Intuitively, let us picture our object terms as naming or ranging
over a fixed domain of objects and picture our relation terms as
naming or ranging over a fixed domain of relations. Let us further
imagine that there is a domain of possible worlds (our primitive
notion `it is necessary that' is, intuitively, a universal quantifier
that ranges over the domain of possible worlds). Since objects can
exemplify properties and stand in relations at this world but not
exemplify those same properties and stand in those same
relations at every possible world, we may suppose that each
*n*-place relation *R* in the domain of relations has an
`exemplification extension' at each possible world. The
exemplification extension of a relation *R* at a possible world
*w* is a set of *n*-tuples, and each member of this set
represents an ordered group of objects that exemplify or stand in the
relation *R* at *w*. Now suppose that the variables
*x_1,...,x_n* denote the objects *o_1,...,o_n*,
respectively, and that the predicate `*F*' denotes the
*n*-place relation *R*. Then we may say that the atomic
sentence `*Fx_1...x_n*' is true at world *w* just in
case the *n*-tuple <*o_1,...,o_n*> is an element of the
exemplification extension of the relation *R* at *w*.
In the 1-place case, where `*x*' denotes the object *o*
and `*F*' denotes the property *P*, we may say that the
atomic sentence `*Fx*' is true at world *w* if and only
if *o* is a member of the exemplification extension of the
property *P*.

Now since atomic encoding formulas are monadic (i.e., they
always have a single object term to the left of a 1-place property
term), we can describe their truth conditions as follows. Suppose
that each 1-place property in the domain of relations receives, in
addition to an exemplification extension, an `encoding extension'.
Each object that is in the encoding extension of property *P*
represents the fact that the object encodes *P*. But whereas
the exemplification extension of a property or relation may vary from
world to world, let us say that the encoding extension of a property
does not vary from world to world. Indeed, the encoding extension of
a property is not relativized to a world. We can, nevertheless,
define world-relative truth conditions for atomic encoding formulas:
when `*x*' denotes *o* and `*F*' denotes
*P*, then the atomic formula `*xF*' is true at world
*w* if and only if *o* is an element of the encoding
extension of *P*.

Notice that, given these truth conditions, the following is a
fact: if an atomic encoding formula is true at any world *w*,
it is true at every world *w*. This fact, when expressed from
the point of view of our language in which modality is primitive, is
just our logical axiom for encoding, for that axiom asserts that if an
atomic encoding formula is possibly true, it is necessarily true.

**Logical Theorem: **

**Logical Theorem: **