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 symbols, combined in certain ways, || yield a symbol of this description; & that they do shews something about these symbols.
     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 obviously says something about the kind of symbol you can get in this way.
     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 felt about types – e.g. as to the difference ¤ between things, facts, properties, relations. That || M is a thing can't be said: it is nonsense: but something is shewn by the symbol M. In the same way that a proposition is a subject-predicate proposition can't be said: but it is shewn by the symbol.
     :. 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’ they would be superfluous (tautologous) because what this tries to say is something which is already seen when you see M.
     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.

     (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 a relation is partly fixed by the type of a symbol of a thing, since a symbol of the latter type must occur in it.

     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 no proposition of the form xRy.
     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 with the proposition. In the case of 2 lines we can compare them in respect of their length without any convention & the comparison is automatic. But in our case the possibility of comparison depends upon the conventions by which we have given meanings to our simples (names & relations).]
     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.)

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 from the other. But the difference between a & b is not one of logical form, so that nothing will follow from this difference alone as to the interpretation of other symbols. Thus, e.g. p.q, p ⌵ q || seem symbols of exactly the same logical form in the a–b notation. Yet they say something entirely different; &, if you ask why, the answer seems to be: In the one case the scratch at the top has the || shape b, in the other the || shape a. ¤ The important thing is that the || interpretation of the form of the symbolism must be fixed by giving an interpretation to its logical properties, not by giving interpretations to particular scratches.

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 symbol, there also the connection is symbolising in the same way.
     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) ≡ the symbol of a property. For the symbol of a property e.g. ψx is that ψ stands to the left of a name form, & another symbol φ cannot possibly stand to the left of such a fact: if it could, we should have an illogical language, which is impossible.


      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 correspondence between the terms of different relations.

     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?)

     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

& space & time forms of intuition.

     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.