1.3.2 |
THE CHARACTERISTICS OF TWO DIFFERENT
INTERPRETATIONS |
Technically speaking, the interpretation of a formal
system is determined by the domain itself and a function
assigning elements of this domain to singular terms and n-tuples
of elements of it to n-place predicate expressions (and
functions to function symbols as well). (One-place predicate
expressions are often said to refer to classes of entities
rather than the singletons thereof or the subclasses collectively.)
The singular terms are proper names, definite descriptions,
pronouns or demonstrative phrases. An interpretation function
thus assigns entities of the domain to singular terms; entities
or singletons to attributive expressions; pairs to two-place
relational expressions; and so on. The question which is
important for us now is what 'kinds' of thing are chosen as
elements of the domain in the ontological sense of kind.
In the standard, 'objectual' interpretation of formal systems the
elements of a domain are interpreted as objects which are not attributes
themselves, which bear relations to one another, and which —
expressing it in extrasystematic language — 'have properties'.
The particular in the domain assigned to a basic, singular term is a
(nonattributive) object.
(Note that an 'object' need not be a concrete thing in this terminology
but it has to be of the same ontological order if it is abstract.)
The two- or more-place predicate expressions to which
n-tuples of particulars are assigned correspond to n-place
'relations between them'. This objectual interpretation is
nominalistic in that it does not treat attributes as individuals;
it is realistic in that it does admit sets of particulars,
and in that it does not reject other abstract entities than
attributes or relations. Furthermore the objectual interpretation of
formal systems tends to be physicalistic. If it is of a
phenomenalist nature, it may be expected to be particularistic
(and in this sense definitely not realistic).
The ontological position we will adopt ourselves fits in
best with an 'attributive' interpretation of formal systems.
This means that we take the elements of the domain to be
attributes which belong to (nonattributive) things, and which
also bear relations to one another. The specific element in the
domain assigned to a basic, singular term is now an attribute
(a 'property' if belonging to a concrete thing or 'object').
In this interpretation the basic two- or more-place predicate
expressions correspond to relations between attributes, not to
relations between objects (or things) which have properties (or
attributes). The difference between the two interpretations of formal
systems is schematically represented in figure I.1.3.2.1.
(In both interpretations a set of elements is ontic, that is,
an existing thing, if between all the elements of the set the
relation of belonging to the same thing holds, as designated by
a two-place predicate expression. Every object which has other
objects or properties as elements should in the first instance
also be represented by a dot in addition to the dots representing
the basic objects or properties it has. Such objects are
shown as closed curves around their elements because this
considerably facilitates the reading of the diagrams.)
In terms of systematics the things in the figure given are
simple, being either basic themselves, or a set of basic things
('objects' in case of the objectual interpretation).
In physical terms the things of the objectual interpretation may be
extremely complex
tho: what they are may
vary from an 'elementary' or smaller particle to a galaxy or bigger whole.
In this interpretation one elementary particle and one galaxy might be
taken as separate entities side by side, and shown as two separate dots
in a diagram; in the attributive interpretation they could at
the most be shown as vague assemblages of dots or of dots and
closed curves (properties or properties and parts) surrounded by
a (bigger) closed curve. Of course, neither interpretation would
in this way represent the structure of the physical universe in
which elementary particles do not exist beside galaxies but
in galaxies as part of them (or part of a part of them, and so
on). A truly ultimate constituent of matter is in the 'objectualist'
scheme a dot and in our 'attributivist' scheme a
closed curve encompassing a number of dots representing properties
(which may be derelativized relations as well). Such a
truly elementary particle is a thing of the first type in the
nonattributive and a thing of the second type in the attributive
interpretation, the properties themselves being things of
the first type. (In the attributive interpretation there are
not 'objects' of the first type, since 'objects' are defined as
'material things', and properties are not material themselves.)
We shall call an element of a domain of discourse "a thing
of the first type", an ontic set of two or more of these
elements "a thing of the second type", an ontic set of two or
more things of the second type "a thing of the third type", and
so on. According to this systematic typification of things, the
elements of a thing of type n are things of type n-1 or of a
lower type. There is no reason why a thing with an element of
the second type, that is, an element which is a thing of the
second type, could not have an element of the first type as
well, that is, an element which is a thing of the first type.
It should be kept in mind that a thing of type n-1 which belongs
to a thing of type n is not a subset of that thing, even not an 'ontic
subset'. This is because the elements of the type n-1 thing are
not elements of the type n thing. Distinguishing elements on a
different level of typification may be unimportant in the
objectual interpretation, it is crucial in the attributive
interpretation. Here it allows complex things to have attributes
which are elements of the domain of discourse, while at the same
time having things as constituent elements which are not elements
of the domain of discourse, that is, not attributes (or relations)
themselves.
Moreover, the attributes of a component part of such a complex thing are
not elements of the complex thing itself either, which makes it possible to
distinguish between 'whole-attributes', that is, the attributes (and
relations) of the whole itself, and 'part-attributes', that is, the
attributes (and relations) of a component part of it.
It is only in this way that we can obtain an insight into the
structure of the concrete world, and also — as we will learn
— an easier and clearer insight into the structure of the world of
attributes and relations.
Figure I.1.3.2.2 shows the difference between the objectual and the
attributive representations of complex objects (ontic collections of
things).
By a whole of nonbasic things / (component) parts /
objects we shall mean an ontic set of which all these things are
members, while no other nonbasic thing is a member, but — and this is
important — of which basic things, that is, attributes, are elements
as well.
The mere collection of component parts
(nonbasic things) is called "the extensionality of the whole",
while the collection of attributes, that is, predicates, which
are an element of the whole is called "the predicament of the
whole". Both these sets are purely conceptual constructs and do
not really exist: solely their elements exist in (nonpropositional)
reality. Extensionalities cannot exist because, if such
sets were ontic, they would be the sole component part of
the whole, and this part would still not be the set of which it
would be the only element. (Set-theoretically one must distinguish
a singleton from the only element it has.) Predicaments (in
the sense used here) cannot exist either because, if a collection
of attributes existed, it would not exist beside the
extensionality but belong to it.
Objectualists have no systematic criterion to distinguish the
sets of objects which exist themselves as objects from those
sets which are extensions of a predicate expression, but can
never be objects themselves. (If a set of objects was an object
itself, it could not be an extension, because even a
predicate expression which is true of one object only would have
at least the singleton of this object as its extension.) In the
attributive interpretation such a criterion can easily be
provided: to be an object a set must have at least one attribute
(as an element), that is, the set should also 'be' something,
possibly in addition to having component parts. The analog of
this criterion in the objectual interpretation would be that an
ontic set needed a basic object as an element to exist. This
seems rather odd tho from a structural point of view. For
example, a material thing, however complex, would only exist if
it had an elementary particle in the direct sense, that is, a
particle which did not belong to any of its component parts.
Being is having, that is, having a (whole-)attribute in the
attributivist ontology. (We shall see that existing attributes
also have attributes.) But having is also being, since it
is either having an attribute and therefore being right away, or
having a component part, and therefore being an existing whole.
Also the whole of one thing (a 'singleton' in some objectualist
sense) must have at least one attribute, or one other attribute,
in order to exist. Hence, such a whole has at least two
elements, and singletons as such do not exist in the attributivist
system. Just as pure collections of nonattributive things
are conceptual constructions which do not exist in reality, so
extensions as sets of objects, or n-tuples of objects, do not
exist 'de re' either; only 'de dicto' may they play a role.
In the objectual interpretation an attribute is instantiated
as it were by the system of things which is its extension.
However, this extension does not determine the meaning of the
corresponding predicate expression, since two or more different
expressions may have the same extension without being synonymous.
Apart from this problem, a result of the objectualist
position seems to be that one first has to know, say, all gray
objects in the universe before one can know what grayness is.
And another problem is that of attribute identity or the
identity of predicate expressions: the objectualist must explain
what happens when one or more temporal objects enter or leave
the predicate extension. These problems cannot be adequately
dealt with without making a distinction between the extension
(or reference or denotation) and the intension (sense,
connotation) of an expression. Being forced or willing to acknowledge
the existence of intensions, intensionalist objectualists might
as well recognize attributes and relations right away. (If
extension is distinguished from denotation, the former
term refers to all the subclasses of the property class collectively,
and the latter one to its members collectively.)
In the attributive interpretation a thing is, roughly, an
instance of a system of attributes. The things are determined by
the attributes which they have themselves or which their
component parts have, or ultimately have. Normally tho, one does
not have to know all attributes of a thing before knowing, or
being able to identify, the thing itself. It depends on the
uniqueness of attributes or of certain combinations of attributes.
As regards concrete things, their spatiotemporal position
is probably one of the most important means of identification.
A special problem the attributivist faces is that of object identity
thru time: the question
whether the object is still the same when it loses or acquires a certain
attribute in the course of time.
This is first of all a problem for an ontology in which
perceptible objects would solely have attributes, and no component
parts as in our own conceptual framework. To overcome
this difficulty a distinction between essential and accidental
predicates might have to be drawn, but only in a conventionalist
sense.
A different problem of object identity arises when two
objects have exactly the same properties and no component
parts. But as speaking of "an object" or "something" itself
presupposes the presence of substance or existence, speaking of
"two objects" itself presupposes that these objects must be
distinguishable in some way. If they cannot be told apart by
the attributes they have (and not by their parts either), then
by means of one or more relations they bear to each other and/or
to other objects. Usually the relationship one has in mind when
speaking of "two objects" is a spatiotemporal one. This being
the case two objects which are 'entirely the same' (with regard
to attributes and parts) will have at least one different
relation with a third object or thing, for example, the person
speaking about these 'two' objects. It is clear that in the
attributive interpretation objects with exactly the same attributes
need not be identical, so long as relationships between
objects are recognized as well. On the other hand, for 'two'
conceptually distinguished objects to be identical 'they' must
have the same attributes, the same component parts (or none in
both cases) and the same relations.
Objectualists who do not attempt to cling to pure extensionalism
say that the extension of the predicate expression < -- is
gray > is something else than its intension. Instead of this we
will say ourselves that the collection of gray things, or of
classes of gray things, is something else than grayness; or
rather that grayness itself is not a gray thing. In this way we do
not only capture the significance of the distinction, but we
also do not unnecessarily deviate from ordinary language, for
we can express ourselves very well in this language in this
case. It is precisely one of the distinctive advantages of the
attributive interpretation of formal systems that the way of
formulating the object/property or thing/attribute relationship
directly mirrors the way of talking about this relationship in
extrasystematic discourse. Objects are not elements of a property
or of a fictional singleton or subclass belonging to a
property or one-place predicate extension: A is strong means
A has strength; strength is a property (or fully derelativized
relation) and thus A has the property of strength is true.
Some confusion might arise sometimes, because things with a certain
quality may on occasion, informally or figuratively, also
be given the name of this quality, for example, beauty for
somebody or something that is beautiful itself, or neutrality
for something that is neutral itself. Yet, even then it is only
the single things (beauties and neutralities) which have that
name, not the set of all things having the quality in question,
let alone the set of all singletons or subclasses of those things.
