4.3 The truth-possibilities of elementary propositions mean [bedeuten] the possibilities of the existence and non-existence of states of affairs.

This just seems to be amplification of 4.27. Nothing new.

4.28 To these combinations there correspond just as many possibilities of truth – and falsehood – for n elementary propositions.

No comment yet.

4.27 As to the existence and nonexistence of n states of affairs, there are possibilities.

All combinations of states of affairs can exist and the others not exist.

K_n has the value 2ⁿ, according to Black (p. 215).
States of affairs are independent of each other is the idea here, presumably.

4.26 The statement of all true elementary propositions describes the world completely. The world is completely described by the statement of all elementary propositions plus a statement as to which of them are true and which are false.

OK. Would giving the false ones and saying that they are false add anything?

4.25 If an elementary proposition is true then the [relevant] state of affairs exists; if an elementary proposition is false, then the state of affairs does not exist.

OK

4.243 Can we understand two names without knowing whether they signify the same thing or two different things? -- Can we understand a proposition, in which two names occur, without knowing if they mean the same or something different?

If I know the meaning [Bedeutung] of an English word and of a synonymous German word, then it is impossible that I do not know that they are both synonymous; it is impossible that I cannot translate them by each other.

Expressions like “a = a,” or those derived from such expressions, are neither elementary propositions nor otherwise signs with sense [sinnvolle Zeichen]. (This will be shown later.)

Black (p. 211): “The questions are, of course, rhetorical (cf. 5.5303).” What is to be shown later is so, he presumes, at 5.531-5.533.

If Black is right, then the intended answer to the questions posed above seems to be “No.” In the case of Tractarian names this would make sense. But the reference to English words and German words complicates things. Couldn't I know the English name of some mythical creature and the German name of a mythical creature without knowing that they are names of one and the same mythical creature? Not if by Bedeutung we mean what Frege means by it.

I'm not entirely sure what would count as expressions like "a = a". Does "a = b" lack sense? Presumably not, given 4.242.

4.242 Expressions of the form “a = b” are thus only aids for presentation; they say nothing about the meaning [Bedeutung] of the signs “a” and “b.”

That is, they tell us nothing about a and b, only about “a” and “b.” They tell us about the signs themselves, but not about what they stand for. Wittgenstein later (see PI §370) seems to have repudiated this distinction. Perhaps we should consider the possibility that he finds it untenable here too. Perhaps not.

4.241 If I use two signs with one and the same meaning [Bedeutung] then I express this by putting between them the sign “=”.

Thus “a = b” means that the sign “a” is replaceable by the sign “b”.

(If I introduce a new sign “b” by an equation, in which I stipulate that it should replace an already known sign “a”, then I write the equation – the definition – (like Russell) in the form: “a = b Def.”. The definition is a rule for signs.)

OK. Just stipulations here. Ogden has ‘symbolic rule,’ P&McG have ‘rule dealing with signs,’ and Black (p. 211) suggests ‘rule about signs.’

4.24 Names are simple symbols. I indicate them by single letters (“x,” “y,” “z”).

I write an elementary proposition as a function of names, in the form: “fx,” “ø(x,y),” etc.

Or else I indicate it by the letters p, q, r.

Black (p. 209) notes that this seems to conflict with 3.202 and 3.26. There, Wittgenstein says that names are signs.

4.23 A name occurs in a proposition only in the context of an elementary proposition.

So are Tractarian names in doubt? Or is this just a quite reasonable expression of the context principle?

4.2211 Even if the world is infinitely complex, so that each fact consists of infinitely many states of affairs and each state of affairs is composed of infinitely many objects, even then there must be objects and states of affairs.

Well yes, there would be. If there are “objects” and “states of affairs” and “facts.”

4.221 It is obvious that by the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination.

Here the question arises of how the combination of propositions comes to be.

Or, if an “elementary proposition” is inconceivable, then it is obvious that “the analysis of propositions” as conceived so far is impossible or inconceivable. The question that “asks itself here” can have no real answer. Complex propositions, the origin of whose complexity we might wonder about, would not be complex (made up of elementary propositions) at all. So it isn’t a real question, by 4.1274. Black (p. 208): “The questions [i.e. ‘How can names combine to form a sentence?’ and ‘How can objects combine to form a state of affairs?’] are nowhere answered and it is hard to see how any answers, in W.’s view, could be expected. Here perhaps we have instances of irredeemable nonsense.”

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

Cf. 2.03.

So can there be such a thing? A proposition that cannot be contradicted, that contains no verb, but somehow asserts the existence of a state of affairs. Something of the form “chair” or “cat, mat” or “red, here, now” or “hard, blue, 5 o’clock”?

4.211 It is a sign of an elementary proposition that no elementary proposition can stand in contradiction to it.

Or perhaps just: can contradict it.

4.21 The simplest proposition, the elementary proposition, asserts the existence of a state of affairs.

OK

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

OK.

4.128 Logical forms are unnumbered [number-less, but not in the sense of too numerous to count].

Therefore there are in logic no pre-eminent numbers, and therefore there is no philosophical monism or dualism, etc.

Black (p. 206) suggests ‘anumerical’ for zahllos (‘unnumbered’). He goes on: “It is nonsense to speak of counting logical forms. It is not clear what W. had in mind here: certainly in a universe containing a finite set of objects and a finite set of their combinations, a list could be made of distinct logical forms, which might then be counted. … Perhaps W. wanted to stress that ‘is a logical form’ is not an authentic predicate such as ‘is a star.’

My first reaction: Are logical forms without number (as Pears and McGuinness have it) because they cannot be counted, since their ‘existence’ is so dubious/problematic, or is it because they are more or less arbitrary inventions/conventions (like mathematical operations and stipulated first terms in series)?

4.1274 The question of the existence of a formal concept is nonsensical [unsinnig]. Because no proposition can answer such a question.

(Thus one cannot ask, e.g.: “Are there unanalysable subject-predicate propositions?”)

A formal concept, after all, is purely formal. If one existed, what would exist? Or not exist, on the contrary?

4.1273 If we want to express in the concept-script the general proposition: “b is a successor of a,” then we need for this an expression for the general term of the formal series: aRb, (Ex):aRx. xRb, (Ex,y):aRx. xRy. yRb, … The general term for a formal series can be expressed only by a variable, because the concept ‘term for this formal series’ is a formal concept. (This has been overlooked by Frege and Russell: because of this the way they want to express general propositions like the one above is false; it contains a vicious circle.)

We can determine the general term of a formal series by giving its first term and the general form of the operation that produces the next term from the proposition that goes before it.

Black (p. 203) points out that Wittgenstein seems to be attacking “Frege and Russell’s definition of the so-called ‘ancestral’ of a relation, which they use in their definition of a natural number.”

4.12721 With an object that falls under it, a formal concept is already given. Thus one cannot introduce as primitive ideas the objects of a formal concept and the formal concept itself. Thus one cannot (like Russell) introduce, e.g., the concept of a function and also special functions as primitive ideas; or the concept of number and specific numbers.

Yes, Russell seems to be making a mistake here about what it makes sense to call “primitive ideas.” So what is (or should be) primitive? The concept under which certain objects fall, or the objects themselves, which perhaps somehow bring the relevant concept with them? Or can there be no true primitiveness or foundation here?

4.1272 Thus the variable name “x” is the proper sign of the pseudo-concept object.

Wherever the word “object” (“thing,” “item,” etc.) is used rightly, it is expressed in the concept-script by a variable name.

For example in the proposition “there are two objects, such that …” by “(Ex,y)…”.

Wherever it is used otherwise, thus as a proper concept word, nonsensical [unsinnige] pseudo-propositions arise.

Thus one cannot, e.g., say “There are objects,” as one says “There are books.” Just as little can one say “There are 100 objects” or “There are א₀ objects.”

And it is nonsensical [unsinnig] to speak of the number of all objects.

The same goes for the words “complex,” “fact,” “function,” “number,” etc.

They all signify formal concepts and are represented in the concept-script by variables, not by functions or classes. (As Frege and Russell believed.)

Expressions like “1 is a number,” “there is only one zero,” and all such are nonsensical [unsinnig].

(It is equally nonsensical [unsinnig] to say “there is only one 1,” as it would be nonsensical [unsinnig] to say: 2+2 is at 3 o’clock equal to 4.)

Quite a bit of Frege- and Russell-bashing going on here, it seems. But what concept-script is he talking about? His own, or the only possible correct one? In other words, is he stipulating how he wants these things to be done, or claiming that Frege and Russell are wrong in some more objective sense? Perhaps it comes to the same thing if he is here developing Frege's and Russell’s ideas as well as they can be developed. But in what sense, if any, is it nonsensical to say there are objects or that 1 is a number? Surely in teaching a child or doing philosophy we make such assertions often. They do not, presumably, count as propositions though for Wittgenstein. They are what he later called grammatical remarks, not (metaphysical) facts. To treat them as facts is to misuse them, i.e. to speak nonsense. Certainly attempts to specify the minimum number of objects there must be are badly mistaken, in Wittgenstein’s view. The Hebrew letter Aleph with the suffix 0 is used in mathematics, including in Principia Mathematica, says Wittgenstein in Letters to Ogden, p. 29.

Ostrow (p. 77): “In acknowledging the weakness of our grasp on “object” we are acknowledging the same about “complex,” “fact,” “function,” “number,” and so on.” On p. 78 he says: “We do not by means of this text arrive at new, superior accounts of “fact,” “object,” and “number.” What the Tractatus seeks instead is to lead us to regard in a new way our attempts to gain clarity about all such notions; it seeks to get us to go on differently in our efforts to know the world. We are called to go on without philosophy.”

4.1271 Each variable is the sign of a formal concept.

Because each variable presents a constant form, which all its values possess, and which can be conceived as a formal property of these values.

So a formal concept is purely formal, not really a concept at all? And formal properties are purely formal, lacking content?

4.127 A propositional variable signifies a formal concept and its values [signify] the objects that fall under this concept.

Black (p. 201) says that this is not strictly correct unless “signify the objects” means “refer to the objects in combination.” 3.313 has propositions as the values of a propositional variable, not names or other referring expressions.

4.126 In the sense of which we speak of formal properties, we can now also speak of formal concepts.

(I introduce this expression in order to make clear the basis of the confusion of formal concepts with proper concepts, which runs through the whole of the old logic.)

That something is an instance of a formal concept cannot be expressed through a proposition. Rather it shows itself in the sign of this object itself. (A name shows that it signifies an object, a numeral that it signifies a number, etc.)

Formal concepts cannot, in the way that proper concepts can, be presented by a function.

Because of their defining characteristics, formal properties are not expressed through functions.

The expression of a formal property is a feature of certain symbols.

The sign for the defining characteristics of a formal concept is therefore a characteristic feature of all symbols whose meaning falls under the concept.

The expression of a formal concept is therefore a propositional variable, in which only this characteristic feature is constant.

In the third sentence here I take Black’s suggestion (p. 199) of saying ‘is an instance of a formal concept’ rather than the more literal ‘falls under a formal concept as an object belonging to it,’ as Ogden has it. It is Black also (same page) who suggests ‘numeral’ instead of ‘numerical sign’ as Ogden has it. Black says that Merkmale should be translated as ‘marks,’ since it means criteria or defining properties. But “Because of their marks, formal properties…” sounds obscure to me.

Cf. 4.122 and note that “the sense in which we speak of formal properties” might be no sense at all. Wittgenstein says that the term “Merkmal” (characteristic) here is taken from Frege’s terminology. See Letters to Ogden, p. 28. See Joan Weiner pp. 47-49 of Future Pasts.

Richard L. Mendelsohn The Philosophy of Gottlob Frege Cambridge University Press, 2005, p. 82: “Wittgenstein clearly had Frege’s predicament about the concept horse in mind when he spoke about ‘formal concepts’… Frege’s concept and object are just such formal concepts.” (Anscombe (p. 122) and Black (p. 198) would seem to agree on that.) According to Mendelsohn (p. 81) “there is just no way of coherently expressing this principle [i.e. “No concept is an object”] in the symbolism.” Frege is thus committed to the view that the principle, which he wants to uphold, is meaningless. Michael Dummett (“Frege on Functions: A Reply,” Philosophical Review 64, pp. 96-107, 1955, reprinted in E. D. Klemke (ed.) Essays on Frege University of Illinois Press, Urbana, 1968, pp. 268-283, p. 269) argues that Frege could have avoided the appearance of paradox if he had talked only about kinds of expression and not the things for which expressions stand, but Mendelsohn objects. Predicates and concepts must have analogous properties, he says, as “an immediate consequence of Frege’s general view that the structure of language mirrors the structure of the world.” (p. 81)

4.1252 Series which are ordered according to internal relations I call formal series.

The series of numbers is ordered not by an external but rather by an internal relation.

Equally the series of propositions:

“aRb,”

“(Ex): aRx. xRb,”

“(Ey): aRx. xRy. yRb,” etc.

(If b stands in one of these relations to a then I call b a successor of a.)

The first part confirms my comment on 4.123. The second introduces a definition of “successor.”

4.1251 Here now the vexed question “whether all relations are internal or external” disappears.

“Relations” seems to have different senses: definitional/necessary/internal and factual/contingent/external. If this is so, then the vexed question looks like “Are all banks financial institutions or sides of rivers?” A question not worth asking. 4.1241 implies an internal connection between questions of sense (logical) and questions of form (metaphysical). The vexed metaphysical question of 4.125 seems to be the result of a mistake about logic/sense. Cf. the second paragraph of the foreword.

Black (p. 198) says that this might be a reference to G. E. Moore’s essay “Relations,” which attacks the views of Bradley and others on internal relations. Hegelian, idealist views on relations certainly were a concern of Russell’s.

Marie McGinn says that “This remark and the one following (4.1252) make a clear reference to Russell. Russell had argued against the intelligibility of internal relations and held that all relations are external.” (p. 178)

4.125 The existence of an internal relation between possible states of things expresses itself linguistically through an internal relation between the propositions that represent them.

OK

4.1241 One cannot distinguish forms from each other by saying that the one has this but the other has that property; because this presupposes that it makes sense to assert either property of either form.

And what’s wrong with that?

4.124 The holding of an internal property of a possible state of things will not be expressed through a proposition, but rather it expresses itself in the proposition that presents the state of things, through an internal property of this proposition.

It would be equally senseless to ascribe a formal property to a proposition as to deny it.

By 4.122, do “formal property” and “internal property” mean the same here? If so, then the second sentence/paragraph implies that the first is senseless. And does the second sentence really have a sense? Has “formal property” been defined?

4.123 A property is internal if it is unthinkable that its object should not possess it.

(This blue color and that stand in the internal relation of lighter and darker eo ipso. It is unthinkable that this pair of objects not stand in this relation.)

(Here the shifting use of the word “object” corresponds to the shifting use of the words “property” and “relation.”)

Marie McGinn (p. 182): “In the later philosophy, it is clear that Wittgenstein thinks that the colour-wheel is itself a part of the symbolism, in the sense that the ordered colour samples of the colour-wheel constitute an instrument of our language, by means of which the logical order of our colour concepts is presented. However, it is not clear that he held this view at the time of writing the Tractatus, where he seems to suggest that the logical order of colour-space will be revealed through the logical analysis of colour terms (see TLP 6.3751).” McGinn also discusses Remarks on Colour p. 34 in connection with this.

So, this is what is special about facial features: they have a kind of essentiality. The reference to shifting uses of words here might alert us to the possibilities that LW’s example does not really tell us anything about what he has been talking about till now (objects, etc. in a different sense) and that nothing really can be identified as what he has been talking about till now (objects, etc.). On the other hand, isn’t there a relation of darker/lighter internal to a pair of shades of blue? And a similar essential relation between earlier/later, left/right, and so on? The very necessity involved (seemingly) here makes them, perhaps, not relations proper (matters of fact), but don’t they indeed seem to be relations of a kind? Isn’t one essentially less than three? Perhaps LW wants us to see such apparent metaphysical truths as misunderstood points of logic, definition, stipulation, convention, or grammar.

4.1221 We can also call an internal property of a fact a feature of that fact. (In the sense in which we speak of facial features.)

Cf. references to physiognomy in the Investigations, perhaps. This sounds as though LW really means it, but we can call anything anything, after all, and is the word ‘features’ used in a special way in connection with faces?

4.122 We can talk in a certain sense of formal properties of objects and states of affairs or of properties of the structure of facts, and in the same sense of formal relations and relations of structures.

(Instead of structural property I say also “internal property;” instead of structural relation, “internal relation.”

I introduce these expressions in order to show the basis of the confusion between internal relations and proper (external) relations, which is very widespread among philosophers.)

The holding of such internal properties and relations, however, cannot be asserted through propositions, but rather it shows itself in the propositions which present the states of affairs and deal with the objects in question.

The last sentence here seems patently absurd (cf. 4.116: there is no excuse for trying to get something out of ‘impermissible’ sentences). And if internal relations are not proper relations, what are they? Not relations at all, one is tempted to think. And perhaps internal properties are not really properties at all, although LW does not say so here. Haven’t we seen already how hard it would be to think what external properties of objects, etc. might be? If now their internal properties and relations go up in smoke, what can be left of the objects and states of affairs themselves? Black (p. 195) points out that, by 3 and 4, formal properties are not really properties at all and cannot be talked about. Hence, in the first sentence, the expression “in a certain sense.” Of that sentence, Black says: “The sentence should be put into apposition with 4.12a, where we are told that logical form belongs to the unspeakable.”

On p. 57 McManus says that “the intent of the qualification ‘internal’ seems to be that it taketh away what the word ‘relation’ giveth.”

4.1213 Now we understand too our feeling that we have a correct logical apprehension only if everything is right in our sign-language.

For Auffassung here I have ‘apprehension,’ although ‘comprehension,’ ‘conception,’ and ‘view’ would also be all right. Ogden has ‘conception.’ P&McG have ‘point of view.’ It means ‘taking in’ or ‘gathering’ in both the literal and the metaphorical, psychological sense. In PI § 4 Wittgenstein talks of Augustine’s Auffassung of language. So perhaps ‘conception’ is best after all.

My first reaction: So we are talking about concept-scripts and, perhaps more importantly, psychology here. LW aims to show us why we feel, mistakenly, that a perfect concept-script is essential to logic. We want a kind of unmistakeable means of presenting thoughts without ambiguity or interpretation. We want to make our meaning visible, so it will be as plain as the nose on one’s face. But meaning, what can be said, cannot be shown. Our quest is forlorn.

4.1212 What can be shown cannot be said.

OK, but is this a stipulation about a possible concept-script or a grammatical truism (like “colors cannot be spoken”)? Or something else? Cf. Schopenhauer on music: We could “just as well call the world embodied music as embodied will.” (WWR, v. 1, §52, pp. 262-3)

Music, like the phenomenal world, shows, as it were, the nature of the thing-in-itself. What this is cannot be said or translated into concepts. Nor can it be demonstrated or proved that music does this. The listener must simply hear the music and agree that it expresses the inner nature of the world. On that idea cf. the preface, where LW talks about being understood only by someone who has already had similar thoughts.

4.1211 Thus the proposition “fa” shows that it is about the object a, two propositions “fa” and “ga” show that they are both about the same object.

If two propositions contradict each other then their structure shows this; the same applies if one follows from the other. And so on.

In what sense is “fa” a proposition? Only according to the conventions of a concept-script. “Fa” stands in need of some interpretation, and only in a context of certain conventional understanding/interpretation does it show anything. Is LW assuming some transparent language of thought here? Is he describing how a possible concept-script might go? Is he talking nonsense? Or is he just talking about a notational system that would work well for us, given our understanding of these conventions?

4.121 Propositions cannot present logical form, it is reflected in them.
What is reflected in language, cannot be presented by it.
What expresses itself in language, we cannot express through it.
Propositions show the logical form of reality.
They display it.

Propositions show something that cannot be presented [darstellen] by language.

Ostrow (p. 107) notes that this “makes clear that it is the genuine proposition that shows logical form,” a job taken by some commentators to be done by the (pseudo-) propositions of logic. But couldn’t both do it?