\documentstyle[marc,12pt,24.151-ps]{article}
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\def\part#1{\begin{center}#1\end{center}}
\def\prob #1.{\item[(#1)]}
\def\uo{\underline{$_1$}}
\begin{document}
\head{March 6, 1992}

\part{II}

\begin{enumerate}
\prob 1.  `\uo{} is small'.  The concept takes objects as arguments, and
returns truth values as values.
\prob 2.  `king of \uo'.  This function takes countries as arguments,
and returns a function which decides if a person is the king of that
country.
\prob 3.  `Something'.  If the argument to `Something' returns
{\bf True} for any object, `Something' returns true.
\prob 4.  `the'.  This is the $\iota$ (iota) function.  It
takes a function, and returns the object for which the function
returns true.
\prob 5.  `not \uo'.  Returns the opposite truth value.
\prob 6.  `Marc Horowitz is the ruler of the Universe.'  This sentence
has no truth value because `the ruler of the Universe' has no
reference.  Therefore, the equivalence relation has only one argument,
and the sentence has no truth value.
\end{enumerate}

\part{III}

The first sentence is of the logical form

\[
(\mbox{queen}(\mbox{France}))((\iota x)(\mbox{queen}(\mbox{England})x))
\]

This sentence has the truth value {\bf False}.  `is queen of France'
denotes a function which takes an object as an argument and returns a
truth value.  The argument refers to an actual person, who may also be
referred to as `Queen Elizabeth II'.

The second sentence is of the logical form

\[
(\iota x)(\mbox{queen}(\mbox{England})x) =
(\iota x)(\mbox{queen}(\mbox{France})x)
\]

This sentence has no truth value at all.  `$=$' denotes a two argument
function.  But, the second argument (the left-hand side) has no
reference (since there is no queen of France), so the $=$ function has
no value.

This is a case where the language of the two sentences masks the
logical forms of the sentences.  Although they are very similar
grammatically, their logical forms are completely different.

\part{IV}

In the first discussion, Frege points out that the sentence expresses
two basic propositions, one derived from the dependent clause, and one
derived from the independent clause.  In the second discussion, Frege
notes that there may be a difference in cognitive significance if one
sentence is replaced by a proposition of similar reference. There is a
third proposition which is expressed by the logical form of the
sentence.  I'm probably misusing some technical term here, but this
third proposition is an implication that the proposition expressed by
the independent clause has a causal relationship to the proposition
expressed by the dependent clause.

\part{V}

Take a sentence like `Dave believes that MIT is in California.'  `MIT
is in California' denotes {\bf False}.  However, this merely
substituting coreferentials here leads to the nonsensical sentence
`Dave believes that {\bf False}'.  Frege gets around this by stating
that in the context of a propositional verb such as `believes', the
reference of a sentence is not its truth value, but its sense.  Then,
we can substitute the sense into the sentence as a coreferential, and
it keeps the same truth value.

\end{document}
