1.3.3 |
SECOND-ORDER PREDICATES ON THE OBJECTUAL VIEW |
Until now we have considered only one domain of discourse in isolation,
namely that of objects, or of objects and attributes of objects.
We have confined ourselves to the attributes and relations of these
objects, and we have compared the capacities of the
objectual and the
attributive interpretations of formal
systems for dealing with the possession of these attributes and the
existence of these relations.
We have not yet considered the attributes and relations of these predicates
themselves.
Doing this will open up a new field, or even a second domain of
discourse.
In the predicate calculus predicates of predicates are called
"second-order".
In principle this hierarchy of orders of predicate expressions is unlimited.
An example of a predicate expression which is taken to be second-order is
< (-- is a) color > 'because green is a color' and green (or
<green>) itself is a first-order predicate expression, 'because grass
is green' and grass is chosen as an entity (or collection of entities) in
the domain.
An expression of the type unhealthy may be treated as a compound
second-order predicate term which results when a productive process is
applied to the second-order predicate term healthy, as in
exercises are healthy.
Unhealthy is, then, analyzed as if it would mean not healthy.
The first flaw in this approach is that the prefix un- in
unhealthy, like in unhappy or unwise , does not
simply mean not.
For example, something that or someone who is unhappy is not happy, and
yet, something that is not happy is not necessarily unhappy at all.
As we will see later (in
2.3.2) no fewer than three
possibilities of not being happy must be distinguished instead of one.
There are other objections that can be made against the above-mentioned
approach, such as treating the same term both as a first- and as a
second-order expression, even
tho it appears this is
done in everyday discourse.
While words such as healthy are also applied to attributive
expressions, no-one could understand their meaning if one did not first
understand what a healthy living being or person is.
And in the latter usage healthy is not of the second but of the
first order.
That exercising is healthy should therefore be read as there is a
causal relationship between somebody's exercising and somebody's being
healthy/becoming healthier.
In this way we still speak of a relation between attributes of the first
order which is of the second order itself.
The difference is that causality is, then, always a second-order relation
and health(iness) always a first-order attribute.
In a hierarchy of orders it is important that expressions or entities of a
different order have different names (something that does not apply to
purely ontological entities without any particular order such as
having-as-an-element or being-an-element-of and existing).
If we think of grass as a basic object, then grass is green —assuming
this is true—, grass is colored and grass has a color.
This means that, if color is an attribute, it is just a
first-order attribute but an indefinite one. Color is certainly
not an ontic property beside being green, being red, and so
on. It is rather a conceptually produced property: if something
is green or red or .., then it is (said to be) colored, and then
it has a color. Hence, being-green or greenness, being-red or
redness, and so on, is having-a-color. Now, green is also said
to be a color, but then green is considered a basic object in
a domain of discourse, not a first-order attribute in such a
domain. Treating color simply as a second-order predicate
expression therefore passes by the tendency in everyday language
to treat phenomena of perception, or objects of experience, as
basic entities of the or a domain of discourse. Thus we speak of
colors, sounds, feelings, smells and flavors as particulars and
not as intensions of first- or second-order predicate expressions
or something of that nature. In this respect everyday
discourse seems to be phenomenalistic (and realistic) instead of
physicalistic. The objectual interpretation of formal systems is
just not very well suited to handle such an ontology in which
both physical things like grass and phenomenal things like
greenness or the color green are recognized as real entities.
The objectual and the attributive views are, strictly speaking,
not dissimilar logical views. Yet, it is not impossible
that the development of certain logical theories only makes
sense in an objectualist frame of mind or, for that matter, in
an attributivist frame of mind. Nevertheless, everything
that can be formulated in attributivist terms can be formulated
in objectualist terms, and vice versa, at least so far as the
essentials of our conceptual framework are concerned.
Given our ontological presuppositions, and with an eye to the
constructional,
normative doctrine to be expounded in
due course, the attributivist instrument is much more adequate, however.
Hence, we shall use this tool from now on and look how it works in more
detail.
