defn of impredicativity (was Re: goals and language?)

["glen e. p. ropella" <***>]

I think both of your descriptions are misleading.  Here's the clearest
definition I've found:

"The Model-Based Mind" by Kercel, VanHoozer, and VanHoozer, citing
Kleene, presents it as:

  "An impredicative property, P(x), of an object x in X, is the property
such that X is the set of objects possessing property P(x).  In other
words, an impredicative property participates in its own definition.
Mathematicians do not deny the existence of impredicativities, but
regard them as a necessary evil.  Impredicativite processes, closed
loops of causality and bizarre systems are equivalent concepts."

The concept is not as strange or bizarre as everyone would make it out
to be.  In everyday language, just imagine looking at an object, say, an
apple, and noticing that it is red.  Then think about the set of all red
objects.  The property of being red is "impredicative" in this use of
the property "red".

The basic concept seems like "lazy evaluation", to me.  Lazy evaluation
is the idea that the interpretation (impact, affect, etc) can be put off
until some other time.  This opens the door to second-order logic
(predicates that are variable), recursion, etc.

Problems with (including people fixated on) impredicativity are just
evidence the person is trying to reduce everything to first order logic
or, as Barwise puts it, the iterative conception, where something can
only be defined in terms of previously defined things.  It seems, to me,
like we have plenty of math and logic to help us handle
impredicativities when we need to.

But, of course, I'm not a logician... so, I'm sure I'm missing some
implications of the definition. [grin]

Judith seems to be citing Russell when she says that impredicativities
are denotational (semantic).  And if you think about lazy evaluation in
the form of the dynamic binding in computer science, you might think
that's a fundamental part of it.  But, I don't think it's that
fundamental.  Syntax can be recursive and self-referential and remain
purely syntactic as long as the things being referred to are,
themselves, just tokens.  Perhaps I'm wrong.

John M wrote:
> > Judith: Impredicativity has to do with an innate, semantic
> > self-reference, either in mathematics or in system organization,
> > which cannot be dispensed with. So a predicative system is one
> > possessing an organization which does not include any of these 
> > kinds of essential semantic qualities and therefore all predicative
> >  systems are reducible to syntax without loss of information. In
> > other words; they are computable. Impredicative systems, then, are
> > systems with organizations which include inherent semantic 
> > information which cannot be replaced by syntax without loss of 
> > essential information.
>
> [JM]: Gee, I always thought the 'impredicative' refers to a natural
> system's unlimited connectivites that make it rather not-predictable
> (not computable). I deducted that sense similarly to what you wrote
> (below) from the "predicate", identifying something (pre-dicere 
> Lat.)in a more 'word-only related' sense. .

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glen e. p. ropella              =><=                Hail Eris!
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