Reading "The philosophy of logical atomism" I come to think it's the must book that you should read to understand Wittgenstein's Tractatus. Russell explains very clearly what seems that he and Wittgenstein discussed about the nature of propositions, facts and so on (main topics of his book) behind the pages of the tractatus. Surprisingly to me he even says "A very great deal of what I am saying in this course of lectures consists of ideas which I derived from my friend Wittgenstein".
But these lectures had been given before the publishing of the tractatus. And I find it very enjoyable to trace the clues developed into the logically mystified book. For example. Russell says "In order to understand a name for a particular, the only thing necessary is to be acquainted with that particular". So, for example only through seeing a white spot on the wall we understand the proposition "This is white". But there happens to be someone arguing that it may be identical with "This and that have the same colour" or "This is identical in colour with that". Russell seems not take it seriously. But this consideration will lead us to think like this: In order to understand a name for a particular(little patch of colour, for example) we know colour in general. That is, if we don't know what colour is in general we don't know what white is. In another words we can say "When we see a colour we see it in the colour space", seemingly leading to Wittgenstein's logical space.
It seems very confusing. Because Russell insists that we understand only symbols(names, for example "white") and only so through being acquainted with particulars. But it will be not possible we are acquainted with, for example a white spot(a particular) unless we know the knowledge of colour, which is expressed in colour names. When you point your finger at a white spot and say "This is white", it's always possible that I response with "White? What do you mean by that?"
Anyway I am just in the middle of Russell's lectures and don't have the tractatus. I'm not sure if I got a point here or not and it doesn't matter for now.
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3. Atomic and molecular proposition
"Knowing a particular" merely means acquaintance with that particular and is presupposed in the understanding of any proposition in which that particular is mentioned.
The only thing you can really understand (in the strict sense of the word) is a symbol, and to understand a symbol is to know what it stands for.
Understanding a predicate is quite a different thing from understanding a name. To understand a name you must be acquainted with the particular of which it is a name, and you must know that it is the name of that particular.
To understand "red" , for instance, is to understand what is meant by saying that a thing is red. You have to bring in the form of a proposition. You do not have to know, concerning any particular "this", that "This is red" but you have to know what is the meaning of saying that anything is red. When you understand "red" it means that you understand propositions of the form that "x is red". It is in the fact that a predicate can never occur except as a predicate(the theory of types).
Exactly the same applies to relations, and in fact all those things that are not particulars. Take, e.g., "before" in "x is before y": you understand "before" when you understand what that would mean if x and y were given.
If anyone is of the opinion that there is reason to try to get on without subject-predicate propositions, all that is necessary is to take some standard red thing and have a relation which one might call "colour-likeness", sameness of colour, which would be a direct relation, not consisting in having a certain colour. You can then define the things which are red, as all the things that have colour-likeness to this standard thing. You can perfectly well do in that way a formal reduction of predicates to relations. There is no objection to that either empirically or logically.
I define an atomic proposition as one which contains a single verb. Now there are two different lines of complication in proceeding from there to more complex propositions: Molecular propositions and one proposion containing two or more verbs.
When the truth or falsehood of the molecular proposition depends only on the truth or falsehood of the propositions that enter into, I call these things truth-functions of propositions.
I do not see any reason to suppose that there is a complexity in the facts corresponding to these molecular propositions, because the correspondence of a molecular proposition with facts is of a different sort from the correspondence of an atomic proposition with a fact.
Are there negative facts? Are there such facts as you might call the fact that "Socrates is not alive?" I have assumed that if you say "Socrates is alive", there is corresponding to that proposition in the real world the fact that Socrates is not alive.
Mr. Demos pointed out:
1). A negative proposition is not in any way dependent on a cognitive subject for its definition. When I say "Socrates is not alive", I am merely expressing disbelief in the proposition that Socrates is alive. You have got to find something or other in the real world to make this belief true, and the only question is what. To this I agree.
2). Negative proposition must not be taken at its face value. You cannot regard the statement "Socrates is not alive" as being an expression of a fact in the same sort of direct way in which "Socrates is human" would be an expression of a fact. His argument for that is soley that he cannot believe that there are negative facts in the world.
3). When the word "not" occurs, it cannot be taken as a qualification of the predicate. In all cases where a "not" comes in, the "not" has to be taken to apply to the whole proposition. "Not-p" is the proper formula. I do not entirely agree with it.
According to Mr. Demos we have come now to the question, how are we really to interpret "not-p", and below is his suggested definition:
"not-p" means "There is a proposition q which is true and is incompatible with q."
I find it very difficult to believe Mr. Demos' theory. Because incompatibility is not between facts but between propositions. I think you will find that it is simpler to take negative facts as facts, to assume that "Socrates is not alive" is really an objective fact in the same sense in which "Socrates is human" is a fact. Otherwise you will find it so difficult to say what it is that corresponds to proposition. When you have a false positive proposition, say "Socrates is alive", it is false because of a fact in the real world.
Q: "Socraetes is dead" is a positive or a negative fact?
Mr. Russell: It is two statements rolled into one: "Socrates was alive" and "Socrates is not alive".