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Topic: universal quantification vs existential quantifier
Replies: 2   Last Post: Jul 28, 2013 5:12 PM

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Posts: 1,968
Registered: 12/4/12
Re: universal quantification vs existential quantifier
Posted: Jul 28, 2013 9:13 AM
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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.

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.

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