Determiners are the basic linguistic expressions relating predicates: ''all P1 are P2'', ''five or more P1 are P2'', ''not enough P1 are P2'', etc.. Within this class are found, in particular, both the traditional and non- standard logical quantifiers. Not all set-theoretically possible relations between predicates are actually expressible in natural language. For a start, a simple proof is presented of the Keenan \& Stavi theorem, characterizing the latter as those satisfying the mathematical condition of conservativity. Next, various additional constraints are proposed to focus on the ''logical'' determiner expressions: notably, postulates of context-neutrality and topic-neutrality (permutation invariance). Classification theorems for the remaining determiners are provided, some of them in terms of various kinds of inferential behaviour. These results are then used to evaluate certain current proposals for ''semantic universals'', i.e., broad linguistic constraints on the distribution of determiner expressions in human languages. Finally, the previous notions and results are extended to non- permutation-invariant determiners, to continuous (as opposed to the usual countable) forms of quantification - and, most generally, the enterprise is generalized so as to apply to arbitrary categories of linguistic expressions.