# I don't understand why Kant's definition of an analytic concept can't be expressed in terms of intensional logic

I've been studying the SEP article about intensional logic, and I think I have a good grasp of what "intension" means. The intension of a term like "bachelor" would be a function that designates a set of objects in each possible world. So bachelor is defined as unmarried men, and so the intension of "unmarried men" would be the intersection of the set of unmarried objects and the set of men, in each possible world.

So from the Critique of Pure Reason, Kant wrote:

In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic.

I don't understand why we can't just understand concepts contained within concepts as intensions contained within intensions, such that an intension B is contained in intension A, if and only if, in every possible world there is no object that B designates that A doesn't also designate in that world.

Does anyone know why this doesn't work? There seems to be a lot of skepticism about the analytic/synthetic distinction online.

Everything you said seems fine. You propose to think of `intensions` as functions from possible worlds to subsets of the domain of individuals. That's a pretty standard way of thinking about intensions (goes back to at least Carnap's work in semantics). That particular criterion of analyticity, namely that:
doesn't require an intensional treatment, because concept(x) could simply denote the `extension` of x in a given world. That being said, I think an intensional interpretation of concepts is much stronger and therefore less susceptible to criticism. The criterion above can be slightly modified to work in a possible worlds framework exactly as you proposed.