You need only read pp. 303-14. We will not discuss the final two sections from Marcus's paper, "The Interpretation of Quantification" and "Semantic Considerations". But if you have some knowledge of and experience with modal logic, you should read the former, at least. (The basic idea here is that Marcus proposes to treat quantification substitutionally.) It is also not essential that you understand the section "Intensional Languages" (pp. 303-7): You should simply skim it if it is too difficult and trust what I say about it below.
You need only read pp. 323-7. The rest is concerned with the parts of Marcus's paper we will not discuss.
There is also a record of the discussion that followed this exchange (JSTOR). It makes for interesting reading: In the audience was a Harvard senior named "Saul Kripke".
Ruth Barcan Marcus was one of the early advocates of quantified modal logic. In this paper, she responds to Quine's arguments against the intelligibility of such systems. It's a difficult paper, but also a very important one.
Marcus begins by explaining how she understands the notion of intensionality. Though she expresses some reservations on this point, she regards identity, strictly speaking, as defined in terms of indiscernibility:
xIy iff ∀F(Fx eq Fy)
Note here that this quantifies over properties, and not just over objects. The relation 'eq' is supposed to be some equivalence relation between propositions. As Marcus says, it is a substantial question what eq should be taken to be: material equivalence (≡) or necessary equivalence (which she writes as a quadruple bar).
Marcus then goes on to introduce what she calls extensionality principles. These are principles of the form: x eq y → xIy. I.e., principles that say that identity may be inferred from some other relation. What Marcus has in mind emerges in her discussion of Quine. In first-order logic, one has the schema:
(p ≡ q) → (p' ≡ q')
where p' and q' are the result of making some substutution in p and q. As Marcus notes, this fails if p and q may contain "believes" (or any other non-truth-functional operator). And the problem, as she sees it, is that the relation "p ≡ q" is just too weak: We need not truth-functional equivalence, but identity of propositions.
Marcus's response to Quine is contained in the section "Identity and Substitution in Modal Logic". She begins with some remarks about quantified modal logic, noting in particular the provablity of the necessity of identity: x = y → □(x = y). Marcus then turns to Quine's worry about this: Surely it isn't necessary that the morning star is the evening star?
Marcus responds that if aIb is true, then a and b are names of the same thing, and so aIb must say the same thing as aIa. Since the latter is a "tautology", then so is the former. But this, she says, is true only if a and b are really names. If, on the other hand, one of a and b isn't a name but instead a description, then aIb need not be necessary.
Marcus does not commit herself to any view of descriptions. Explain why, if a and b are descriptions (to be treated a la Russell), then aIb can uncontroversially express a contingent truth.
On the other hand, if a and b are descriptions understood as Strawson would have us understand them, then can aIb express a contingent truth?
Marcus then explains that she regards names as "mere tags", which have no meaning of their own, but merely serve to "tag" things. It follows that, if two expressions "really are names for the same thing, then they must be intersubstitutable in every context" (p. 309). (Quotational contexts are presumably excepted, on the ground that the names do not really occur there at all.) Crucially, this includes epistemic contexts. Thus, Marcus explicitly denies that it could ever be "an empirical fact" that aIb, if a and b are really names. So, for example, if "Hesperus" and "Phosphorous" really are names, then Marcus is denying that it is an empirical fact that Hepserus is Phosphorous.
Marcus's views commit her to the claim that it is impossible for "N believes that a is F" to be true but "N believes that b is F" to be false, if aIb and if a and b are really names. Why? And what options does Marcus have in the face of the obvious alleged counter-examples?
The discussion on pp. 311-4 is confusing, and I confess that I am not sure that I understand it. Marcus seems to be suggesting that, no matter how we handle descriptions, if we regard (say) "the evening star" and "the morning star" as being descriptions, then we should not regard "the evening star is the morning star" as a true identity, since the two expressions are not intersubstitutable in all contexts. Rather, we should regard this as being a "weaker" form of equivalence than identity, one that permits substitution only in some contexts. (It might be defined as: ∀F(Fx ≡ Fy).)
This seems to be the key to Marcus's response to Quine's example about the number of planets, on pp. 314-5: Since "the number of planets" is a description, "9 = the number of planets" cannot be regarded as a true identity, and hence the substitution Quine wants to make is invalid.
For what it's worth, it seems to me that Marcus would be best off by far simply adopting Russell's view of descriptions and insisting, on that ground, that the substitution is not permitted. How would that argument go?