions of any sign-language whatsoever in such a way that every possible sense can be expressed....

Cf., 6.0, 5.471 and 5.4716! Cf., also, Philosophical Investigations I, 114, 134-135.

4.52 Propositions comprise all that follows from the totality of all elementary propositions (and, of course, from its being the totality of them all). (Thus, in a certain sense, it could be said that all propositions were generalizations of elementary propositions.)

This may be called his Thesis of Extensionality—and it says that propositions are truth functions of more simple propositions. Its “metaphysical” correlate is that there are no real relations amongst states of affairs (and this yields a contingent world). According to Rom Harre, “the extensionalist position is based on a simple principle. Meanings are, in the end, reducible to the sets of objects denoted by a concept.”[23]

The most important passages from 5.0 to 5.555 are the following two, and concentrating on them can help clarify all the remaining ones:

5.3 All propositions are results of truth-operations on elementary propositions.

A truth-operation is the way in which a truth-function is produced out of elementary propositions.

It is of the essence of truth-operations that, just as elementary propositions yield a truth function of themselves, so too in the same way truth-functions yield a further truth-function. When a truth operation is applied to truth functions of elementary propositions, it always generates another truth-function of elementary prepositions, another proposition....

-That is, however logically complex some [complex] proposition is, and whatever logical symbolization or notation we employ (whether we use ‘and’, ‘or’, ‘if/then’, and ‘not’, or use the “Scheffer stroke,” whether we use “brackets” or “periods”), truth-tables and logical analysis can tell us all we need to know about the complex propositions by talking about the elementary propositions and the truth-operators and operations.

5.526 We can describe the world completely by means of fully generalized propositions, i.e. without first correlating any name with a particular object.

Then, in order to arrive at the customary role of expression, we simply need to add, after an expression like, ‘There is one and only one x such that...” the words, ‘and that x is a’.

-That is, all that remains for us to do when we have reached such a level of “logical understanding” of the complex propositions, is for us to add the “names” (“simples”).

For Wittgenstein, elementary propositions comprise the basic unit of construction for the truth-table analysis of the meaning of propositions. [Complex] propositions are “constructed” out of them.

5.101 The truth-functions of a given number of elementary propositions can always be set out in a schema of the following kind:

5.131 If the truth of one proposition follows from the truth of others, this finds expression in relations in which the forms of the propositions stand to one another; nor is it necessary for us to set up these relations between them, by combining them with one another in a single proposition; on the contrary, the relations are internal, and their existence is an immediate result of the existence of propositions.

5.135 There is no possible way of making an inference from the existence of one situation to the existence of another, entirely different situation.

-5.1362 The freedom of the will consists in the impossibility of knowing actions that still lie in the future. We could know them only if causality were an inner necessity like that of logical inference.—The connexion between knowledge and what is known is that of logical necessity.

5.143 Contradiction is that common factor of propositions which no proposition has in common with another....

Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them.

Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their center.

5.15 If T_ris the number of truth-grounds of a proposition r, and if T_rs is the number of the truth-grounds of a proposition s that are at the same time truth-grounds of r, then we call the ratio T_rs: T_r the degree of probability that the proposition r gives to the proposition s.

5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way.

5.156 ....We use probability only in default of certainty—if our knowledge of a fact is not indeed complete, but we do know something about its form.

5.21 ...we can represent a proposition as the result of an operation that produces it out of other propositions (which are the bases of the operation).

5.234 Truth-functions of elementary propositions are results of operations with elementary-propositions as bases. (These operations I call truth-operations.)

5.2523 The concept of successive applications of an operation is equivalent to the concept ‘and so on’.

Propositions are results of truth operations on elementary propositions: [the 5.3s]

5.3 All propositions are results of truth-operations on elementary propositions.

A truth-operation is the way in which a truth-function is produced out of elementary propositions.

5.4 At this point it becomes manifest that there are no ‘logical objects’ or ‘logical constants’ (in Frege’s and Russell’s sense).

5.42 ....The interdefinability of Frege’s and Russell’s ‘primitive signs’ of logic is enough to show that they are not primitive signs, still less signs for relations.

5.43 Even at first sight it seems scarcely credible that there should follow from one fact p infinitely many others, namely ~~p, ~~~~p, etc. And it is no less remarkable that the infinite number of propositions of logic (mathematics) follow from a half a dozen ‘primitive’ propositions.

....The proposition ‘~~p’ is not about negation, as if negation were an object: on the other hand, the possibility of negation is already written into affirmation.

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.

5.452 The introduction of any new device into the symbolization of logic is necessarily a momentous event.

5.47 ....An elementary proposition really contains all logical operations in itself. For ‘fa’ says the same thing as:

5.4711 To give the essence of a proposition means to give the essence of all description, and thus the essence of the world.

What he means here, I believe, is that nothing stands outside logic and, thus, is able to explain it. Logic is the scaffolding of the world, and stands at its limit.

5.474 The number of fundamental operations that are necessary depends solely on our notation.

That is, we can use ‘|’ [ p|q = ~p·~q ]; ‘~’ and one of ‘·‘, ‘V‘, ‘®‘; all of ‘~’, ‘·‘, ‘V‘,

5.5 Every truth-function is the result of successive applications to elementary propositions of the operation

The stipulation is a description of the propositions that have the variable as their representative.

....We can distinguish three kinds of description: 1. direct enumeration...2. giving a function fx whose values for all values of x are the propositions to be described; 3. giving a formal law that governs the construction of the propositions....

-5.511 How can logic—all-embracing logic, which mirrors the world—use such peculiar crotchets and contrivances?

Only because they are all connected with one another in an infinitely fine network, the great mirror.

5.514 Once a notation has been established, there will be in it a rule governing the construction of all propositions that negate p, a rule governing the construction of all propositions that affirm p or q; and so on. These rules are equivalent to the symbols; and in them their sense is mirrored.

-5.5151 ....The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.

5.526 We can describe the world completely by means of fully generalized propositions, i.e. without first correlating any name with a particular object.

Then, in order to arrive at the customary role of expression, we simply need to add, after an expression like, ‘There is one and only one x such that...” the words, ‘and that x is a’.

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.

5.54 In the general propositional form propositions occur in other propositions only as bases of truth-operations.

-5.542 It is clear that ‘A believes that p.’ ‘A has the thought p’, and ‘A says p’ are of the form ‘“p” says p’: and this does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects.

5.55 We now have to answer a priori the question about all the possible forms of elementary propositions.

Elementary propositions consist of names. Since, however, we are unable to give the number of names with different meanings, we are also unable to give the composition of elementary propositions.

5.552 The ‘experience’ that we need in order to understand logic is not that something or other is the state of things, but that something is: that, however, is not an experience.

The limit also makes itself manifest in the totality of elementary propositions.

-5.5563 In fact, all the propositions of our everyday language, just as they stand, are in perfect logical order.—That utterly simple thing, which we have to formulate here, is not an image of the truth, but the truth in its entirety.

-5.557 The application of logic decides what elementary propositions there are.

-5.5571 If I cannot say a priori what elementary propositions there are, then the attempt to do so must lead to obvious nonsense.

Brockhaus maintains that: “every thought is a proposition, that is, a logical picture of a possible Sachverhalt [atomic fact, state of affairs]. The world is a contingent aggregate of Sachverhalte, and, since every elementary proposition pictures its sense with perfect clarity and precision, the world allows of being represented in speech with such clarity and precision. But behind each of these Gedanken [experiences] must lie the intending metaphysical subject, and thus the world—everything that I can represent in language or thought—is conditioned by this intending metaphysical ego and thus is “mine.””[24]

5.62 This remark provides the key to the problem, how much truth there is in solipsism.

For what the solipsist means is quite correct; only it cannot be said, but makes itself manifest.

The world is my world; this is manifest in the fact that the limits of language (of that language which alone I understand) mean the limits of my world.

-Early in 1920, Wittgenstein tells Frege that there are deep grounds for idealism, and here he says that there is “much truth” in solipsism.[25] Clearly, there was as much disagreement between the logical positivists and the early Wittgenstein as there was agreement. Both contended that the metaphysical statements of idealists and solipsists were nonsensical. The positivists left it at this, however, while Wittgenstein found that it is the saying which was flawed. As David Pears notes, “...when Wittgenstein excludes the solipsist’s claim from factual discourse, he implies that it literally lacks sense, but he does not imply that it is rubbish. On the contrary, he allows that among the theses of metaphysics, all of which are literally senseless, there are some that are acceptable for a deeper and more interesting reason than that they make successful claims to factual truth.”[26]

5.631 There is no such thing as the subject that thinks or entertains ideas.

If I wrote a book called The World As I Found It, I should have to include a report on my body, and should have to say which parts were subordinate to my will, and which were not, etc., this being a method of isolating the subject, or rather of showing that in an important sense there is no subject; for it alone could not be mentioned in that book.—

5.632 The subject does not belong to the world: rather, it is a limit of the world.

In his “Lecture On Ethics” [~1929], Wittgenstein says that: “suppose one of you were an omniscient person and therefore knew all the movements of all the bodies in the world dead or alive and that he also know all the states of mind of all human beings that had ever lived, and suppose this man wrote all that he knew in a big book, then this book would contain the whole description of the world [“all that is the case”]; and what I want to say is, that this book would contain nothing that we would call an ethical judgment or anything which would logically entail such a judgment.”[27]

In his Wittgenstein’s Metaphysics John Cook cites Wittgenstein’s Notebooks 1914-1916, p. 80: “The I is not an object” and then maintains: “...meaning that in a fully analyzed version of the sentences in question there will be nothing corresponding to the first person pronoun. In the Tractatus he allows himself, at one point, to put this matter in the material mode:[28] “There is no such thing as the subject that thinks or entertains ideas” (TLP, 5.631), but in another passage he avoids the material mode, saying of the sentences in question:

“It is clear, however, that “A believes that p”, “A has the thought p”, and “A says p” are of the form “p says p”: and this does not involve a correlation of a fact with an object [a self], but rather the correlation of facts by means of the correlation of objects” (TLP, 5.542).

“This shows too that there is no such thing as the soul—the subject, etc.—as it is conceived in the superficial psychology of the present day. A composite soul would of course no longer be a soul” (TLP, 5.45421).

Here Wittgenstein is saying that once we realize that the true logical form of the sentences in question does not involve a subject we will also see the form of the facts in question, will see the essence of the world. And yet he here again resorts to the material mode to say what is shown, for he says: “This shows that there is no such thing as the soul.” It is this sort of thing that, at the end of the Tractatus, he declares to be nonsensical (TLP, 6.54).”[29]

5.6331 Wittgenstein offers a metaphor (the eye not being in the visual field) and a drawing which is of some use in explicating his views. I will modify the drawing several times as I use it to elaborate some basic elements of his overall metaphysical view.

Note that this is a comment on the claim (5.6) that the limits of my language are the limits of my world. Cf., 6.124: “the propositions of logic...describe the scaffolding of the world....” Also cf., Wittgenstein’s Notebooks 1914-1916, pp. 72-73:

Therefore that good and evil are somehow connected with the meaning of the world.

I cannot bend the happenings of the world to my will: I am completely powerless.

I can only make myself independent of the world—and so in a certain sense master it—by renouncing any influence on happenings.[30]

5.641 What brings the self into philosophy is the fact that ‘the world is my world.’

The philosophical self is not the human being, not the human body, or the human soul, with which psychology deals, but rather the metaphysical subject, the limit of the world—not a part of it.

This Wittgenstenian proposition is different from 4.5 however. The latter is restricted to the discussion of the propositional calculus, while this one extends Wittgenstein’s discussion of logic to cover predicate logic with identity. The Wikipedia article on Wittgenstein’s Tractatus is helpful here:

What proposition 6. really says is that any logical sentence can be derived from a series of nand operations on the totality of atomic propositions. This is in fact a well-known logical theorem produced by Henry M. Sheffer, of which Wittgenstein makes use. Sheffer's result was, however, restricted to the propositional calculus, and so, of limited significance. Wittgenstein's N-operator is however an infinitary analogue of the Sheffer stroke, which applied to a set of propositions produces a proposition that is equivalent to the denial of every member of that set. Wittgenstein shows that this operator can cope with the whole of predicate logic with identity, defining the quantifiers at 5.52, and showing how identity would then be handled at 5.53-5.532. [31]

For additional insight into the formula here, cf., Richard Brockhaus’s Pulling Up the Ladder, op. cit., pp. 175-176 (esp., footnotes 44 and 45). Cf., also Wittgenstein’s Philosophical Investigations I, 114.

6.001 What this says is that every proposition is a result of successive applications to elementary propositions of the operation [...].

In 6.02-6.031 he explains the concept of “number” which he derives through the application of this operation. I will skip over these passages altogether.

The Propositions of Logic are Tautologies Which Say Nothing, But Show Much: [the 6.1s]

Note that we know that they don’t “say” anything! [6.11] That is, they lack sense (cf., 4.461 and 5.124).

6.12 The fact that the propositions of logic are tautologies shows the formal—logical—properties of language and the world.

6.1222 ....Not only must a proposition of logic be irrefutable by any possible experience, but it must also be unconfirmable by any possible experience.

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 connexion 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. We have said that some things are arbitrary in the symbols that we use and that some things are not. In logic it is only the latter that express; but that means that logic is not a field in which we express what we wish with the help of signs, but rather one in which the nature of the natural and inevitable signs speaks for itself. If we know the logical syntax of any sign-language, then we have already been given all the propositions of logic.

-Here we can use the following modified diagram from 5.6311 to help us see what he is trying to “show” us here:

The perimeter of the figure includes the propositions of logic”—the “limits of the world” are their limits. They are necessary.

The logical propositions (which are tautologies [6.1], and which are without sense [6.11, 4.461, and 5.142]) do not (“strictly speaking” say anything because they are not about the world—at least not in the way that ordinary (contingent) propositions are. Instead, they provide the scaffolding which makes ordinary language (and ordinary thought and facts) possible. Logic provides the structure which is necessary for there to be facts, language, and thought. Strictly speaking, then, logic can’t be said but only shown—it is like the eye in 5.6311—it can’t be part of the “field” and, so, it is represented as at the “limit”—thus [5.6], it is the limit (or, more properly, as the following will “show,” one of the limits) of the world and of language and thought.

6.127 All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions.

The propositions of mathematics are equations, and therefore pseudo-propositions.

6.211 Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics.

(In philosophy the question, ‘What do we actually use this word or this proposition for?’ repeatedly leads to valuable insights).

This last statement is particularly suggestive of the views of the middle and later Wittgenstein!

The Laws of Causation, Sufficient Reason, Induction, etc. are “Forms of Laws:” [the 6.3]s

6.3 The exploration of logic means the exploration of everything that is subject to law. And outside logic everything is accidental.

6.31 He says that the “law of induction” is not a law of logic, is not an a priori law, but, instead, is a “form of a law.”

He wants to distinguish between (a) empirical [meaningful] propositions (which say how things are); (b) general statements about the empirical statements—that is, laws (“induction,” “sufficient reason,” “causation”); and (c) the statements of logic.

6.34 All such propositions, including the principle of sufficient reason, the laws of continuity in nature and of least effort in nature, etc. etc—all these are a priori insights about the forms in which the propositions of science can be cast.

-6.341 He uses the metaphor of laying a “square mesh” over a white surface with irregular black spots on it to describe it (and of a triangular mesh, etc.) to clarify the place of logic, laws relating to contingent propositions, etc.:

-6.342 ....The possibility of describing a picture like the one mentioned above with a net of a given form tells us nothing about the picture. (For that is true of all such pictures.) But what does characterize the picture is that it can be described completely by a particular net with a particular size of mesh.

Similarly the possibility of describing the world by means of Newtonian mechanics tells us noting about the world: but what does tell us something about it is the precise way in which it is possible to describe it by these means. We are also told something about the world by the fact that it can be described more simply with one system of mechanics than with another.

6.35 ....Laws like the principle of sufficient reason, etc. are about the net and not about what the net describes.

If there were a law of causality, it might be put in the following way: there are laws of nature.

Here a modification of the earlier diagram [5.6311 and 6.124] can help us see what he is saying:

The perimeter of the figure includes the propositions of logic”—the “limits of the world” are their limits. They are necessary.

The interior slanted lines represent the laws of induction, causality, sufficient reason, etc., and their placement inside the world indicates their relative position vis-a-vis the laws of logic. While the latter are necessary, the former are not. But while they are contingent [6.3], they provide a priori insights [6.34] about forms in which the laws of science (etc.) can be cast. Strictly speaking, these propositions here are about the net and not about what the net describes [6.35]. These “laws,” like the laws of logic, “cannot be said: [they] make [themselves] manifest” [6.35].

-6.363 The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences.

-6.3631 This procedure, however, has no logical justification but only a psychological one.

6.37 There is no compulsion making one thing happen because another has happened. The only necessity that exists is logical necessity.

6.371 The whole modern conception of the world is founded on the illusion that the so-called laws of nature are the explanations of natural phenomena.

-Cf., 5.1362—note that the world can’t depend on the will as the objects are independent of one another (and anything else, and the facts are simply truth-functions of the independent states of affairs.

-Cf., 6.13 Logic is transcendental; and 5.5632 The subject does not belong to the world.

6.423 It is impossible to speak about the will in so far as it is the subject of ethical attributes.

6.43 If the good or bad exercise of the will does not alter the world, it can alter only the limits of the world, not the facts—not what can be expressed by means of language.

In short the effect must be that it becomes an altogether different world. It must, so to speak, wax and wane as a whole.

Here the diagram from 5.6311, 6.124, and 6.35 can help us again. This time I will draw your attention to the “dot” in the middle of those diagrams (and this represents the final version of this complex diagram).

The perimeter of the figure includes the propositions of logic”—the “limits of the world” are their limits. They are necessary.

The interior dot [·] doesn’t really belong in the interior—as 6.41 makes clear this dot represents (as does the exterior square of logic) a “limit” (it is “outside” the world). I place it inside for two reasons: first because in truth there is no “outside”—the world is all that is the case, and Wittgenstein doesn’t want an ontology which allows for nonexistent objects. Second, the perimeter is already taken up with logic, and it is clear that the metaphysical self, which the dot is to represent is a different limit from logic. In Notebooks 1914-1916 he says: “I know that this world exists....I am placed in it like my eye in its visual field.”[32] There he also says: “there are two independent godheads: the world and my independent I.”[33]

-Cf., 6.41 Ethics is transcendental; 6.13 Logic is transcendental; and 5.5632 The subject does not belong to the world.

In his Pulling Up The Ladder, Richard Brockhaus maintains that: “my fundamental thesis is that Wittgenstein holds an extremely rarefied version of the Schopenhauerian view that there is an irreducibly human world, although we must carefully limit the meaning of ‘human’ to exclude the psychological or biological. There are many such “human worlds,” each composed of two indivisibly linked although wholly disparate components, one of the greatest possible generality (a generality so great that its assertion is impossible), the other of the greatest possible uniqueness (so unique that it too escapes language). The first is so general that it represents the bare possibility of a world, the latter so particular that its duplication is inconceivable. These two elements—respectively Logic and the Metaphysical Ego—are the elements that Wittgenstein terms “my World.”

Importantly, these claims transcend the bounds of language insofar as they serve as necessary conditions for any symbolic representation. The ensuing ineffability complicates the exposition of the major themes of the Tractatus, since so much of what is central cannot be said. But it also leads straight to Wittgenstein’s fundamental concern, namely the extent to which linguistic representation—thought—and with it reason are connected with what I shall loosely call value.”[34]

6.44 It is not how things are in the world that is mystical, but that it exists.

6.45 To view the world sub specie aterni is to view it as a whole—as a limited whole.

6.5 When the answer cannot be put into words, neither can the question be put into words.

For doubt can exist only where a question exists, a question only where an answer exists, and an answer only where something can be said.

6.52 We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.

6.521 The solution of the problem of life is seen in the vanishing of the problem.

6.522 There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical.

6.53 The correct method in philosophy would really be the following: to say nothing expect what can be said, i.e. propositions of natural science...and then whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions. Although it would not be satisfying to the other person—he would not have the feeling we were teaching him philosophy—this method would be the only strictly correct one.

6.54 My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them—as steps—to climb up beyond them. (He must...throw away the ladder after he has climbed up it.)

-What do we do after we throw away the ladder? Cora Diamond discusses this in her “Throwing Away the Ladder.”[36]

Notes: (click on note number to return to the text for the note)

[1] Ludwig Wittgenstein, Tractatus Logico-Philosophicus [1921 in German, 1922 doe English translation], trans. D.F. Pears and B.F. McGuinness (London: Routledge, 1961). All further citations to the Tractatus in these lecture notes will be identified by the relevant section number.

[2] Norman Malcolm, Wittgenstein: A Religious Point of View? (Ithaca: Cornell U.P., 1994), p. 31.

[3] Ibid.

[4] Ibid.

[5] Richard Brockhaus, Pulling Up the Ladder, (LaSalle: Open Court 1991), p. 143.

[6] Cited by Brockhaus, op. cit., in his discussion on p. 142.

[7] Cf., David Pears, The False Prison: A Study of the Development of Wittgenstein’s Philosophy v. 1, (Oxford: Oxford U.P., 1987), pp. 27-29.

[8] Norman Malcolm, Wittgenstein: A Religious Point of View?, op. cit., p. 32. Malcolm cites “Appendix III” of Wittgenstein’s Notebooks 1914-1916, eds. G.H. von Wright and G.E.M. Anscome, trans. G.E.M. Anscome (N.Y.: Harper & Row, 1961). While he indicates that the citations are on p. 130 and 131, they are on p. 129 and 130.

[9] Ibid., pp. 32-33.

[10] Ludwig Wittgenstein, The Blue Book, in The Blue and Brown Books (N.Y.: Harper and Row, 1958), p. 3.

[11] Brockhaus uses the notion of a grapheme like that of a phoneme to mark out a unit of graphicality.

[12] Richard Brockhaus, Pulling Up the Ladder, op. cit., p. 163.

[13] Ibid., pp. 190-191. Emphasis added to passage.

[14] Ibid., p. 164.

[15] Ibid., p. 165.

[16] Ibid.

[17] Ibid., pp. 171-172.

[18] Ibid., pp. 18-19.

[19] Cf., ibid., pp. 175-177.

[20] Richard Fumerton, Metaepistemology and Skepticism (Lanham: Rowman, 1995), p. 22.

[21] Cf., ibid., p. 213.

[22] David Pears, The False Prison: A Study of the Development of Wittgenstein's Philosophy v. 1, op. cit., pp. 22-23.

[23] Rom Harre, Varieties of Realism: A Rationale for the Natural Sciences (Oxford: Basil Blackwell, 1986), p. 319.

[24] Richard Brockhaus, Pulling Up the Ladder, op. cit., p. 292.

[25] Cf. Ray Monk, Ludwig Wittgenstein, (N.Y.: Penguin, 1990, pp. 190-191. ‘Solipsism’ if from solus ipse (“the self alone”).

[26] David Pears, The False Prison v. 1, op. cit., p. 164.

[27] Ludwig Wittgenstein, “Wittgenstein’s Lecture on Ethics,” Philosophical Review v. 74 (1965), pp. 3-12, p. 6. The lecture was delivered by Wittgenstein in Cambridge between 1929 and 1930.

[28] The “material mode” of speech is used to speak about things; the “formal mode” is employed when we speak about language, propositions, etc. For example “do physical objects exist while unperceived” is in the material mode, while “what is the correct analysis of propositions about physical objects” is in the formal mode.

[29] John W. Cook, Wittgenstein's Metaphysics (Cambridge: Cambridge U.P., 1994), pp. 59-60. Cf., also pp. 48-52.

[30] Ludwig Wittgenstein, Notebooks 1914-1916, ed. G.H. von Wright and G.E.M. Anscome, trans. G.E.M. Anscome (N.Y.: Harper, 1961), pp. 72-73.

[31] Wikipedia, “Tractatus Logico-Philosophicus, https://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicus accessed on 02/18/14. A yet more careful discussion of his views and predicate logic can be found in the Stanford Encyclopedia of Philosophy essay of “Wittgenstein’ Logical Atomism”: https://plato.stanford.edu/entries/wittgenstein-atomism/ accessed on 02/18/14.

[32] Ibid., pp. 72-73.

[33] Ibid., p. 74. This occurs in the context of a longer discussion (pp. 72-75) which will be important as we discuss the metaphysical self (or metaphysical ego).

[34] Richard Brockhaus, Pulling Up The Ladder, op. cit., pp. 14-15.

[35] Ludwig Wittgenstein, On Certainty (N.Y.: Harper, 1969).

[36] Cora Diamond, “Throwing Away the Ladder,” Philosophy v. 63 (1988), pp. 5-27.