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?

There's no guarantee that the race was run, or that someone won - perhaps everyone wanted to win, but the race was called off due to rain. Perhaps every runner crossed the line at the same time, and the officiator decided that no-one would be declared winner. Perhaps no-one actually signed up for the race - typically if a set $S$ is empty, then a statement like $(\forall x \in S) P(x)$ is vacuously true, but any attempt to say something about a particular element of $S$ will fail to hold any meaning.
