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C00002 00002	tower[f83,jmc]		axioms for towers
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tower[f83,jmc]		axioms for towers

	A structure is a set of objects and relationships among them.
We build new structures by adding objects of given kinds in given
relations to the objects already there.  We suppose that objects
may be removed from structures, and when an object is removed, all
its relationships are deleted.  The operation of adding an object
to a structure is not necessarily physically possible.  Thus we may
imagine a new structure consisting of a hollow sphere with another
object added in its center.  The construction operations are
a subset of the design operations.  Scaffolding may be required
in order to perform certain construction operations.  Both
design and construction are constrained by rules.  The rules
constraining design prevent, for example, objects with interpenetrating
parts.  Typically they will be expressed by expressions forbidding
certain combinations of objects, i.e. substructures.  These will
usually involve a fixed number of objects and relations.  For
example,

	block a ∧ block b ∧ block c ∧ on(a,b) ∧ on(a,c)

may be forbidden, assuming cubical blocks, on the grounds that this
would require  a  and  c  to coincide.

Painting a color and painting with paint

	Leslie Pack points out that a necessary condition for painting
something a given color is that paint of that color be available, whereas
I had not been thinking at all about the availability of paint.  If
we go into the matter there are many other necessary conditions.
Much of it can be handled with circumscription, but the contrast
between painting a color and painting with paint takes a linguistic
form.  Let's write some formulas.

paint(x,c) is the action of painting  x  the color  c.

paint1(x,p) is the action of painting  x  with paint  p.

We can use the same function  color  to talk about  color(x)
and to talk about  color(p).  Painting a color is less specific
than painting with paint, because there could be several cans of
paint available.  This suggests the following somewhat drastic
revision of situation calculus.

In situation calculus the situation is thought of as encompassing
infinite detail and never fully describable.  We only know facts
about situations.  However, actions were considered discrete
objects and leading to definite new situations, e.g.  result(a,s).
Suppose now we regard actions and events as also infinitely
detailed and we only know facts about actions and events.  We
still have  result(a,s),  but now  a  cannot be  paint(x,c)  or
paint1(x,p).  Instead we write  paint(x,c,a)  or  paint1(x,p,a)
and the new  paint  and  paint1  are predicates.  The fact
that  paint1  is more specific than  paint  is now expressed by

	%2∀x p c a.paint1(x,p,a) ∧ color(p) = c ⊃ paint(x,c,a)%1.

If we later have to reify  paint  and  paint1,  we'll have to
write something like  applies(paint(x,c),a).  This revised situation
calculus may solve a lot of problems.

states and sits, continuous action