Potted summary: “Reasoning About Categories in Conceptual Spaces”

What follows is a short summary of the main elements of a paper written by Peter Gardenfors (Lund) & Mary-Anne Williams (Newcastle) in their paper from 2001, “Reasoning About Categories in Conceptual Spaces”. It contains a way of thinking about concepts and categorization that seems quite lovely to me, as it captures something about the meat and heft of discussions of cognition, ontology, and lexical semantics by deploying a stock of spatial metaphors that is accessible to most of us. I confess that I cannot be sure I have understood the paper in its entirety (and if I have not, feel free to comment below). But I do think the strategy proposed in their paper deserves wider consideration in philosophy. So what follows is my attempt to capture the essential first four sections of the paper in Tractarian form.

1. An object is a function of the set of all its qualities. (For example, a song is composed of a set of notes.)
   1. Every quality occurs in some domain(s) of evaluation. (e.g., each tone has a pitch, frequency, etc.)
   2. A conceptual space is a set of evaluative domains or metrics. (So, the conceptual space around a particular song is the set of metrics used to gauge its qualities: pitch, frequency, etc.)
   3. Just like ordinary space, a conceptual space contains points and regions. Hence, an object is a point in conceptual space.
   4. We treat some objects as prototypes with respect to the part of conceptual space they are in (e.g., the prototype of a bird is a robin.)
      1. Those objects which have been previously encountered (e.g., in inductive fashion), and their location registered, are exemplars.
         
2. A concept is a region in conceptual space.
   1. Some of those regions are relatively amorphous, reflecting the fact that some concepts are not reliable and relevant in the judgments we make. (e.g., a Borgesian concept.)
   2. Categorization identifies regions of conceptual space with a structure. e.g., in our folk taxonomy, we have super-ordinate, basic, and sub-ordinate categories.
      • Super-ordinate categories are abstract (fewer immediate subcategories, high generality, e.g., ‘furniture’); basic categories are common-sense categories (lots of immediate subcategories, medium generality; e.g., ‘chairs’); and sub-ordinate categories are detail-oriented (few immediate subcategories, low generality; e.g., ‘Ikea-bought chaise-longue’).
   3. The boundaries of a category are chosen or “built”, depending on the structure that is identified with the concept in the context of the task. They can be classical (“discrete”) boundaries, or graded, or otherwise, depending on the demands of content, context, and choice.
   4. The structure of a conceptual space is determined by the similarity relations (“distances“) between points (or regions) in that space.
   5. One (but only one) useful way of measuring distance in a conceptual space is figuring out the distance between cases and prototypes, which are especially salient points in conceptual space.
      • Every prototype has a zone of influence. The size of that zone is determined by any number of different kinds of considerations.
3. There are at least three kinds of structure: connectedness, projectability (“star-shapedness”), and perspicuity (“convexity”).
   1. A conceptual region is connected so long as it is not the disjoint union of two non-empty closed sets. By inference, then, a conceptual region is disconnected so long as its constituents each contain a single cluster, the sets intersect, but the intersection is empty. For example, the conceptual region that covers “the considered opinions of Margaret Wente” is disconnected, since the intersection of those sets is empty.
   2. Projectability (they call it ‘star-shapedness’) means that, for a particular given case, and all points in a conceptual space, the distance between all points and the case do not exit the space.
      1. For example, consider the concept of “classic works of literature”, and let “For Whom the Bell Tolls” be a prototype; and reflect on the aesthetic qualities and metrics that would make it so. Now compare that concept and case to “Naked Lunch”, which is a classic work of literature which asks to be read in terms of exogenous criteria that have little bearing on what counts as a classic work of literature. There is no straight line you can draw in conceptual space between “For Whom the Bell Tolls” and “Naked Lunch” without wandering into alien, interzone territory. For the purposes of this illustration, “For Whom…” is not projectable.
   3. Perspicuity (or contiguity; they call it ‘convexity’) means all points of a conceptual space are projectable with each other.
      • By analogy, the geography of the United States is not perspicuous, because there is no location in the continental United States that is projectable (given that Puerto Rico, Hawaii, and Alaska all cross spaces that are not America).
      • According to the authors, the so-called “natural kinds” of the philosopher seem to correspond to perspicuous categories. Presumably, sub-ordinate folk categories are more likely to count as perspicuous than basic or super-ordinate ones.
4. One mechanism for categorization is tessellation.
   1. Tessellation occurs according to the following rule: every point in the conceptual space is associated with its nearest prototype.
   2. Abstract categorizations tessellate over whole regions, not just points in a region. (Presumably, this accounts for the structure of super-ordinate categorizations.)
      1. There are at least two different ways of measuring distances between whole regions: additively weighted distance and power distance. Consider, for example, the question: “What is the distance between Buffalo and Toronto?”, and consider, “What counts as ‘Toronto’?”
         1. For non-Ontarian readers: the city of Toronto is also considered a “megacity”, which contains a number of outlying cities. Downtown Toronto, or Old Toronto, is the prototype of what counts as ‘Toronto’.
         2. Roughly speaking, an additively weighted distance is the distance between a case and the outer bounds of the prototype’s zone of influence.
            • So, the additively weighted distance between Buffalo and Toronto is calculated between Buffalo and the furthest outer limit of the megacity of Toronto, e.g., Mississauga, Burlington, etc.
            • The authors hold that additively weighted distances are useful in modeling the growth of concepts, given an analogy to the ways that these calculations are made in biology with respect to the growth of cells.
            • In a manner of speaking, you might think of this as the “technically correct” (albeit, expansive) distance to Toronto.
         3. Power distance measures the distance between a case and the nearest prototype, weighted by the prototype’s relative zone of influence.
            • So, the power distance between Buffalo and Toronto is a function of the distance between between Buffalo, the old city of Toronto, and the outermost limit of the megacity of Toronto. Presumably, in this context, it would mean that one could not say they are ‘in Toronto’ until they reached somewhere around Oakville.
            • This is especially useful when the very idea of what counts as ‘Toronto’ is indeterminate, since it involves weighting multiple factors and points and triangulating the differences between them. Presumably, the power distance is especially useful in constructing basic level categories in our folk taxonomy.
            • In a manner of speaking, you might think of this as the “substantially correct” distance to Toronto.
         4. So, to return to our example: the additively weighted distance from Buffalo to Toronto is relatively shorter than when we look at the power distance, depending on our categorization of the concept of ‘Toronto’.
   3. For those of you who don’t want to go to Toronto, similar reasoning applies when dealing with concepts and categorization.

Non-classical conceptual analysis in law and cognition

Some time ago I discovered a distaste for classical conceptual analysis, with its talk of individually-necessary-and-jointly-sufficient conditions for concepts. I can’t quite remember when it began — probably it was first triggered when reading Lakoff’s popular (and, in certain circles of analytic philosophy, despised) Women, Fire, and Dangerous Things; solidified in reading Croft and Cruse’s readable Cognitive Semantics; edified in my conversations with neuroscientist/philosopher Chris Eliasmith at Waterloo; and matured when reading Elijah Millgram’s brilliantly written Hard Truths. In the most interesting parts of the cognitive science literature, concepts do not play an especially crucial role in our mental life (assuming they exist at all).

Does that mean that our classic conception of philosophy (of doing conceptual analysis) is doomed? Putting aside meta-philosophical disagreements over method (e.g., x-phi and the armchair), the upshot is “not necessarily”. The only thing you really need to understand about the cognitive scientist’s enlarged sense of analysis is that it redirects the emphasis we used to place on concepts, and asks us to place renewed weight on the idea of dynamic categorization. With this slight substitution taken on board, most proposition-obsessed philosophers can generally continue as they have.

Here is a quick example. So, classical “concepts” which ostensibly possess strict boundaries — e.g., the concept of number — are treated as special cases which we decide to interpret or construe in a particular sort of way in accordance with the demands of the task. For example, the concept of “1” can be interpreted as a rational number or as a natural one, as its boundaries are determined by the evaluative criteria relevant to the cognitive task. To be sure, determining the relevant criterion for a task is a nigh-trivial exercise in the context of arithmetic, because we usually enter into those contexts knowing perfectly well what kind of task we’re up to, so the point in that context might be too subtle to be appreciable on first glance. But the point can be retained well enough by returning to the question, “What is the boundaries of ‘1’?” The naked concept does not tell us until we categorize it in light of the task, i.e., by establishing that we are considering it as a rational or a natural.

Indeed, the multiple categorizability of concepts is familiar to philosophers, as it captures the fact that we seem to have multiple, plausible interpretations of concepts in the form of definitions, which are resolved through gussied-up Socratic argument. Hence, people argue about the meaning of “knowledge” by motivating their preferred evaluative criteria, like truth, justification, belief, reliability, utility, and so on. The concept of knowledge involves all the criteria (in some amorphous sense to be described in another post), while the categorization of the concept is more definite in its intensional and extensional attributes, i.e., its definition and denotation.

The nice thing about this enlarged picture of concepts and category analysis is that seems to let us do everything we want when we do philosophy. On the one hand, it is descriptively adequate, as it covers a wider range of natural language concepts than the classical model, and hence appeals to our sympathies for the later Wittgenstein. On the other hand, it still accommodates classical categorizations, and so does not throw out the baby with the bathwater, so not really getting in the way of Frege or Russell. And it does all that while still permitting normative conceptual analysis, in the form of ameliorative explications of concepts, where our task is to justify our choices of evaluative criteria, hence doing justice to the long productive journey between Carnap and Kuhn described in Michael Friedman’s Dynamics of Reason.

While that is all nice, I didn’t really start to feel confident about the productivity of this cognitivist perspective on concepts until I started reading philosophy of law. One of the joys of reading work in the common-law tradition is that you find that there is a broad understanding that conceptual analysis is a matter of interpretation under some description. Indeed, the role of interpretation to law is a foundational point in Ronald Dworkin, which he used it to great rhetorical effect in Law’s Empire. But you can find it also at the margins of HLA Hart’s The Concept of Law, as Hart treats outlying cases of legal systems (e.g., international law during the 1950’s) as open to being interpreted as legal systems, and does not dismiss them as definitely being near-miss cases of law. Here, we find writers who know how to do philosophy clearly, usefully, and (for the most part) unpretentiously. The best of them understand the open texture of concepts, but do not see this as reason to abandon logical and scholarly rigor. Instead, it leads them to ask further questions about what counts as rigor in light of the cognitive and jurisprudential tasks set for them. There is a lot to admire about that.

Compatibilist free will and quasi-perceptual intentions

If the only plausible compatibilist idea of freedom of the will demands that our conscious faculties be capable of exerting control at least sometimes over our behaviour, then it is hard for me to escape the inference that our intentions (or, anyway, our “free” intentions) are quasi-perceptual. That is, it is hard to avoid saying that quasi-perceptuality is a necessary but not sufficient condition for the free will, if there is such a thing.

[Updated Nov. 16.15]

Very quickly, there are at least two different kinds of accounts of intentions. Many agree that intentions are causally self-referential: that is, they are the sort of thing that you represent as true, and become true by representing as true, and thereby causing it to happen. I say to myself, “I will lift my arm”, and then that saying-to-myself makes the thing happen. I describe a state of affairs, and then it happens. Such intentions are cases of de dicto reference, meaning they involve descriptions that refer.

Unfortunately, that makes it seem as though intentions are contingent on our capacity to introspectively verbalize. But (some complain): can’t children intend? What about non-human animals, like corvids? It seems they can intend to do stuff even if they can’t verbalize. What’s up with that? Hence the alternative — proposed by Tyler Burge, endorsed by AA Roth (and others) — is that our intentions only have de re contents. That is, our intentions have the structural feature of causal self-reference — they function in such a way as to refer, but don’t involve any inner second-order faculty that is capable of introspecting on and verbalizing the thoughts that refer.

That all sounds great, except for one thing: the free will drops out. Right or wrong, few people have traditionally wanted to say that non-human animals have a free will. The free will is supposed to be a function of deliberation and our capacity for conscious control. And it seems to me that the connection between such a faculty and the notion of a de dicto intention should be obvious enough to be suggestive.

Of course, it remains to be seen if any such thing as the de dicto intentions actually exist. In all probability, I think, even mature adults will not have rich internal descriptions that they could report as reasons, even after deliberation. For it seems to me that de dicto intentions may have gaps in their content. But this is no reason to suppose that they do not exist, or that our account of intentions must be replaced by the de re account, which is tailored to fit children and corvids.