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(i)
 Domain: something suitably restricted, so as to make the translation plausible a: John Wx: x is a winner

If the truth of the sentence does not require the existence of the winner (so this occurrence is not a p.r.o), we could translate it thus: ¬[WaÙ"x[Wx®x=a]].

If, on the other hand, it is a p.r.o., we could paraphrase it: \$x[[WxÙ"y[Wy®x=y]]Ù¬x=a]

(ii)
 Domain: something suitably restricted, so as to make the translation plausible a: John Wx: x is a winner Rx: x is a rich man

If the truth of the sentence does not require the existence of the winner (so this occurrence is not a p.r.o), we could translate it thus: [[WaÙ"x[Wx®x=a]]®Ra]

If, on the other hand, it is a p.r.o., we could paraphrase it: \$x[[WxÙ"y[Wy®x=y]]Ù[x=a®Ra]]

(iii)

We could translate what Peter thinks ("John is the winner") using Russell's theory of descriptions, but not the sentence as a whole.

(iv)
 Domain: Holders of UK offices Px: x is Prime Minister Lx: x is the First Lord of the Treasury

We'll take two steps.

First step: "x[x is the Prime Minister ® Lx]

Second step: "x[[PxÙ"y[Py®y=x]]®Lx]

(v)

This is a bit trickier. We can't translate "x is the winner of a competition" as "Wx", because we want something for "it" to refer to.

 Domain: competitors and competitions Cx: x is a competition Wxy: x wins y Hxy: x is the highest scorer in y.

Take things slowly.

First step: "x"y[[Cx Ù y is the winner of x]®Hyx]

Second step: "x"y[[CxÙ[WyxÙ"z[Wzx®z=y]]®Hyx]

(vi)

We could try this:

 Domain: everything Cx: x is Father Christmas

Formalisation: ¬\$x[CxÙ"y[Cy®x=y]] (and not, of course, \$x[[CxÙ"y[Cy®x=y]]Ù¬\$yy=x])

This answers the question, but it doesn't do something else we might like. Our sentence poses a certain problem. On the one hand it is obviously true; but on the other hand "Father Christmas" appears to be a referring expression. But, if it were a referring expression, it seems that the sentence would not be true.

So what is the role of the expression? One possible answer is this. It is really a disguised definite description, and if we paraphrase the sentence using Russell's theory of descriptions being careful about scope, we will see how the sentence can be unproblematically true. (This was Russell's own answer.)

Good. But what is the definite description which is disguised as "Father Christmas"? Evidently we can't answer this question by saying, "the thing which is Father Christmas", because that repeats the expression "Father Christmas", with the disguise still in place. In fact the question of how to understand sentences such as this (innocent as it looks) has a long history and no simple answer.

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