# The Proper Axioms of the
Theory

Examples
In the monograph *Principia Metphysica*, there are three
proper axioms. The first two tell us: (1) ordinary objects, i.e.,
objects that possibly have a location in spacetime, necessarily do not
encode properties, and (2) objects *x* and *y* bear the
E-identity relation iff *x* and *y* are both ordinary
objects and (necessarily) they exemplify the same properties. These
two axioms may be formally represented as follows:

The third proper axiom is a
comprehension principle that asserts conditions under which abstract
objects exist: We complete this tutorial with
4 example sentences that are instances of this axiom schema. Of course,
there are a denumerably infinite number of other sentences which
are axioms as well.

## Examples of the Comprehension Axiom for Abstract Objects

1) There is an abstract object that encodes all and only the
properties Clinton exemplifies.

If we suppose that the constant symbol
`*c* ' denotes Bill Clinton, then sentence is instance of the
comprehension principle which asserts that there exists an abstract
object which encodes exactly the properties that Clinton exemplifies.
This is a perfect, abstract model of Clinton. If we use Leibniz's
terminology, we might say that this object is an individual concept
that contains all the general concepts that apply to Clinton. In
other words, this is Clinton's monad. There are a lot of interesting
theorems that govern monads, and these are described in the section
of *Principia Metaphysica* entitled The Theory of Monads.
2) There is an abstract object that encodes all and only the
properties that Holmes exemplifies in the story *s*.

Let us assume for the moment that the phrase `In the story
*s*' is well-defined. That is, let us assume: (1) that there
is a story *s* which is the classical source of information
about Sherlock Holmes (we may think of this as the conjunction of
Conan Doyle stories about Holmes, conceived as one lengthy story), and
that (2) there is some determinate set of properties *F* such
that the proposition *Holmes exemplifies F* is true in the
story *s*. At the very least, we take as *data* that
there is a set of ordinary English sentences of the form `According to
the story *s*, Holmes is *F* ' which are true. Then the
above instance of the comprehension axiom for abstract objects asserts
that there is an abstract object that encodes just those properties
*F* that satisfy these sentences. Let us assume further that
there is a unique object that encodes just those properties (this is
true and can be proven). Then we claim that Sherlock Holmes just is
this abstract object. Holmes encodes the properties attributed to him
in the story, and so is an incomplete object (with respect to the
properties he encodes). That is, there are properties *F* such
that neither *F* nor the negation of *F* are attributed
to Holmes in the story. However, like every object whatsoever, no
matter whether ordinary or abstract, Holmes is complete with respect
to the properties he exemplifies. For every property *F*,
either Holmes exemplifies *F* or exemplifies the negation of
*F*. He exemplifies such properties as *not being a
detective*, *not living in London*, *not being Watson's
friend*, etc., because he is an abstract object and abstract
objects do not exemplify such ordinary properties. However, Holmes
will exemplify such properties as *being admired by some real
detectives*, *being conceived by Conan Doyle*, *being
famous among devotees of detective novels*, etc.
3) There is an abstract object that encodes just the two
properties *being round* and *being square*.

Let `*R* ' denote the property of
being round and `*S* ' denote the property of being square.
Then this instance of comprehension asserts that there is an abstract
object that encodes both of those properties and no others. Why is it
that a disjunction (`or') is used in the right side of the
biconditional rather than a conjunction (`and')? Well, consider the
following description of a set: {x | x=1 V x=2}. How many elements
does this set have? The answer is: two. It describes the set {1,2},
for only the numbers 1 and 2 satisfy the defining condition `x=1 V
x=2'. Similarly, the set {F | F=R V F=S} is a description of the set
containing both the properties of being round and being square and no
others; i.e., {R,S}. So if we take the condition `F=R V F=S' and use
it as the right condition of the biconditional in an instance of the
comprehension axiom, we assert the existence of an abstract object
encoding two properties. NOTE: This abstract object is not
inconsistent with the law of geometry that says: necessarily
everything round fails to be square. This law is a law that governs
objects that *exemplify* roundness, not those that encode it.
It is rendered into formal notation as: The present instance of comprehension
merely asserts that there is an abstract object that encodes roundness
and squareness; it is logically compatible with the above law of
geometry for it does not follow from the fact that an object encodes a
property that it also exemplifies that property.
4) There is an abstract object that encodes just the propositional
properties *F* that are constructed out of true propositions.

We say that a property *F* is
a propositional property constructed out of proposition *p*
just in case *F* just is the property *being such that
p*. The property *being such that p* is represented by the
following predicate: An object *x* exemplifies this
property iff *p* (is true). So *being such that p* is a
propositional property constructed out of a true proposition whenevern
*p* is true. Our instance of comprehension asserts that there
is an abstract object that encodes all and only such properties. In
the section entitled The Theory of Worlds, this object will provably
satisfy the definition of an `actual world'. We shall extend our
notion of encoding so that we may say that an object *x*
encodes proposition *p* whenever *x* encodes the
property being such that *p*. So the above object encodes all
and only true propositions: it encodes all (and only) that (which) is
the case.