Why is this inference invalid? So I purchased a book on logic (for beginners) as the subject interests me, and the author presents the following statement as an example of an invalid inference:
"Everyone wanted to win the prize; so the person who won the race wanted to win the prize."
Symbolised as follows, where xP is 'x wanted to win the prize' and xR is 'x won the race': $$\frac{\forall x\;xP}{(|x\;xR)P }$$ where |x means "the object x, such that"  (a notation I cannot seem to find anywhere else?).
The author states this is invalid because there is potentially a situation s in which everyone satisfies P but nobody satisfies R. But I do not understand how this could be true, due to the structure of the sentence - surely, xR in the conclusion means the race was ran and there was a winner?

The notation is difficult to understand, the way I read it is:
$$\frac{\forall x\;xP}{(|x\;xR)P }$$
is equivalent to:
$$\forall x P(x) \rightarrow \exists x (R(x)\land P(x))$$
Which implies:
$$\forall x P(x) \rightarrow \exists x R(x)$$
And this is indeed not the case as $R(x)$ could never be true, for any x.
