Vol. I.
Wittgenstein
on Logic, April
1914.
G.E.
Moore |
Moore
good “Moore” ◇◇◇
Logical so-called propositions shew logical properties of language & therefore of the Universe, but say nothing. This is shewn by fact || means that by merely looking at them you can see these properties; whereas, in a proposition proper, you cannot see what it says || is true by looking at it. 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 be a proper language. Impossible to construct an 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 which it must have; & logical so-called propositions shew in a systematic way those properties. How, usually, logical propositions 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. Every real proposition 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 logical 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 proposition, namely of the proposition (φa . φa ⊃ ψa : ⊃ : ψa). But this is not a proposition; but by seeing that it is a tautology I can see what I already saw by looking at the 3 propositions: the difference is that I now see that it is a tautology. We want to say, in order to understand the above, what properties a symbol must have, in order to be a tautology:– Many ways of saying this are possible: (1) One way is to give certain symbols; then to give a set of rules for combining them; & then to say any symbol formed 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 propositions is a logical proposition that you have proved. The truth is it tells you something about the kind of proposition 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 proposition follows logically from another, this means something quite different from saying that a real proposition follows logically from another. For a so-called proof of a logical proposition does not prove its truth (logical propositions are neither true nor false) but proves that it is a logical proposition = is a tautology. Logical propositions are forms of proofs: they shew that one or more propositions follow from one (or more). |
Logical
propositions shew something,
because the language in which 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 the same reason: but you say simply This is 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 symbolises & not that. This seems again to make the 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 propositions of different type, connected by ‘and’. It is obvious that, e.g. with a subject-predicate proposition, if it has any sense at all, you see the form, as soon as you understand the proposition, in spite of not knowing whether it is true or false. Even if there were propositions of the form ‘M is a thing’ In the above expression ‘aRb’, we were talking only of this particular R, whereas what we want to do is to talk of all similar symbols. We have to say: in any symbol of this form what corresponds to R is not a proper name, & the fact that … expresses a relation. This is what is sought to be expressed by the nonsensical assertion: Symbols like this are of a certain type. This you can't say, because in order to say it you must first know what the symbol is: & in knowing this you see the types, & therefore also the types of what is 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 propositions (1) “What symbolises here is a thing”, (2) “What symbolises here is a relational fact (or relation || = relation)”. These are nonsensical for 2 reasons: (a) because they mention ‘thing’ & ‘relation’ (b) because they mention them in propositions of the same form. The 2 propositions 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 propositions in which ‘thing’, ‘relation’, etc. occur. |
⍈
N.B. “x”
can't be the name of this actual scratch
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, which did actually consist in
one thing's being to the left of
another, that in it a thing symbolised.
|
(1) Take φx. We want to explain the meaning of “In “φx” a thing symbolises.” The analysis is:– (∃ y) . y symbolises . x || y = “x” .“φx”. [“x” is the name of y: “φx” = ‘“φ” is at the left of “x”’, & says φx.] ¥ |
⟵ [N.B. In the 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 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, || of the same form as the one of which we are
speaking; & hence it will be at once obvious that we
cannot get the one kind of
proposition from the other by
substitution. 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 the left of “good”; &
here what is shewn can be said by
another proposition. 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 proposition of form aRb
that what symbolises is that ‘a’
“R” is between “a”
& “b”, it must be remembered that in fact
the proposition 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
propositions 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 unanalysable propositions in which particular names & relations occur? What justifies us in doing this is that though we don't know any unanalysable propositions of this kind yet we can understand what is meant by a proposition of the form (∃x,y,R). xRy (which is unanalysable), even though we know If you had any unanalysable proposition in which particular names & relations occurred (and an unanalysable proposition = one in which only fundamental symbols = ones not capable of definition, occur) then you always can form from it a proposition of the 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) (∃x,ψξ). φA = ψx . φx |
Use of logical
propositions¤ 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 || certain other
propositions
which according to our rule for
constructing tautologies; & hence you are enabled to
see that one thing follows from another, when you would not have been
able to see it otherwise. E.g. if our
tautology is of the form p ⊃ q, you
can see that q follows from p; & so on.
|
The Bedeutung of a
proposition is the fact that corresponds
to it, e.g., if our
proposition 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 proposition 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 propositions to reality. The relation is as follows: Its simples have meaning = are names of simples; & its relations have a quite different relation to relations; & these 2 facts already establish a sort of correspondence between a proposition which contains them || these & only these & reality: i.e. if all the simples of a proposition are known, we already know that we can describe reality by saying that it behaves in a certain way to the whole proposition. [This amounts to saying that we can compare reality It only remains to fix the method of comparison, by saying what || about our simples is to say what about reality. E.g. suppose we take 2 lines of unequal length; & say that the fact that the shorter is of the length it is is to mean that ¤ the longer is of the length it is: we should then have established a convention as to the meaning of the shorter of the sort we are now to give it. From this it results that ‘true’ & ‘false’ are not accidental properties of a proposition, 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. the being true or false actually constitutes the relation of the proposition to reality, which 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
proposition is ‘true’,
owing to the fact that it seems as if in the case of different
propositions the way in
which they correspond to the facts
to which 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
proposition 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 proposition is
true; & you must do it once for all. To say
“This proposition has
sense” means ““This
proposition is true” means
… ” (“p” is true =
“p” . p .
Def. :
only instead of “p”, we must have introduced
the general form of a
proposition.) |
Vol. II.
W.
on L., April
1914.
G.E.
Moore |
[﹖
G.E. Moore (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. It must be possible to see from the
symbols themselves that there is some essential difference
between the poles, if the notation is to be right; & it seems
as if in fact this was impossible.
[True] How asymmetry is introduced is by giving a description of a particular form of symbol which we call a ‘tautology’. The description of the a–b symbol alone is symmetrical with respect to a & b; but this description & the fact that what satisfies the description of the || a tautology is a tautology is asymmetrical with regard to them. (To say that a description was symmetrical with regard to 2 symbols, would mean that || we could substitute one for the other, & yet the description remain the same, i.e. mean the same.) |
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
|| interpret the
true-pole as the false, the same symbol
would stand for p ⌵ q, from
which q doesn't
follow. But the moment you say
which symbols are tautologies, it
at once becomes possible to see from the
fact, that they are & the original symbol that
q does follow. |
Logical propositions,
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 versa,
they are
different in logical form, because 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 is: that the process of reasoning by which we || arrive at the result that a – b – a – p … is the same symbol as a – p … , is exactly the same as that by which we discover that its meaning is the same, viz. where we reason if b – apb – a then not apb, if a – b – ap then not b – apb, :. if a – b – ap¤ then apb. |
Logical constants
can't be made into variables: because in them
what symbolises is not the same;
all symbols for which
a variable can be substituted symbolise in the
same way. We describe a symbol, & say arbitrarily ‘A symbol of this description is a tautology’. And then, it follows at once, both that any other symbol which answers to the same description is a tautology; & that any symbol which does not isn't. That is, we have arbitrarily fixed that || any symbol of that description is to || be a tautology; & this being fixed it is no longer arbitrary with regard to all other symbols whether they are a tautologies || any other symbol whether it is a tautology or not. Having thus fixed what is a tautology & what is not, we can then, having fixed arbitrarily again that the relation a – b is transitive ◇◇◇ get from the 2 facts together that ‘p ≡ ~(~p)’ is 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 c–true, we get that a–true implies c–true. And we shall be able to see, having fixed the description of a tautology that p ≡ ~(~p) is a tautology. That, when a || certain rule is given, a symbol is tautological shews a logical truth. This symbol might be interpreted either as a tautology or as a contradiction. 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 say that the way in which the a–pole is connected with the whole 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 || any a – b function without using 2 outside poles at all, merely, e.g., omitting the b–pole; & here what would symbolise would be that the 3 pairs of inside poles of the propositions were connected in a certain way with the a–pole, while the other pair was not connected with it. And thus the difference between the scratches a & b, where we do use them, merely shews that it is a different state of things that is symbolising in the one case & the other: in the one case that certain inside poles are connected in a certain way with an outside pole, in the other that they're not. The symbol for a tautology, in whatever form we put it, e.g. whether by omitting the a–pole or by omitting the b, 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
~ξ.
What symbolises in φξ is that φ
stands to the left of a proper name:
& obviously this is not so in ~p.
What is common to all
propositions 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 ‘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 the left of a name should. The reason why ‘The property of not being green is not green’ is nonsense, is because we have 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 of (or in any other relation to) p is false = ~(p is true) Def. |
It is very important that the apparent logical
relations ⌵ , ⊃ etc. need
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 which makes it so important.
|
There are internal
relations between one proposition
& another; but a proposition
cannot have to another the internal relation
which a name has to the
proposition of
which it is a constituent, &
which ought to be meant by saying it
‘occurs’ in it. In this sense one
proposition can't
‘occur’ in another.
Internal relations are relations between types, which can't be expressed in propositions, but are all shewn in the symbols themselves, & can be exhibited systematically in tautologies. Why we come to call them ‘relations’ is because logical propositions have || an analogous relation to them, to that which properly relational propositions have to relations. Propositions can have many different internal relations to one another. The one which entitles us to deduce one from another, is that if say, they are φa & φa ⊃ ψa, then φa . φa ⊃ ψa : ⊃ : ψa is a tautology. The symbol of identity expresses the internal relation between a function & its argument: i.e. φa = (∃x)φx.x = a. |
The proposition
(∃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;
& this again is true, if so
& so¤ etc. for
(∃x).φx . x = a;
and then also for φy. To
express the matter in this way is itself a cumbrous notation,
which is what is expressed more
neatly || of
which the
a–b notation is a neater
translation. |
What
symbolises in a symbol, is that which is common to all the symbols
which could in accordance with the
rules of logic = syntactical rules for
manipulation of symbols, be substituted for it. |
The question whether a
proposition has
sense || Sinn can never depend on the
truth of another proposition
about a constituent of the first.
E.g. the question whether
(x). x = x has meaning || Sinn
can't depend on the question whether
(∃x). x = x is
true. It doesn't describe reality at
all, & deals therefore solely with symbols;
it says that they must symbolise,
but not what they symbolise. 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 = the general form of a proposition You thus introduce both a–b functions, identity & universality (the 3 fundamental constants) simultaneously. |
The
variable-proposition
p ⊃
tildap ¤ is not identical
with the
variable-proposition
~(p
. ~p).
The corresponding
universals would be identical. But the variable
proposition
~(p
. ~p) 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,
& b &
c: there is no correlation
whatsoever thus established. Of course, in the case of 2
pairs of terms united by the 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 proposition (or
function) occur is in another
proposition? The
proposition or function itself
can't possibly stand in relation to the other
symbols. For
this reason we must introduce functions as well as
|| names at once in our general
form of a proposition; explaining what is
meant, by assigning meaning to the fact that
the names stand between the |, & that the function stands
on the left of the names. |
It is true, in a sense, that logical
propositions are
‘postulates’– something which we
‘demand’; for we do demand a satisfactory
notation. |
A tautology
(not a logical
proposition) is not nonsense in the
same sense in which
e.g. a proposition
in which 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 or 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 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
the left of another, or that one colour is darker
|| than another it seems
to follow that it is so; & if
so this can only be if there is an internal relation
between the two; & we might express this by saying that the
form of the latter is part of the form of the
former. We might thus give a sense to the assertion that
logical laws are forms of thought |
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
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
which depends on experience. This
answers the question how we can know that we have
really got the most general form of a
proposition. 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”
says || besagt p’ to
p: it is just as impossible that
I should be a simple as that “p” should be. |
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