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\title{Intensional Platonism}
%plato[w88,jmc]		Empirical Platonism

\noindent Abstract: We propose to formalize concepts as abstract,
almost Platonic entities, neither sets nor predicates.  The advantage
is {\it elaboration tolerance}, i.e. it permits elaboration without
losing existing formalism.

\section{Introduction}

	Consider a concept such as {\it mother} or {\it chair}.  We ask
what is it in humans and what should it be in AI systems.  In principle,
these are different questions, but it may be that they have the same
answer, i.e. what works in humans may be the same thing that will work for
computer programs.

	From the mathematical logical point of view, the simplest answer
is that they are unary predicates $mother(x)$ and $chair(x)$.  However,
this doesn't have enough elaboration tolerance and modification tolerance.
A child goes through several stages with regard to the concept {\it
mother}.  In the simple logical treatment, it would first be a proper name
$mother$ for the child's own mother.  Then it would become a unary
predicate $mother(x)$ designating adult women.  (This is indicated by the
common childish mistake of referring to a man's wife as his mother).
Finally (perhaps), it would become a binary relation $mother(x,y)$.

	However, these changes in the way a child uses the word occur
rather smoothly and suggest a different way of doing it.

	Early on a concept appears as some kind of abstract identifier.
Even in a prelinguistic state it is attached to observations of the
mother, thoughts about the mother, missing the mother, etc.  When the
child begins to talk, its name, say ``mama'', for the mother becomes
attached to the concept as a word for it.  The unary and binary
predicates later are also attached.

	This resembles Platonism, or at least some popular view of it,
in certain respects.  Namely, an ideal entity is formed to which the
more concrete individuals, functions and predicates and distinguishing
criteria are attached.  It differs from Platonism in that the ideal
entity is purely internal to the child although it is likely to resemble
the corresponding entities in the minds of other people, thus justifying
some name for the equivalence class.

	The general hypothesis is that concepts in people's minds
are primarily a kind of identifier, e.g. in LISP terminology a GENSYM,
to which other entities are attached analogously to the property
list of a LISP symbol.

	Maybe there is something to this microtheory of human concepts
and maybe it can be used to invent some empirically testable propositions.
However, I won't pursue the psychological idea further in this note.
Instead let's consider it as an AI idea, a proposal about how
intelligent computer programs should be constructed.  If it works
out as an AI idea, i.e. programs using it are more powerful than
those that don't, there will be more point in suggesting it to
the psychologists who will have to figure out how to get testable
hypotheses.

	As an AI idea, we shall consider it epistemologically rather
than heuristically.  Specifically, we are concerned with its use in
the {\it ontology} of the logical languages we use.
Thus we introduce constants $mother$ and $chair$.  These aren't names
for anything in particular.  However, they are to be used to construct
logical entities with a more concrete interpretations.  For example,
we can have $is\hbox{-}a(x,mother)$ to say that $x$ is a mother.
This doesn't require us to identify the object $mother$ in the formula
with the set of mothers, and I advocate not doing so.  If we are
interested in the set of mothers regarded extensionally, we would
refer to it as $set\hbox{-}of(mother)$ and use $x ε set\hbox{-}of(mother)$ as
synonymous with $is\hbox{-}a(x,mother)$.  $set\hbox{-}of(mother)$ might be equal to a
set defined in some other way.  Another notation is $unary\hbox{-}pred(mother)$
designating the unary predicate of being a mother and admitting
the notation $unary\hbox{-}pred(mother)(x)$ as another synonym for
$is\hbox{-}a(x,mother)$.  We aren't interested in choosing a language for now
but will continue with considerations independent of it.

	The binary predicate offers some further problems requiring
further abstraction.  If we have an arbitrary concept, there isn't
necessarily a unique binary predicate associated with it.  Indeed the
treatment of the previous paragraph may be inadequate in that we
may need more than one way of associating unary predicates with
concepts.  Therefore, perhaps we shouldn't regard $is\hbox{-}a(x,mother)$
as generated from the concept $mother$ by a universal process but
rather as a particular notation that has to be axiomatized separately.
Concretely, the issue comes down to whether $is\hbox{-}a(x,y)$ satisfies
some axioms involving $∀y$ or whether $is\hbox{-}a(x,mother)$ and $is\hbox{-}a(x,chair)$
are to be axiomatized entirely separately.

	Returning to the binary predicate, we can get by for now with
something like $is\hbox{-}in\hbox{-}relation(x,y,mother)$ or perhaps
$is\hbox{-}in\hbox{-}relation17(x,y,mother)$ if we want to allow other relations
determined from $mother$.

	There are certain advantages to treating $mother$ as a {\it rich
entity}.  The idea is that rich entities are never completely defined.
We don't know them; we only know some facts about them.  In previous
work, situations were proposed as rich entities.  Having rich entities
permits elaboration tolerance in that more facts about a rich entity
can always be added, including more facts about its definition.  For example,
a child or a naive program can treat $mother$ as a natural kind and
expect other people or programs to know more about what defines it.

	This gives another resemblance between what we are proposing
and Plato's platonism.  Allowing that we might learn more about what
defines a chair but never but never understand the concept totally
agrees with Plato's idea of an ideal chair.

\section{Relation to Formalized Contexts}

	Adhering fully to this kind of platonism is likely to result
in rather long formulas.  This can be mitigated by using formalized
contexts.  For example, one can enter a context in which $Mother$
is indeed a proper name for what is called $belonging\hbox{-}to(timothy,mother)$
in an outer context in which it is defined which Timothy is meant.

\smallskip\centerline{Copyright \copyright\ \number\year\ by John McCarthy}
\smallskip\noindent{This draft of PLATO[W88,JMC]\ TEXed on \jmcdate\ at \theTime}
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%\hbox{-} or \mathord{-} or define a macro \isa

Conjecture on formalized contexts:  Using them, one need never get
stuck.  But isn't that true already?  Only by thinking at the
meta-level.  With contexts one might hope to reason about
generalizations within the original language.  We win on the Platonism
also, because that permits us to say that associated with $mother$
is a binary relation.