Re: Intensions vs. extensions

Nicola Guarino (
Thu, 30 May 1996 18:21:17 +0100 (MET)

At 6:52 28/5/96, John F. Sowa wrote:
>A relation is defined as a set of lists, each of which represents some
>n-tuple of entities for which the relation is true. Everything in the
>KIF semantics is based on the assumption of a fixed universe of discourse
>in which the relation can be identified with the set of lists that define
>its current extension.
>But then, I would like to say something like the following:
> A conceptual schema is a set of type definitions and relation definitions
> together with a set of constraints expressed in terms of those types
> and relations.
>But since a type cannot be distinguished from a set and a relation cannot
>be distinguished from a set of lists, a definition of the type or relation
>must be equivalent to a specification of some set in the current U of D.
>In purely extensional terms, a conceptual schema and a universe of
>discourse collapse into exactly the same thing.
>With an intensional semantics, I could say something like the following:
> The definitions determine all the kinds of entities and relations that
> may exist in some family of universes of discourse. The constraints
> assert propositions that must be true of every one of those UoDs.
>Once we admit multiple UoDs, we are outside the scope of the KIF semantics.
>For such a construct, we need an intensional definition that defines a
>relation as a rule rather than a set: A relation R is a rule, which
>for any universe of discourse U, determines a set of lists of entities
>taken from U. The rule is an algorithm that allows us to determine
>whether a particular list of entities in a particular UoD happen to
>be related by the given relation R.

I noticed exactly the same problem when looking at the notion of
"conceptualization" introduced by Genesereth and Nilsson's in their
well-known textbook on AI. They give to this term an extensional semantics,
defining a conceptualization to be to a set of relations on a fixed UoD.
Together with Pierdaniele Giaretta, I have extensively discussed this issue
in two papers presented at AAAI94 and KB&KS 95 [1, 2]. I report here some
of this discussion:

----- (more or less) literal citation from [2] ----

Nilsson and Genesereth take into account a situation where two piles of
blocks are resting on a table (Fig. 1a).

b d
a d
c e
b e

Fig. 1a Fig. 1b.
The same conceptualization?

According to them, a possible conceptualization of this scene is given by
the following structure:

<{a, b, c, d, e}, {on, above, clear, table}>

where {a, b, c, d, e} is the universe of discourse, consisting of the five
blocks we are interested in, and {on, above, clear, table } is the set of
the relevant relations among this blocks, of which the first two, 'on' and
'above', are binary and the other two, 'clear' and 'table', are unary . The
authors make clear that objects and relations are extensional entities. For
instance, the 'table' relation, which is understood as holding of a block
if and only if that block is resting on the table, is just equal to the set
{c, e}. It is exactly such an extensional interpretation that originates
our troubles.
Let us notice first that Genesereth and Nilsson used natural language terms
(like 'on', 'above') in the metalanguage chosen to describe a
conceptualization. This could perhaps be seen as nothing more than a
didactical device. However, these linguistic terms do convey essential
information in order to understand the criteria used to consider some sets
of tuples as the relevant relations. Such an extra information cannot be
accounted for by the conceptualization itself.

Referring to the example given, consider a different arrangement of blocks,
where c is on the top of d and a and b form a separate stack standing on
the table (Fig. 1b). The corresponding structure would be different from
the previous one, generating therefore a different conceptualization. Of
course there is nothing wrong in such a view, if one is only interested in
isolated snapshots of the block world. But the meanings of the terms used
to denote the relevant relations are still the same, since they are
invariant with respect to the possible configurations of blocks. In fact,
in the metalanguage adopted in their book, Genesereth and Nilsson would
adopt the same symbols (on, above, clear, table) to denote the new
conceptualization. We prefer to say in this case that the states of affairs
are different, but the conceptualization is the same. The structure
proposed by Genesereth and Nilsson seems to be more apt to represent a
*state of affairs* rather than a conceptualization.

In order to capture such intuitions, the linguistic terms we have used to
denote the relevant relations cannot be thought of as mere comments,
informal extra-information. Rather, the formal structure used for a
conceptualization should somehow account for their *meaning*. As the
logico-philosophical literature teaches us, such a meaning cannot coincide
with an extensional relation.

Sticking to a set-theoretical framework, a possible way capture this
meaning is to consider our relations as *intensional relations*, taking
inspiration from Montague's semantics. Formally, an intensional relation r
of arity N on a domain D can be seen as a function from a set W of possible
worlds to the set 2**(D**N) of all N-ary relations on D:

r: W -> 2**(D**N)

This function specifies a set of admissible extensions, relative to the
domain and the set of possible worlds considered. This means that not only
the extension in the actual world, but also those relative to the other
possible worlds are specified. We can therefore represent a
conceptualization by the following *intensional structure*:

<W, D, R>

where W is a set of possible worlds, D is a domain of objects, and R is a
set of intensional relations on D.

According to this intensional interpretation, a conceptualization accounts
for the intended meanings of the terms used to denote the relevant
relations. These meanings are supposed to remain the same if the actual
extensions of the relations change due to different states of affairs. This
means that, for instance, the actual extensions of the relation 'on' in the
two examples of Fig. 1a and 1b belong to the same conceptualization.
Intuitively, we can see a conceptualization as a set of informal rules that
constrain the structure of a piece of reality, which an agent uses in order
to isolate and organize relevant objects and relevant relations: the rules
which tell us whether a certain block is on another one remain the same,
independently of the particular arrangement of the blocks.

---- end of citation from [2] -----

The intensional definitions proposed by John Sowa:

> A conceptual schema is a set of type definitions and relation definitions
> together with a set of constraints expressed in terms of those types
> and relations.
> The definitions determine all the kinds of entities and relations that
> may exist in some family of universes of discourse. The constraints
> assert propositions that must be true of every one of those UoDs.

can fit very well with the semantics described above [I have something to
say on the definition of the term 'type' (see our KR94 paper), but this is
another story...]. It also captures the notion of a relation as a rule.

Notice that the problem is not so much having a fixed UoD as having a fixed
*state of affairs*: an extensional interpretation of relations fixes in
fact a state of affairs.

Regarding constraints, we need to express their validity across all *states
of affairs* (not UoDs as suggested by John). In a recent paper on physical
objects [3], we have introduced states of affairs in our domain, as usual
in situation semantics, expressing constraints on the meaning of physical
properties like "rigidity" by means of a universal quantification on all

References (all accessible via the WWW page reported below):

[1] Guarino, N., Carrara, M., and Giaretta, P. 1994. Formalizing
Ontological Commitment. In Proceedings of National Conference on Artificial
Intelligence (AAAI-94). Seattle, Morgan Kaufmann.
[2] Guarino, N. and Giaretta, P. 1995. Ontologies and Knowledge Bases:
Towards a Terminological Clarification. In N. Mars (ed.) Towards Very Large
Knowledge Bases: Knowledge Building and Knowledge Sharing 1995. IOS Press,
Amsterdam: 25-32.
[3] Borgo, S., Guarino, N., and Masolo, C. 1996. Stratified Ontologies:
the Case of Physical Objects. In Proceedings of ECAI-96 Workshop on
Ontological Engineering. Budapest.

-- Nicola


Nicola Guarino
National Research Council phone: +39 49 8295751
LADSEB-CNR fax: +39 49 8295778
Corso Stati Uniti, 4 email:
I-35127 Padova
(*** UPDATED 29 May, 1996 ***)