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:
- BaC (All cats are black): Universal affirmative (the lower case 'a' comes from the first vowel of the latin verb 'adfirmo'= to affirm)
- BiC (Some cats are black): Particular affirmative (the lower case 'i' comes from the second vowel of the latin verb 'adfirmo'= to affirm)
- BeC (No cats are black): Universal negative (the lower case 'e' comes from the first vowel of the latin verb 'nego'= to deny)
- BoC (Some cats are not black): Particular negative (the lower case 'o' comes from the second vowel of the latin verb 'nego'= to deny)
This way of formalizing Aristotelian logic was invented during the Middle Ages and was common until the XIXth century.
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'.
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.