| | Vol. I.
Wittgenstein
on Logic, April
1914.
G.E.
Moore |
| | ⌊⌊
Moore
good “Moore” ◇◇◇ ⌋⌋
Logical so called props. shew logical properties of language & therefore of Universe, but say nothing. This
It is impossible to say what these properties are, because in order to do so, you would need a language, which hadn't got the properties . in question, & it is impossible that this should need a be a proper language. Imposs. to construct illogical language. In order that you should have a language which can express or say everything that can be said, this language must have certain properties; & when this is the case that it has them can no longer be said in that language or any language. An illogical language would be one in which e.g. you could put an event into a hole. Thus a language which can express everything mirrors certain properties of the world by those properties wh. it must have; & logical so called props. shew in a systematic way those properties. In Ordinary logic How, usually, logical props. do shew these properties is this. We give a certain description of a kind of symbol; we find that other
As a rule the description given in ordinary Logic is the description of a tautology; but others might shew equally well, e.g.: a contradiction. Everyc prop. real prop. shews something, besides what it says, about the Universe: for, if it has no sense, it can't be used; &, if it has a sense, it mirrors some ˇlogical property of the Universe. E.g. take φa, φa ⊃ ψa, ψa. By merely looking at these 3, I can see that 3 follows from 1 & 2: i.e. I can see what's called the truth of a logical prop., namely of prop. (φa . φa ⊃ ψa : ⊃ : ψa) But this is not a prop.; but by seeng seeing that it is a tautology I can see what I already saw by looking at the 3 props.: the difference is that I now see that it is a tautology. We want to say, in order to understand above, what properties a symbol must have, in order to be a tautology:– Many ways ˇof saying this are possible: (1) One way is to say: Given certain symbols, give certain symbols; then to give a set of rules for combining them; & then to say any symbol ˇformed from those symbols, by combining them according to one of the given rules, is a tautology. This is the actual procedure of old Logic: It gives so called primitive propositions; so called rules of deduction; & then says that what you got by applying the rules to the props. is a logical prop. that you have proved. The truth is it tells you something about the kind of prop. you have got, viz that it can be derived from the first symbols by these rules of combination = ? is a tautology. :. if we say one logical prop. follows logically from another, this means something quite different from saying that a real prop. follows logically from another. For so called proof of a logical prop. does not prove its truth (logical props. are neither true nor false) but proves that it is a logical prop. = is a tautology. Logical props. are forms of proofs: they shew that one ˇor more props. follow from one (or more). |
| | Logical
props. shew something,
because the language in wh. they
are expressed can say everything that can be
said. |
| | This same
distinction between what can be shewn by the language but
not said, explains the difficulty that is
:. a theory of types is impossible. It tries to say something about the types, when you can only talk about the symbols. But what you say about the symbols is not that this symbol has that type, which would be nonsense for same reason: but you say simply This is symb the symbol, to prevent a misunderstanding. E.g. In „aARbB’, R is not a symbol, but that R is between one name & another symbolises. Here we have not said this symbol is not of this type but of that, but only: This is the symbol & symbolises & not that. This seems again to make same mistake, because ‘symbolises’ is ‘typically ambiguous’. The true analysis is: R is no proper name, &, that R stands between a & b ,(expresses a relation). Here are 2 props. of different type, connected by ‘and’ It is obvious that, e.g. with a subject-predicate prop., if it has any sen sense at all, you see the form, as soon as you understand the prop., in spite of not knowing whether it is true or false. Even if there were props. of form ‘M is a thing’ In the above expression ‘aRb’, we were talking only of this particular R, whereas what we want to do to is to talk of all similar symbols. We have to say: inc any symbol of this form R ˇwhat corresponds to R is not a proper name, & fact that … expresses a relation. This is what is sought to be expressed by ˇthe nonsensical assertion: [s|S]ymbols like this are of a certain type. This you can't say, bec. in order to say it you must first know what the symbol is: & in knowing this you see the types, & therefore also types of symbolised. I.e. in knowing what symbolises, you know all that is to be known; you can't say anything about the symbol. For instance: Consider the 2 props. (1) “What symbolises here is a thing”, (2) “What symbolises here is a ˇrelational fact (or relation = relation)”. These are nonsensical for 2 reasons: (a) bec. they mention ‘thing’ & ‘relation’ (b) bec. they mention them in props. of same form. But The 2 props. must be expressed in entirely different forms, if properly analysed; & neither the word ‘thing’ nor ‘relation’ must occur. Now we shall see how properly to analyse props. in wh. ‘thing’ ‘relation’, etc. occur. (1) analysed = &(∃x). x symbolises.
N.B. here ‘thing’
doesn't occur |
| | ⍈
„x” is the Name of
y |
| | ⍈
N.B. “x”
can't be the name of this actual scratch x
y; because this isn't a thing: but it can
be the name of a thing; & we must understand that
what we are doing is to explain what would be meant by saying of an
ideal symbol, wh. did actually consist in
one thing's being to left of
another, that in it a thing symbolised.
|
| |
and instead of ‘thing’, what occurs is
the form of a name (1) Take φx. We want to explain the meaning of in “In “φx” there a thing symbolises.” The analysis is:– (∃ y) . y symbolises . [x| y] = “x” .“φx”. [“x” is the name of xy: “φx” says that = ‘“φ” is at left of “x”’, & says φx.] ¥ |
| |
⟵ [N.B. In expression (∃y). φy, one is apt to say this means ‘There is a thing such that …’. But in fact, we should say ‘There is a y, such that … ’; the forms of the y being what expressing fact that the y symbolises, expressing what we mean.] |
| | In
general: When such propositions are analysed, while the
words ‘thing’, ‘fact’
etc. will disappear, there will appear instead of them
a new symbol,
In our language names are not things: we don't know what they are: all we know is that they are of a different type from relations etc. etc. … The type of a symbol of |
| |
p. 1, N.B. In any
ordinary proposition, e.g.
“Moore good”, this shews
& does not say that “Moore” is
to left of “good”; &
here what is shewn can be said by
another prop.. But this
only applies to that part of what is shewn which is
arbitrary. The logical properties which it shews,
are not arbitrary, & that it has these cannot be said in any
proposition. |
| | When
we say of a prop. of form aRb
that what symbolises is that ‘a’
“R” is between “a”
& “b”, it must be remembered that in fact
the prop. is capable of further
analysis because a, R, & b are not
simples. But what seems certain is that when we
have analysed , it, we shall in the end come to
props. of the same form in respect
of the fact that they do consist in one thing being between 2
others. How can we talk of the general form of a proposition, without knowing any simples or any unanalysable props. ˇin which particular names & relations occur? What justifies us in doing this is that though we don't know any unanalysable props. ˇof this kind yet we can understand ˇwhat is meant by a prop. of the form (∃x,y,R). xRy ˇ(which is unanalysable), even though we know If you had any unanalysable prop. in which particular names & relations occurred (and unanalysable prop. = one in which the only fundamental symbols = ones not capable of definition, occur) then you always can form from it a prop. of form (∃x,y,R). xRy, which though it contains no particular names & relations, is unanalysable. |
| |
(2) The point can here be brought out as
follows. Take φa, & φA: , where & ask what is meant by saying “There is a thing in φa, & a complex in φA”? (1) means: (∃x). φx . x = a (2) means φ [aRb] = ψa . ψb . aRb Def. x
A = aRb
Def.
We then get (∃x,y,(ξRη) . φx . φy . xRy (∃x,ψξ). φA = ψx . φx |
| | Use of logical
props.. You may
have one so complicated that you cannot, by looking at it, see that it
is a tautology; but you have shewn that it can be derived by certain
operations from
|
| | The Bedeutung of a
prop. is the fact that corresponds
to it, e.g. aRb, if our
prop. be aRb, if
it's true, the corresponding fact would be the fact
aRb, if false, the fact ~aRb.
But both “the fact aRb” &
“the fact ~aRb” are incomplete symbols,
which must be analysed. That a prop. has a relation (in wide sense) to Reality, other than that of Bedeutung, is shewn by the fact that you can understand it when you don't know the Bedeutung, i.e. don't know whether it's true or false. Let us express this by saying “It has sense” (Sinn). In analysing Bedeutung, you come upon Sinn, as follows:– We want, to explain the relation of props. to reality. The relation is as follows: It's simples have meaning = are names of simples; & it's relations have a quite different relation to relations; such that, if all the fact that these things are true & these 2 facts already establishes a sort of correspondence between a prop. wh. contains th[em|ese] & reality. & only these & reality: i.e. if all the simples of a prop. are known, we already k already know that Reality “behaves” in a certain way towards these. we can describe reality by saying that it behaves in a certain way to the whole proposition. All that remains to be done is to state in what way reality must behave to the prop., if the prop. is to be called ‘true’. [This amounts to saying that we can compare reality It only remains to fix the method of comparison, by saying what
From this it results that ‘true’ & ‘false’ are not accidental properties of a prop., such that, when it has meaning, we can say it is also true or false: on the contrary to have meaning means to be true or false; i.e. that reality is true or false to it. bei the being true or false actually constitutes the relation of the prop. to reality, wh. we mean by saying that it has meaning. (Sinn) |
| | There seems at first sight to be a certain
ambiguity in what is meant by saying that a
prop. is ‘true’,
owing to the fact that it seems as if in the case of different
props. the way in
wh. they correspond to the facts
which co to wh. they
correspond is quite different. But what is really common to
all cases is that they must have the general form of a
proposition. In giving the general form of a
prop. you are explaining what kind
of way of putting together the symbols of things & relations
will correspond to (be analogous to) the things having those
relations in reality. In doing this you are saying what is
meant by saying that a prop. is
true; & you must do it once for all. To say a
prop. f
“This prop. has
sense” means ⌊“⌋“This
prop. is true” means
… ” (“p” is true =
“p” . p .
Def. ):
only instead of “p”, we must have introduce
the general form of a
prop..) |
| | Vol. II.
W.
on L., April
1914.
G.E.
Moore |
| | ﹖
G.E.M. (used in
Tractatus, p.
158–9, but with W–F instead of
a–b)
It seems at first sight as if the a–b notation must
be wrong, because it seems to treat true & false as on exactly
the same level. The symbolism must its
S Some more fact about the
symbo It must be possible to see from the
symbols themselves that there is some ˇessential difference
between the poles, if the notation is to be right; & it seems
as if in fact this was impossible.
The interpretation of a symbolism must not depend upon giving a different interpretation to symbols of the same types. True How asymmetry is introduced is by giving a description of a particular form of symbol which we call a ‘tautology’, & thus, the fact that this The description of the a–b symbol alone is symmetrical with respect to a & b; but this description & the ˇfact that what satisfies the description of
|
| | Take p.q &
q. When you write p.q in the a–b notation,
it is impossible to see from the symbol alone that q follows
from it; for if you were to
|
| |
Logical props.,
of course, all shew something different: all of them
shew, in the same way viz. by the fact that
they are tautologies, but they are different tautologies &
therefore shew each something different. |
| | What is unarbitrary about our symbols, is not
them, nor the rules we give; but the fact that, having given certain
rules, others are fixed = follow logically. |
| | Thus, though it would be possible to interpret the
form which we take as the form of a tautology as that of a
contradiction & vice vers,
it would be impossib they are
different in ˇlogical form, ˇbecause though the apparent
form of the symbols is the same, what symbolises in them is
different. & hence what follows about the symbols
from the one interpretation will be different from what follows
|
| | The point of this illustration is that the process of reasonings by wh. we discover that p ≡ ~(~p) is a tautology, is exactly the same by which we reason “if a–true implies b–false, etc..” The The point is: that the process of reasoning by wh. we
|
| | Logical constants
can't be made into variables: because in them
what symbolises is not the same; ◇◇◇
ˇall symbols ◇◇◇ for wh.
a variable can be substituted symbolises in the
same way. We describe a symbol, & say This is a ˇarbitrarily ‘A symbol of this description is a tautology’. And then, it follows at once, both that any other symbol wh. answers to the same description is a tautology; & that any symbol which does not isn't. That is, we have arbitrarily fixed that that ˇany symbol of that description is to
Having thus fixed what is a tautology & what is not, we can then, having fixed arbitrarily again that if the relation the relation a – b is transitive ◇◇◇ get from the 2 facts together that e.g. ‘p ≡ ~(~p)’ ˇis a taut.. For ~(~p) = a – b – apb – a – b. It follows from the fact that a – b is transitive, that where we have a – b – a, the first a has to the second the same relation that it has to b. It is just as from the fact that a–true implies b–false, & b–false implies ctrue, we get that a–true implies c–true. And we shall be able to see, having fixed the meaning description of a tautology that, p ≡ ~(~p) is a tautology. That, when a
) But now it is obvious that the a In settling that it is to be interpreted as a tautology & not as a contradiction, I am not assigning a meaning to a & b; i.e. saying that they symbolise different things but in the same way. What I am doing is to to say that the method way in wh. the a–pole is connected with the whole system symbol symbolises in a different way from that in which it would symbolise if the symbol were interpreted as a contradiction. And I add the scratches a & b merely in order to shew in which way the connection is symbolising, so that it may be evident that wherever the same scratch occurs in the corresponding place in another We could, of course, symbolise
Th connected in a certain way with an outside pole, in the other that they're not. The symbol for a tautology, whatever we take to in whatever form we find put it, e.g. whether by omitting the a–pole or by omitting the ub, would always be capable of being used as the symbol for a contradiction; only not in the same language. |
| |
The reason why ~x is
meaningless, is simply that we have given no meaning to the symbol
~ξ.
I.e. whereas
φx &
~p look as if they were of the
same type, they are not so because in order to give a meaning to
~x you would have to have
some property φξ
~ξ.
What symbolises in φξ is that φ
stands to left of a proper name:
& obviously this is not so in ~p.
All What is common to all
props. in which the name of a
property (to speak loosely) occurs is that this name stands to
the left of a name-form. The reason why e.g. it seems as if ~x or ‘Plato Socrates’ might have a meaning, while ‘Abracadabra Socrates’ would never be suspected to have one, is because we know that Plato has one, & do not observe that in order that the whole phrase should have one, what is necessary is not that Plato should have one, but that the fact that Plato is to left of a name should. The reason why ‘thThe property of not being green is not green’ is nonsense, is because we have given no meaning to only given meaning to the fact that green stands to the right of a name; & ‘the property of not being green’ is obviously not that. φ cannot possibly stand to the left ˇof (or in any other relation ˇto) p is false = ~(p is true) Def. |
| | It is very important that the apparent logical
relations ⌵ , ⊃ etc. need
brak brackets, dots etc.,
i.e. have “ranges”; which by
itself shews they are not relations. It This
fact has been overlooked, because it is so universal – the very
thing wh. makes it so important.
|
| | There are internal
relations between one prop.
& another; but a prop.
cannot have to another the internal relation
wh. a name has to the
prop. it o of
wh. it is a constituent, &
wh. ought to be meant by saying it
‘occurs’ in it. In this sense one
prop. can't
‘occur’ in another.
Internal relations are relations between types, wh. can't be expressed in props., but are all shewn in the symbols themselves, & can be exhibited systematically in tautologies. Why we come to call them ‘relations’ is because logical props. have
Props. can have many different internal relations to one another. The one wh. entitles us to deduce the one from another, is that if put in expressed in say, they are φa & , φa ⊃ ψa, then φa . φa ⊃ ψa : ⊃ : ψa is a tautology. Ide The symbol of identity expresses the internal relation between a function & its argument: i.e. φa = (∃x)φx.x = a. |
| | The prop.
(∃x)φx . x = a
: ≡ : φa can be seen to be a
tautology, if one expresses the conditions of the truth of
(∃x).φx .
x = a, successively, e.g. by
saying: This is true if so & so;
that & this again is true, if so
& so. ˇetc. for
(∃x).φx . x = a;
and then also for φy. To
express the matter in this way is itself a cumbrous notation,
|
| | What
symbolises in a symbol, is that which is common to all the symbols
wh. could in accordance with the
rules of logic = syntactical rules for
manipulation of symbols, be substituted for it. |
| | The question whether a
prop. has
It's obvious that the dots & brackets are symbols, & obvious also that they haven't any independent meaning. You must therefore, in order to introduce so called “logical constants” properly, introduce the general notion of all possible combinations of them –t = the general form of a prop. You thus introduce both a–b functions, identity & universality (the 3 fundamental constants) simultaneously. |
| | The
variable-prop.
p ⊃
tildap is. is not identical
with the
variable-prop.
~(p
. ~p).
The corresponding
universals would be identical. But the one
variable prop. says that out of
e.g. p ⊃ q you get a tautology
by substituting the variable
prop.
~(p
. ~p) says shews that out of
~(p
. q) you get a tautology by substituting
~p for q, whereas
the other does not say shew this. |
| | It's very important to realise that
when you have 2 different relations (a,b)R
(c,d)S this does not establish a
correlation between a &
c,
& b &
d,
or a &
d,
[or|&] b &
c: there is no correlation
whatsoever thus established. Of course, in the case of 2
pairs of terms united by same relation, there
is a correlation. This shews that the theory which held
that a relational fact contained the terms & relations united
by a copula (ε2) is
untrue; for if this were so there would be a |
| | The question
arises how can one prop. (or
function) stand occur is in another
prop.? The
This can only be by its The
prop. or function itself
can't possibly stand in relation to the other
symbols. It must therefore be that we For
this reason we must introduce functions as well as
|
| |
It is true, in a sense, that logical
props. are
‘postulates’– something which we
‘demand’; for we do demand a satisfactory
notation. |
| | A tautology
(not a logical
prop.) is not nonsense in the
same sense in wh.
e.g. a prop.
in wh. words which have no meaning occur
is nonsense. What happens in it is that all its simple
parts have meaning, but it is such that the connections between these
paralyse ˇor destroy one another, so that they are all
connected only in an irrelevant manner. ( = one such
as to have no sense?) |
| | Vol. III.
W.
on L., April
1914.
G.E.
Moore |
| | Logical functions all presuppose one
another. Just as we can see ~p has no sense, if
p has none; so we can also say p has none, if
~p has none.
The case is quite different with φa, &
a: since here
a
has a meaning independently of φa, though
φa presupposes it.
|
| | The logical constants seem to be
complex-symbols, but on the other
hand, they can be interchanged with one another. They are
not therefore really complex; what symbolises is simply the way
in which general way in which they are combined. |
| | The combination of symbols in a
tautology cannot possibly correspond to any one particular combination
of their meanings – it corresponds to every possible combination;
& therefore what symbolises can't be the
connection of the symbols. |
| |
From the fact that I see that one spot is to
left of another, or that one colour is darker
|
| | Different
logical types can have nothing whatever in common. But the
same fact that we can talk of the possibility of a relation of
n places, or of an analogy between one with
2-places & one with 4, shews that th
relations with different numbers of places have something in common,
that therefore the difference is not one of type but like the
difference between different names – something
wh. depends on experience. This
answers the question how we can know that the we have
really got the most general form of a
prop.. We have only to
introduce what is common to all relations of whatever number
of places. |
| | The relation
of ‘I believe p’ to
‘p’ can be compared to the
relation of ‘“p”
|