The other day in my logic class
I had some rather odd reactions to the classic Geach
sentence about donkeys:
(1) Every man who owns a donkey beats it.
I thought, and I
thought everyone who understood the sentence thought, that its truth conditions
were
(2) "x"y ((Man(x) Ù Donkey(y) Ù Owns(x, y)) ® Beats(x, y))
This isnt a
translation, I think, but I think (1) and (2) are logically equivalent. I
mentioned that its kind of odd that these are the truth conditions, because it
looks like (a) there is an existential quantifier in there, and (b) a pronoun
in the last spot that should be bound by that existential quantifier. But its
hard to get an analysis of the sentence in FOL that respects that. I mentioned
that the closest you could get is (3):
(3) "x ((Man(x) Ù $y (Donkey(y) Ù Owns(x, y)) ® $z (Donkey(z) Ù Owns(x, z) Ù Beats(x, z)))
But (3) doesnt sound
like a translation of (1), because (3) doesnt require that every donkey owner
beats every donkey he owns, only that
he beats some of the donkeys he owns,
and (1) does require that. Then, just to be on the safe side, I checked what
the class thought. And most people agreed, (2) gives the truth conditions for
(1), and (3) doesnt. But not everyone. About a quarter of the class thought
that the truth conditions for (1) were given by (3) not (2).
I should note here
that this wasnt, or at least wasnt entirely, because they didnt understand
the formalism. I spelled out in English just what each of (2) and (3) meant,
and went over the kinds of situations in which they had different truth values,
before taking the poll. And since I explicitly noted that I thought (2) was
right, I was hardly biasing the sample in favour of (3). Anyway, it seems that
either many people were making mistakes about their own language, or they speak
a dialect in which (1) only requires that every donkey-owner is a donkey-beater
(and not just beating any donkey, but one of the donkeys they own). There could
be such a language, I think, and maybe more people than I realised speak it.