Consider the two arguments:All A's are B's
All parts of A's are parts of B'sMost C's are A's
Most C's are B's
Some A's are B'sTextbooks often charge that traditional logic is "inadequate" because it can- not accommodate patently valid arguments like the first. But this holds equally true of modern quantificational logic itself, which cannot accommodate the second. Powerful tool though it is, quantificational logic is unequal to certain childishly simple valid arguments, which have featured in the logical literature for over a century (i.e., since the days of De Morgan and Boole). Plurative syllogisms afford an interesting instance of an inferential task in which the powerful machinery of quantificational logic fails us, but to which the humble technique of Venn diagrams proves adequate.
[Nicholas Rescher and Neil Gallagher, "Venn Diagrams for Plurative Syllogisms", Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 16, No. 4 (Jun., 1965), p. 55.]
Rescher elsewhere notes (in "Plurality-Quantification", if I am recalling correctly) that plurative syllogisms are even more difficult for standard predicate calculus than they seem. You might think, for instance, that you could solve the problem (as you would with syllogistic) by adding a 'Most' quantifier, (Mx), which in the predicate calculus would have to mean something like 'For most of the individuals x of the universe of discourse'. But it turns out, given how standard predicate calculus structures propositions, that simply adding a Most quantifier to the predicate calculus doesn't make it possible to say that Most S are P. You get something that looks superficially like it, but doesn't have the right logical properties; in other words, the standard pred. calc. rules for quantifiers and how they are used are tailored specifically for universal quantifiers and existential quantifiers and how those quantifiers, specifically, relate to each other, so merely adding a plurative quantifier doesn't get you something that works right. You'd have to rebuild the system from the ground up to get things right.
As Rescher and Gallagher note, people have a long history of criticizing basic syllogistic and class logics for not having an immediately obvious way to handle relational arguments (like the first in Rescher's and Gallagher's comment above), while at the same time just ignoring the basic kinds of argument that the predicate calculus doesn't directly accommodate. In reality, of course, this is a childish way of arguing; your logical system always has a purpose, and it just doesn't have to deal with things that are not part of the purpose; that you might use a different logical system for a different thing has no bearing on the value of any logical system. But people often seem allergic to this sort of logical pluralism; they really want there to be a ONE TRUE LOGIC, in the sense of a logical system that covers absolutely everything logical that you might want to do, or to which everything logical could be cleanly reduced. But there isn't one, and even if there were, we don't have it.
This, of course, is different from holding that all logical systems are equally good. For one, given a particular logical purpose, not all logical systems are equally good means to that end. And, perhaps more importantly, not all logical purposes are equally important. It would be entirely possible to argue that one logical system, or one family of logical systems, is the primary logical system, in the sense of being the best logical system for the most basic or the most important thing logic can be used for. No doubt there would be some controversy about it, but you could very well argue it. But saying that it is the best instrument for the most important things is not the same as saying that it can do everything important.
