103 lines
2.7 KiB
TeX
103 lines
2.7 KiB
TeX
\section{Bonus Problems}
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\definition{}
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The identity bird has sometimes been maligned, owing to
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the fact that whatever bird x you call to $I$, all $I$ does is to echo
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$x$ back to you.
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\vspace{2mm}
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Superficially, the bird $I$ appears to have no intelligence or imagination; all it can do is repeat what it hears.
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For this reason, in the past, thoughtless students of ornithology
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referred to it as the idiot bird. However, a more profound or-
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nithologist once studied the situation in great depth and dis-
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covered that the identity bird is in fact highly intelligent! The
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real reason for its apparently unimaginative behavior is that it
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has an unusually large heart and hence is fond of every bird!
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When you call $x$ to $I$, the reason it responds by calling back $x$
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is not that it can't think of anything else; it's just that it wants
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you to know that it is fond of $x$!
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\vspace{2mm}
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Since an identity bird is fond of every bird, then it is also
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fond of itself, so every identity bird is egocentric. However,
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its egocentricity doesn't mean that it is any more fond of itself
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than of any other bird!.
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\problem{}
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The laws of the forest no longer apply.
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Suppose we are told that the forest contains an identity bird
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$I$ and that $I$ is agreeable. \
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Does it follow that every bird must be fond of at least one bird?
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\vfill
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\problem{}
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Suppose we are told that there is an identity bird $I$ and that
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every bird is fond of at least one bird. \
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Does it necessarily follow that $I$ is agreeable?
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\vfill
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\pagebreak
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\problem{}
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Suppose we are told that there is an identity bird $I$, but we are
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not told whether $I$ is agreeable or not.
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However, we are told that every pair of birds is compatible. \
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Which of the following conclusiens can be validly drawn?
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\begin{itemize}
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\item Every bird is fond of at least one bird
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\item $I$ is agreeable.
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\end{itemize}
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\vfill
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\problem{}
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The identity bird $I$, though egocentric, is in general not hope-
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lessly egocentric. Indeed, if there were a hopelessly egocentric
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identity bird, the situation would be quite sad. Why?
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\vfill
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\definition{}
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A bird $L$ is called a lark if the following
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holds for any birds $x$ and $y$:
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\[
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(Lx)y = x(yy)
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\]
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\problem{}
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Prove that if the forest contains a lark $L$ and an identity bird
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$I$, then it must also contain a mockingbird $M$.
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\vfill
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\pagebreak
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\problem{}
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Why is a hopelessly egocentric lark unusually attractive?
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\vfill
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\problem{}
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Assuming that no bird can be both a lark and a kestrel---as
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any ornithologist knows!---prove that it is impossible for a
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lark to be fond of a kestrel.
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\vfill
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\problem{}
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It might happen, however, that a kestrel is fond of a lark. \par
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Show that in this case, \textit{every} bird is fond of the lark.
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\vfill
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