1
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(i)
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$x[[MxÙ"y[My®x=y]]ÙLxb]
(There is exactly one man and he likes Mary.)
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(ii)
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First
step: There is exactly one person who likes Mary and he likes everyone.
So:
$x[[[HxÙLxb]Ù"y[[HyÙLyb]®x=y]]Ù"z[Hz®Lxz]]
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(iii)
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First
step: $x$y[[x is the man who
likes Mary Ù y is the woman
who likes John] Ù Lxy].
x is
the man who likes Mary: [[MxÙLxb]Ù"z[[MzÙLzb]®z=x]]
y is
the woman who likes John: [[WyÙLya]Ù"u[[WuÙLua]®u=y]]
So:
$x$y[[[[MxÙLxb]Ù"z[[MzÙLzb]®z=x]]Ù[[WyÙLya]Ù"u[[WuÙLua]®u=y]]]ÙLxy]
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(iv)
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[[MaÙ"x[Hx®Lax]]Ù"y[[MyÙ"x[Hx®Lyx]]®y=a]].
(Notice that it is all right to use "x twice, because the second occurrence
is not in the scope of the first.)
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(v)
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¬$z [HzÙ[[MzÙ"x[Hx®Lzx]]Ù"y[[MyÙ"x[Hx®Lyx]]®y=z]]].
(We don't, of course, want to say that there is something which
is the man who likes everyone, and there is no such person as it.)
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(vi) |
This
is perhaps ambiguous. It could be committed to the existence of
the man who like everyone, and be saying that if John is he, John
likes Mary. But (more likely?) it is not committed to the existence
of such a man, but is saying that if John is such a man, he likes
Mary. Taken the second way we have:
[[[MaÙ"x[Hx®Lax]]Ù"y[[MyÙ"x[Hx®Lyx]]®y=a]]®Lab]
Taken
the first way we would have:
$z[[[MzÙ"x[Hx®Lzx]]Ù"y[[MyÙ"x[Hx®Lyx]]®y=z]]Ù[a=z®Lab]]
(Consider
instead, "If John is the person who took my milk, he is no
friend of mine". Here, most likely, I do mean to commit myself
to the existence of the culprit.)
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(vii) |
$xx=x.
(Or we could have, say, $x[MxÚ¬Mx].
But using "=" seems neater.)
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(viii) |
$xx=x
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(ix) |
$x$y¬x=y
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(x) |
$x$y[¬x=yÙ"z[z=xÚz=y]]
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2
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(i) |
"x[Ux®$yTyx]
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(ii) |
¬$x[$yTxyÙ$zTzx].
Or, "x[$yTxy®¬$zTzx]. (If you really want to have all the quantifiers
at the beginning, you would have ¬$x$y$z[TxyÙTzx];
or, "x"y"z[Txy®¬Tzx].
But there is absolutely no need, and it is easy to make mistakes
trying to achieve this bit of regimentation.)
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(iii) |
$x[UxÙ$y$z[[TyxÙTzx]Ù¬y=z]]
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(iv) |
First
step: ¬$x[UxÙ x is a member
of more than one college]
x is
a member of more than one college: $y$z[[[CyÙMxy]Ù[CzÙMxz]]Ù¬y=z]
So:
¬$x[UxÙ$y$z[[[CyÙMxy]Ù[CzÙMxz]]Ù¬y=z]]
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(v) |
First
step: $x$y[[¬x=yÙx
and y are members of different colleges]Ù$z[TzxÙTzy]].
(We need "¬x=y", because otherwise what is said could
be made true by one person who is a member of more than one college
– supposing such a thing to be possible.)
x and
y are members of different colleges: $u$v[[[CuÙCv]Ù¬u=v]Ù[MxuÙMyv]]
So:
$x$y[[¬x=yÙ$u$v[[[CuÙCv]Ù¬u=v]Ù[MxuÙMyv]]]Ù$z[TzxÙTzy]]
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(vi) |
First
step: ¬$x$y[[¬x=yÙ$u$v[[[CuÙCv]Ù¬u=v]Ù[MxuÙMyv]]]Ù
x and y have exactly the same tutors] (The first part is taken from
the answer to (v) above.)
x and
y have exactly the same tutors: "z[Tzx«Tzy]
So:
¬$x$y[[¬x=yÙ$u$v[[[CuÙCv]Ù¬u=v]Ù[MxuÙMyv]]]Ù"z[Tzx«Tzy]]
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3
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(i) |
There
is an undergraduate reading Philosophy and Economics and one reading
Philosophy but not Economics.
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(ii) |
This
says that there is exactly one thing which… which what? That is
given by [CxÙ¬$y[[MyxÙUy]Ù¬Py]].
This means: x is a college and there is nothing which is a member
of it and is an undergraduate and is not reading Philosophy. That
is, x is a college, all of whose undergraduate members are reading
Philosophy.
So:
There is just one college all of whose undergraduate members are
reading Philosophy.
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(iii) |
This
says that, for any x, if there is someone (strictly something)
reading Philosophy whose tutor is x, then no one (thing) whose tutor
is x is reading Economics. So: No one (thing, strictly) tutors both
Philosophy and Economics pupils.
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(iv) |
Some
tutors are members of more than one college.
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(v) |
Sometimes
tutors who are members of one college teach pupils who are members
of another.
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(vi) |
Anyone
(thing) who tutors some pupils who are reading Philosophy also tutors
some who are reading Economics.
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(vii) |
This
says that there is exactly one thing of which something is true
and that something else is true of it. The first something is given
by "z[Pz®Txz],
which means: x is tutor to everyone (everything) reading Philosophy.
The something else is given by "z[Ez®¬Txz], which means: x is tutor to no one reading
Economics. So the whole formula means: there is exactly one thing
which is tutor of everyone reading Philosophy and it is tutor of
no one reading Economics.
So:
The person who tutors everyone reading Philosophy tutors no one
reading Economics.
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