So even if our brains don't naturally think in terms of necessary-and-sufficient, it's not immediately obvious that it's stupid and impossible to try to come up with necessary-and-sufficient conditions for our categories.

I haven't claimed this, and in fact have specifically denied it. But it is apparently a common reading of my post, so I've added a sentence toward the end to make this clear. Sorry about that.

maybe that's not what conceptual analysts are doing.

I think it is, in many cases. Maybe the clearest argument for this is from Ramsey (1992). I'll quote an extended passage below, though you may want to skip to the part that reads: "At first blush, it might seem a little odd to suppose that conceptual analysis involves any presuppositions about the way our minds work..."

[Discussions of the conflict between conceptual analysis and the psychology of concepts] have been floating around philosophical circles for some time. Perhaps the best known expression of these sentiments is Wittgenstein's discussion of family resemblance concepts in the Investigations, though similar ideas can be found in the writings of other philosophers, including Hilary Putnam (1962), Peter Achinstein (1968), Harold Brown (1988), Terence Horgan (1990), and in particular, Stephen Stich (1990, [1992])...

Conceptual analysis and its underlying assumptions

It would be a bit of an understatement to claim that conceptual analysis has been an important aspect of Western philosophy. Since the writings of Plato, in which Socrates and his cohorts repeatedly attempt to discern the true essence of matters such as piety and justice, philosophers have been in the business of proposing and (more typically) attacking definitions for a huge range of abstract notions. These include such concepts as knowledge, causation, rationality, action, belief, person, justification and morality (to name just a few)... But how does this enterprise get carried out and, perhaps more importantly, what are its underlying assumptions about the way we represent concepts?

Two criteria for definitions

Answering the first question -- i.e., how does conceptual analysis get done? -- is, at first glance, relatively easy: philosophers propose and reject definitions for a given abstract concept by thinking hard about intuitive instances of the concept and trying to determine what their essential properties might be. However, this characterization is really too vague to tell us anything useful. Perhaps a better way to gain insight into conceptual analysis is to consider what is normally expected of the definitions put forth. By looking at the criteria philosophers use for definitions, we can get a firmer grasp on what philosophers are up to and perhaps uncover some of the presuppositions lurking behind this enterprise.

Naturally, there are a number of different criteria commonly invoked by philosophers searching for definitions. Here, I'll focus upon only two... The first of these requirements is that the definitions be relatively straightforward and simple. Indeed, a popular syntactic form assumed for definitions is that of a small set of properties regarded as individually necessary and jointly sufficient for the concept in question. Hence, more often than not philosophical definitions take a syntactic form in which the notorious (at least among copy-editors) "iff" is followed by a short conjunction of properties. Thus, X is knowledge if and only if X is justified, true belief or X is acting freely if and only if X is doing what he or she wants. As with explanatory theories in science, a popular underlying assumption of conceptual analysis is that overly complex and unwieldy definitions are defective, or ad-hocish, even when no better definition is immediately available. If an analysis yields a definition that is highly disjunctive, heavily qualified or involves a number of conditions, a common sentiment is that the philosopher hasn't gotten it right yet. Accordingly, different analyses are typically regarded as competitors, and, for the most part, few people take seriously the idea that the correct analysis might be one involving a disjunctive combination of these alternate definitions. To borrow a technical phrase from Jerry Fodor, analyses of this complex sort are commonly regarded as "yucky'. For many philosophers, a proposed definition should be short and simple.

A second criterion definitions are generally expected to meet is a concern not about their form, but their degree of robustness. If a definition is to count as a real definition, then it is generally assumed that it cannot admit of any intuitive counterexamples. Hence, as we all learned in introductory philosophy, the standard way to gun down a proposed analysis is to find either a noninstance of the concept that possesses the definitional properties in question -- thereby showing that the defining properties are insufficient to capture the concept -- or an instance of the concept that doesn't possess the definitional properties -- thereby showing the defining properties aren't necessary. If counterexamples of this sort can be found, then the proposed definition is typically regarded as inadequate...

Hence, definitions sought by philosophers engaged in conceptual analysis typically must pass at least two tests: they must be relatively simple -- generally a conjunction of individually necessary and jointly sufficient properties -- and it must not admit of any intuitive counterexamples. With this in mind, we can now turn to the question of psychological presuppositions.

Psychological presuppositions of conceptual analysis

At first blush, it might seem a little odd to suppose that conceptual analysis involves any presuppositions about the way our minds work. After all, if people are interested in defining notions like justice or causation, then it's justice or causation that they are concerned with -- not human psychology. Nonetheless, when we look more closely at the criteria for definitions I've just sketched, we can indeed find lurking in the background certain assumptions about human cognition. Perhaps the easiest way to see this is to consider the significant role intuitive categorization judgments play in this type of philosophy. Notice, for example, that for either type of counterexample to actually count as a counterexample, there are going to have to be fairly strong and widely shared intuitions that some particular thing or event either is or is not an instance of the concept in question. In other words, the process of appraising definitions requires comparing and contrasting the definitional set of properties with intuitively judged instances and non-instances of the target concept. Without these intuitive categorization judgments, conceptual analysis as a practice could never get off the ground.

Because of this important role of intuitive judgments, conceptual analysis can't avoid being committed to certain assumptions about the nature of our cognitive system. One such assumption is that there is considerable overlap in the sorts of intuitive categorization judgments that different people make. Without this consensus, an intuitive counterexample for one individual would fail to be an intuitive counterexample for another individual, and no single definition could be agreed upon. Moreover, given that definitions are expected to express simple conjunctions of essential properties and allow no intuitive counterexamples, there also appears to be the fairly strong presumption that our intuitive categorization judgments will coincide perfectly with the presence or absence of a small but specific set of properties. In other words, lurking in the background of this enterprise is the assumption that our intuitions will nicely converge upon a set whose members are all and only those things which possess some particular collection of features. Given that philosophers expect to find tidy conjunctive definitions, and given that they employ intuitions as their guide in this search, the presupposition seems to be that our intuitive categorization judgments will correspond precisely with simple clusters of properties.