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Notes on Logic by Ludwig
Wittgenstein
September
1913.
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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.
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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?
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The verb of a proposition cannot be “is true” or
“is false”,- but whatever is true or false must already
contain the verb.
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Deductions only proceed according to the laws of deduction,- but these
laws cannot justify the deduction.
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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.
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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.
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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.
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In a proposition convert all its indefinables into- variables; there
then remains a class of propositions which- is not all propositions but a
type.
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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.
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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.
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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.
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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.
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The meaning of a proposition is the fact which actually- corresponds to
it.
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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.
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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.
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[Note by Bertrand Russell ab means the
same as WF, which means true-false.]
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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.
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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.
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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.
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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.
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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.
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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 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.
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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.
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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.
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The notation is:
for
(x)
φx : a -
(x) - a φx b - (∃
x) - b
and
for (∃x)
φx : a -
(∃x) - a φx b -
(x) - b
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Old definitions now become tautologous.
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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.
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Judgment, question and command are all on the same level. -
What interests logic in them is only the unasserted proposition.
-
Facts cannot be named.
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A proposition cannot occur in itself.
This is the fundamental- truth of the theory of types.
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Every proposition that says something indefinable about one- thing is a
subject-predicate proposition, and so on.
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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.
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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.
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[Components are forms and constituents.]
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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).
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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.
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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.
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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.
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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.”
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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?
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The reason why “~Socrates” means nothing is that
“~x” does- not express a
property of x.
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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.)
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Positive and negative facts there are, but not
true and- false facts.
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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.
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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.
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Whatever corresponds in reality to compound propositions- must not be
more than what corresponds to their several atomic- propositions.
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Not only must logic not deal with [particular] things, but- just
as little with relations and predicates.
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There are no propositions containing real variables.
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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.
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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.
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Propositions are not names.
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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.
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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.
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Molecular propositions contain nothing beyond what is- contained in
their atoms; they add no material information- above that contained in
their atoms.
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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].
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Alternative indefinability shows that the indefinables- have not been
reached.
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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.
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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”.
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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.
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Propositions are always complex even if they contain no- names.
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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.
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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?
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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 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.
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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.
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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.
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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.
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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..
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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
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.
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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.
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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.
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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.
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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.
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If we change a constituent a of a proposition
φ(a) into- a
variable, then there is a class
This class in general still depends upon what, by an
arbitrary- convention, we mean by
“φ(x)”.
But if we change into- variables all those symbols whose significance
was arbitrarily- determined, there is still such a class.
But this is now not- dependent upon any convention, but only upon the
nature of the- symbol “φ(x)”.
It corresponds to a logical type.
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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.
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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?”
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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.
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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.
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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”.
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In place of every proposition “p”, let us write
“
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 “
p” is the same
symbol as “
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.
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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.
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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.
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Neither the sense nor the meaning of a proposition is a- thing.
These words are incomplete symbols.
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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.
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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.
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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
“ ⌵ ” 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.
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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.
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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.
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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.
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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].
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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.]
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The chief characteristic of my theory is that, in it,
p-
has the same meaning as not-p.
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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.
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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.
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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.
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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”.
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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.
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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.
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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.
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The sense of an ab-function of
p
is a function of the- sense of p.
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The ab-functions use the discrimination of facts,
which- their arguments bring forth, in order to generate new-
discriminations.
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Only facts can express sense, a class of names cannot. -
This is easily shown.
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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.
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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.
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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.
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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
justness- of our procedure is proved.
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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.
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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.
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Cross-definability in the realm of general propositions- leads to
quite similar questions to those in
the realm- of ab-functions.
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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.
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The ab-notation shows the dependence of or
and not, and- thereby that they are not to be employed as
simultaneous- indefinables.
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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.
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Preliminary
In philosophy there are no deductions: it is purely-
descriptive.
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Philosophy gives no pictures of reality.
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Philosophy can neither confirm nor confute scientific-
investigation.
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Philosophy consists of logic and metaphysics: logic- is its
basis.
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Epistemology is the philosophy of psychology.
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Distrust of grammar is the first requisite for-
philosophizing.
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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.
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Only the doctrine of general indefinables permits us to- understand the
nature of functions.
Neglect of this doctrine- leads to an impenetrable thicket.
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Preliminary
Philosophy is the doctrine of the logical form of scientific-
propositions (not only of primitive propositions).
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The word “philosophy” ought always to designate
something- over or under but not beside, the natural
sciences.
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Judgment, command and question all stand on the same level;- but all
have in common the propositional form, which does interest-
us.
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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..
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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.
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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.
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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.
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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.
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Preliminary
A correct explanation of logical propositions must give- them a unique
position as against all other propositions.
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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.
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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-
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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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.
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This is the symbol for ~p
⌵ ~q
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