In which way has the modern advancement in science and humanities conflicted with Aristotelianism and, generally, what is a lowdown of Russell's opposition to it?
First, this claim:
I conclude that the Aristotelian doctrines with which we have been concerned in this chapter are wholly false, with the exception of the formal theory of the syllogism, which is unimportant.
This comes at the end of the chapter on Aristotle's logic. So, he is saying that Aristotle's logic is wholly false with the exception of the formal theory of the syllogism.
This claim is based on what Russell say are "formal defects" in Aristotle's logic. One example in this chapter of a formal defect is explained by Russell as follows:
(1) Formal defects. (…) Some of Aristotle's syllogisms are not valid (…) If I were to say: 'All golden mountains are mountains, all golden mountains are golden, therefore some mountains are golden,' my conclusion would be false, though in some sense my premisses would be true. -- Bertrand Russell, History of Western philosophy (1946)
My conclusion would be false, though in some sense my premisses would be true.
Russell's claim here is patently fallacious. He chooses to assess the premises as true "in some sense" and to assess the conclusion as false as a matter of fact. Thus, Russell's fallacy here is a shameless equivocation on the word "true", which certainly comes on top of all logical fallacies.
I conclude that Russell's criticism of Aristotle's logic was seriously biased.
Remember that Russell was one of the most brilliant intellect of the time. This is no mistake on Russell's part. It had to be a deliberate choice.
Any person in the present day who wishes to learn logic will be wasting his time if he reads Aristotle or any of his disciples.
The reality is that Russell has been largely ignored by scholars and logicians alike. Logicians keep coming back to Aristotle's logic essentially because it makes sense. It makes sense because it is simple enough that we can check for ourselves that his reasonings are logical, something no one can do in mathematical logic because it is too complicated.
Aristotle's logic is simple, but modern logicians often don't have the time to read it properly. It is frequent to find mistakes in modern commentaries on Prior Analytics for example. My personal experience is that every time I checked a claim by some modern professor of philosophy or some mathematician, including Russell, that Aristotle had made some mistake or committed some fallacy, it turned out that Aristotle was right and the modern logician was wrong. Russell is consistently slipshod about his criticisms of Aristotle. Not just me saying this, read John Corcoran, Robin Smith or Strawson.
Aristotle kept his logic simple. His syllogistic is really simple. All his reasonings can be shown to be trivial, somewhat on a par with the modus ponens or the modus tollens, including his reductio ad impossibile reasonings (always the same throughout Prior Analytics). And it is formally impeccable, notwithstanding Russell's bilious criticisms.
We should also keep in mind that Russell worked for a very long time on mathematical logic and was unable to extend Aristotle's syllogistic so as to be able to apply it to mathematics. He didn't extend it because he didn't even try. However, the result is that the kind of mathematical logic that Boole, Frege and Russell invented is contradictory with Aristotle's logic. Contradictory, not just something else. As such, Russell had to claim that Aristotle was plain wrong. The problem for Russell is that anybody can verify for themselves that Aristotle's logic is good. 100% good.
As to its limitations, it is funny to say that since it is really Russell who failed to extend Aristotle's logic to mathematics. Aristotle's logic as it is applies to mathematics just as well as to anything else. Limited, sure, wrong, no.
Here is a complement where modern logicians express the same point as I do here, only more diplomatically:
A major objection to this modeling of Aristotelian syntax is that it does not exactly reproduce the Aristotelian theorems; more specifically: By the rules of First-Order Predicate Logic one cannot, for example, prove the Law of Subalternation, A(S, P) → I(S, P), which plays a central role in Aristotle’s theory. Whereas our undergraduate texts today still use to offer a simple “solution” to this problem, called existential import, we know now that such auxiliary constructs have nothing to do with problems of Aristotelian logic, but solely of its inadequate translation into a modern framework. I agree with Nedzynski (1979):¹ “The problem of existential import developed along with the development of modern symbolic logic during the nineteenth century. The problem is peculiar to the standard predicate calculus. There never was a real problem of existential import within the traditional syllogistic logic—it was placed there in retrospect by the modern logicians.” — Klaus Glashoff (University of Lugano), An intensional Leibniz semantics for Aristotelian logic (The Review of Symbolic Logic, June 2010)
1 Nedzynski, T. G. (1979). Quantification, domains of discourse, and existence. Notre Dame Journal of Formal Logic
Academics are almost always very diplomatic, but here this is Bertrand Russell which is criticised:
"* (…) we know now that such auxiliary constructs have nothing to do with problems of Aristotelian logic, but solely of its inadequate translation into a modern framework*".
You have to wonder why my answers on logic and incidentally on mathematical logic are systematically either voted down or "closed" by some ayatollah. Maybe I am not sufficiently diplomatic, or perhaps diplomacy is only for show.