Vanishing Intersections

From: Stevan Harnad (
Date: Sun Feb 25 1996 - 14:38:40 GMT

Hi Psych-Thinkers:

Emma asked me a question about the last seminar, and I couldn't for the
life of me retrieve what the point had been: The keywords were: "blue,
overlapping circles, vectors": What was I talking about?

It wasn't, perchance, the issue of "vanishing intersections" was it?
Chomsky pointed out the special case of the poverty of the stimulus:
The child could not be learning the very abstract rules of Universal
Grammar (UG) by trial and error with feedback because the "stimulus" --
the sentences the child hears and speaks -- is too impoverished; the
basis for coming up with the right rule from the examples by trial and
error just isn't there. Mistakes just don't get made, so they never get
corrected. So the child must already have had UG in his head all along.

Now I went on to say that Fodor and others had generalised (in my view,
OVERgeneralised) the problem of the poverty of the stimulus from
children's early grammar learning to concepts in general, and part of
the generalisation was based on the so-called "vanishing intersections"
argument, according to which: It cannot be true that you learn a
concept by trial and error and feedback from a lot of examples, because
the examples don't have anything in common!

The concepts people cite here are usually abstract ones like
"goodness," "truth" or "beauty," and the argument applies mainly
to SENSORY features: It may be that all examples of "blue" that you
ever see do have something in common, namely, that they have light of a
certain wave length (blue) reflected from them. So the "intersection"
of all these examples, like the intersection of a lot of overlapping
circles -- which means the area they all have in common -- the
intersection of all the examples of "blue" you ever see is NOT empty;
it does not vanish. What all examples of blue have in common is that
they are all blue.

But what about something more abstract, like "colour"? What do all
examples of "colour" have in common? And if you don't think the
intersection vanishes there, then go even more abstract: what about
"sensory quality" (which includes shapes, color, sound, smell)? Or
"property" (which includes ANY property of things, whether sensory or
not, for example, mass, momentum, electric charge)? What do all things
that are "true" or "good" have in common? If the answer is "nothing,"
-- i.e., if the intersection, the overlap, of all examples of them is
empty, vanishes -- then how do we know what they mean at all? We
couldn't have learned it from sensory examples by trial and error and
feedback, eventually figuring out what they have in common, because
they have nothing in common. (Wittgenstein made a similar argument
about "games").

Fodor wants to conclude from the vanishing intersections argument that
concepts, or at least abstract concepts, are, like UG, innate, inborn,
unlearnt, already there in our heads. (Wittgenstein instead thinks
concepts are just arbitrary rules that we simply agree amongst ourselves
to abide by: the rules of the "language game".)

If this is the point Emma meant (about overlapping circles and blue),
then the conclusion I wanted to draw was that the
vanishing-intersections argument does not have the force of the
poverty-of-the-stimulus argument because it is still very possible that
although we cannot SAY what the common sensory properties are that are
shared by members of abstract categories, they may still EXIST, and be
detected by our sensory system, and used to ground the meanings of
abstract terms, which can then be used to DEFINE further abstract terms
by merely combining them into sentences (e.g., a "colour" is what every
visible object surface -- blue, green, red, grey, white -- all have in
common). It is not at all obvious that all of this cannot be learnt by
trial and error and feedback, in a perfectly Skinnerian way. The same
is NOT true for the rules of UG...

Besides, it makes sense to say grammatical rules are inborn; they are
just rules for what you can and cannot say. But does it make sense to
say "meanings" are inborn? Meanings are not just rules about what you
can say. We all know what "colour" means, and that DOESN'T just mean
that we know which sentences we can and cannot use it in (although
Wittgenstein may perhaps think that's all there is to it: that meaning
is just a kind of social grammar). We know what "colour" means in
EXACTLY the same sense that we know what "blue" means: There are
examples that we can point to that are examples of "colour," just as
there are examples of things we can point to of "blue." More important,
there are plenty of examples to point to that are NOT "colour" (and not
"blue"). Chances are, the child encountered those examples, and sorted
them out, by trial and error (and later, by instruction, by definitions
made out of the words he alreday knew). So there is no poverty of the
stimulus there.

Chrs, S

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