How would one go about symbolizing the sentence "some cats are black?"
Would I use A: All cats are black and N: No cats are black, in the notation [~A & ~N] so as to say "It is not the case that all cats are black and it is not the case that no cats are black"?
Does that sentence sufficiently imply "some cats are black"?
In Propositionals Logic 'Some cats are black', since it is atomic, will be formalised like this: 'P'.
In Predicate Logic 'Some cats are black' will be formalised like this: '∃x (Fx & Gx)', which should be read as 'There is at least one thing that is both a cat and black.
Yes, in propositional logic you could use ¬ A ∧ ¬ B where A = 'no cats are black' and B = 'all cats are black', but that's hardly an improvement from C = 'some cats are black', which is an atomic sentence (can't be split up) as well.
For things like this it's much easier to use first-order predicate logic and quantifiers: ∃c ∈ C [B(c)], where C is the set of all cats and B(c) ⇔ c is black. You can read this as 'there exists at least one element c in the set C of all cats for which B(c), that is, the cat being black, holds'.
Abstract: ∃v∈V [P(v)] ⇔ there exists an (at least one) v in V such that P(v).
Other quantifiers are ∃! (there exists exactly one ... such that ...) and ∀ (for all ... holds ...).
Commonest types of formalization of the sentence 'Some cats are black'
The Aristotelian way
One way - historically, the first one - to formalize your sentence apart from those already given above (i.e. 1.through atomic or negated atomic propositions and 2. through quantifiers in predicate logic, which I will re-sketch below) is the one which is commonly used in the particular case of dealing with syllogistic logic in the Aristotelian tradition. It amounts to the following:
If 'C' stands for the term 'cat' and 'B' for the term 'black', then 'BiC' stands for 'some cats are black' or, equivalently, '"black" is predicated of some of "cat"' (this is Aristotle's preferred formulation, which explains the subject-predicate inversion in the formalization: but some commentators maintain the more natural order, whereby your sentence would be formalized as 'CiB' -- personally I prefer to stick more closely to Aristotle's formulations).
To explain why the 'i' between 'C' and 'B', a sketch will help of the commonly used abbreviations within Aristotelian syllogistic:
This way of formalizing Aristotelian logic was invented during the Middle Ages and was common until the XIXth century.
Predicate logic
Today, if one is not dealing with Aristotelian logic, the most natural way to formalize this sort of sentences is through the quantificational machinery of predicate logic: as suggested by others, the resulting formalization would then be '∃x (Cx & Bx)', where in this case 'C...' and 'B...' are predicative letters, i.e. stand for predicates and not for terms (as in the Aristotelian logic), and these two predicates are respectively assigned the interpretations '...is a cat' and '...is black'.
Propositional logic
If instead you choose to formalize it as a "one-block proposition", then you'll have 'P' (atomic sentence letter), where 'P' directly stands for an atomic (i.e. unanalyzable) sentence, and the interpretation assigned to it will be 'Some cats are black'. By so doing, however, you'll lose in expressivity, e.g. you won't be able to define rules of inference from a sentence like 'All men are mortal' to a sentence like 'Socrates is mortal'.
Remaining within propositional calculus, I think you shouldn't use the formalization '~A & ~N', where 'A' stands for 'All cats are black' and 'N' for 'No cats are black'.
First reason: to get a logical equivalent of 'Some cats are black' you don't need to negate 'All cats are black' (unless you read 'some' as implying the negation of 'all', which is not the commonest view: when you say that some cats are black you are not necessarily excluding the possibility that all cats be black). Indeed, it is enough to negate 'No cats are black', namely to write '~N'.
Second reason: even in this latter case, I would say that what you'd be formalizing is not the English sentence 'Some cats are black', but the English sentence 'It is not the case that no cats are black'. But a formalization should be as closest as possible to the sentence it formalizes. Thus, formalizing 'Some cats are black' as '~N' would be justified only if you were forced to use a hyper-regimented language: for example, a language with only one atomic sentence letter, namely 'N' itself, and the interpretation 'No cats are black' forcefully assigned to it.
"Some cats are black" is equivalent to "Not (no cats are black)". It doesn't tell you which one of "All cats are black" or "Not all cats are black" is true.
So to answer your question: No, "Some cats are black" does not correspond to the formula ~A & ~N; instead it corresponds to the formula ~N.
However, if someone tells you "Some cats are black", you are going to be tempted to conclude that probably not all cats are black, and you would be right. But this is not a consequence of the logic proposition "Some cats are black".
In real life, when someone pronounces a sentence, you get two pieces of information: (1) The logic proposition intrinsic to this sentence; (2) The fact that this person chose to pronounce this sentence, rather than a different sentence, in this context.
People tend to be precise about some things, so if someone chooses to tell you "Some cats are black", then you can deduce the extra information "Probably not all cats are black, otherwise this person would have chosen to use the words 'All cats are black', and not the words 'Some cats are black'". But that is an extra piece of information, and is not implied by the logic proposition "Some cats are black".
