As the examples of the previous sections have shown, admitting concepts as objects and introducing standard concept functions makes ``quantifying in'' rather easy. However, forming propositions and individual concepts by quantification requires new ideas and additional formalism. We are not very confident of the approach presented here.

We want to continue describing concepts within first order
logic with no logical extensions. Therefore, in order to form
new concepts by quantification and description, we introduce
functions *All*, *Exist*, and *The* such that *All*(*V*,*P*) is
(approximately) the proposition that *``for all values of V,
P is true''*,

Since *All* is to be a function, *V* and *P* must be objects in the
logic. However, *V* is semantically a variable in the formation of
*All*(*V*,*P*), etc., and we will call such objects *inner variables*
so as to distinguish them from variables in the logic. We will use
*V*, sometimes with subscripts, for a logical variable ranging over
inner variables. We also need some constant symbols for inner
variables (got that?), and we will use doubled letters, sometimes with
subscripts, for these. *XX* will be used for individual concepts,
*PP* for persons, and *QQ* for propositions.

The second argument of *All* and friends is a ``proposition with
variables in it'', but remember that these variables are inner
variables which are constants in the logic. Got that? We won't
introduce a special term for them, but will generally allow concepts
to include inner variables. Thus concepts now include inner variables
like *XX* and *PP*, and concept forming functions like *Telephone* and
*Know* take as arguments concepts containing internal variables in
addition to the usual concepts.

Thus

is a proposition with the inner variable *PP* in it to the effect that
if *PP* is a child of Mike, then his telephone number is the same as
Mike's, and

is the proposition that all Mike's children have the same telephone
number as Mike. Existential propositions are formed similarly to
universal ones, but the function *Exist* introduced here should not be confused
with the function *Exists* applied to individual concepts introduced
earlier.

In forming individual concepts by the description function *The*,
it doesn't matter whether the object described exists. Thus

is the concept of Mike's only child. is the proposition that the described child exists. We have

but we may want the equality of the two propositions, i.e.

This is part of general problem of when two logically equivalent concepts are to be regarded as the same.

In order to discuss the truth of propositions and the denotation
of descriptions, we introduce *possible worlds* reluctantly and
with an important difference from the usual treatment.
We need them to give values to the
inner variables, and we can also use them for axiomatizing the
modal operators, knowledge, belief and tense. However, for axiomatizing
quantification, we also need a function such that

is the possible world that is the same as the world except that
the inner variable *V* has the value *x* instead of the value it
has in . In this respect our possible worlds resemble the *state
vectors* or *environments* of computer science more than the
possible worlds of the Kripke treatment of modal logic.
This Cartesian product structure on the space of possible
worlds can also be used to treat counterfactual conditional sentences.

Let be the actual world. Let mean that the
proposition *P* is true in the possible world . Then

Let mean that *X* denotes *x* in , and let
mean the denotation of *X* in when that is
defined.

The truth condition for *All*(*V*,*P*) is then given by

Here *V* ranges over inner variables, *P* ranges over propositions,
and *x* ranges over things. There seems to be no harm in making the
domain of *x* depend on .
Similarly

The meaning of *The*(*V*,*P*) is given by

and

We also have the following *syntactic* rules governing
propositions involving quantification:

and

where *absent*(*V*,*X*) means that the variable *V* is not present in
the concept *X*, and *Subst*(*X*,*V*,*Y*) is the concept that results
from substituting the concept *X* for the variable *V* in the
concept *Y*.
*absent* and *Subst* are characterized by the following axioms:

axioms similar to (92) for other conceptual functions,

axioms similar to (98) for other functions,

and corresponding axioms to (100) for *Exist* and *The*.

Along with these comes an axiom corresponding to -conversion,

The functions *absent* and *Subst* play a ``syntactic'' role
in describing the rules of reasoning and don't appear in the
concepts themselves. It seems likely that this is harmless
until we want to form concepts of the laws of reasoning.

We used the Greek letter for possible worlds, because we did not want to consider a possible world as a thing and introduce concepts of possible worlds. Reasoning about reasoning may require such concepts or else a formulation that doesn't use possible worlds.

Martin Davis (in conversation) pointed out the advantages of an alternate treatment avoiding possible worlds in case there is a single domain of individuals each of which has a standard concept. Then we can write

Tue May 14 16:07:43 PDT 1996