Shape

281 I Don’t Like Rules—They’re Too Rigid
is an example of a rule defined in a schema x fi y that produces the shape
and others like it from a given quadrilateral. An assignment g gives values to the variables
x and y according to this predicate—
x is a quadrilateral, and y ¼ x þ z, where z is a quadrilateral inscribed in x.
The predicate is framed in terms of the schema x fi A þ B in the special case where the
shapes A and B are quadrilaterals. Further elaboration is possible, but this is already
familiar from part I. Nonetheless, some things are worth reinforcing.
Rules have an advantage over schemas because I don’t have to say anything definite
about the shapes they contain—the intuitive idea is to draw them and calculate in
terms of anything I see. But for schemas, I have to constantly remind myself that there
are indefinitely many predicates equivalent to any one I’ve got. Shapes made up of
quadrilaterals may contain triangles, etc. This defines the same rules, and it confirms
the key idea that you can say almost anything you want to about shapes without it making
the slightest difference when you calculate. And in particular, the description of a
shape in a rule needn’t match the description of the shape to which the rule is applied.
These are things I’ve shown. But there are some details I didn’t cover in part I. It’s
also worth noting that my schema for quadrilaterals produces the shape
and others like it, because the schema can be applied to a quadrilateral more than once
under different assignments. This is easy to avoid in a number of ways. For example, I
can put a notch in the most recent quadrilateral, so that the schema applies only to it,
with rules like this