SUO: Re: SUOP Topic :> Definition Of Quantifier
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SUOPT :> Quantifier. Note 2
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SUOPT Outline. http://suo.ieee.org/email/thrd1.html#11635
Chris,
I am getting ahead of myself, and taking the risk
of posting a postmidnight brainstorm, but for the
sake of providing a short answer to your previous
question about quantifiers, I think that maybe it
will be worth a shot.
Let B = {0, 1},
where "0" is interpreted as "false" and
where "1" is interpreted as "true", and
let N = {0, 1, 2, 3, ...}, a set I call
the "additive natural numbers".
Let X be a set of "issues", the definition thereof to be enunciated
sometime in the future, and let f : X -> B be a "predicate", in one
2-gone sense of the word that is still commonly used in math, stats,
engin, and the crustier crusts of AI. By way of an object instance
that prospectively falls under the "in futuro" (IF) description, IF
the uncertainty about the definition of the domain is ever lessoned,
the predicate u : X -> B might be contemplated, where u(x) = 1 says
that the issue x is unresolved and u(x) = 0 says that x is resolved.
Let |X| be the cardinality of X, presumed for facility to be finite.
Let t in N be a "threshold" value such that 0 =< t =< |X|.
Let Sum_(x in X) f(x) be notationally the sum of f(x) over x in X.
Let (X -> B) be the set of all predicates f : X -> B.
Let E : (X -> B) -> N be defined by E(f) = Sum_(x in X) f(x).
Then E(f) is the number of elements x in X where f(x) is true.
For example, E(u) is the number of unresolved issues in X.
Let E_t : (X -> B) -> B be the predicate on the predicates
in this "predicate space" (X -> B) that is defined like so:
E_t (f) = 1 if and only if E(f) >= t.
By way of interpretation, E_t (f) is true iff
the number of elements in X where f is true
is >= the threshold value t in N.
For example, E_1 : (X -> B) -> B
is a predicate on the predicates
f : X -> B that is true iff f is
true for some x in X.
By way of interpretation, one says that
E_1 (f) is true iff Some x has true f.
These types of "measures on predicates", like E : (X -> B) -> N, and
these types of "predicates on predicates", as E_t : (X -> B) -> B, can
be said, sans or with threshold, to "quantify" the argument predicates.
One observes that E_1 does the work of the
usual "existential quantifier" in the more
"ordinary formulations of logic" (OFOL's).
Thus, one may call the E_t, in this context,
a type of "generalized quantifier", or else
one may reserve the unadorned nomen for the
general case, rather more general than this,
and recognize the garden variety quantifier
as an extremely specialized species in kind.
Well, I tried to B wise and rewrite it in the morning,
but still, I'll have to think about it for a few days.
E_t, phone home ...
Jon Awbrey
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