The Logic of Complex Predicates
Examples
For convenience, we reproduce the item Logic/Logic of Complex
Predicates from Principia Metaphysica: In this tutorial, we give examples of the
axioms and then draw out some consequences.
Examples of the Axioms
Instances of Axiom 1:
The first example asserts: objects c and a
exemplify being an x and y such that x, b, and y exemplify S
if and only if c, b, and a exemplify S. As an intuitive example, we
might say: John and Betty stand in the relation being an x and y
such that x gives Fido to y if and only if John gives Fido to
Betty. The second example asserts: objects b and a exemplify
being an x and y such that necessarily, x fails to bear P to y if
and only if necessarily, b fails to bear P to a. As a more concrete
example: Sherlock Holmes and Gladstone exemplify being an x and y
such that necessarily x and fail to meet each other if and only
if necessarily, Holmes fails to meet Gladstone (the right condition in
this biconditional is in fact true---an abstract object like Holmes is
not the kind of thing that could meet an ordinary object such as
Gladstone, and so given this principle, these two objects exemplify
the complex relation denoted by the complex predicate; note, however,
that the right condition of the biconditional is consistent with the
claim: according to the Conan Doyle novels, Holmes met Gladstone).
The third example asserts: object b exemplifies being an object x
such that, necessarily, if x exemplifies Q, then x exemplifies having
a spatiotemporal location if and only if necessarily, if b
exemplifies Q, then b has a spatiotemporal location. As a concrete
instance: an object y exemplifies being something such that,
necessarily, if it exemplifies being a plant, it has a spatiotemporal
location iff necessarily, if y exemplifies being a plant, y has a
spatiotemporal location (the right condition of this biconditional is
true for every object y whatsoever). Exercise:
Produce specific examples of the other instances of Axiom 1.
Remark: Notice that in the last instance of Axiom 1,
we have an example of a 3-place relation that is exemplified just in
case a 2-place relation is exemplified. The objects a, b, and c
exemplify the relation being an x, y, and z such that x bears R to
y just in case a bears R to b. Such relations will play an
interesting role in developing the theory of possible worlds. In
particular, we will focus on 1-place properties that objects exemplify
just in case a proposition (0-place relation) is true. For example,
by Axiom 1, we know that an object x exemplifies the property
being such that Clinton is President if and only if Clinton
is President. By Universal Generalization, it follows that every
object x exemplifies being such that Clinton is President if
and only if Clinton is President. And by the Rule of Necessitation,
it follows that necessarily, every object x exemplifies being such
that Clinton is President if and only if Clinton is President.
We call these properties that objects exemplify whenever a proposition
is true `propositional properties'. For a more precise definition,
see the item Logic/Propositional Properties.
Instance of Axiom 2:
This instance asserts: the 3-place
relation being an x, y, and z such that x, y, and z exemplify the
relation G is identical to the 3-place relation G. In other
words, the simplest possible complex predicates, consisting of an
atomic exemplification formula in which all the object terms are
variables bound by a lambda, denote the same relation as the simple
predicate symbol involved in the atomic formula.
Instance of Axiom 3:
The first example asserts: the relation
being an x and y such that necessarily, it is not the case that x
bears P to y is identical with the relation being a y and z
such that necessarily, it is not the case that y bears P to z.
In other words, interchange of bound variable makes no difference to
the relation denoted by the complex predicate.
Some Consequences:
Logical Theorem: Comprehension Principle for Relations
This tells us that for any
exemplification-based formula phi (containing no quantifiers over
relations or definite descriptions), there is a relation F which
is necessarily such that objects x_1,...,x_n exemplify F if and only
if x_1,...,x_n are such that phi. This is a logical theorem
schema that can be derived from Axiom 1 in n+2 simple steps:
apply Universal Generalization to Axiom 1 n times, beginning
with the variable x_n and ending with the variable
x_1; then apply the Rule of Necessitation; finally apply the
rule of Existential Generalization to the complex predicate. EG can
be applied to the complex predicate because the latter is an
n-place term that denotes a relation.
Here are some examples of this logical theorem schema---these
correspond to the examples of Axiom 1 above:
In each example, the existence of the
complex relation in question is explicitly asserted. It is important
to recall now that there is a definition of identity for relations.
In the section Language/Relation Identity in Principia
Metaphysica, one finds the following two definitions, the first
of which gives conditions under which properties (1-place relations) are
identical and the second of which gives conditions under which
n-place relations (n > 1) are identical: The first definition tells us that
properties F and G are identical if and only if,
necessarily, they encode the same properties. The second definition
tells us that, for n > 1, n-place relations
F and G are identical iff for each way of plugging
n - 1 objects in the same order into F and
G, the resulting 1-place properties are identical. These
definitions, together with the comprehension principle for relations,
constitute a mathematically precise theory of relations and
properties, for they are explicit existence and identity conditions
for these entities.
Notice that these definitions allow us to consistently assert
that there are properties and relations which are necessarily
exemplified by the same objects but which are nevertheless distinct.
For example, the properties being equiangular and being
equilateral are distinct properties, and so it is natural to
suppose, therefore, that the property being an equiangular
Euclidean triangle is distinct from the property being an
equilateral Euclidean triangle. However, these two complex
properties are exemplified by the same objects at every possible world
(it is necessarily the case that anything exemplifying the one
exemplifies the other). This is a virtue of the theory---many
theories and/or treatments of properties identify properties that are
necessarily equivalent. We identify properties only when they are
necessarily equivalent with respect to the objects that encode them,
not when they are necessarily equivalent with respect to the objects
that exemplify them. The intuition here is that abstract objects
represent possible objects of thought. If properties F and
G are distinct, then it is possible to conceive of an object
having (i.e., encoding) F and not G (and vice
versa). So the `encoding extensions' of F and G are
distinct when F and G are distinct. However, if
F and G are not distinct properties, one couldn't
conceive of an object having (i.e., encoding) F and not
G. The encoding extensions of identical properties are the
same.
Logical (Axioms and) Theorem: Comprehension Principle for
Propositions
This is actually the `degenerate' case of
Axiom 1 when n = 0. It can be read as follows: the
proposition that-phi is true if and only if phi. Here are
some examples: We may read the first example as
follows: the proposition that a exemplifies P and it is not the
case that b exemplifies Q is true if and only if a exemplifies P
and it is not the case that b exemplifies Q.
From the Axiom Schema, we may derive the following theorem schema that
constitutes a comprehension principle for propositions:
Here are two instances of the theorem
schema that correspond to the two instances (above) of the axiom
schema: These examples explicitly assert
the existence of complex propositions.
Here, too, it is important to recall the definition for the
identity of propostions in the item Language/Relation Identity:
This tells us that propositions p
and q are identical if and only if the properties being
such that p and being such that q are identical. The
conditions under which properties are identical has already been
defined, and so propositional identity is hereby reduced to property
identity. Consequently, we now have a precise theory of propositions:
the comprehension axiom and the above definition give us explicit
existence and identity conditions for propositions.
This definition also allows us to consistently assert that
certain necessarily equivalent propositions are nevertheless distinct.
For example, both of the following propositions are necessarily false:
Fido is a dog and it is not the case that Fido is a dog and
There is a barber who shaves only those people who don't shave
themselves. In formal terms, these propositions would be
denoted by the following complex 0-place predicates:
Though the two propositions are
necessarily equivalent (i.e., true in the same possible worlds), we
may consistently assert that they are distinct. This stands in
contrast to treatments of propositions on which necessarily equivalent
propositions are identified, contrary to intuition.
