Prototypes and Compositionality 101

In short, prototypes don’t compose. Since this is the heart of the case
against statistical theories of concepts, I propose to expatiate a bit on the
examples.
(i) The Uncat Problem
For indefinitely many “Boolean” concepts,13 there isn’t any prototype even
though:
—their primitive constituent concepts all have prototypes,
and
—the complex concept itself has definite conditions of semantic
evaluation (definite satisfaction conditions).
So, for example, consider the concept NOT A CAT (mutatis mutandis, the
predicate ‘is not a cat’); and let’s suppose (probably contrary to fact) that
CAT isn’t vague; i.e. that ‘is a cat’ has either the value S or the value U for
every object in the relevant universe of discourse. Then, clearly, there is a
definite semantic interpretation for NOT A CAT; i.e. it expresses the
property of not being a cat, a property which all and only objects in the
extension of the complement of the set of cats instantiate.
However, although NOT A CAT is semantically entirely well behaved
on these assumptions, it’s pretty clear that it hasn’t got a stereotype or an
exemplar. For consider: a bagel is a pretty good example of a NOT A
CAT, but a bagel couldn’t be NOT A CAT’s prototype. Why not? Well, if
bagels are the prototypic NOT A CATs, it follows that the more a thing is
like a bagel the less it’s like a cat; and the more a thing isn’t like a cat, the
more it’s like a bagel. But the second conjunct is patently not true. Tuesdays
and erasers, both of which are very good examples of NOT A CATs, aren’t
at all like bagels. An Eraser is not more a Bagel for being a bad Cat. Notice
that the same sort of argument goes through if you are thinking of
stereotypes in terms of features rather than exemplars. There is nothing
that non-cats qua non-cats as such are likely to have in common (except,
of course, not being cats).14
Prototypes and Compositionality 101
13 To simplify the exposition, I’ll use this notion pretty informally; for example, I’m
glossing over the distinction between Boolean sentences and Boolean predicates. But none
of this corner-cutting is essential to the argument.
14 This is not to deny that there are typicality effects for negative categories; as Barsalou
remarks, “with respect to birds, chair is a better nonmember than is butterfly” (1987: 101).
This observation does not, however, generalize to Boolean functions at large. I doubt that
there are more and less typical examples of if it’s a chair, then it’s a Windsor or of chair or
butterfly.
The moral seems clear enough: the mental representations that
correspond to complex Boolean concepts specify not their prototypes but
their logical forms. So, for example, NOT A CAT has the logical form
not(F), and the rule of interpretation for a mental representation of that
form assigns as its extension the complement of the set of Fs. To admit
this, however, is to abandon the project of using prototype structure to
account for the productivity (/systematicity) of complex Boolean
predicates. So be it.
(ii) The Pet Fish Problem
Prototype theories want to explicate notions like falling under a concept by
reference to notions like being similar to the concept’s exemplar. Correspondingly,
prototype theories can represent conceptual repertoires as
compositional only if (barring idioms) a thing’s similarity to the exemplar
of a complex concept is determined by its similarity to the exemplars of its
constituents.However, this condition is not satisfied in the general case. So,
for example, a goldfish is a poorish example of a fish, and a poorish
example of a pet, but it’s a prototypical example of a pet fish. So similarity
to the prototypic pet and the prototypic fish doesn’t predict similarity to
the prototypical pet fish. It follows that if meanings were prototypes, then
you could know what ‘pet’ means and know what ‘fish’ means and still
not know what ‘pet fish’ means.Which is just to say that if meanings were
prototypes, then the meaning of ‘pet fish’ wouldn’t be compositional.
Various solutions for this problem are on offer in the literature, but it
seems to me that none is even close to satisfactory. Let’s have a quick look
at one or two.
Smith and Osherson (1984) take prototypes to be matrices of weighted
features (rather than exemplars). So, for example, the prototype for
APPLE might specify a typical shape, colour, taste, size, ripeness, . . . etc.
Let’s suppose, in particular, that the prototypical apple is red, and consider
the problem of constructing a prototype for PURPLE APPLE. The basic
idea is to form a derived feature matrix that’s just like the one for APPLE,
except that the feature purple replaces the feature red and the weight of
the new colour feature is appropriately increased. PET FISH would
presumably work the same way.
It’s pretty clear, however, that this treatment is flawed. To see this, ask
yourself how much the feature purple weighs in the feature matrix for
PURPLE APPLE. Clearly, it must weigh more than the feature red does
in the matrix for APPLE since, though there can be apples that aren’t red,
there can’t be purple apples that aren’t purple; any more than there can be
red apples that aren’t red, or purple apples that aren’t apples. In effect,
102 Prototypes and Compositionality
purple has to weigh infinitely much in the feature matrix for PURPLE
APPLE because purple apples are purple, unlike typical apples are red, is a
logical truth.
So the Smith/Osherson proposal for composing prototypes faces a
dilemma: either treat the logical truths as (merely) extreme cases of
statistically reliable truths, or admit that the weights assigned to the
features in derived matrices aren’t compositional even if the matrices
themselves are. Neither horn of this dilemma seems happy. Moreover, it’s
pretty clear what’s gone wrong: what really sets the weight of the purple in
PURPLE APPLE isn’t the concept’s prototype; it’s the concept’s logical
form. But prototypes don’t have logical forms.
Another way to put the pet fish problem is that the ‘features’ associated
with the As in AN constructions are not, in the general case, independent
of the features associated with the Ns. So, suppose that the prototype for
NURSE includes the feature female. Pace Smith and Osherson’s kind of
proposal, you can’t derive the prototype for MALE NURSE just by
replacing female with male; all sorts of other things have to change too.
This is true even though the concept MALE NURSE is ‘intersective’; i.e.
even though the set of male nurses is the overlap of the set of males with
the set of nurses (just as the set of pet fish is the overlap of the set of pets
with the set of fish). I want to stress this point because prototype theorists,
in their desperation, are sometimes driven to suggest that MALE NURSE,
PET FISH, and the like aren’t compositional after all, but it’s all right that
they aren’t, since they are idioms. But surely, surely, not. What could be
stronger evidence against PET FISH being an idiom or for its being
compositional than that it entails PET and FISH and that {PET, FISH}
entails it?
It’s perhaps worth mentioning the most recent attempt to salvage the
compositionality of prototypes from pet fish, male nurses, striped apples,
and the like (Kamp and Partee 1995). The idea goes like this: maybe good
examples of striped apples aren’t good examples of striped things tout
court (compare zebras). But, plausibly, a prototypic example of a striped
apple would ipso facto be as good an example of something striped as an
apple can be. That is a way of saying that the relevant comparison class for
judging the typicality of a sample of apple stripes is not the stripes on
things at large but rather the stripes on other apples; it’s these that typical
apple stripes are typical of. In effect, then, what you need to do to predict
whether a certain example of apple stripes is a good example of apple
stripes, is to “recalibrate” STRIPES to apples.
A fair amount of algebra has recently been thrown at the problem of
how, given the appropriate information about a reference set, one might
calculate the typicality of one of its members (for discussion, see Kamp