Ts-201a1,TP Wittgenstein Summary Ts-201a1,a1

Summary

One reason for thinking the old notation wrong is that it is- very unlikely that from every proposition p an infinite number of- other propositions not-not-p, not-not-not-not-p, etc., should follow.

If only those signs which contain proper names were complex- then propositions containing nothing but apparent variables would- be simple. Then what about their denials?

The verb of a proposition cannot be “is true” or “is false”,- but whatever is true or false must already contain the verb.

Deductions only proceed according to the laws of deduction,- but these laws cannot justify the deduction.

One reason for supposing that not all propositions which have- more than one argument are relational propositions is that if they were, the relations of judgement and inference would have to hold- between an arbitrary number of things.

Every proposition which seems to be about a complex can be- analysed into a proposition about its constituents and about- the proposition which describes the complex perfectly; i.e., that- proposition which is equivalent to saying the complex exists.

The idea that propositions are names of complexes [L.W.]- suggests that whatever is not a proper name is a sign for a- relation. Because spatial complexes – you for instance imagine every fact as a- spatial complex- – consist of Things- & Relations only & the idea of a complex is taken -from space.

In a proposition convert all its indefinables into variables;- there then remains a class of propositions which is not all- propositions but a type. Ts-201a1,a2

There are thus two ways in which signs are similar. The- names Socrates and Plato are similar: they are both names. But- whatever they have in common must not be introduced before Socrates- and Plato are introduced. The same applies to subject-predicate- form etc. Therefore, thing, proposition, subject-predicate form,- etc., are not indefinables, i.e., types are not indefinables.

When we say A judges that etc., then we have to mention a- whole proposition which A judges. It will not do either to- mention only its constituents, or its constituents and form, but- not in the proper order. This shows that a proposition itself- must occur in the statement that it is judged; however, for- instance, “not-p” may be explained. The question „What is negated” must have a meaning.

To understand a proposition p it is not enough to know that- p implies ‘“p” is true’, but we must also know that ~p implies “p is false”. This shows the bipolarity of the proposition.

W-F = Wahr-Falsch      To every molecular function a WF scheme corresponds. Therefore- we may use the WF scheme itself instead of the function. Now- what the WF scheme does is, it correlates the letters W and F- with each proposition. These two letters are the poles of atomic- propositions. Then the scheme correlates another W and F to these poles. - In this notation all that matters is the correlation of the- outside poles to the pole of the atomic propositions. Therefore- not-not-p is the same symbol as p. And therefore we shall never get- two symbols for the same molecular function. Ts-201a1,a3

The meaning of a proposition is the fact which actually- corresponds to it.

As the ab functions of atomic propositions are bi-polar- propositions again we can perform ab operations on them. We- shall, by doing so, correlate two new outside poles via the old- outside poles to the poles of the atomic propositions.

The symbolising fact in a-p-b is that, say a is on the- left of p and b on the right of p; then the correlation of new- poles is to be transitive, so that for instance if a new pole a in whatever- way i.e. via whatever poles is correlated to the inside a, the symbol is not changed- thereby. It is therefore possible to construct all possible ab- functions by performing one ab operation repeatedly, and we can- therefore talk of all ab functions as of all those functions which- can be obtained by performing this ab operation repeatedly. [This is quite arbitrary but if we such have fixed- on which sides the poles have to stand we must of- course stick to our convention. If for instance „apb”- says p then bpa says nothing. (It does not say- ~p.) But a-apb-b is the same symbol as apb- (here the ab function vanishes automatically for here the new poles are- related to the same side of p as- the old ones. The question is always: how are the- new poles correlated to p compared with the way the- old poles are correlated to p.]

[Note by Bertrand Russell] [NB. ab means the same as WF, which means true-false.]

Naming is like pointing. A function is like a line dividing- points of a plane into right and left ones; then “p or not-p” has no meaning- because it does not divide the plane.

But though a particular proposition “p or not-p” has no- meaning, a general proposition “for all p's, p or not-p” has a- meaning because this does not contain the nonsensical function “p- or not-p” but the function “p or not-q” just as “for all x's xRx” contains the function “xRy”. Ts-201a1,a4

A proposition is a standard to which all facts behave, with- names it is otherwise; it is thus bi-polarity and sense comes in;- just as one arrow behaves to another arrow by being in the same- sense or the opposite, so a fact behaves to a proposition.

The form of a proposition has meaning in the following way. - Consider a symbol “xRy”. To symbols of this form correspond- couples of things whose names are respectively “x” and “y”. The- things x y stand to one another in all sorts of relations, amongst- others some stand in the relation R, and some not; just as I- single out a particular thing by a particular name I single out- all behaviours of the points x and y with respect to the relation R.- I say that if an x stands in the relation R- to a y the sign “x R y” is to be called true to the fact and- otherwise false. This is a definition of sense.

!      In my theory p has the same meaning as not-p but opposite- sense. The meaning is the fact. The proper theory of judgment- must make it impossible to judge nonsense.

It is not strictly true to say that we understand a- proposition p if we know that p is equivalent to “p is true”- for this would be the case if accidentally both were true or- false. What is wanted is the formal equivalence with respect- to the forms of the proposition, i.e., all the general- indefinables involved. The sense of an ab function of a proposition- is a function of its sense. There are only unasserted propositions. - Ts-201a1,a5 Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among- the facts which make “p or q” true there are also facts which make “p- and q” true; if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical, but in fact,- their sense is different for we have introduced sense by talking- of all p's and all q's. Consequently the molecular propositions- will only be used in cases where there ab function stands under- a generality sign or enters into another function such as “I- believe that, etc.,” because then the sense enters.

In “a judges p” p cannot be replaced by a proper name. This- appears if we substitute “a judges that p is true and not p is- false”. The proposition “a judges p” consists of the proper name a, the proposition p with its 2 poles, and a being related to both- of these poles in a certain way. This is obviously not a- relation in the ordinary sense.

The ab notation makes it clear that- not and or are dependent on one another and we can therefore not- use them as simultaneous indefinables. | Same objections in the case of apparent variables to old indefinables, as in the case of molecular functions: The application- of the ab notation to apparent-variable propositions becomes- clear if we consider that, for instance, the proposition “for all x, φx” is to be true when φx is true for all x's and false- when φx is false for some x's. We see that some and all- occur simultaneously in the proper apparent variable notation. Ts-201a1,a6

The Notation is: for (x) φx; a ‒ (x) ‒ a φ x b ‒ (∃ x) ‒ b and for (∃x) φx : a ‒ (∃x) ‒ a φ x b ‒ (x) ‒ b

Old definitions now become tautologous.

In aRb it is not the complex that symbolises but the fact- that the symbol a stands in a certain relation to the symbol b. - Thus facts are symbolised by facts, or more correctly: that- a certain thing is the case in the symbol says that a certain- thing is the case in the world.

Judgment, question and command are all on the same level. - What interests logic in them is only the unasserted proposition. - Facts cannot be named.

A proposition cannot occur in itself. This is the fundamental- truth of the theory of types.

Every proposition that says something indefinable about one- thing is a subject-predicate proposition, and so on.

Therefore we can recognize a subject-predicate proposition- if we know it contains only one name and one form, etc.. This- gives the construction of types. Hence the type of a proposition- can be recognized by its symbol alone. Ts-201a1,a7

What is essential in a correct apparent-variable notation is this:– - (1) it must mention a type of propositions; (2) it must show- which components of a proposition of this type are constants.

[Components are forms and constituents.]

Take (φ).φ!x. Then if we describe the kind of symbols, for which φ! stands- & which, by the above, is enough to determine the type, then- automatically “(φ).φ!x” cannot be fitted by this description, because it contains „φ!x” & the- description is to describe all- that symbolizes in symbols- of the φ! – kind. If the description- is thus complete vicious circles- can just as little occur as- for instance in (φ).(x)φ (where- (x)φ is a subject-predicate proposition). Ts-201a1,b1

Wittgenstein First MS.

Indefinables are of two sorts: names, & forms. Propositions cannot- consist of names alone; they cannot be classes of names. A name- can not only occur in two different propositions, but can occur in the- same way in both.

Propositions [which are symbols having reference to facts] are- themselves facts: that this inkpot is on this table may express- that I sit in this chair.

It can never express the common characteristic of two- objects that we designate them by the same name but by two different- ways of designation, for, since names are arbitrary, we might- also choose different names, & where then would be the common- element in the designations? Nevertheless one is always tempted,- in a difficulty, to take refuge in different ways of designation.

Frege said “propositions are names”; Russell said- “propositions correspond to complexes”. Both are false; & especially- false is the statement “propositions are names of complexes”.

It is easy to suppose that only such symbols are complex as contain- names of objects, & that accordingly “(∃x,φ).φx” or “(∃x,R,y).xRy” must be- simple. It is then natural to call the first of these the name of a form,- the second the name of a relation. But in that case what is the meaning- of (e.g.) “~(∃x,y).xRy”? Can we put “not” before a name? Ts-201a1,b2

Wittg. –      The reason why “~Socrates” means nothing is that “~x” does- not express a property of x.

There are positive & negative facts: if the proposition “this rose is not- red” is true, then what it signifies is negative. But the occurrence of the- word “not” does not indicate this unless we know that the- signification of the proposition “this rose is red” (when it is true) is positive. It is- only from both, the negation & the negated proposition, that we can- conclude to a characteristic of the significance of the whole proposition.- (We are not here speaking of negations of general propositions, i.e.- of such as contain apparent variables. Negative facts only justify- the negations of atomic propositions.)

Positive & negative facts there are, but not true &- false facts.

If we overlook the fact that propositions have a sense which- is independent of their truth or falsehood, it easily seems as if- true & false were two equally justified relations between the sign- & what is signified. (We might then say e.g. that “q” signifies- in the true way what “not-q” signifies in the false way). But- are not true & false in fact equally justified? Could we not express- ourselves by means of false propositions just as well as hitherto with- true ones, so long as we know that they are meant falsely?- Ts-201a1,b3 No! For a proposition is then true when it is as we assert in this- proposition; & accordingly if by “q” we mean “not-q”, & it is as we- mean to assert, then in the new interpretation “q” is actually true- & not false. But it is important that we can mean the same- by “q” as by “not-q”, for it shows that neither to the symbol- “not” nor to the manner of its combination with “q” does a- characteristic of the denotation of “q” correspond. Ts-201a1,b4

2nd MS.

We must be able to understand propositions which we have never heard- before. But every proposition is a new symbol. Hence we must have- general indefinable symbols; these are unavoidable if propositions are not- all indefinable.

Whatever corresponds in reality to compound propositions must not be- more than what corresponds to their several atomic propositions.

Not only must logic not deal with [particular] things, but just as- little with relations & predicates.

There are no propositions containing real variables.

What corresponds in reality to a proposition depends upon whether it is- true or false. But we must be able to understand a proposition without- knowing if it is true or false.

What we know when we understand a proposition is this: We know- what is the case if the proposition is true, & what is the case if it is false.- But we do not know [necessarily] whether it is true or false.

Propositions are not names.

We can never distinguish one logical type from another by- attributing a property to members of the one which we deny to members of- the other.

Symbols are not what they seem to be. In “aRb”, “R” looks- like a substantive, but is not one. What symbolizes in “aRb” is- that R occurs between a & b. Hence “R” is not the indefinable- in “aRb”. Similarly in “φx”, “φ” looks like a substantive- but is not one; in “~p”, “~” looks like “φ” but is not like it.- This is the first thing that indicates that there may not be logical constants.- A reason against them is the generality of logic: logic cannot treat a- special set of things. Ts-201a1,b5

Wittg.–      Molecular propositions contain nothing beyond what is contained in their- atoms; they add no material information above that contained in their- atoms.

All that is essential about molecular functions is their T-F schema- [i.e. the statement of the cases when they are true & the cases when they are false].

Alternative indefinability shows that the indefinables have not- been reached.

Every proposition is essentially true-false: to understand it, we- must know both what must be the case if it is true, & what must be- the case if it is false. Thus a proposition has two poles, corresponding- to the case of its truth & the case of its falsehood. We call this the- sense of a proposition.

In regard to notation, it is important to note that not every- feature of a symbol symbolizes. In two molecular functions which have- the same T-F schema, what symbolizes must be the same. In- “not-not-p”, “not-p” does not occur; for “not-not-p” is the same- as “p”, & therefore, if “not-p” occurred in “not-not-p”, it would- occur in “p”.

Logical indefinables cannot be predicates or relations, because- propositions, owing to sense, cannot have predicates or relations. Nor are- “not” & “or”, like judgment, analogous to predicates or relations,- because they do not introduce anything new.

Propositions are always complex even if they contain no names. Ts-201a1,b6

A proposition must be understood when all its indefinables are understood.- The indefinables in “aRb” are introduced as follows:      “a” is indefinable;      “b” is indefinable;       Whatever “x” & “y” may mean, “xRy” says something- indefinable about their meanings.

A complex symbol must never be introduced as a single indefinable.- (Thus e.g. no proposition is indefinable.) For if one of its parts occurs- also in another connection, it must there be re-introduced. And would- it then mean the same?

The ways by which we introduce our indefinables must permit us- to construct all propositions that have sense [[?] meaning] from these- indefinables alone. It is easy to introduce “all” & “some” in- a way that will make the construction of (say) “(x,y).xRy”- possible from “all” & “xRy” as introduced before. Ts-201a1,b7

Wittg.– 3rd MS.

An analogy for the theory of truth: Consider a black patch on- white paper; then we can describe the form of the patch by mentioning, for- each point of the surface, whether it is white or black. To the fact that a- point is black corresponds a positive fact, to the fact that a point is- white (not black) corresponds a negative fact. If I designate a- point of the surface (one of Frege's “truth-values”), this is as if I set up an- assumption to be decided upon. But in order to be able to say of a point- that it is black or that it is white, I must first know when a point is- to be called black & when it is to be called white. In order to be- able to say that “p” is true (or false), I must first have- determined under what circumstances I call a proposition true,- & thereby I determine the sense of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper that is- white & black, but to a proposition without sense nothing corresponds,- for it does not designate a thing (truth-value), whose- properties might be called “false” or “true”; the verb of a- proposition is not “is true” or “is false”, as Frege believes,- but what is true must already contain the verb.

The comparison of language & reality is like that of retinal image & visual- image: to the blind spot nothing in the visual image seems to correspond,- & thereby the boundaries of the blind spot determine the visual image – as- true negations of atomic propositions determine reality. Ts-201a1,b8

Wittg.–      Logical inferences can, it is true, be made in accordance with- Frege's or Russell's laws of deduction, but this cannot justify the inference;- & therefore they are not primitive propositions of logic. If p follows from- q, it can also be inferred from q, & the “manner of deduction”- is indifferent.

Those symbols which are called propositions in which “variables- occur” are in reality not propositions at all, but only schemes of- propositions, which only become propositions when we replace the- variables by constants. There is no proposition which is expressed- by “x = x”, for “x” has no signification; but there is a proposition- “(x).x = x” & propositions such as “Socrates = Socrates” etc..

In books on logic, no variables ought to occur, but only the general- propositions which justify the use of variables. It follows that the- so-called definitions of logic are not definitions, but only schemes of- definitions, & instead of these we ought to put general propositions;- & similarly the so-called primitive ideas (Urzeichen) of logic are not primitive- ideas, but the schemes of them. The mistaken idea that there are- things called facts or complexes & relations easily leads to the opinion- that there must be a relation of questioning to the facts, & then the question- arises whether a relation can hold between an arbitrary number of things,- since a fact can follow from arbitrary cases. It is a fact that the proposition- which e.g. expresses that q follows from p & p ⊃ q is this: p.p ⊃ q. ⊃ p.q.q. Ts-201a1,b9

Wittg.–      At a pinch, one is tempted to interpret “not-p” as- “everything else, only not p”. That from a single fact p an- infinity of others, not-not-p etc., follow, is hardly credible.- Man possesses an innate capacity for constructing symbols with- which some sense can be expressed, without having the- slightest idea what each word signifies. The best example of- this is mathematics, for man has until lately used the- symbols for numbers without knowing what they signify or- that they signify nothing.

Russell's “complexes” were to have the useful property of being compounded,- & were to combine with this the agreeable property that they could be- treated like “simples”. But this alone made them unserviceable as- logical types, since there would have been significance in asserting,- of a simple, that it was complex. But a property cannot be a- logical type.

Every statement about apparent complexes can be resolved into the- logical sum of a statement about the constituents & a statement- about the proposition which describes the complex completely. How,- in each case, the resolution is to be made, is an important- question, but its answer is not unconditionally necessary for the- construction of logic. Ts-201a1,b10

Wittg.–      That “or” & “not” etc. are not relations in the same sense- as “right” & “left” etc., is obvious to the plain man. The possibility- of cross-definitions in the old logical indefinables shows, of itself,- that these are not the right indefinables, &, even more conclusively,- that they do not denote relations.

If we change a constituent a of a proposition φ(a) into a- variable, then there is a class ^ p {(∃x).φ(x) = p}. This class in general still depends upon what, by an arbitrary convention,- we mean by “φ(x)”. But if we change into variables all- those symbols whose significance was arbitrarily determined, there is- still such a class. But this is now not dependent upon any convention,- but only upon the nature of the symbol “φ(x)”. It corresponds- to a logical type.

Types can never be distinguished from each other by saying (as is often- done) that one has these but the other has that properties, for this presupposes- that there is a meaning in asserting all these properties of both types. But- from this it follows that, at best, these properties may be types, but certainly- not the objects of which they are asserted. Ts-201a1,b11

Wittg.–      At a pinch, we are always inclined to explanations of logical functions of- propositions which aim at introducing into the function either only the constituents of these propositions,- or only their forms, etc. etc.; & we overlook that ordinary language- would not contain the whole propositions if it did not need them:- However, e.g., “not-p” may be explained, there must always be- a meaning given to the question “what is denied?”

The very possibility of Frege's explanations of “not-p” &- “if p then q”, from which it follows that not-not-p denotes- the same as p, makes it probable that there is some method of- designation in which “not-not-p” corresponds to the same- symbol as “p”. But if this method of designation suffices for- logic, it must be the right one.

Names are points, propositions arrows – they have sense. The- sense of a proposition is determined by the two poles true & false. - The form of a proposition is like a straight line, which divides all- points of a plane into right & left. The line does this automatically,- the form of proposition only by convention. Ts-201a1,b12

Wittg.–      Just as little as we are concerned, in logic, with the relation of a- name to its meaning, just so little are we concerned with the- relation of a proposition to reality, but we want to know the- meaning of names & the sense of propositions – as we- introduce an indefinable concept “A” by saying: “‘A’ denotes- something indefinable”, so we introduce e.g. the form of propositions- aRb by saying: “For all meanings of “x” & “y”, “xRy”- expresses something indefinable about x & y”.

In place of every proposition “p”, let us write “ a b p”. Let every- correlation of propositions to each other or of names to propositions- be effected by a correlation of their poles “a” & “b”. Let this- correlation be transitive. Then accordingly “ a ‒ a b ‒ b p” is the same- symbol as “ a b p”. Let n propositions be given. I then call a- “class of poles” of these propositions every class of n members, of which- each is a pole of one of the n propositions, so that one member corresponds- to each proposition. I then correlate with each class of poles one of two- poles (a & b). The sense of the symbolizing fact thus constructed I- cannot define, but I know it.

If p = not-not-p etc., this shows that the traditional method of- symbolism is wrong, since it allows a plurality of symbols with the same- sense; & thence it follows that, in analyzing such propositions, we- must not be guided by Russell's method of symbolizing. Ts-201a1,b13

Wittg.–      It is to be remembered that names are not things, but classes:- “A” is the same letter as “A“. This has the most important- consequences for every symbolic language.

Neither the sense nor the meaning of a proposition is a thing. These- words are incomplete symbols.

It is impossible to dispense with propositions in which the same- argument occurs in different positions. It is obviously useless to- replace φ(a,a) by φ(a,b).a = b.

Since the ab-functions of p are again bi-polar propositions, we- can form ab-functions of them, & so on. In this way a series of- propositions will arise, in which in general the symbolizing facts will- be the same in several members. If now we find an ab-function of such- a kind that by repeated application of it every ab-function can be- generated, then we can introduce the totality of ab-functions as the totality of- those that are generated by application of this function. Such a function is ~p ⌵ ~q. Ts-201a1,b14

Wittg.–      It is easy to suppose a contradiction in the fact that on the one hand- every possible complex proposition is a simple ab-function of simple- propositions, & that on the other hand the repeated application of one- ab-function suffices to generate all these propositions. If e.g. an- affirmation can be generated by double negation, is negation in any- sense contained in affirmation? Does “p” deny “not-p” or assert “p”,- or both? And how do matters stand with the definition of “ ⊃ ” by- “ ⌵ ” & “.”, or of “ ⌵ ” by “.” & “ ⊃ ”? And how e.g. shall- we introduce p|q (i.e. ~p ⌵ ~q), if not by saying that this- expression says something indefinable about all arguments p & q?- But the ab-functions must be introduced as follows: The function- p|q is merely a mechanical instrument for constructing all possible- symbols of ab-functions. The symbols arising by repeated application- of the symbol “|” do not contain the symbol “p|q”. We need a rule- according to which we can form all symbols of ab-functions, in order- to be able to speak of the class of them; & we now speak of them e.g. as- those symbols of functions which can be generated by repeated application- of the operation “|”. And we say now: For all p's & q's, “p|q” says- something indefinable about the sense of those simple propositions- which are contained in p & q. Ts-201a1,b15

Wittg.–      The assertion-sign is logically quite without significance. It only shows, in- Frege & Whitehead & Russell, that these authors hold the propositions so- indicated to be true. “⊢” therefore belongs as little to the proposition as- (say) the number of the proposition. A proposition cannot possibly assert- of itself that it is true.

Every right theory of judgment must make it impossible for me- to judge that this table penholders the book. Russell's theory does not- satisfy this requirement.

It is clear that we understand propositions without knowing whether- they are true or false. But we can only know the meaning of a proposition- when we know if it is true or false. What we understand is the sense- of the proposition.

The assumption of the existence of logical objects makes it- appear remarkable that in the sciences propositions of the form “p ⌵ q”, “ p ⊃ q”,- etc. are only then not provisional when “ ⌵ ” &” ⊃ ” stand within the- scope of a generality-sign [apparent variable]. Ts-201a1,b16

Wittg.– 4th MS.

If we formed all possible atomic propositions, the world would be completely- described if we declared the truth or falsehood of each. [I doubt this.]

The chief characteristic of my theory is that, in it, p has the same- meaning as not-p.

A false theory of relations makes it easily seem as if the relation- of fact & constituent were the same as that of fact & fact which follows- from it. But the similarity of the two may be expressed thus: φa. ⊃ .φ,a a = a.

If a word creates a world so that in it the principles of logic- are true, it thereby creates a world in which the whole of mathematics holds;- & similarly it could not create a world in which a proposition was- true, without creating its constituents.

Signs of the form “p ⌵ ~p” are senseless, but not the proposition- “(p).p ⌵ ~p”. If I know that this rose is either red or not red, I- know nothing. The same holds of all ab-functions.

To understand a proposition means to know what is the case- if it is true. Hence we can understand it without knowing if it is true.- We understand it when we understand its constituents & forms.- If we know the meaning of “a” & “b”, & if we know what “xRy”- means for all x's & y's, then we also understand “aRb”.

I understand the proposition “aRb” when I know that either the fact that- aRb or the fact that not aRb corresponds to it; but this is not to be- confused with the false opinion that I understand “aRb” when I know that- “aRb or not-aRb” is the case. Ts-201a1,b17

Wittg.–      But the form of a proposition symbolizes in the following way: Let us- consider symbols of the form “xRy”; to these correspond primarily pairs of- objects, of which one has the name “x”, the other the name “y”. The x's & y's- stand in various relations to each other, among others the relation R holds- between some, but not between others. I now determine the sense- of “xRy” by laying down: when the facts behave in regard to “xRy”- so that the meaning of “x” stands in the relation R to the meaning of “y”,- then I say that they [the facts] are “of like sense” [“gleichsinnig”] with- the proposition “xRy”; otherwise, “of opposite sense” [“entgegengesetzt”];- I correlate the facts to the symbol “xRy” by thus dividing them into- those of like sense & those of opposite sense. To this correlation corresponds- the correlation of name & meaning. Both are psychological. Thus I- understand the form “xRy” when I know that it discriminates the- behaviour of x & y according as these stand in the relation R or not.- In this way I extract from all possible relations the relation R,- as, by a name, I extract its meaning from among all possible things.

Strictly speaking, it is incorrect to say: We understand the proposition- p when we know that ‘“p” is true’ ≡ p; for this would naturally always- be the case if accidentally the propositions to right & left of the symbol “≡”- were both true or both false. We require not only an equivalence, but a- formal equivalence, which is bound up with the introduction of the- form of p.

The sense of an ab-function of p is a function of the sense of p. Ts-201a1,b18

Wittg.–      The ab-functions use the discrimination of facts, which their- arguments bring forth, in order to generate new discriminations.

Only facts can express sense, a class of names cannot.- This is easily shown.

There is no thing which is the form of a proposition, & no- name which is the name of a form. Accordingly we can also- not say that a relation which in certain cases holds between- things holds sometimes between forms & things. This goes- against Russell's theory of judgment.

It is very easy to forget that, though the propositions of a form- can be either true or false, each one of these propositions can only- be either true or false, not both.

Among the facts which make “p or q” true, there are some- which make “p & q” true; but the class which makes “p or q” true- is different from the class which makes “p & q” true; & only this is what- matters. For we introduce this class, as it were, when we introduce- ab-functions.

A very natural objection to the way in which I have introduced e.g. propositions- of the form xRy is that by it propositions such as (∃x,y).xRy & similar ones- are not explained, which yet obviously have in common with aRb what- cRd has in common with aRb. But when we introduced propositions of the form- xRy we mentioned no one particular proposition of this form; & we only need to- introduce (∃x,y).φ(x,y) for all φ's in any way which makes the sense of these- propositions dependent on the sense of all propositions of the form φ(a,b), & thereby the- justification of our procedure is proved. Ts-201a1,b19

Wittg.–      The indefinables of logic must be independent of each other. If an indefinable- is introduced, it must be introduced in all combinations in which it can occur.- We cannot therefore introduce it first for one combination, then for another;- e.g., if the form xRy has been introduced, it must henceforth be- understood in propositions of the form aRb just in the same way as in- propositions such as (∃x,y).xRy & others. We must not introduce it- first for one class of cases, then for the other; for it would remain doubtful if- its meaning was the same in both cases, & there would be no ground for- using the same manner of combining symbols in both cases. In short, for- the introduction of indefinable symbols & combinations of symbols the same holds,- mutatis mutandis, that Frege has said for the introduction of symbols- by definitions.

It is a priori likely that the introduction of atomic propositions is- fundamental for the understanding of all other kinds of propositions.- In fact the understanding of general propositions obviously depends on- that of atomic propositions.

Cross-definability in the realm of general propositions leads to the- quite similar questions to those in the realm of ab-functions. Ts-201a1,b20

Wittg.–      When we say “A believes p”, this sounds, it is true, as if here- we could substitute a proper name for “p”; but we can see that here- a sense, not a meaning, is concerned, if we say “A believes that- ‘p’ is true”; & in order to make the direction of p even more- explicit, we might say “A believes that ‘p’ is true & ‘not-p’ is false”.- Here the bi-polarity of p is expressed, & it seems that we shall only- be able to express the proposition “A believes p” correctly by the- ab-notation; say by making “A” have a relation to the poles “a”- & “b” of a-p-b. The epistemological- questions concerning the nature of judgment & belief cannot be solved- without a correct apprehension of the form of the proposition.

The ab-notation shows the dependence of or & not, & thereby- that they are not to be employed as simultaneous indefinables.

Not: “The complex sign ‘aRb’” says that a stands in the- relation R to b; but that ‘a’ stands in a certain relation to ‘b’- says that aRb.

In philosophy there are no deductions: it is purely descriptive.

Philosophy gives no pictures of reality.

Philosophy can neither confirm nor confute scientific investigation. Ts-201a1,b21

Wittg.–      Philosophy consists of logic & metaphysics: logic is its basis.

Epistemology is the philosophy of psychology.

Distrust of grammar is the first requisite for philosophizing.

Propositions can never be indefinables, for they are always complex.- That also words like “ambulo” are complex appears in the fact that- their root with a different termination gives a different sense.

Only the doctrine of general indefinables permits us to- understand the nature of functions. Neglect of this doctrine leads- to an impenetrable thicket.

Philosophy is the doctrine of the logical form of scientific- propositions (not only of primitive propositions).

The word “philosophy” ought always to designate- something over or under, but not beside, the natural sciences.

Judgment, command & question all stand on the same level; but- all have in common the propositional form, which does interest us.

The structure of the proposition must be recognized, the- rest comes of itself. But ordinary language conceals the- structure of the proposition: in it, relations look like predicates,- predicates like names, etc..

Facts cannot be named. Ts-201a1,b22

Wittg.–      It is easy to suppose that “individual”, “particular”, “complex”- etc. are primitive ideas of logic. Russell e.g. says “individual”- & “matrix” are “primitive ideas”. This error presumably- is to be explained by the fact that, by employment of variables- instead of the generality-sign, it comes to seem as if logic dealt- with things which have been deprived of all properties except- thing-hood, & with propositions deprived of all properties except- complexity. We forget that the indefinables of symbols [Urbilder- von Zeichen] only occur under the generality-sign, never outside it.

Just as people used to struggle to bring all propositions into the- subject-predicate form, so now it is natural to conceive every- proposition as expressing a relation, which is just as incorrect.- What is justified in this desire is fully satisfied by Russell's theory- of manufactured relations.

One of the most natural attempts at solution consists in- regarding “not-p” as “the opposite of p”, where then “opposite” would be- the indefinable relation. But it is easy to see that every such attempt- to replace the ab-functions by descriptions must fail. Ts-201a1,b23

Wittg.–      The false assumption that propositions are names leads us to- believe that there must be logical objects: for the meanings of logical- propositions will have to be such things.

A correct explanation of logical propositions must give them a unique- position as against all other propositions.

No proposition can say anything about itself, because the symbol- of the proposition cannot be contained in itself; this must be the- basis of the theory of logical types.

Every proposition which says something indefinable about a- thing is a subject-predicate proposition; every proposition which- says something indefinable about two things expresses a dual relation- between these things, & so on. Thus every proposition which contains- only one name & one indefinable form is a subject-predicate- proposition, & so on. An indefinable simple symbol can only be a- name, & therefore we can know, by the symbol of an atomic- proposition, whether it is a subject-predicate proposition.