As shown in the Configurations section, a contingent property (monadic predicate) P is predicable of an object x if some state of x is in the extension of P, or more simply, is an element of P. Since we are here only interested in contingent properties, we can exclude {S} and the empty set {O} from the extension of a property -- they can never be states of any object --yielding the definition:
a contingent property of objects is a proper non-empty subset of the set of non-trivial states of affairs.
Predicability serves as a way of classifying predicates based on what objects they are predicable of, and, more importantly, given the philosophers' interest in the question of "natural kinds", objects can be classified on the basis of what predicates are predicable of them. The predicates predicable of an object x are all those predicates which contain as elements at least one but not all states of x, or more formally:
P is predicable of x iff (P intersect x) is neither empty nor x.
If S is finite, we can count the number of properties of x, based on its size (number of states.) The number of properties of even very small objects is quite large -- the object {12-3} in a 3-state microcosm has 32 contingent properties!
Are there "natural kinds" of objects based on the properties they have? What sort of categorial structure does the predicability relation induce? One particular structure has been rediscovered several times: the lattice of closed sets of objects induced by the relation "P is predicable of x".
The technical concept of closure and the closure lattice is discussed in [Birkhoff, 1948, pp. 54-56]. The relationship between object categories and their common properties is roughly the relationship between the classical notions of "extension" and "intension" of a concept, treated mathematically under the label Galois connection. (See [Wille, 1982] for a detailed lattice-theoretic discussion of concepts).
For a single object, we can make the following definition:
The category (closed set) to which x belongs is the set of all objects which share all the properties which are predicable of x.
This idea, in a somewhat murky form, crops up as the basis for defining ontological categories in Sommers' "Types and Ontology" [1963]. ( See also [Roosen-Runge, 1973] for other categorial applications of the closure lattice.) In Sommers' formulation, x is in the span of P (i. e., P is predicable of x) if P can be "either truly or falsely but not absurdly" predicated of x, leaving open the issue of what counts as an object, and what constitutes truth/falsity for non-contingent predicates. He calls the set of objects spanned by a predicate P the alpha-type defined by P, and attributes the concept to Russell. Similarly, the set of predicates which have the object x in their span is called the B-type defined by x, a concept he attributes to Ryle. (A Rylean category mistake is the application of a predicate to x which is not in the B-type associated with x.) From this, Sommers defines a beta-type for objects:
"a set of things all of whose members are spanned by predicates of some B-type and none of whose members is spanned by any predicate outside of that B-type."
Alternatively, a beta-type is constructed from a set X of objects as the set of all objects which are in the span of every B-type defined by elements of X. This is the closure of X with respect to the predicability relation. To each object x is associated a unique beta-type ( its "natural kind") = the closure of {x} = the set of all objects which define the same B-type as x.
Unfortunately, the structure of the beta-types or closed classes derived from single objects is quite uninteresting, for the closure of {x} is just {x}, for any object x. This stems from the extensional nature of objects -- an object is defined by the set of its states. For if X is a state of x, then {X} is a contingent property of x, so X is in the B-type of x. But if z is in the closure of {x}, then z defines the same B-type as x, so {X} must be a contingent property of z. However, z can only have the property {X} in states which are in {X}, i. e., the state X, so X must be a state of z. By a parallel argument, every state of z must be a state of x, and it follows that z = x, and the closure of {x} = {x}.
So each object is sui generis; its "natural kind" is just the class consisting of itself.
But what about beta-types in general? What is the structure of the closures of arbitrary collections of objects? Is each collection of objects closed, so that like singleton collections, every collection is its own "natural kind"? If so, the lattice of kinds is again not very interesting -- it is just the Boolean algebra generated by the set of singleton sets {x}. My suspicion is that this is indeed the case but I am unable to construct a proof.
A parallel closure operation is definable for a set of properties, which Sommers labels the A-type of the set. The A-type of a set of properties K is the set of all properties which span all those objects spanned by all properties in K (the "intension" of the extension of K). Given the parallelism in the definitions of A-type and beta-type, the following fundamental theorem on the structure of closure lattices should not be surprising:
The lattice of the closures of sets of objects is "dual-isomorphic" to the lattice of the closures of sets of predicates; each lattice has the structure of the other one, with the partial-ordering inverted. (See Birkhoff for details.)So if we know the categorial structure of predicates, we know the categorial structure of objects, and vice-versa -- object categories and predicate categories, extension and intension, are dual concepts.
"The basic combination in which general and singular terms find their contrasting roles is that of predication . . . . Predication joins a general term and a singular term to form a sentence that is true or false according as the general term is true or false of the object, if any, to which the singular term refers."
So for Quine, predication is a grammatical relation, in which the roles of the subject and predicate terms are distinguished by what they refer to -- in particular, the subject term must refer to an object in order for the sentence as a whole to have a truth-value. Quine, as a platonist, does not recognize that the general term can only be true or false of the way the object is (as opposed to describing a general form of the object or including the object in its number), and so misses that the grammatical roles he describes are only applicable in contingent propositions. ( In mathematical propositions, we may find the grammatical roles of subject and predicate, but there are no referents for the singular terms.)
Strawson in his critique of this passage, argued that "the distinction drawn remains inadequately explained" by grammatical role, and, after lengthy argument, comes to a generalization of predication as a construction in which "what the [subject] designates or signifies is an instance of what the [predicate] signifies." But this too is incorrect-- Strawson's example, "Betty is pretty", does not say that Betty herself is an instance of prettiness (see below); it is the way Betty looks that may or may not be such an instance.
One consequence of our analysis is that the obvious difference between the grammatical roles of subject and predicate in ordinary language does not match the actual difference between (metaphysical) objects and what is predicable of them. Whether the distinction is cast as one between singular and general terms (Quine) or between particulars and terms which group particulars (Strawson), it misses the mark. Both object and predicate terms are general terms; both identify groupings of particulars. In set-theoretic language, both objects and properties are sets of states of affairs; both object and property references are instantiated by the same kind of particular, a reference to a state of affairs.
Nonetheless there is a difference between properties and objects: a property need not partition the states of the World into disjoint states of affairs; an object always does. But we cannot tell this from the logical form-- in the proposition P x, it is possible that both P and x refer to objects, that one object is predicated of another. Of course, this seems odd; the objects of experience and ordinary language are phenomenological objects, not objects in metaphysical terms, and in the way that we ordinarily think of phenomenological objects, they are not predicable of each other.
In order for one metaphysical object to be predicable of another, the predication must be contingent. For example, an object can never be predicated of itself, since its states are its elements, hence always instances of itself considered as a predicate.
A toy example in a 5-state microworld shows how one object can be predicated of another:
Let x be the object a-b-c-d-e and y the object a-bc-d-e. Then the proposition y x (or x y) is contingently true in the state of affairs {d, e} and is false otherwise. In {d, e}, the predicate y holds of x, and conversely, the predicate x holds of y.
s <= A where A is some state of affairs such that x is true of A (i. e., A is a state of x), and A has the property (or is in the extension of) P.
For any property Q, we can define the attribute Q-ness as a collection of subsets (often a partition) of Q . The elements of Q-ness are collections of different manifestations or values of Q.
Thus, dark red is a property which happens to be an element of the partition of the property red into dark and not dark red. The distinction between dark red as a subset of red: "if x is dark red, then it is red", and dark red as an element of redness: "if x is dark red, it has redness" is purely formal, and has no metaphysical significance.
Similarly, the property red is a value of the attribute 'colour-ness', and so whatever is red is coloured, or has the attribute of colour.
Last updated: August 19, 2008