While the relation denotes(X,x) between concepts and things is many-one, functions going from things to certain concepts of them seem useful. Some things such as numbers can be regarded as having standard concepts. Suppose that Concept1 n gives a standard concept of the number n, so that
We can then have simultaneously
and
(We have bravely used Knew instead of Know, because we are not now concerned with formalizing tense.) (31) can be condensed using Composite1 which takes a number into the proposition that it is composite, i.e.
getting
A further condensation can be achieved using Composite2 defined by
letting us write
which is true even though
is false. (36) is our formal expression of ``Kepler knew that the number of planets is composite'', while (31), (33), and (35) each expresses a proposition that can only be stated awkwardly in English, e.g. as ``Kepler knew that a certain number is composite, where this number (perhaps unbeknownst to Kepler) is the number of planets''.
We may also want a map from things to concepts of them in order to formalize a sentence like, ``Lassie knows the location of all her puppies''. We write this
Here Conceptd takes a puppy into a dog's concept of it, and Locationd takes a dog's concept of a puppy into a dog's concept of its location. The axioms satisfied by Knowd, Locationd and Conceptd can be tailored to our ideas of what dogs know.
A suitable collection of functions from things to concepts
might permit a language that omitted some individual concepts like
Mike (replacing it with 
) and wrote many sentences
with quantifiers over things rather than over concepts. However, it
is still premature to apply Occam's razor. It may be possible to
avoid concepts as objects in expressing particular facts but impossible
to avoid them in stating general principles.
