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Restricted quantification (ex: Subjunctives)
- Subject: Restricted quantification (ex: Subjunctives)
- From: Pycyn@aol.com
- Date: Mon, 31 Jan 2000 05:20:13 EST
I don't know exactly why nobody else likes restricted quantification, but
since it tends to get shot down in the context of unicorns and the like, I
suspect it is that many like it that statements about the member of the empty
class are always true in standard (20th century) logic and people want to be
able to say true things about unicorns (I suggested a different way to deal
with that a while back, using another favorite bugbear, intentional context
and/or subject raising). The present situation in fact evolved in a complex
way from restricting all restricted quantifiers (the original sort) to the
universe of discourse and then doing the sorting (restricted quantifiers are
also called sortl quantifiers) by conditionals in the predicate place. This
left the universal quantifers with residual existential import, which bugs
some logicians still (who want empty universe logics of one sort or another
-- what happens when there is nothing in the world to which quantifiers and
terms generally allude).
Restricted quantifiers are the original (i.e., Aristotelian and, covertly,
Akshapadan) version, realized in modern form with each bound variable
restricted to members of a given set, which is assumed to be non-empty.
(There is a version with empty sets permitted, but it reduces on the one hand
to the usual system and on the other to an empty-universe mishmash, which no
one likes.) Restricted quantifiers also seem to be underlie quantifier ((and
determiner generally) structures in natural languages, so that the usual
quantifiers really are as artifical as they often feel. So, given that we
have lebbenty-lebben ways to do quantifiers, I always thought it made sense
to allow at least one of them to be this natural and ancient logical sort,
most conveniently the da poi format. This suggestion always gets
pooh-poohed, though I conveniently forget why each time. (Oh yes, it allows
for the differences between each & every against any & all in English).
Universals take the Aristotelian A form seriously, rather than degenerately
(the empty-class version uses the obverse No S is non-P, which under the best
version of A's system lacks existential import). The particular forms don't
usually make much difference in interpretation, and the other determiners are
dealt with in all the weird ways Lojban has.
pc