Just think of the universal quantifier "forall" (every $x in X
satisfies P) as a conjunction over zero or more elements. Its
initialization must be true (as the initial value for a product must
be 1). Therefore, the "forall" for an empty set is true.
The quantifier "exists" (some $x in X satisfies P) is a disjunction
over elements, and its initialization must be false (as the initial
value for a sum must be 0). Hence, the "exists" for an empty set X is
false.
If you negate a "forall" over an empty set you get the "exists" for
the negated proposition (de Morgan's rule), still over the empty set,
which is false.
-W
On 3 October 2011 15:10, Andrew Welch
<andrew(_dot_)j(_dot_)welch(_at_)gmail(_dot_)com> wrote:
It's a common misunderstanding about universal quantification. The
proposition
every S satisfies P
is always true when S is empty, regardless of P.
For example, the statement "every hotel on St Kilda is fully booked" is
true, as is the statement "every hotel on St Kilda has vacancies" (there are
no hotels on St Kilda).
Heh, nice. So:
every hotel on St Kilda is fully booked, yet some hotel on St Kila
isn't fully booked.
In a potential summary then:
() eq () returns () because... it's a special case where the
atomisation of () returns () and not the empty string?
() = () returns false because its based on 'some', and there are no
items to compare, so false is implied.
deep-equal((), ()) returns true because its based on every, which in
universal quantification is a "vacuous truth" (I googled it :)
--
Andrew Welch
http://andrewjwelch.com
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