1 |
(i)
|
$xWx
|
| |
(ii) |
$x$y[[WxÙWy]Ù¬x=y]
(for example)
|
| |
(iii)
|
$x$y$z[[WxÙ[WyÙWz]]Ù[¬x=yÙ[¬y=zÙ¬z=x]]] |
| |
(iv)
|
¬$x$y[[WxÙWy]Ù¬x=y];
or, "x"y[[WxÙWy]®x=y] |
| |
(v)
|
¬$x$y$z[[WxÙ[WyÙWz]]Ù[¬x=yÙ[¬y=zÙ¬z=x]]];
or
"x"y"z[[WxÙ[WyÙWz]]®[x=yÚ[y=zÚz=x]]]
|
|
(vi) |
[$xÙ¬$x$y[[WxÙWy]Ù¬x=y]];
or,
[$xÙ"x"y[[WxÙWy]®x=y]];
or
$x[WxÙ¬$y[WyÙ¬x=y]];
or,
$x[WxÙ"y[Wy®x=y]];
or
(but with a health warning!) $x"y[Wy«x=y].
|
|
(vii) |
$x[[WxÙLxa]Ù"y[[WyÙLya]®x=y]]
(say) |
| |
(viii) |
$x[[HxÙ[LxaÙLxb]]
Ù ¬$y[[HyÙ[LyaÙLyb]]Ù¬x=y]]
(say) |
| |
(ix) |
Answer.
Best to take it slowly.
| First
step: |
"x[Hx®x likes at
most one woman who likes only one man] |
| Next
step: |
"x[Hx® there is
at most one woman-who-likes-only-one-man who x likes] |
| Next
step: |
"x[Hx® ¬$y$z[[[y is a woman-who-likes-only-one-man Ù Lxy] Ù
[z is a woman-who-likes-only-one-man Ù Lxz]] Ù
¬y=z]] |
| Next
step: |
y
is a woman-who-likes-only-one-man becomes [WyÙ$u[[MuÙLyu]Ù"v[[MvÙLyv]®u=v]]];
so we plug in this (and the formula that says the same thing
about z) to give finally: |
"x[Hx®¬$y$z[[[[WyÙ$u[[MuÙLyu]Ù"v[[MvÙLyv]®u=v]]]ÙLxy]Ù
[[WzÙ$u[[MuÙLzu]Ù"v[[MvÙLzv]®u=v]]]ÙLxz]]Ù¬y=z]]
|
2
|
(i) |
There
is at least one man. (Or just, there is a man.) |
| |
(ii) |
There
is at least one man. |
| |
(iii) |
There
are least two men. |
| |
(iv) |
There
is at most one man. |
| |
(v) |
There
is at most one man. |
| |
(vi) |
There
are at least two men. |
| |
(vii) |
Exactly
one woman likes John. |
| |
(viii) |
There
is exactly one woman and she likes John. |
| |
(ix) |
There
is a man.
|
| |
(x) |
There
is at most one man. (It says that there is something to which any
man is identical - i.e. is one and the same thing as. So there can't
be more than one man.)
|
| |
(xi) |
Exactly
one woman likes John.
|
| |
(xii) |
There
is a man. (It says that there is something such that, if anything
is identical to it, it (the first thing) is a man. Since there is
something identical to the first thing (namely it!), this means that
the first thing is a man.)
|
| |
(xiii) |
There
is something, and, if it is man, it is the only thing there is.
|
| |
(xiv) |
This
means something rather peculiar. It is equivalent to "y$x[¬x=yÚMx].
It is compatible with there being nothing at all (since it begins
with """ and
contains no names); it implies that, if there is just one thing, it
is a man (since, if there is just one thing, there won't be anything
which is not identical to it; so Mx will have to be true of it). But,
if there is more than one thing, each could be a man, but none would
have to be. So that is what it means: either there is nothing, or
there is just one thing and it is a man, or there is more than one
thing. (Not something one is likely to find oneself wanting to say!)
|
| |
(xv) |
There
is exactly one man whom every woman likes – i.e. the same man in each
case. |
| |
(xvi) |
Every
man likes just one woman (but not necessarily the same one). |