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.: RA1.710.057824.

This is a typescript of two parts (8ff.+25ff.).

The first part is a copy of The Bertrand Russell

Archives item no. RA1.710.057822 incorporating

the handwritten corrections. The second part is

a copy of The Bertrand Russell Archives item no.

RA1.710.057823. The whole has handwritten Roman

numerals in the left margin against each

paragraph indicating the sections of Russell´s

subsequent rearrangement of Wittgenstein´s text

(the so-called Costello Version, von Wright´s

catalogue no. 201b). 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

and

S1 = Bertrand Russell

Item 201a-2 Recto Page TP

~~SUMMARY~~^{2}

‹Notes on Logic

by

Ludwig Wittgenstein

September 1913.›^{1}

Item 201a-2 Recto Page A1

__1__

SUMMARY.

‹III› 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.

‹IV› 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?

‹I› 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.

‹I› 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 judgment and inference would have to

hold between an arbitrary number of things.

‹II› 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.

Item 201a-2 Recto Page A2

2

‹I› The idea that propositions are names of

complexes suggests that whatever is not a proper

name is a sign for a relation. Because spatial

complexes* * consist~~s~~of Things and Relations only

and the idea of a complex is taken from space.

‹VI› In a proposition convert all its indefinables

into variables; there then remains a class of

propositions which is not all propositions but a

type.

‹VI› 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 a

subject-predicate form etc. Therefore, thing,

proposition, subject-predicate form, etc., are not

indefinables, i.e., types are not indefinables.

‹I› 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.

-----------

<* Russell - for instance imagines every fact as a

spatial complex.>^{23}

Item 201a-2 Recto Page A3

3

‹I› 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 bi-polarity of the proposition.

‹III› 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 poles 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.

‹I› The meaning of a proposition is the fact

which actually corresponds to it.

‹III› As the ab functions of atomic propositions

are bi-polar propositions again we can performab

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.

----------

<* W-F = Wahr-Falsch.>^{23}

Item 201a-2 Recto Page A4

4

‹III› The symbolising fact in a-p-b is that, say*a

is on the left ofpandbon the right ofp; then

the correlation of new poles is to be transitive,

so that for instance if a new poleain whatever

way i.e. via whatever poles is correlated to the

insidea, the symbol is not changed thereby. It is

therefore possible to construct all possibleab

functions by performing oneaboperation

repeatedly, and we can therefore talk of allab

functions as of all those functions which can be

obtained by performing thisaboperation

repeatedly.

[Note by B.R.abmeans the same as WF, which

means true-false.]

‹III› 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.

‹III› 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".

----------

<* This is quite arbitrary but, if we once have

fixed on which order the poles have to stand we

must of course stick to our convention. If for

instance "a p b" says p then b p a saysnothing.

(It does not say ˜p) But a - a p b - 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.>^{23}

Item 201a-2 Recto Page A5

5

‹I› 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.

‹II› 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 thingsxy

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 "xRy" 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.

‹I› 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 ofanab

function of a proposition is a function of its

sense. There are only unasserted propositions.

Item 201a-2 Recto Page A6

6

‹I› Assertion is merely psychological. Innot-p,pis

exactly the same as if it stands alone; this point

is absolutely fundamental. Among the facts that

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 theirabfunction stands under a generality

sign or enters into another function such as "I

believe that, etc"., because then the sense

enters.

‹I› 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, anda

being related to both of these poles in a certain

way. This is obviously not a relation in the

ordinary sense.

‹III›‹V› Theabnotation makes it clear thatnotand

orare dependent on one another and we can

therefore not use them as simultaneous

indefinables. Same objections in the case of

apparent variables to^{the usual}^{1}old indefinables,

as in the case of molecular functions. The

application of theabnotation to apparent

variable propositions becomes clear if we consider

that, for instance, the proposition "for all x,

x" is to be true when x is true for all x´s and

false when x is false for some x´s. We see that

someandalloccur simultaneously in the proper

apparent variable notation.

Item 201a-2 Recto Page A7

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‹IV› The notation is:

for (x) x : a - (x) - a x b - ( x) - b

and

for (x) x : a - (x) - a x b - (x) - b

Old definitions now become tautologous.

‹V› 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.

‹I› Judgment, question and command are all on the

same level. What interests logic in them is only

the unasserted proposition. Facts cannot be named.

‹VI› 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.

‹VI› Therefore we can recognise 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-2 Recto Page A8

8

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.

‹VI› [Components are forms and constituents.]

Take (). !x. Then if we describe thekind

of symbols, for which ! stands and which, by the

above, is enough to determine the type, then

automatically "(). ! x" cannot be fitted by this

description, because itCONTAINS"!x" and the

description is to describeALLthat symbolises in

symbols of the ! kind. If the description isthus

complete vicious circles can just as little occur

as for instance (). (X) (where (X) is a

subject-predicate proposition).

Item 201a-2 Recto Page B1

1

First MS.

‹II› Indefinables are of two sorts: names, and

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.

‹V› 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, and 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.

‹I› Frege said "propositions are names"; Russell

said "propositions correspond to complexes". Both

are false; and especially false is the statement

"propositions are names of complexes."

‹IV› It is easy to suppose that only such symbols

are complex as contain names of objects, and that

accordingly " (x,). x" or "(x,y). x R y" 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). x R y"? Can we put "not" before a

name?

Item 201a-2 Recto Page B2

2

‹III› The reason why "˜Socrates" means nothing is

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

‹I› There are positive and 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 and 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 atomic propositions.)

Positiveandnegativefacts there are, but

nottrueandfalsefacts.

‹I› If we overlook the fact that propositions

have asensewhich is independent of their truth

or falsehood, it easily seems as if true and false

were two equally justified relations between the

sign and what is signified. (We might then say

e.g. that "q"^{24}signifiesin the true way what

"not-q"signifiesin the false way). But are not

true and 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? No! For a proposition is then^{25}true when

Item 201a-2 Recto Page B3

3

it is as we assert in this proposition; and

accordingly if by "q" we mean "not-q", and it is

as we mean to assert, then in the new

interpretation "q" is actually true andnotfalse.

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-2 Recto Page B4

4

Second MS.

‹II› 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.

‹III› 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 and predicates.

‹IV› There are no propositions containing real

variables.

‹I› 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, and what is the case if it is

false. But we do not know [necessarily] whether it

is true or false.

Propositions are not names.

‹VI› 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.

Item 201a-2 Recto Page B5

5

‹II› Symbols are not what they seem to be. In "a R

b", "R" looks like a substantive, but is not one.

What symbolizes in "aRb" is that R occurs

betweenaandb. Hence "R" isnotthe indefinable

in "aRb". Similarly in "x", "" looks like a

substantive but is not one; in "˜p", "˜" looks

like "" 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.

‹III› 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 and the cases when

they are false].

‹V› Alternative indefinability shows that the

indefinables have not been reached.

‹I› Every proposition is essentially true-false:

to understand it, we must know both what must be

the case if it is true, and what must be the case

if it is false. Thus a proposition has twopoles,

corresponding to the case of its truth and the

case of its falsehood. We call this thesenseof a

proposition.

Item 201a-2 Recto Page B6

6

‹V› 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", and therefore, if

"not-p" occurred in "not-not-p", it would occur in

"p".

‹III› Logical indefinables cannot be predicates or

relations, because propositions, owing to sense,

cannot have predicates or relations. Nor are "not"

and "or", like judgment,analogousto predicates

or relations, because they do not introduce

anything new.

‹IV› Propositions are always complex even if they

contain no names.

‹II› A proposition must be understood whenallits

indefinables are understood. The indefinables in

"a R b" are introduced as follows:

"a" is indefinable;

"b" is indefinable;

Whatever "x" and "y" may mean, "x R y" says

something indefinable about their meaning.

‹V› 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?

Item 201a-2 Recto Page B7

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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" and "some" in a way that will make the

construction of (say) "(x, y). x R y" possible

from "all" and "x R y"as introduced before.

Item 201a-2 Recto Page B8

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3rd. MS.

‹I› 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 and 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, and

thereby I determine thesenseof a proposition.

The point in which the analogy fails is this: I

can indicate a point of the paper what is white^{26}

and 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.

Item 201a-2 Recto Page B9

9

‹I› The comparison of language and reality is

like that of retinal image and visual image: to

the blind spot nothing in the visual image seems

to correspond, and thereby the boundaries of the

blind spot determine the visual image - as true

negations of atomic propositions determine

reality.

‹III› 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;

and therefore they are not primitive propositions

of logic. Ifpfollows fromq, it can also be

inferred fromq, and the "manner of deduction" is

indifferent.

‹IV› 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" and propositions such as "Socrates =

Socrates" etc.

‹IV› 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, and

instead of these we ought to put general

propositions; and similarly the so-called

primitive ideas (Urzeichen) of logic are not

primitive ideas, but the schemes of them. The

mistaken idea that there

Item 201a-2 Recto Page B10

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are things called facts or complexes and relations

easily leads to the opinion that there must be a

relation ofquestioning^{10}<(?)>^{27}to the facts, and

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

thatqfollows frompand p q is this: p. p q.

_{p.q}.q.

‹I› 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.

‹II› Russell´s "complexes" were to have the useful

property of being compounded, and 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 apropertycannot

be a logical type.

Item 201a-2 Recto Page B11

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‹II› Every statement about apparent complexes can

be resolved into the logical sum of a statement

about the constituents and 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.

‹III› That "or" and "not" etc. are not relations in

the same sense as "right" and "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, and, even more conclusively, that

they do not denote relations.

‹VI› If we change a constituentaof a proposition

(a) into a variable, then there is a class

˜p {(x). (x) = p }.

This class in general still depends upon what, by

anarbitraryconvention, we mean by "(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 "(x)". It corresponds to a logical

type.

Item 201a-2 Recto Page B12

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‹VI› Types can never be distinguished from each

other by saying (as is often done) that one has

thesebutthe other has those properties, 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.

‹I› 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 form, etc. etc.; and 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?"

‹III› The very possibility of Frege´s explanations

of "not-p" and "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.

‹I› Names are points, propositions arrows - they

havesense. The sense of a proposition is

determined by the two polestrueandfalse. The

form of a proposition is like a straight line,

which divides all points of a plane into right and

left. The line does this autom«^{at}»ically, the form

of proposition only by convention.

Item 201a-2 Recto Page B13

13

‹II› 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 and 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" and "y", "x R y"

expresses something indefinable about x and y".

‹III› 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" and

"b". Let this correlation be transitive. Then

accordingly "^{a-a}_{b-b}p" is the same symbol as "^{a}_{b}p".

Letnpropositions be given. I then call a "class

of poles" of these propositions every class ofn

members, of which each is a pole of one of then

propositions, so that one member corresponds to

each proposition. I then correlate with each class

of poles one of two poles (aandb). The sense of

the symbolizing fact thus constructed I cannot

define, but I know it.

‹III› 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;

and thence it follows that, in analyzing such

propositions, we must not be guided by Russell´s

method of symbolizing.

Item 201a-2 Recto Page B14

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‹V› 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.

‹I› Neither the sense nor the meaning of a

proposition is a thing. These words are incomplete

symbols.

‹V› It is impossible to dispense with

propositions in which the same argument occurs in

different positions. It is obviously useless to

replace (a, a) by (a, b). a = b.

‹III› Since theab-functions ofpare again

bi-polar propositions, we can formab-functions of

them, and 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

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.

‹III› It is easy to suppose a contradiction in the

fact that on the one hand every possible complex

proposition is a simpleab-function of simple

propositions, and 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

Item 201a-2 Recto Page B15

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matters stand with the definition of "" by "v"

and ".", or of "v" by "." and ""? 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 argumentspandq? But the

ab-functions must be introduced as follows: The

function p[~~"~~||]q is merely a mechanical instrument

for constructing all possiblesymbolsof

ab-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; and 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 and

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

sense of those simple propositions which are

contained in p and q.

‹I› The assertion-sign is logically quite without

significance. It only shows, in Frege and

Whitehead and 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.

Item 201a-2 Recto Page B16

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‹I› 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.

‹III› The assumption of the existence of logical

objects makes it appear remarkable that in the

sciences propositions of the form "pvq", "p

q", etc. are only then not provisional when "v"

and "" stand within the scope of a

generality-sign [apparent variable].

Item 201a-2 Recto Page B17

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4th. MS.

‹II› 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.]

‹I› The chief characteristic of my theory is

that, in it,phas the samemeaningas not-p.

‹II› A false theory of relations makes it easily

seem as if the relation of fact and constituent

were the same as that of fact and fact which

follows from it. But the similarity of the two may

be expressed thus: a. ._{,a}a = a.

‹II› 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; and

similarly it could not create a world in which a

proposition was true, without creating its

constituents.

‹III› 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.

‹I› 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

and forms. If we know the meaning of "a" and "b",

and if we know what "x R y" means for all x´s and

y´s, then we also understand "a R b".

Item 201a-2 Recto Page B18

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‹I› I understand the proposition "a R b" when I

know that either the fact that a R b or the fact

that not a R b corresponds to it; but this is not

to be confused with the false opinion that I

understood "a R b" when I know that "a R b or not

a R b" is the case.

‹II› But the form of a proposition symbolizes in

the following way: Let us consider symbols of the

form "x R y"; to these correspond primarily pairs

of objects, of which one has the name "x", the

other the name "y". The x´s and 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 "x R y" by

laying down: when the facts behave in regard to "x

R y" 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 "x R y";

otherwise, "of opposite sense" [entgegengesetzt"];

I correlate the facts to the symbol "x R y"by

thus dividing them into those of like sense and

those of opposite sense. To this correlation

corresponds the correlation of name and meaning.

Both are psychological. Thus I understand the form

"x R y" when I know that it discriminates the

behaviour of x and 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.

Item 201a-2 Recto Page B19

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‹I› 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 and 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.

‹III› The sense of anab-function ofpis a

function of the sense ofp.

‹III› Theab-functions use the discrimination of

facts, which their arguments bring forth, in order

to generate new discriminations.

‹V› Only facts can express sense, a class of

names cannot. This is easily shown.

‹II› There is no thing which is the form of a

proposition, and 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 and things.

This goes against Russell´s theory of judgment.

^{28}[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.]^{28}

Item 201a-2 Recto Page B20

20

‹III› Among the facts which make "p or q" true,

there are some which make "p and q" true; but the

class which makes "p or q" true is different from

the class which makes "p and q" true; and only

this is what matters. For we introduce this class,

as it were, when we introduceab-functions.

‹IV› A very natural objection to the way in which

I have introduced e.g. propositions of the form x

R y is that by it propositions such as (. x. y).

x R y and similar ones are not explained, which

yet obviously have in common with a R b what c R d

has in common with a R b.Butwhen we introduce

propositions of the form x R y we mentioned no one

particular proposition of this form; and we only

need to introduce ( x, y). (x, y) for all ´s in

any way which makes the sense of these

propositions dependent on the sense of all

propositions of the form (a, b), and thereby the

just~~ification~~^{2}^{ness}^{1}of our procedure is proved.

‹V› 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 x R y has been introduced, it must

henceforth be understood in propositions of the

form a R b just in the same way as in propositions

such as (x, y). x R y and 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

Item 201a-2 Recto Page B21

21

both cases, and there would be no ground for using

the same manner of combining symbols in both

cases. In short, for the introduction of

indefinable symbols and combinations of symbols

the same holds, mutatis mutandis, that Frege has

said for the introduction of symbols by

definitions.

‹III› 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.

‹IV› Cross-definability in the realm of general

propositions leads to~~the~~^{2}quite similar questions

to those in the realm ofab-functions.

‹I› 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"; and in order to make the

direction ofpeven more explicit, we might say "A

believes that ´p´ is true and ´not-p´ is false".

Here the bi-polarity ofpis expressed and 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" and "b" of a-p-b.

The epistemological questions concerning the

nature of judgment and belief cannot be solved

without a correct apprehension of the four^{10}^{rm}^{1}of

the proposition.

Item 201a-2 Recto Page B22

22

‹III› Theab-notation shows the dependence ofor

andnot, and thereby that they are not to be

employed as simultaneous indefinables.

‹V›Not: "The complex sign ´a R b´" says thata

stands in the relation R tob; butthat´a´ stands

in a certain relation to ´b´ saysthata R b.

(^{29}[<Preliminary>^{6}In philosophy there are no

( deductions:itis purely descriptive.

( Philosophy gives no pictures of reality.

( Philosophy can neither confirm nor confute

( scientific investigation.

( Philosophy consists of logic and metaphysics:

( logic is its basis.

( Epistemology is the philosophy of psychology.

( Distrust of grammar is the first requisite

( for philosophizing.]^{29}

^{28}[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.]^{28}

‹II› Only the doctrine of general indefinables

permits us to understand the nature of functions.

Neglect of this doctrine leads to an impenetrable

thicket.

Item 201a-2 Recto Page B23

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(^{29}[<Preliminary>^{6}

( 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.]^{29}

‹I› Judgment, command and question all stand on

the same level; but all have in common the

propositional form, which does interestus.

‹I› 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 benamed.

‹VI› It is easy to suppose that "individual",

"particular", "complex" etc. are primitive ideas

of logic. Russell e.g. says "individual" and

"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, and 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.

Item 201a-2 Recto Page B24

24

‹IV› 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.

‹I› 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 theab-functions by

descriptions must fail.

‹I› 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.

<Preliminary>^{6}

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.

‹VI› 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, and so on.

Thus every proposition which contains only one

name and one indefinable form is a

subject-predicate proposition, and so on. An

Item 201a-2 Recto Page B25

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indefinable simple symbol can only be a name, and

therefore we can know, by the symbol of an atomic

proposition, whether it is a subject-predicate

proposition.

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