COMMENT ⊗   VALID 00002 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.require "memo.pub[let,jmc]" source
C00005 ENDMK
C⊗;
.require "memo.pub[let,jmc]" source;
.bb ABSTRACT FORMS


	This note should be read in conjunction with the FOL axioms
in the file ABSTRA.AX[S76,JMC].  The ideas of abstract forms are a
development from my earlier ideas on %2abstract syntax%1, %2extensional
forms%1 and %2concepts as objects%1.

	An %2abstract form%1 ⊗u contains variables such as ⊗V, ⊗V0, ⊗V1, etc.
and has a value %2value(u,e)%1 in an environment ⊗e.  In some cases,
this value may be the special element ⊗UU, and in this case the value
is regarded as undefined.  The abstractness comes from the fact that
abstract forms will not be defined as strings of symbols, although often
they may be regarded as the equivalence classes determined by some
equivalence relation on strings of symbols.  Even this is not always
possible when, for example, we are dealing with real numbers and take
each number as a ⊗constant (i.e. a particular kind of form), so that
there is a non-denumerable number of forms.  In the cases we have in
mind, classes of abstract forms will be built up inductively from
constants and variables.

	In general the value of an abstract form is another abstract
form, but often its value is a constant.  A constant is an abstract
form ⊗c satisfying

!!a1:	%2∀e.(value(c,e) = c)%1.

.item←0;
Definition #.  An abstract form is called ⊗simple if its value
in an environment ⊗e depends
only on the values of its variables in ⊗e.  Non-simple forms are useful
for merging the idea of an environment or state vector with the idea
of possible world from modal logic.  A concept may have no variables, but
its value may depend on the environment.