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. 2
          • 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.

Premier Ford and the rule of law

Recently, a constitutional challenge arose in the province of Ontario, as the newly elected Conservative Premier sought to pass a Bill to interfere with Toronto municipal elections mid-cycle to settle a few scores in his old stomping grounds. Problems arose when the judiciary told him he was violating the Charter. Tensions ratcheted up when he invoked the little-used notwithstanding clause — section 33 — in order to overcome the decision of the Court, resulting in widespread dissent from legal professionals and from the official opposition. Just recently, Ford’s party won an appeal, as a stay was placed on the Court verdict that blocked the Bill.

For now, let’s put aside the merits of the stay or the claimed violation of the Charter. Instead, zoom in on Ford’s reason for opting out of the Canadian constitution. Focus on the rationale: “unelected judges”, filed under apparent threats to democracy. Pin this little offering to a corkboard. Put a light on it. Study under glass.

The invocation of section 33 was argued on ostensibly democratic grounds. Compare specimen to encyclopedia of modern conservative thought. The pattern of argumentation that could have been reminiscent of Jeremy Waldron’s majoritarianism, if done thoughtfully. Admittedly, it’s a weird species of argument to us, we the complacent and diffident Canadians. But the world is weird. That’s why we keep reference manuals. Gotta keep an index of all the weirds.

Now turn back to the corkboard. The actual arguments presented were a mixture of Pravda and Powerpoint. Mutant variation. Pull out the red pen.


  1. At bottom, a nation of laws is a nation that makes sense, whose stability can be taken for granted. We can only get the first glimmer of a sense of obligation to such institutions when we see their rules as a going concern. The stability of law is primarily achieved through judicial review, an institution where governmental rules are deliberately and carefully interpreted and maintained. The judges are curators and stewards.
  2. When we talk about our favorite form of government— democratic, monarchical, or whatever— we are tacitly making an assumption that the rulers are not being systematically misled. The sovereign’s affirmation of counsel implies they have *informed consent*. So, if a monarch is constantly fooled by a Rasputin, then it is not strictly speaking a monarchy. Similarly, if a population is fed on a diet of lies, then strictly speaking we cannot say there is a democracy. Every form of government depends on faithful expertise.

So, a democratic nation of laws presupposes judicial review in two ways. First, because judicial review produces stability that makes it possible to talk about true and false claims of legality. Second, because it provides people with informed consent to past and future rules. You can criticize or condemn the operations of the courts for all sorts of reasons. But complaining they are not elected, is not a good reason. Quite the opposite: by challenging them, you undermine democracy.