What is the logic behind uncountable nouns?

Solution 1:

As Gregory Bateson put it in "Every Schoolboy Knows",

The division of the perceived universe into parts and wholes
is convenient, and may be necessary, but
no necessity determines how it shall be done.

It is tricky, and it's not ideal, but there you are. That's language.

In English, nouns that normally refer to objects that can be identified as units have different rules (singular indefinite article, pluralization) than other nouns that refer mostly to

• abstractions (leadership, competence, darkness, death)
• emotions (fear, joy, surprise -- really just one popular subset of abstractions)
• fluids (water, gin, lava)
• aggregates of small granularity (oats, wheat, rice, sand, gravel).

These are "mass nouns", and can be pluralized, or otherwise used like count nouns. As noted,
this means different kinds (types, varieties, brands, categories) of the otherwise mass noun.

In addition to this conventional countification of mass nouns, indicating varieties,
there is also a conventional massification of count nouns, indicating essence.

• You get a lot of car for your money when you buy X.
• After the bomb he ate went off, there was Godzilla all over Times Square.

This is English. But in many languages that use numeric classifiers (Malay, Burmese, Mandarin, Japanese), there's no distinction to speak of between mass and count -- all nouns are mass nouns, and if you need to count or otherwise distinguish what English would use a count noun for, you simply use a classifer instead. For instance, Japanese

• ni-hon no enpitsu ‘two pencils’
• ni-mai no kami ‘two sheets of paper’
• ni-ko no ringo ‘two apples’
• ni-ko no nasu ‘two [round] eggplants’
• ni-hon no nasu ‘two [long thin] eggplants’

English doesn't have numeric classifiers, so we can't use this strategy.

Solution 2:

In fact, mass nouns are routinely, and quite acceptably, treated as countable when they are taken to refer to individual items, collections, or quantities that are conceptually separable from one another:

Diner: What kinds of water do you have?

Server: In addition to tap water, we have three different bottled waters: Perrier, Aquafina, and Dasani.

Diner: My wife and I will have tap water, and the kids will each have a Coke.

Server: So that's two Cokes and two waters.

Similarly, signs offering meats and breads are commonplace at grocery stores, by the same rationale.

Solution 3:

There are some nouns whose plurality (or singularity) is expressed in an auxiliary fashion (this is, by counting an auxiliary noun). For example if there is corn on the table:

There are twenty kernels of corn on the table. There are twelve cobs of corn on the table. There are seven ears of corn on the table.

If there is bread on the table:

There are nines slices of bread on the table. The are six loaves of bread on the table.

So you see the counting is performed in an auxiliary fashion, but we can still count!