Feature Centrality as Conceptual Immutability

Previously I wrote that, given an object category defined over object representations, a prototype might be a set of features distinguished by some nice semi-formal properties:

1.  Prototype feature values are individually typical of the category, so that p(feature value is true of object | object is a category member) is high, and
2. A prototype feature value set is jointly diagnostic of category membership, so that p(object is category member | all the prototype feature values are true of the object ) is high.

This was meant as a plausible weakening of the folk intuition of category essentialism, wherein categories are defined by individually necessary and jointly sufficient criteria.

I also had some underwhelming speculation about replacing typicality with average values for graded and scalar-valued features (meant to capture the notion of "feature centrality"), and a little rant about fuzzy\graded category membership judgements being the result of individuals equivocating between similar but distinctly-motivated categories when they don't have domain expertise or introspective fluency to articulate the criteria distinguishing membership.

I've since mostly given up on the conception of prototypes as weakened-essence feature sets for categories. The experience that motivates people to talk about prototypes of object categories is just mental imagery of category members, which from the outside view is something like inferences about perceptual features of category members. Since these inferences aren't necessarily tied to memories of the situations from which the category's concept was learned, we usually think of prototypes as being unnamed and source-unsituated.

I've also since found a much more persuasive notion of feature centrality: the authors of "Feature Centrality and Conceptual Coherence" (Sloman et al., 1998) give an account of central features of a concept as being those features which are less mutable, in the senses that other object-features depend upon the central ones (within the subject's intuitive explanations of category feature co-occurrence), such that imagining mutations to central features (while maintaining explanatory coherence over the object features) may result in large downstream changes of dependent\peripheral features, with the possible effect of excluding the hypothetical object from category membership. Breathing is a central feature of the concept of birds, because we have difficulty imagining a bird which has never taken breath, except for example if it is stillborn, or if it's an ornament that looks like a bird, et cetera.

Fuzzy membership is still dumb though.

Dyadic Belief Modelling

Let A be a person, and A(B) be a description of person B as given by A.

A(A) will be very close to A, because of A's familiarity with and privileged access to particulars of A's character. Often, deviations between A(A) and A seem like evolutionarily adaptive non-conscious self-deception, producing mistaken inferences which support systematically distorted conclusions of one's own competence or status, which mistaken beliefs make belief-aligned social impression management efforts more persuasive in the minds of others.

Differences between A(B) and (B) will be mostly due to A's unfamiliarity with B, and access to particulars of B's character being costly for A, but there are also distortion effects from evolutionarily-politically-motivated social perception of others,  and distortions of convenience from using resource-inexpensive mental models.

At two levels deep, A(B(A)) corresponds to this situation: we go to A and say, "We asked B for a description of you. What do you think they said?".  A's answer, A(B(A)), might be thought of as properly contained within A(B) - since B's beliefs about A are features of B, but it's also sometimes useful to maintain the distinction. The distortions of A(B(A)) from A(A) will stem from A's guesses of what particulars B knows, or does not know, or misbelieves about A, relative to A(A). Of course, we expect that A(B(A) will be a multiply distorted version of A, but there are also a few improbable opportunities for A(B(A)) to be a more accurate account of A than either A(A) or B(A): A could inaccurately fill in details from A(A) which are unfamiliar to B, and A could incorporate B's presumed-mistaken but actually-accurate beliefs about A, correcting for A's self-misperception.

A(B(B)) is A's account of B's self-misperceptions. It's also properly contained within A(B). Not much to see here.

A(A(A)) is A's account of A's self-misperceptions, properly contained within A(A). Differences between A(A(A)) and A(A) should probably all be due to failures of logical closure; a deductively productive mind can be mistaken about many of their attributes (so that A(A) differs from A), but their beliefs should probably not be mistaken about those beliefs themselves (so that belief in belief contracts to belief, and A(A(A)) becomes equivalent to A(A), and not merely contained within it). Likewise, A(A(B)) should contract to A(B).

At 3rd level, if A thinks that B can contract belief in belief, then A(B(B(A))) becomes A(B(A)), and A(B(B(B) becomes A(B(B)). Of A's eight possible 3rd-level representations of the two people, the only ones which are not trivially contracted are A(B(A(A))) and A(B(A(B))), both of which are properly contained within A(B).

 The first kind is useful for correcting self misperception: Alan muses to himself, "Bettie and Barb say that I under-estimate myself as a programmer, so maybe I'm better than I think," and then contracts his belief in belief with an update.

The second kind, A(B(A(B))) could be elicited in this way: we go to Alan and say, "We asked Bettie to describe herself as you would describe her. What did she say?". Alan thinks of Bettie, A(B), and how she thinks of herself, A(B(B)), and what she knows about Alan, or does not know, or misbelieves (relative to A(A)), which is A(B(A)), ... and then he segfaults and asks you to repeat the question.

"We asked Bettie to describe herself as you would describe her. What did she say?"

Alan starts describing Bettie as he would normally describe her, making small omissions where he thinks Bettie thinks he doesn't know about her, or small substitutions where he thinks Bettie misperceives herself and doesn't know what Alan thinks, and other substitutions where he thinks that Bettie thinks his beliefs are mistaken about her relative to her B(B), when in fact he and Bettie agree, and probably other things, I don't know. This is probably useful for impression management, but I don't want to think about it any more today.

Prototypes

What is a prototype? One hypothesis: given a category defining membership over fully specified things, a prototype is a certain underspecified object whose features are individually typical and jointly discriminative\diagnostic of objects within the category, so that each prototype feature probably holds for an object drawn from the category, and category membership is probable for any object having all the prototype features.

This view of prototypes is just a less committal version of the idea that a category is defined by necessary and sufficient criteria; now a category has (or is associated with a prototype object having) individually kind-of-necessary (i.e. typical) features and jointly kind-of-sufficient (i.e. discriminative) features.

I. Centrality
When I posted this on #slatestarcodex at FreeNode, several people said that a definition of prototypy\prototypicality should make reference to "centrality". What does "centrality" introduce that isn't captured by typicality? Here's a guess: central position refers to average feature values for scalar\graded features of a category. Thus instead of saying that a prototypical willow tree has typical proportions, so that most willow trees have those proportions, we now say that prototypical willow trees have average proportions, whether or not most willow proportions are close to the average value. I don't think this is a very good guess! It would probably help if I had intuitions about the importance of centrality to prototypes.

II.  Underspecification and graded membership
Another comment, this one from grognor alone, was that underspecification wasn't playing a well motivated role in my definition of prototypes. To grognor, a prototype can be a fully specified object, and graded judgements of category membership can be made by gauging the similarity of objects to the prototype.

I'm receptive to the idea that prototypes can be fully specified objects, but I don't think I like the second part. People don't say "This tangelo is 70% an orange", and rarely say things like "Tangelos are more oranges than are lemons." Surely they can make judgements of relative similarity, but those judgements aren't then stated as relative partial memberships.

III. Articulating membership
Or consider, if you ask an articulate person to make a hard-ish categorization, like whether curtains are furniture, they're often capable of providing fine introspective criteria distinguishing category membership definitely. They might say, "Curtains are furniture in the sense of semi-functional, semi-stationary materials which I use to decorate my house, but they are not furniture in the sense of bulky objects that I use to support things off the ground in my house." Or they'll say, "Curtains, like pillows and clocks and dishes, are furnishings, not furniture. That's not a hard categorization. Who do you think you're talking to? I am the queen of home decor categorization judgements. Get out of my kitchen, you uncultured swine."

So some people can figure out distinguishing criteria on-the-fly for different categories, and other people who are familiar with a domain might have a remembered repository of more and finer categories with distinguished labels for objects in the domain. I think it's mostly the less articulate people who are sweating about having to make a binary judgement: the ones who know that words are rough equivocations of categories across minds, but who don't have lots of finely distinguished expert category labels or the ability to invent longer phrases on the fly to explain the motivations behind potentially conflicting membership judgements; to them, hard categorization prompts are a trap. Some people recognize the trap and stop sweating: they give up on reductionism and say, "Curtains are kind of furniture. I'm sure there's no binary answer. Fuzzy concepts. Get with the psychological research zeitgeist." I guess that's fine, and I don't want to trap people. But I don't think I want to make "partial membership" part of my own ontology either.

IV. Uncertain membership
We just saw a fictional case where someone was uncertain about category membership because they didn't know what judgements were relevant to another person's label applications. This is maybe a special case of a category judgement depending on unknown empirical features. Another source of membership uncertainty: the person might not be able to make a confident binary judgement because of logical uncertainty, like if the category is not specified directly in terms of objects having membership\non-membership. Most aren't!

Maybe someday I'll work these sources of membership uncertainty into my definition of prototypes, along with an improved understanding the centrality caveat. And maybe a prototype is still a thing which has individually typical (or central) and jointly discriminative features, but it's fine for any other features to be specified also, in a less principled way, like just cutting and pasting from the population as they do in instance-based statistical learning techniques.

My next post will be about stereotypes as privileged objects of categorical cognition. It's gonna be so sweet.