Notes on Logic
by Ludwig Wittgenstein September
1913. 1 |
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. 2 |
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.
* Russell – for instance imagines every fact as a spatial complex. 3 |
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. |
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.
* W-F = Wahr-Falsch. 4 |
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 Bertrand Russell: 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”.
* 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 a p b (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. 5 |
A proposition is a standard to which all facts behave, with names
it is otherwise; it is thus bi-polarity and sense comes in; just as
one arrow behaves to another arrow by being in the same sense or the
opposite, so a fact behaves to a proposition. |
The form of a proposition has meaning in the following way.
Consider a symbol “xRy”.
To symbols of this form correspond couples of things whose names are
respectively “x” and
“y”.
The things x y stand to one another in
all sorts of relations, amongst others some stand in the relation
R, and some not; just as I single out a particular thing by a
particular name I single out all behaviours of the points x and
y with respect to the relation R.
I say that if an x stands in the relation R to a
y the sign “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.
6
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 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 old indefinables, as in the
case of molecular functions.
The application of the ab notation to apparent
variable propositions becomes clear if we consider that, for instance,
the proposition “for all x, φx” is to be
true when φx is true for all
x's and false when
φx is false for some
x's.
We see that some and all occur simultaneously in
the proper apparent variable notation. 7 |
The Notation is: for (x) φx : a - (x) - a φx b - (∃ x) - b and for (∃x) φx : a - (∃x) - a φx b - (x) - b |
Old definitions now become tautologous. |
In aRb
it is not the complex that symbolises but the fact that the symbol
a stands
in a certain relation to the symbol b.
Thus facts are symbolised by facts, or more correctly: that a
certain thing is the case in the symbol says that a certain thing is the
case in the world. |
Judgment, question and command are all on the same level.
What interests logic in them is only the unasserted proposition.
Facts cannot be named. |
A proposition cannot occur in itself.
This is the fundamental truth of the theory of types. |
Every proposition that says something indefinable about one thing is a
subject-predicate proposition, and so on. |
Therefore we can recognize a subject-predicate proposition if we
know it contains only one name and one form,
etc.
This gives the construction of types.
Hence the type of a proposition can be recognized by its symbol
alone. 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. |
[Components are forms and constituents.] |
Take (φ).
φ!x.
Then if we describe the kind of symbols, for which
φ! stands and
which, by the above, is enough to determine the type, then automatically
“(φ).
φ! x” cannot be fitted by this
description, because it contains
“φ!x” and
the description is to describe all that symbolises in
symbols of the φ!
kind.
If the description is thus complete vicious circles can just
as little occur as for instance (φ).
(X)φ (where
(X)φ is a
subject-predicate proposition). 1 |
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. |
Propositions [which are symbols having reference to facts] are
themselves facts: that this inkpot is on this table may express
that I sit in this chair. |
It can never express the common characteristic of two objects that we
designate them by the same name but by two different ways of designation,
for, since names are arbitrary, we might also choose different names, 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,φ).
φx” 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? 2 |
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
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 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.4 |
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. |
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. |
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. |
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. 5 |
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 “φx”,
“φ” looks like a substantive
but is not one; in “~p”,
“~”
looks like “φ” but is not
like it.
This is the first thing that indicates that there may not be
logical constants.
A reason against them is the generality of logic: logic cannot
treat a special set of things. |
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. |
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 two poles, corresponding to the case
of its truth and the case of its falsehood.
We call this the sense of a proposition. 6 |
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. |
Propositions are always complex even if they contain no names.
|
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. |
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? 7 |
The ways by which we introduce our indefinables must permit us to
construct all propositions that have sense [﹖ meaning]
from these indefinables alone.
It is easy to introduce “all” 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. 8 |
3rd. MS. |
An analogy for the theory of truth: Consider a black patch
on white paper; then we can describe the form of the patch by mentioning,
for each point of the surface, whether it is white or black.
To the fact that a point is black corresponds a positive fact, to the
fact that a point is white (not black) corresponds a negative
fact.
If I designate a point of the surface (one of
Frege's
“truth-values”), this is as if I set up an
assumption to be decided upon.
But in order to be able to say of a point that it is black or that it
is white, I must first know when a point is to be called black 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 what 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. 9 |
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. |
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 10 are things called facts or complexes and
relations easily leads to the opinion that there must be a relation of
questioning(﹖)
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 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
“everything else, only not p”.
That from a single fact p an infinity of others,
not-not-p etc., follow, is hardly
credible.
Man possesses an innate capacity for constructing symbols with which
some sense can be expressed, without having the slightest idea
what each word signifies.
The best example of this is mathematics, for man has until lately used
the symbols for numbers without knowing what they signify or that they
signify nothing. |
Russell's
“complexes” were to have the useful property of being
compounded, 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 a property cannot be a logical type. 11 |
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. |
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. |
If we change a constituent a of a proposition
φ(a) into a
variable, then there is a class
12 |
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 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?” |
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. 13 |
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 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”, let us write
“
|
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. 14 |
It is to be remembered that names are not things, but classes:
“A” is the same letter as
“A”.
This has the most important consequences for every symbolic
language. |
Neither the sense nor the meaning of a proposition is a thing.
These words are incomplete symbols. |
It is impossible to dispense with propositions in which the same
argument occurs in different positions.
It is obviously useless to replace φ(a, a) by
φ(a, b). a = b. |
Since the ab-functions of
p
are again bi-polar propositions, we can form
ab-functions of them, 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 ⌵
~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 15 matters stand with the definition of
“ ⊃ ” by
“ ⌵ ” and
“.”, or of
“ ⌵ ” by “.” and
“ ⊃ ”?
And how e.g. shall we introduce
p❘q
(i.e. ~p ⌵
~q), if not by saying that this expression
says something indefinable about all arguments
p
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. |
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. 16 |
It is clear that we understand propositions without knowing whether
they are true or false.
But we can only know the meaning of a proposition when we
know if it is true or false.
What we understand is the sense of the proposition.
|
The assumption of the existence of logical objects makes it appear
remarkable that in the sciences propositions of the form
“p
⌵ q”,
“p
⊃ q”, etc. are only
then not provisional when “ ⌵ ” and
“ ⊃ ” stand within the scope of a
generality-sign [apparent variable]. 17 |
4th. MS. |
If we formed all possible atomic propositions, the world would be
completely described if we declared the truth or falsehood of
each.
[I doubt this.] |
The chief characteristic of my theory is that, in it,
p
has the same meaning as not-p. |
A false theory of relations makes it easily seem as if the relation of
fact 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:
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;
and similarly it could not create a world in which a proposition was
true, without creating its constituents. |
Signs of the form “p ⌵
~p” are senseless, but not the
proposition “(p). p ⌵
~p”.
If I know that this rose is either red or not red, I know
nothing.
The same holds of all ab-functions. |
To understand a proposition means to know what is the case if it is
true.
Hence we can understand it without knowing if it is true.
We understand it when we understand its constituents 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”. 18 |
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. 19 |
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. |
The ab-functions use the discrimination of facts,
which their arguments bring forth, in order to generate new
discriminations. |
Only facts can express sense, a class of names cannot.
This is easily shown. |
There is no thing which is the form of a proposition, 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. |
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. 20 |
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. |
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.
But when 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). 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), and thereby the
justification || 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
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. |
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
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.
The epistemological questions concerning the nature of judgment and belief
cannot be solved without a correct apprehension of the
¤
form
of the proposition. 22 |
The ab-notation shows the dependence of or
and not, and thereby that they are not to be employed as
simultaneous indefinables. |
Not: “The complex sign
‘a R
b’” says that
a stands in the relation
R to
b; but that
‘a’ stands in a certain relation
to ‘b’ says that
a R
b. |
[Preliminary](Ƒ)
In philosophy there are no deductions: it is 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. |
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. 23 |
[Preliminary](Ƒ)
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 and question all stand on the same level; but all
have in common the propositional form, which does interest
us. |
The structure of the proposition must be recognized, the rest comes of
itself.
But ordinary language conceals the structure of the proposition:
in it, relations look like predicates, predicates like names,
etc. |
Facts cannot be named. |
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. 24 |
Just as people used to struggle to bring all propositions into the
subject-predicate form, so now it is natural to conceive every
proposition as expressing a relation, which is just as incorrect.
What is justified in this desire is fully satisfied by
Russell's theory of
manufactured relations. |
One of the most natural attempts at solution consists in regarding
“not-p” as “the
opposite of p”, where then
“opposite” would be the indefinable relation.
But it is easy to see that every such attempt to replace the
ab-functions by descriptions must fail. |
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]
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, and so on.
Thus every proposition which contains only one name and one
indefinable form is a subject-predicate proposition, and so
on.
An 25 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. |
Wittg.– I. Bi-polarity of propositions: sense & meaning, truth & falsehood. II. Analysis of atomic propositions: general indefinables, predicates, etc. III. Analysis of molecular propositions: ab-functions. IV. Analysis of general propositions. V. Principles of symbolism: what symbolizes in a symbol. Facts for facts. VI. Types. |
1) For the dating of Ts-201a2, see Biggs 1996 and Potter 2009.
To cite this element you can use the following URL:
BOXVIEW: http://www.wittgensteinsource.org/BTE/Ts-201a2_n