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 a closed curve 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 the attributes (and relations) of a whole and the attributes (and relations) of a part. 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 were 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 besides 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 an 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 would 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.