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

Torkel Franzen wrote: 
"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."



  I guess it's inevitable that the logical term "impredicative" should
inspire all sorts of extended or analogous uses. The above, however,
is not at all correct applied to the use of "impredicative" in logic.
An impredicative definition, in logic, is one that defines an object -
typically a set - by means of quantification over a totality to which
that object itself belongs.

Sorry for my incompetence, here; but, that's not very clear. Are you talking about things like "the set of all sets"? Or are you talking about objects that can predicate themselves (e.g. "unity of unity")?
These are what I thought were impredicative objects in logic and below, I'll try to show where Kercel's jabber about properties is related.

It's hard for me to parse Russell, himself... [grin] So, I rely on others to interpret him for me. This example was taken from Kline's "Mathematics: The Loss of Certainty":

"On the other hand Russell's paradox, Cantor's paradox of the set of all sets, and the Burali-Forti paradox are considered logical contradictions. Russell himself did not make this distinction. He believed that all the paradoxes arose from one fallacy which he called the vicious circle principle and which he described thus: 'Whatever involves all of a collection must not be one of the collection.' Put otherwise, if to define a collection of objects one must use the total collection itself, then the definition is meaningless. This explanation given by Russell in 1905 was accepted by Poincare' in 1906, who coined the term impredicative definition, that is, one wherein an object is defined (or described) in terms of a class of objects which contains the object being defined." [Sorry for any typos.]

All these descriptions of impredicativity seem, to me, to agree with the one Kercel gives. The only difference is that Kercel gives it as an impredicative property rather than an impredicative object (to which a predicate might be applied). Classes and sets can be defined by properties, hence an impredicative property would be one that results in an impredicative set or class. Perhaps my example of apples and redness are not appropriate; but, that doesn't necessarily mean that Kercel is misapplying the term.

Predicates are operators where their subject is the thing being operated on. In Kercel's definition, P(x) is a predicate. However, it's not a clearly defined predicate because it's definition depends on the definition of the objects over which it can operate and the definition of the objects depends on the predicate that operates over them.

The reason (I infer) he calls these impredicative properties is because, in cases like this, the collection of objects can be operated on by P(), thereby satisfying the logical definition of an impredicative set. (I don't know if this proposition is true... but it's how I interpret Kercel.)

For example, if P(x) = "x is red" and x,y in X such that P(x) and P(y), then P(x+y) = "x+y is red". In english, you might say, all the apples in that basket are red or that collection of apples is red. The same property can be true of both the single object and the collection. And, yet, the property is defined by referring to the members of the collection (or the whole collection since the whole collection is defined by the members of the collection to begin with).

Since the collection possesses the same property as the objects in the collection, the collection is an impredicative object. Kercel is simply being more specific about the way these impredicative objects are constructed.

Now, that leaves the following proposition to be proven or refuted: Objects defined by predicates which are defined by their subjects are impredicative objects.

I'm not capable of proving or refuting that proposition... But, it sure feels true. [grin]

That "mathematicians regard the existence
of impredicativities as a necessary evil" is an arbitrary statement.
Mathematicians in general pay no attention to whether or not a
definition is impredicative, and are unaware of the concept.

I agree with you here. As long as a concept is useful, mathematicians will use it.

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.


  Since you refer to Barwise, I suspect that what you have in mind here
is rather the distinction between well-founded and non-well-founded
sets. This is a different matter. The classical iterative conception
of the set-theoretical universe is profoundly impredicative, i.e.
presupposes that impredicative definitions make good sense.

Well, again, I'm struggling against my own ignorance, here. So, I was not explicitly referring to the foundation axiom. But, according to Barwise:

"Proposition 2.6  Assuming the Axiom of Foundation:
   1. For all a, a is not in a.
..."

This proposition that "there can be no a that is a member of itself" seems to directly address the idea of "the set of all sets" or collections that have the same properties as their members. As such, it seems directly related to impredicativity, to me.

But, later in "Vicious Circles" Barwise talks about the iterative conception, if one assumes the foundation axiom, all of the elements of a set must be pre-defined (without referring to sets). This is what leads me to the analogy with dynamic binding in computer science. Static binding is analogous to the iterative conception, where one starts with an initial _hard_ definition and defines everything else in terms of that, building up definitions. Dynamic binding allows you to refer to something even though it hasn't been defined, yet. And this also seems related to "predicates as variables" or, at least, predicates that are not yet fully defined, thereby allowing a collection to be defined by its members and its members being defined by the collection.

So, again, all these things seem intimately related to me even though you say they are different or not correctly applied. I'd appreciate any disambiguation you can present! [grin] Perhaps, then we can get back to RR and his claim that "organisms are full of such impredicativities".

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