Transcriber(s): Michael Biggs, Peter Cripps
Proofreader(s): Peter Cripps
Comments:
Transcriptions of material from The Wittgenstein
Archives at the University of Bergen. Item
referred to by Michael Biggs as TSx (related to
201a). This is a typescript of 15ff. The first 2
folios are unnumbered, the foliation then runs
from 2 to 14. Some logical notation symbols added
by hand, others omitted. References to the
published text of Tractatus Logico-Philosophicus
are added by hand. Fully typed pages have between
61 and 65 lines of type; each new sentence is
separated from the last by three spaces. Blank
lines between paragraphs.
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 
S2 = handwritten insertions presumably made D.
Schwayder.
Item 201a-3 Recto Page TP
Notes on Logic
by
Ludwig Wittgenstein
September 1913.
Item 201a-3 Recto Page 1
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 judgment 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 suggests that whatever is not a proper
name is a sign for a relation. Because spatial
complexes* consist of Things and Relations only
and 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.
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.
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 bi-polarity of the proposition.
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.
--------
<* Russell - for instance imagines every fact as a
spatial complex.
* W-F = Wahr-Falsch.>23
Item 201a-3 Recto Page 2
- 2 -
‹×› 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.
(Note by B.R. 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".
‹×› A proposition is a standard to which 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 "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.
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.
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 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
--------
<* 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 says nothing.
(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-3 Recto Page 3
- 3 -
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 their 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 the usual 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,
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
some and all occur simultaneously in the proper
apparent variable notation. 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.
¦ 30[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.]30
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 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 recognised by its symbol
alone.
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.)
Tale (
). []!x. Then if we describe the
kind 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 it CONTAINS "
! x" and
the description is to describe ALL that symbolises
in symbols of the
! kind. If the description is
thus complete vicious circles can just as little
occur as for instance (
). (X) (where (X) is
a subject-predicate proposition).
Item 201a-3 Recto Page 4
- 4 -
First MS
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.
¦ 30[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.]30
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.
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."
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?
The reason why "˜Socrates" means nothing is
that "˜x" does not express a property of x.
‹×› 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 of general propositions, i.e. of such as
contain apparent variables. Negative facts only
justify the negations of atomic propositions.)
‹×› Positive and negative facts there are, but
not true and 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 and false
were two equally justified relations between the
sign and 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 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 true when
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 and 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. ‹Cf. 4.061, 4.062,
4.0621›31
Item 201a-3 Recto Page 5
- 5 -
Second 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. ‹Cf 4.02,
4.021, 4.027›31
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.
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. ‹cf. 4.024›31
‹×› What we know when we understand a
proposition is this: We know [
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. ‹cf. 4.024›31
Propositions are not names. ‹cf. 3.144›31
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 "a
R b", "R" looks like a substantive, but is not
one. What symbolizes in "a R b" is that R occurs
between a and b. Hence "R" is not the indefinable
in "a R b". 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 the re may not be logical
constants. A reason against them is the generality
of logic: logic cannot treat a special set of
things. ‹Cf. 3.1432›31
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).
Alternative indefinability shows that the
indefinables have not been reached.
‹×›¦ 30[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.]30 Thus a proposition
has two poles, corresponding to the case of its
truth and 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", and 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"
and "or", like judgment, analogous to predicates
or relations, because they do not introduce
anything new.
Item 201a-3 Recto Page 6
- 6 -
Propositions are always complex even if they
contain no names.
¦ 30[A proposition must be understood when all
¦ its 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.]30
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 sence
these indefinables alone. It is easy to introduce
"all" and "some" in a way that will make the
construction of [
from "all" and "x R y" as introduced before.
Item 201a-3 Recto Page 7
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Third 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 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 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
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. ‹See 4.063›31
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.
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. If p follows from q, it can also be
inferred from q, and the "manner of deduction" is
indifferent. ‹Cf 5.132›31
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.
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 are things called facts
or complexes and relations easily leads to the
opinion that there must be a relation of
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 and p
q is this: p.p q.
p.q q.
At a pinch, one is tempted to interpret
"not-p" as "everything34 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. ‹Cf. 5.43›31
Russell´s "complexes" were to have the
useful property of being compounded, and were to
combine with this the agreeable
Item 201a-3 Recto Page 8
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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 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. ‹Cf 2.0201›31
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. ‹See31 5.42›31
If we change a constituent a35 of a
proposition
(a) into a variable, then there is a
class
p^ [(
x). (x) = p] .
This class in general still depends upon what, by
an arbitrary convention, 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. ‹Cf 3.315›31
Types can never be distinguished from each
other by saying (as is often done) that one has
these but the other has those 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. ‹[
At a pinch we are always inclined to
explanations of logical functions of propositions
which aim at introducing into the functions 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?"
The very possibility of Frege´s explanations
of "not-p" and "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 and false. 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 automatically, the form
of proposition only by convention. See 3.14431
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 ind[
saying: "´A´ denotes something indefinable", so we
introduce e.g. the form of propositions a R b by
saying: "For all meanings of "x" and "y", "x R y"
expresses something indefinable about x and y".
In place of every proposition "p[
us write "ab p". Let every correlation of
propositions to each other or of names to
Item 201a-3 Recto Page 9
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propositions be effected by a correlation of their
poles "a" and "b". Let this correlation be
transitive. Then accordingly "a-ab-b p" is the
same symbol as "ab 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 and 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;
and thence it follows that, in analyzing such
propositions, we must not be guided by Russell´s
method of symbolizing.
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. ‹See 3.203›31
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
(a, a) by (a, b). a = b.
Since the ab-functions of p are again
bi-polar propositions, we can form ab-functions of
them, and 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 v ˜q.
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, and 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
"v" and "[.|˜]", or of "v" by "[.|˜]" and "
"? And
how e.g. shall we introduce p/q (i.e. ˜p v ˜q) if
not by saying that this expression says something
indefinable about all arguments p and 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; 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. ‹See 5.44›31
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.
‹See 4.42›31
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. ‹5.5422›31
‹×› It is clear that we understand propositions
without knowing
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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. ‹Cf 4.024›31
The assumption of the existence of logical
objects makes it appear remarkable that in the
sciences propositions of the form "p v q", "p
q", etc. are only then not provisional when " v "
and "
" stand within the scope of a
generality-sign (apparent variable).
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Fourth 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. R? DS. ‹See31 4.26.›31
‹×› The chief characteristic of my theory is
that, in it, p has the same meaning as not-p. ‹Cf
4.0621›31
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.
If a word create
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. Cf 5.12331
Signs of the form "p v ˜p" are senseless,
but not the propositions "(p). p v ˜p". If I know
that this rose is either red or not red, I know
nothing. The same holds of all ab-functions. Cf
4.46131
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". Cf 4.02431
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.
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.
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 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 of p.
The sense of an ab-function of p is a
function of the sense of p. ‹See 5.2341›31
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. ‹See 3.142›31
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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.
‹X› 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[
This was typed in but has excesses through
it. (D.S.)
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 introduce ab-functions. ‹Cf
5.1241›31
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 commonx with a R b. But when we introduced
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
justness of our procedure is proved.
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
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. Cf 5.45131
It is a priori likely that the introduction
of atomi[x|c] 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 quite similar questions to
those in the realm of ab-functions.
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"; and in order to make the
direction of p even more explicit, we might say "A
believes that ´p´ is true and ´not-p´ is false".
Here the bi-polarity of p is expressed and 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" and "b" of a-p-b.
Franz Hespe