Exercise 4.1 - Answers

1	(i)	(a) "x[x is man ® x is human]; (b) ¬$x[x is a man Ù ¬x is human]
	(ii)	(a) "x[x is woman ® ¬x likes John]; (b) ¬$x[x is a woman Ù x likes John]
	(iii)	(a) $x[x is a woman Ù John likes x]; (b) ¬"x[x is a woman ® ¬John likes x]
	(iv)	(a) $x[x is a woman Ù x likes John]; (b) ¬"x[x is a woman ® ¬x likes John]
	(v)	This is perhaps ambiguous. We could translate it thus: (a) "x[x likes John ® x is a woman], or (equivalently) "x[¬x is a woman ® ¬x likes John] (b) ¬$x[x likes John Ù ¬x is a woman] Note that these translations will be true if no one likes John. If one thinks that the original implies that some women do like John, one will need to add something to that effect: e.g. $x[x is woman Ù x likes John] . Actually it will be enough to add something which says that something likes John (e.g. $x x likes John, or ¬"x ¬x likes John), because, of course, if something likes John and nothing apart from women do, it will follow that some woman does. So here are alternative translations which imply that some woman likes John. (a) ["x[x likes John ® x is a woman] Ù ¬"x¬ x likes John] (b) [¬$x[x likes John Ù ¬x is a woman] Ù $x x likes John]
	(vi)	(a) "x[x is a person ® [x is a man Ú x is a woman]]; (b) ¬$x[x is a person Ù [¬ x is a man Ù ¬ x is a woman]]
	(vii)	(a) ¬"x[x is a human ® x is man]; (b) $x[x is human Ù ¬ x is a man]
	(viii)	(a) "x[x is a woman ® x likes John]; (b) ¬$x[x is a woman Ù ¬x likes John] Notice that it would be wrong to translate it as "[$x x is a woman ® x likes John]", because the "x" in "x likes John" would be free. It would also be wrong to translate it as "$x[x is a woman ® x likes John]", which means quite the wrong thing. It is equivalent, as we have seen, to "$x[¬x is a woman Ú x likes John]", which would be made true, for instance, by the mere existence of  something other than a woman.
2	(i)	(a) "x[x is a woman Ú x is a man]. (It would also be perfectly correct to translate it as "x[x is a person ® [x is a man Ú x is a woman]]); (b) ¬$x[¬x is a woman Ù ¬x is a man]
	(ii)	(a) ¬"x x is a man; (b) $x ¬ x is a man.

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