Some sentences that recur as examples in the philosophical literature will be expressed in our notation so the treatments can be compared.
First we have ``The number of planets = 9'' and ``Necessarily 9 = 9'' from which one doesn't want to deduce ``Necessarily the number of planets = 9''. This example is discussed by Quine (1961) and (Kaplan 1969). Consider the sentences
and
Both are true. (56) asserts that it is not necessary that the number of planets be 9, and (57) asserts that the number of planets, once determined, is a number that is necessarily equal to 9. It is a major virtue of our formalism that both meanings can be expressed and are readily distinguished. Substitutivity of equals holds in the logic but causes no trouble, because ``The number of planets = 9'' may be written
or, using concepts, as
and ``Necessarily 9=9'' is written
and these don't yield the unwanted conclusion.
Ryle used the sentences ``Baldwin is a statesman'' and
``Pickwick is a fiction'' to illustrate that parallel sentence
construction does not always give parallel sense.
The first can be rendered in four ways, namely
or 
or 
or 
where the last
asserts that the concept of Baldwin is one of a statesman. The
second can be rendered only as
as 
or 
.
Quine (1961) considers illegitimate the sentence
obtained from ``Philip is unaware that Tully denounced Catiline'' by existential generalization. In the example, we are also supposing the truth of ``Philip is aware that Cicero denounced Catiline''. These sentences are related to (perhaps even explicated by) several sentences in our system. Tully and Cicero are taken as distinct concepts. The person is called tully or cicero in our language, and we have
and
We can discuss Philip's concept of the person Tully by introducing a function Concept2(p1,p2) giving for some persons p1 and p2, p1's concept of p2. Such a function need not be unique or always defined, but in the present case, some of our information may be conveniently expressed by
asserting that Philip's concept of the person Tully is Cicero. The basic assumptions of Quine's example also include
and
From (63), 
, (67) we can deduce
from (67) and others, and
using the additional hypotheses
and
Presumably (68) is always true, because we can always construct a concept whose denotation is Cicero unbeknownst to Philip. The truth of (69) depends on Philip's knowing that someone denounced Catiline and on the map Concept2(p1,p2) that gives one person's concept of another. If we refrain from using a silly map that gives something like Denouncer(Catiline) as its value, we can get results that correspond to intuition.
The following sentence attributed to Russell is is discussed by Kaplan: ``I thought that your yacht was longer than it is''. We can write it
where we are not analyzing the pronouns or the tense, but are using denot to get the actual length of the yacht and Concept1 to get back a concept of this true length so as to end up with a proposition that the length of the yacht is greater than that number. This looks problematical, but if it is consistent, it is probably useful.
In order to express ``Your yacht is longer than Peter thinks it is.'', we need the expression Denot(Peter,X) giving a concept of what Peter thinks the value of X is. We now write
but I am not certain this is a correct translation.
Quine (1956) discusses an example in which Ralph sees Bernard J. Ortcutt skulking about and concludes that he is a spy, and also sees him on the beach, but doesn't recognize him as the same person. The facts can be expresed in our formalism by equations
and
where P1 and P2 are concepts satisfying denot P1 = ortcutt and denot P2 = ortcutt. P1 and P2 are further described by sentences relating them to the circumstances under which Ralph formed them.
We can still consider a simple sentence involving the persons as things--write it believespy(ralph,ortcutt), where we define
using suitable mappings Concept1 and Concept7 from persons to
concepts of persons. We might also choose to define believespy in
such a way that it requires 
for several concepts P of p2, e.g. the concepts arising from all
p1's encounters with p2 or his name. In this case
will be false and so would a corresponding
. However, the simple-minded predicate believespy, suitably defined, may be quite useful for expressing the facts necessary to predict someone's behavior in simpler circumstances.
Regarded as an attempt to explicate the sentence ``Ralph believes Ortcutt is a spy'', the above may be considered rather tenuous. However, we are proposing it as a notation for expressing Ralph's beliefs about Ortcutt so that correct conclusions may be drawn about Ralph's future actions. For this it seems to be adequate.
