Language and Ontology
Russell's model for a "logically perfect language" is the language of symbolic logic. I will use the "blocks language" in Barwise and Etchemendy, Language, Proof, and Logic, to illustrate.
|names: a, b, c, d, e, f||particulars|
(arity 1: Tet, Dodec, Large, Small, etc.
arity 2: Larger, LeftOf, etc.)
|atomic sentences (Tet(a), Larger(a, b))||atomic facts (positive)|
|negations of atomic sentences (¬Tet(a), ¬Larger(a, b))||atomic facts (negative)|
|truth-functional compounds (Tet(a) & Large(a), Tet(a) v Large(a))||no new facts needed|
|belief sentences (Othello believes that Desdemona loves Cassio)||a new kind of fact; not a relation to a fact (since they can be false); not a relation to a proposition (since these don't really exist) [e.g. lecture 4, p. 58]|
|propositional functions (Tet(x) ∧ Large(x)) [lecture 5, p. 192]||don't express facts (incomplete)|
|universally quantified sentences ∀x (Tet(x) → Large(x))||general facts [argument that these are irreducible: lecture 5, pp. 198-199]|
|existentially quantified sentences ∃x (Tet(x) ∧ Large(x))||existence facts|
Necessity, possibility, impossibility
Russell says these notions only apply to propositional functions, and identifies them with "true of everything," "true of something," and "true of nothing" [lecture 5, p. 193]. I think this is a serious mistake. (He also says they mean the same as "always true," "sometimes true," and "never true," but that compounds the mistake with still another! Being always true is different from being true of every object, and both of them are different from being necessary.) See pp. 96-97. (The confusion causes further problems in the discussion at p. 108, where Russell says that 'existence' can't be a predicate because there's no point in having a predicate that can't be false of anything. But it's simply not the case that 'exists' cannot be false of anything! It isn't false of any (existing!) thing, but it could be: for instance, I might never have existed.)
Russell claims that it makes no sense to say that a particular exists. The concept of existence only makes sense in relation to propositional functions, when we say there exists something that satisfies the function. (What's a propositional function? Something like "Tet(x)" i.e. "x is a tetrahedron." You can think of this as a function that takes particulars as arguments and returns a proposition as value. A particular satisfies a propositional function if the proposition which is the value of the propositional function for that argument is true.)
Ordinary vs. Ideal Language
Note the critique of ordinary language at lecture 5, p. 197 [Open Court ed. p. 100] (and other places).
Names and Descriptions
In lecture VI, Russell offers an interesting and very influential account of names and descriptions.
The first point he makes is that descriptions work in a way completely unlike the way names work. Names just pick out a particular individual, and that's all they do. In a logically perfect language, there won't be any names that don't refer to anything. In an ordinary language, if we have a name that doesn't refer to anything, it's meaningless: its meaning is just the object it picks out, so if there's no object, there's no meaning.
Descriptions are different. The description "the author of 'Waverly'" is not a name: it doesn't simply pick out an object.
Russell's main argument for this conclusion is the possibility of informative identity sentences. I can meaningfully and informatively say "Scott is the author of Waverly." But Russell claims that if "the author of Waverly" were a name, then this sentence would have to be either tautological or false (if it were meaningful at all). (See lecture 6, p. 212 [open court ed. pp. 111-113].)
How do definite descriptions work, then? Russell's view is that they are similar to the general propositions he has discussed earlier: a sentence containing a definite description really says something about a propositional function. In the case of "the author of Waverly" the propositional function is "x wrote Waverly." To say that Scott is the author of Waverly is to say: (1) at least one object makes the propositional function "x wrote Waverly" true; (2) at most one object makes "x wrote Waverly" true; (3) Scott makes the propositional function "x wrote Waverly" true.
This is Russell's famous analysis of definite descriptions. Symbolically, Russell analyses "Scott is the author of Waverly" like this: ∃x (WroteWaverly(x) ∧ ∀y (WroteWaverly(y) → y = x) ∧ x = scott)."
In addition to this analysis of descriptions, Russell makes an extremely interesting and important proposal about proper names. That is, he says that most of the "proper names" in ordinary language are not names in the logical sense at all! The main argument for this has to do with the possibility of true negative existentials. (See lecture 6, p. 208.)
A "negative existential" is a sentence like "Romulus does not exist." Russell argues that if 'Romulus' were a name in the logical sense, such a sentence could never be true. It would be either false (if the name denoted something) or meaningless (if it didn't denote anything). So the only way to understand such a sentence is to take 'Romulus' to be, not a name, but a disguised description. Then we can apply his analysis of descriptions to it.
On this analysis, "Romulus does not exist" can be analyzed, if the description we associate with 'Romulus' is F, as ¬Ex F(x).