1. Analysis. This is one of the earliest texts in what has come to be called "analytic philosophy," so it is of interest to notice that Russell says that what he is doing is "analysis."
The contrast between analysis and synthesis has been important in philosophy ever since the ancient Greeks, and became especially important in the early modern period. (It is discussed by Hobbes, Descartes, and Kant, among others. Think of Descartes's "analytic geometry" and Kant's distinction between analytic and synthetic judgments.)
The contrast has been drawn in at least two different ways. (See the article on Analysis in the Stanford Encyclopedia of Philosophy.)
a. Analysis as regression. Analysis attempts to start with derived propositions and find the more basic principles from which they are derived. In geometry, for instance, we might start with a generalization, e.g. the Pythagorean theorem, and try to discover axioms from which it can be derived. (Since we don't know what the axioms are when we begin, analysis in this sense is also described as a method of discovery.) Once we know what the axioms are, we can use them to prove less basic propositions. This is the process of synthesis, which moves in the opposite direction from that of analysis.
b. Analysis as decomposition. Analysis of a concept consists of trying to discover the more basic concepts of which it is composed, i.e. trying to find a definition of the concept in terms of more basic concepts. This seems to be the sense of analysis that Russell primarily has in mind, except that he is interested in decomposing not only individual concepts but also sentences or propositions to try to discover the most basic elements of which they are composed.
2. Atoms. Russell hopes that analysis will lead to discovery of the most basic, irreducible components out of which any judgment or proposition is built. These will be the atoms out of which propositions are constructed. But he stresses that these atoms are the result of a logical (i.e. philosophical, non-empirical) investigation, so they are logical atoms rather than physical atoms. Thus the name logical atomism.
3. Atomism vs. monism. It's a little hard to remember now, but at the time Russell was writing, the dominant intellectual movement in Britain was Hegelianism or absolute idealism. One thesis of the Hegelians was monism: when you properly understand the world, you see that it consists of only one thing; any attempt to identify parts of the whole inevitably falsifies it. Russell explicitly rejects this view. His starting point is the assumption that there are in fact many things.
4. Logic and ontology. Although Russell spends a good deal of time on issues in philosophical logic, i.e. roughly the philosophy of language, his ultimate aim is to illuminate ontology. His starting point is the analysis of the way we think and talk about the world, but in the end he thinks that this is the best guide we have to the actual nature of the world. So the logical atoms of our thought and speech about the world will correspond to the most basic ontological categories of the real world. So we have an early example of linguistic philosophy, i.e. of the idea that philosophical questions reduce to questions of language.
5. Ordinary vs. ideal language. It is important, though, that Russell doesn't think we should simply take language as we find it as the basis of our ontology. Rather, we need to construct a better language, one that makes the structure of thought clearer than ordinary language does. Note his discussion of a "logically perfect language" (lecture 2, p. 520). Only such a logically perfect language will be a reliable guide to ontology. (In this he is very different from his student and later close associate Ludwig Wittgenstein, who is often called an "ordinary-language philosopher." Later in the semester we will read Wittgenstein and also J. L. Austin, another ordinary-language philosopher.)
6. A side note. Lecture 1, p. 498: "I should like, if time were longer and if I knew more than I do, to spend a whole lecture on the conception of vagueness. I think vagueness is very much more important in the theory of knowledge than you would judge it to be from the writings of most people." It's been almost a hundred years since Russell wrote these words, and in the last decade or two, vagueness has become a very hot topic! Here is the first paragraph of Richard Rorty's review of Scott Soames' two-volume history of analytic philosophy:
‘I had hoped my department would hire somebody in the history of philosophy,’ my friend lamented, ‘but my colleagues decided that we needed somebody who was contributing to the literature on vagueness.’
‘The literature on what?’ I asked.
‘Dick,’ he replied, exasperated, ‘you’re really out of it. You don’t realise: vagueness is huge.’
1. One kind of atom in language and thought: names. The corresponding ontological atom: particulars.
2. The other kind of atom in language and thought: predicates (one-place, two-place, etc.), which correspond to properties and relations.
A problem: the things we ordinarily think of as particulars aren't really. People, animals, rocks, rivers, etc. are not simple enough to be particulars in Russell's sense. (lecture 2, pp. 511-12: Russell's interesting argument that "Socrates" and "Picadilly" do not name particulars.) Similarly, the things we ordinarily call proper names aren't really (because they don't refer to real particulars). (Lecture 2, p. 524: "The only words one does use as names in the logical sense are words like 'this' or 'that'.")
3. The simplest kind of sentence, an atomic sentence, is the result of combining n names with an n-place predicate. This corresponds to the simplest kind of fact.
The world can't just consist of things. It also has to contain facts. (Compare Wittgenstein: "The world is all that is the case.") Argument: propositions are true or false. They must be made true or false by something in the real world. But they can't be made true or false by particulars. (It isn't just Socrates that makes true the proposition that Socrates was obnoxious: rather, it is a fact about Socrates.)
4. So we have linguistic atoms and ontological atoms. [Terminology could be a bit confusing here: an "atomic sentence" is not a linguistic atom, and facts are not ontological atoms. Rather, an atomic sentence can be decomposed into more basic parts; and Russell things that correspondingly, facts can be decomposed into more ontologically basic elements, namely particulars, properties, and relations.] What is the relation between the two? Russell calls it meaning: "the word 'Socrates', you will say, means a certain man; the world 'mortal' means a certain quality; and the sentence 'Socrates is mortal' means a certain fact" (p. 46). More generally, names mean particulars; predicates mean properties; propositions mean facts. But Russell points out that these three kinds of "meaning" are radically different from one another. In particular, propositions are not names for facts. Argument: each fact corresponds to two propositions, one that it makes true and one that it makes false. (Example: the fact that it is raining makes the proposition that it is raining true; it also makes the proposition that it is not raining false.) There is nothing that corresponds to this duality in the case of names. Therefore the relation between a name and the particular it names is fundamentally different from the relation between a proposition and the fact that makes it true or false.
[A side comment: if you read chapter one of Lycan, note that Russell's treatment of names apparently assumes that meaning = reference.]
1. There are logical compounds of atomic propositions: the sky is blue and grass is green. According to Russell, there are not conjunctive facts that correspond to such propositions. Rather, "The sky is blue and grass is green" is made true by two facts, namely that the sky is blue and that grass is green. There is no need to posit a third, conjunctive fact in addition to those two.
2. On the other hand, Russell does think that there must be negative facts. (He mentions that this view almost started a riot when he presented it at Harvard!) His reasons are interesting -- basically, that we need negative facts to explain what makes true negative propositions true. (See pp. 74-48, including R's criticism of the rival account of Demos.)
Russell's arguments here are a bit obscure. A primary assumption is that for each true sentence there must be a fact that makes it true, and for each false sentence there must be a fact that makes it false. Then the question is: what fact makes a true negative sentence true, and what fact makes a false positive proposition false?
I think Russell's argument is something like this. The most obvious possible answer to the question is: negative facts. (In this case, the fact that Socrates is not alive, however exactly that is to be analyzed.) Now, we should only reject this answer if we can find an alternative account. The only alternative account Russell is aware of or can think of is something along the lines of Demos's account. But Russell thinks that Demos's account is less parsimonious than his own. Demos's account is roughly: "Not P" is true [and P is false] iff there is a fact Q such that Q is incompatible with P.
One thing Russell points out is (more or less) that we're being careless with our Ps and Qs here, not distinguishing clearly between propositions (which are linguistic) and facts (which are in the world). Suppose P and Q are propositions. Then Demos thinks that "Not P" is true iff there is a proposition Q such that Q is true and Q is incompatible with P. But that doesn't say anything about what in the world makes "Not P" true: it's just about linguistic entities and relations. What we want to know is what fact makes "Not P" true.
On Demos's account, this fact is whatever fact makes Q true. Call the fact that makes Q true f(Q) for convenience. Then the account has to be something like: The fact that makes "Not P" true is f(Q), together with the fact that f(Q) is incompatible with . . . ??? But as Russell argues, incompatibility is a relation between propositions, not between facts.
This leads Russell to (I think) two criticisms of Demos. First, it's not clear that his account can be developed systematically; the objective incompatibility relation would need to be explained clearly. But second, supposing that the account could be made clear, Russell thinks that it will be at least as complex as his own, because it involves not only simple facts but also objective relations of incompatibility. He sees this as being at least as undesirable as recognizing negative facts in the first place.
Two questions about Russell's view of negative facts. (1) I'm not sure what the process of analysis is supposed to reveal when turned on negative facts. The fact that Socrates is alive can, Russell thinks, be decomposed into a property and a particular: <Aliveness, Socrates>. But I'm not sure what he thinks the components of a simple negative fact are. A property, a particular, and negation??
(2) What is wrong with this alternative account? The alternative would say: there isn't a fact that makes negative sentences true. Rather, "Socrates is not alive" is made true by the absence of a fact to make "Socrates is alive" true. What makes "Socrates is not alive" true is not anything the world contains, but rather that it fails to contain something, namely the fact that Socrates is alive.
However, Russell might respond that it's hard to see how to avoid facts about the absence of facts. Isn't it a fact that there is no fact that Socrates is alive? And isn't that a negative fact? One possible response here might be to treat such facts as meta-facts that are defined in terms of more basic facts, so that they don't add any metaphysical bloat. Perhaps something similar could be done with things like conjunctive and disjunctive facts.
1. Possibility of reducing all the propositional connectives to "nand" or the "Scheffer stroke." (Russell doesn't use either of these names, but does explain in some detail how to define "and," "not," "if . . . then" etc. in terms of a single connective.)
2. Distinction between definition and analysis (and the claim that you can't analyze "red"): lecture 2, pp. 515-516.
3. Quotable quote defining philosophy: "[T]he point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it" (lecture 2, p. 514).