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.
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 orof 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 ofinterpretation for a mental representation of that form assigns as its extension the complement of the set of Fs. Toadmit 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 theexemplars of its constituents. However, this condition is not satisfied in the general case. So, for example, a goldfish isa poorish example of a fish, and a poorish example of a pet, but it’s a prototypical example of a pet fish. So similarity tothe prototypic pet and the prototypic fish doesn’t predict similarity to the prototypical pet fish. It follows that ifmeanings 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 becompositional. Various solutions for this problem are on offer in the literature, but it seems to me that none is evenclose 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, inparticular, that the prototypical apple is red, and consider the problem of constructing a prototype for PURPLEAPPLE. The basic idea is to form a derived feature matrix that’s just like the one for APPLE, except that the featurepurple replaces the feature red and the weight of the new colour feature is appropriately increased. PET FISH wouldpresumably 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 APPLEsince, though there can be apples that aren’t red, there can’t be purple apples that aren’t purple; any more than therecan be red apples that aren’t red, or purple apples that aren’t apples. In effect,
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’tcompositional even if the matrices themselves are. Neither horn of this dilemma seems happy. Moreover, it’s pretty clearwhat’s gone wrong: what really sets the weight of the purple in PURPLE APPLE isn’t the concept’s prototype; it‘s theconcept’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.Jean-marc pizano