Transcriber(s):Michael Biggs, Peter Cripps

Proofreader(s):Peter Cripps

Comments:Transcriptions of material from The Bertrand

Russell Archives, McMaster University, Canada;

their catalogue no.s:

TS section (pp. numbered 1-7) - RA1.710.057822,

MS section (pp. numbered 1-23 plus 3 unnumbered

folios) - RA1.710.057823.

This item consists of two parts: a typescript of

7ff. with corrections in Russell´s hand and

additions in Wittgenstein´s hand, and a manuscript

of 26ff. in Russell´s hand. In the TS part:

non-typewriter characters are inserted by hand;

fully typed pages have 24 or 25 lines of type;

each new sentence is separated from the last by

three spaces. True double spaced blank lines used

occasionally but not consistently.

Hands:s = handwritten insertions that belong to first

pass, ie generally the insertion of symbols which

are not found on the typewriter such as φ and &Exist;,

H5 = Ludwig Wittgenstein, or cases where the

transcriber has felt unable to decide whether

Ludwig Wittgenstein or Bertrand Russell

S1 = Bertrand Russell

Item 201a-1 Recto Page A1

1

‹

Summary›^{1}

~~The~~^{2}^{One}^{1}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 th[~~e~~|o^{3}]se 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.

~~The~~^{2}[~~d~~|D^{3}]eductions only proceed according

to the laws of deduction‹,›^{4}but these laws cannot

justify the deduction.

~~The~~^{2}^{One}^{1}reason for supposing that not all

propositions which have more than one argument are

relational propositions is that «^{if}»^{1}they

were‹,›^{1}the relations of judgement and inference

~~that~~^{2}^{would}^{1}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

~~those~~^{2}«^{its}»^{1}cons[~~i~~|t]ituents and about the

proposition which describes~~a~~^{2}^{the}^{1}complex

perfectly; i.e., that proposition which is

equivalent to saying~~a~~^{2}^{the}^{1}complex exists.

The idea that propositions are names of

complexes~~between~~^{5}L.W.^{4}^{ suggest}^{ions}^{5}‹s›^{4}<?>^{6}

L.W.^{4}^{ that whatever is not a proper name is a sign }for a relation. ‹Because spatial complexes*

consist of Things & Relations only & the idea of a

complex is taken from sp›^{4}

In a proposition convert all its indefinables

into variables; there then remains a class of

propositions which~~has~~^{2}^{is}^{1}not all propositions

but a type.

<*you^{7}- for instance imagine every fact as a

spatial complex>^{8}

Item 201a-1 Recto Page A22

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~~^{2}^{A}^{1}judge‹s›^{1}~~is~~^{2}that etc.,

then we have to mention a whole proposition which

~~a~~^{2}^{A}^{1}judge‹s›^{1}~~is~~^{2}. It will not do either to

mention only its constituent‹s,›^{1}or its

constituent‹s›^{1}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[~~.~~|,^{9}]‹p^{10}

~~must occur in it.~~›^{1}^{5}<[~~t~~|T^{9}]he question, "What is

negated" must have a meaning>^{11}

~~Always a que~~[~~x~~|~~s~~]~~tion that is negated must~~

~~have a meaning.~~^{5}<Rott!>^{12}

To understand a proposition p it is not

enough to know that~~"~~^{5}p implies ‹´"›^{4}p" is

true‹´›^{4}, but we must also know that~~p also~~

~~implies~~‹~~´~~›^{4}~~"not-p" is false~~‹~~´~~›^{4}^{5}^{˜p implies "p}

^{is false"}^{4}. This shows the «^{bi}»^{13}polarity of the

proposition.

<W-F = Wahr-Falsch>^{6}

To every molecular function «^{a}»^{1}[~~wf~~|WF^{3}]

scheme corresponds. Therefore we may use the

[~~wf~~|WF^{3}]scheme itself instead of the function.

Now what the[~~wf~~|WF^{3}]scheme does is, it

correlates the letters[~~w~~|W^{3}]and[~~f~~|F^{3}]with each

proposition. These two letters are the poles of

atomic propositions. Then «^{the scheme}»^{1}

corre~~sponds~~^{2}^{lates}^{1}another[~~f~~|W^{3}]and[~~w~~|F^{3}]to

these poles. In this notation all that~~it~~^{2}matters

is the correlation of the outside poles to the

pole of~~its~~^{2}«^{the}»^{1}atomic propositions. Therefore

not-not-p is the same symbol as p. ‹And›^{1}

Therefore we shall never get two symbols for the

same molecular function~~s~~^{5}.

Item 201a-1 Recto Page A3

3

The meaning of a proposition is the fact

which actually corresponds to it.

As the ab functions of atomic propositions

are b~~y~~^{2}^{i}^{1}‹-›^{1}polar propositions again we can

performab^{7}operations on them. We[~~wi~~|sha^{9}]ll‹,›^{4}

b[~~e~~|y^{9}]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^{7}*

a^{7}is on the left ofp^{7}andb^{7}on the right of

p^{7}[~~,~~|;^{9}]then the correlation of new poles is to

be transitive,~~such~~^{2}^{so}^{1}that «^{for instance}»^{4}if a

new polea^{7}in whatever way «^{i.e. via whatever}

^{poles}»^{4}is correlated to the insidea^{7}, the symbol

is not changed thereby. It is therefore possible

to construct all possibleab^{7}functions by

performing oneab^{7}operation repeatedly, and we

can therefore talk of allab^{7}functions as of all

th[~~e~~|o^{9}]se functions which can be obtained by

performing thisab^{7}operation repeatedly.

<[Note by B.R.]>^{6}

[NB.ab^{7}means the same as[~~wf~~|WF3], which

means true-false.]

Naming is like pointing. A function is like a

line dividing points «^{of a plane}»^{4}into right and

left ones; then ‹"›^{1}p or not-p‹"›^{1}has no meaning

because it does not divide[~~a~~|the^{9}]plane.

But though a particular proposition "p~~"~~^{5}or~~a~~

~~"~~^{5}not-p" has no meaning‹,›^{4}a general proposition

‹"›^{4}for all p´s,~~"~~^{5}p~~"~~^{5}or~~"~~^{5}not-p" has a meaning

because this does not contain[~~a~~|the^{9}]nonsensical

function ‹"›^{4}p[~~n~~|o]r not-p‹"›^{4}but[~~a~~|the^{9}]

function ‹"›^{4}p or~~"~~^{5}not-q" just as ‹"›^{4}for all

~~"~~^{5}x´s xRx‹"›^{1}contains the function ‹"›^{4}xRy".

<* 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 saysnothing^{7}. (It

does not say ˜p) But a-apb-b is the same symbol as

apb>^{8}

<the ab function vanishes automatically) for here

the new poles are>^{14}<related to the same side of

p as the old ones. The question is allways: how

are the new poles correlated to p compared with

the way the old poles are correlated to p.>^{15}

Item 201a-1 Recto Page A4

4

A proposition is a standard to which all

facts behave,~~that~~^{2}^{with}^{1}names it «^{is}»^{1}

otherwise; it is th~~en~~^{2}^{us}^{1}b~~y~~^{2}^{i}^{1}‹-›^{1}polarity and

sense comes in‹;›^{1}just as one~~error~~^{2}^{arrow}^{1}

behaves to another~~error~~^{2}^{arrow}^{1}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 ‹"›^{4}xRy‹"›^{4}. To

symbols of this form correspond couples of things

whose names are respectively ‹"›^{4}x‹"›^{4}and

‹"›^{4}y‹"›^{4}. The thingsx^{7}‹/›^{4}y^{7}stand to one

another in all sorts of relations‹,›^{4}amongst

others some stand in the relatio[~~j~~|n]~~of~~^{2}^{R}^{1}, and

some not; just as I single out a particular thing

by a particular name I single out all behaviours

of the point‹s›^{1}x and y~~the one between~~^{2}^{with}

^{respect to}^{1}the relation ‹R.›^{1}~~of the other~~^{2}. I

say that if an x stands in the relation~~of~~^{2}^{R}^{1}to

a y the sign ‹"›^{4}x~~of~~^{2}^{R}^{1}y‹"›^{4}is to be called

true to the fact and otherwise false. This is a

definition of sense.

<!>^{16}

In my theory p has the same meaning as not-p

but opposite snese. 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[~~.~~|,^{9}]i.e.,

[~~A~~|a^{9}]ll the general indefinables involved.The

sense ofanab^{7}function of a proposition is a

function of its sense[~~:~~|.^{9}][~~t~~|T^{9}]here are only

unasserted propositio‹ns.›^{4}

Item 201a-1 Recto Page A5

5

Assertion is merely psychological. I~~f~~^{2}^{n}^{1}not-p‹,›^{1}

«p^{10}»^{1}is exactly the same as if it stands

alone‹;›^{1}this point is absolutely fundamental.

Among the facts which make "p or q"^{true}^{1}there

are also facts which make "p and q" true‹;›^{1}if

propositions~~do only mean~~^{2}«^{have only}

^{meaning}»^{1}‹,›^{1}we ought‹,›^{1}~~to know~~^{2}«^{in}»^{1}such a

case,^{to}^{1}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

thereab^{7}function stands under a generality sign

or enters into another function such as ‹"›^{4}I

believe that, etc.,‹"›^{4}because^{then}^{4}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[~~.~~|,^{9}][~~T~~|t^{9}]he proposition p with its 2

poles‹,›^{4}anda^{7}being related to both of these

poles in a certain way. This is obviously not~~the~~^{5}

‹a›^{4}relation in the ordinary sense.

The ab notation~~and~~^{5}~~for~~^{4}~~apparent variables~~^{5}

make‹s›^{4}it clear thatnotandorare dependent on

one another and we can therefore not use them as

simultaneous indefinables. ‹|^{17}›^{4}~~Some~~^{5}~~‹~~~~Same~~›^{4}

~~objections~~«~~in the case of app. var.~~»^{4}~~to old~~

~~indefinables,~~[~~a~~|~~A~~^{9}]~~s~~^{5}~~as~~^{4}^{18}in the case of

molecular functions^{19}[~~,~~|:^{9}][~~t~~|T^{9}]he application

of the ab notation to apparent~~ly~~^{5}‹-›^{4}variable

propositions become‹s›^{4}clear if we consider that,

for instance, the proposition ‹"›^{4}for all~~"~~^{5}x‹,›^{4}

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

someandalloccur simultaneously in the proper

apparent variable notation.

Item 201a-1 Recto Page A6

6

The Notation is:

for (x) φx ; a - (x) - a φ x b - (&Exist; x) - b

and

for (~~φ~~^{5}^{&Exist;}x) φ x : a - (&Exist;x) - a φ x b - (x) -[~~v~~|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~~the~~^{5}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

~~important~~^{5}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.

Item 201a-1 Recto Page A7

7

What is essential in a correct

apparent«^{-variable}»^{1}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 thekind^{7}

ofsymbols^{20}«^{, for which }φ! stands»^{4}‹&›^{4}which,

by the above, is enough to determine the type,

then automatically "([x|φ]).φ!x" cannot be

fi[~~ll~~|tt]ed by this descri[~~l~~|p]tion[~~.~~|,^{9}]‹because

it contains^{7}"φ!x" & the description is to

describe all^{7}that symbolizes in symbols of the φ!

- kind. If the description isthus^{7}complete

vicious circles can just as little occur as~~if~~^{5}

for instance in~~(φ).φ(x)~~^{5}(φ).(x)φ^{4}(where (x)φ is

a subject-predicat prop) ›^{4}

Item 201a-1 Recto Page B1

Wittgenstein1

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~~denote~~

^{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~~

objects, & «^{that accordingly}» "(&Exist;x,φ).φx" or

"(&Exist;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.)

"˜(&Exist;x,y).xRy"? Can we put "not" before a name?

Item 201a-1 Recto Page B2

Wittg.-2

The reason why "˜Socrates" means nothing is

that "˜x" does not express a property ofx.

There are positive & negative facts: if the

proposition "this rose is not red" is true, then

~~its~~«^{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 ofgeneralpropositions,

i.e. of such as contain apparent variables.~~)~~

Negative facts only justify the negations of

~~simpl~~atomic propositions.)

Positive&negativefacts there are, but

nottrue&falsefacts.

If we overlook the fact that propositions

have asensewhich 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"signifiesin the true way what

"not-q"signifiesin 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?

Item 201a-1 Recto Page B3

__3__

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 ¬false. But it is important

that wecanmean 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.

Item 201a-1 Recto Page B4

__4__

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

generalindefinable 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

betweena&b. Hence "R" isnotthe 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 theremaynot be

logical constants. A reason against them is the

generality of logic: logic cannot treat a

special set of things.

Item 201a-1 Recto Page B5

Wittg.-5

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 twopoles, corresponding to the case of its

truth & the case of its falsehood. We call this

thesenseof 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,analogousto

predicates or relations, because they do not

introduce anything new.

Propositions are always complex even if

they contain no names.

Item 201a-1 Recto Page B6

__6__

A proposition must be understood whenall

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^{21}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 indefinablesalone. 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.

Item 201a-1 Recto Page B7

Wittg.-7

3rd MS.

~~A comparis~~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 thesenseof a

proposition. The point[~~on~~|in]which the analogy

~~depends~~^{fails}is this: I can indicate a point of

the paper what 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.

Item 201a-1 Recto Page B8

Wittg.-8

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. Ifpfollows fromq, it

can also be inferred fromq, & 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 thatq

follows fromp& p⊂q is this: p.p⊂q.⊂_{p.q}.q.

Item 201a-1 Recto Page B9

Wittg.-9

At a pinch, one is tempted to interpret

"not-p" as "everything else, only notp". That

from a single factpan infinity of others,

not-not-p etc., follow, is hardly credible. Man

possesses an innate capacity for constructing

symbols with whichsomesense 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~~as~~«^{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 apropertycannot 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.

Item 201a-1 Recto Page B10

Wittg.-10

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 constituentaof a

proposition φ(a) into a variable, then there is

a class

˜p {(&Exist;x).φ(x) = p}.

This class in general still depends upon what,

by anarbitrary convention, we~~have~~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

th[~~i~~|e]s‹e›^{@ }butthe other has th[~~at~~|ose]

propert[~~y~~|i]‹es›, for this presupposes that

there is ameaningin 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.

Item 201a-1 Recto Page B11

Wittg.-11

At a pinch, we are always inclined to

explanations of «^{logical}» functions of

propositions which «^{aim at introducing into the}

^{function}» either only~~contain~~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" & "ifpthenq", from

which it follows that~~"~~not-not-p~~"~~denotes the

same asp, 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,~~sentences~~^{propositions}

arrows - they havesense. The sense of a

proposition is determined by the two polestrue

&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.

Item 201a-1 Recto Page B12

Wittg.-12

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

propositionsaRbby saying: "For all meanings of

"x" & "y", "xRy" expresses something indefinable

aboutx&y".

In place of every proposition "p", let us

write~~@~~"abp". 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-ab-bp" is the

same symbol as "abp". Let^{ }npropositions be

given. I then call a "class of poles" of these

propositions every class ofnmembers, of which

each is a pole of one of thenpropositions, 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.

Item 201a-1 Recto Page B13

Wittg.-13

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 theab-functions ofpare again

bi-polar propositions, we can formab-functions

of them, & so on. In this way a series of

propositions will arise, in which in general the

symbolizingfacts will be the same in several

members. If now we find anab-function of such a

kind that by repeated application of it every

ab-function can be generated, then we can~~define~~

^{introduce}the totality of ab-functions as the

totality of those that are generated by

application of this function. Such a function is

˜pv˜q.

Item 201a-1 Recto Page B14

Wittg.-14

It is easy to suppose a contradiction in

the fact that on the one hand~~all~~^{every}possible

complex proposition is a simpleab-function of

simple propositions, & that on the other hand

the repeated application of oneab-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 "v" &~~"˜"~~".", or of "v" by

"." & "⊂"? And how e.g. shall we introduce p|q

(i.e. ˜pv˜q), if not by saying that this

expression says something indefinable about all

argumentsp&q? But theab-functions must be

introduced as follows: The function p|q is

merely a mechanical instrument for constructing

all possiblesymbolsofab-functions. The

symbols arising by repeated application of the

symbol "|" donotcontain the symbol "p|q". We

need a rule according to which we can form all

symbols ofab-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 allp´s &

q´s, "p|q" says something indefinable about the

sense of those simple propositions which are

contained inp&q.

Item 201a-1 Recto Page B15

Wittg.-15

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 themeaningof a

proposition when we know if it is true or false.

What we understand is thesenseof the

proposition.

The assumption of the existence of logical

objects makes it appear remarkable that «^{in the}

^{sciences}» propositions of the form "p[~~or~~|v]q",

"p⊂q", etc. «^{are}» only then not provisional when

"v" & "⊂" stand within the scope of a

generality-sign [apparent variable].

Item 201a-1 Recto Page B16

Wittg.-16

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,phas the samemeaningas 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 "pv˜p" are senseless, but

not the proposition "(p).pv˜p". If I know that

this rose is either red or not red, I know

nothing. The same holds of allab-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.

Item 201a-1 Recto Page B17

Wittg.-17

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~~know~~^{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"bythus 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 ofx&yaccording

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 propositionpwhen 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 ofp.

The sense of an ab-function ofpis a

function of the sense ofp.

Item 201a-1 Recto Page B18

Wittg.-18

Theab-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, tho´ 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

(&Exist;x,y).xRy & similar ones are not explained,

which yet obviously have in common with aRb what

cRd has in common with aRb.Butwhen we

introduced propositions of the form xRy we

mentioned no one particular proposition of this

form; & we only need to introduce (&Exist;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.

Item 201a-1 Recto Page B19

Wittg.-19

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

(&Exist;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 &

~~classes~~^{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 ofab-functions.

Item 201a-1 Recto Page B20

Wittg.-20

When we say "A believesp", this sounds, it

is true, as if here we could substitute a proper

name for "p"; but we can see that here asense,

not a meaning, is concerned, if we say "A

believes that ´p´ is true"; & in order to make

the direction ofpeven more explicit, we might

say "A believes that ´p´ is true & ´not-p´ is

false". Here the bi-polarity ofpis expressed,

& it seems that we shall only be able to express

the proposition "A believesp" 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.

Theab-notation shows the dependence ofor

¬, & thereby that they are not to be

employed as simultaneous indefinables.

Not: "The complex sign ´aRb´" says thata

stands in the relation R tob; butthat´a´

stands in a certain relation to ´b´ saysthat

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.

Item 201a-1 Recto Page B21

Wittg.-21

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[~~&~~|a]ll

stand on the same level; but all have in common

the propositional form, which does interestsus.

The~~construction~~^{structure}of the~~sentence~~

^{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 benamed.

Item 201a-1 Recto Page B22

Wittg.-22

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~~s~~, 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 ofp", 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.

Item 201a-1 Recto Page B23

Wittg.-23

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~~sign~~^{symbol}can only be a

name, & therefore we can know, by the symbol of

an atomic proposition, whether it is a

subject-predicate proposition.

Item 201a-1 Recto Page B24

Wittg.-

I. Bi-polarity of propositions: sense & meaning,

truth & falsehood.

II. Analysis of atomic propositions: general

indefinables, predicates, etc.

III. Analysis of molecular~~fu~~propositions:

ab-functions.

IV. Analysis of general propositions^{22}

[~~IV~~|V]. Principles of symbolism: what symbolizes

in a symbol. Facts for facts.

V‹I›. Types

Item 201a-1 Recto Page B25

This is the symbol for

˜pv˜q

Item 201a-1 Recto Page B24