Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


fom
Posts:
1,968
Registered:
12/4/12


Re: universal quantification vs existential quantifier
Posted:
Jul 28, 2013 9:13 AM


On 7/27/2013 11:16 PM, shaoyi he wrote: > i delete former post, beacuse it's format. > > > if set S(X) is 'x is the class's student', C(x) is 'x has learned math'. > > so we cannot use ?x(S(x)>C(x)) (domain: x is all people) shows some students of class have learned math. because S(x) is false (x is not the class's student), S(x)>C(x) is true. > > but why can i use ?x(S(x)>C(x)) (domain: x is all people) shows all the class's student have learned math, > beacuse S(x) is false (x is not the class's student), S(x)>C(x) is also true? >
You are asking about what is called "restricted quantification" or "bounded quantification".
Ax(S(x) > phi(x)) :=> A(x in S)[phi(x)]
Ex(S(x) /\ phi(x)) :=> E(x in S)[phi(x)]
The two quantifiers have slightly different forms. The existential quantifiers uses a conjunction.
http://en.wikipedia.org/wiki/Bounded_quantifier
In the case of the universal quantifier, the antecedent of the conditional asserts a partition on the domain by specifying a relevant species. The truth or falsity with respect to this species then depends wholly upon the consequent. Of the three possible satisfying situations for the conditional, only one corresponds with a true antecedent. This is the situation made relevant by the domain partition.
In the case of the existential quantifier, the one satisfying situation of the conditional asserted by a restricted universal quantifier must be immediately asserted for an individual. Since this is a single situation, it may be expressed by the connective which conveys the semantics of this condition directly. That would be conjunction since what must be asserted is the truth of the antecedent and the truth of the consequent. The domain partition is implicit in what is specified about the individual.
Plato is a philosopher. All philosophers are wordy.  Plato is wordy.
Ex(philosopher(x) /\ wordy(x)) Ex(x in philosophers)[wordy(x)]
Ax(philosopher(x) > wordy(x)) Ax(x in philosophers)[wordy(x)]
What vanishes in the modern compositional form of restricted existential quantification is the explicit reference to a named individual. It affects the choice of connective instead.

Try not to use control characters. If you are not purposely doing it, see if your newsreader has a 'plain text' setting. Even though I can read your quantifiers, one may expect that others still cannot.



