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

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

Torkel Franzen wrote:
> > I have to paraphrase this to see if I understand.  An impredicative 
> > definition (of an object, set, predicate, whatever) is one that relies 
> > on multiple collections of objects, including the object being defined, 
> > where all those collections relate in some way.
>
>   No, an impredicative definition is quite specifically a definition
> that uses quantification (the phrases "for all X", "for some X") over
> a collection containing the object being defined.

OK.  Then it sounds like you're saying that Kline's description (which 
he attributes to Russell) is wrong or incomplete.  I'll have to dig 
further into the source material, I suppose.

> There are
> problematic aspects of this explanation, but they haven't been touched
> on in our exchange.

Right.  I'm not particularly interested in any problems with the 
definition, except if they relate to why it's so hard to find a clear 
and consensual definition of the term.

> It's unclear to me what you intend by "that
> definition depends on some R such that aRb, xRa, and xRb, for
> collections x, a, and b".

My point was that a quantification of a collection of sets is part of 
the definition for a relation amongst those sets.  Rather than say: "for 
all x in X", I explicitly listed all the sets I was talking about and 
treated them explicitly.  This removed the extra element of 
"quantification" from your definition. in an attempt to clarify what you 
meant.  But, you're not playing along. [grin]

I can't get any help from you in reconciling the fact that I see gross 
similarities between Kline, Barwise, Russell, and Kercel and you see 
gross dis-similairty between all of them and your own.  So, I'll just 
have to quit annoying you and hunt down the source material myself.

>   As regards your further remarks, I can only reiterate that Kercel's
> supposed explanation is useless if you're interested in what "impredicative"
> means in logic and philosophy.

I'm not so sure.  I think you're taking a _very_ strict attitude towards 
the way language is used.  And for you, that might work.  Since I'm 
forced (by my choice of profession) to constantly work with people from 
various different disciplines, including biology, math, and computer 
science, I don't have that privilege.  Unfortunately, the only reason I 
care at all about the term "impredicativity" is because RR used it in 
some places.  Hence, I'd like to know where the word came from, what it 
meant to the people who coined it, what it means to the people who coopt 
and abuse it, and where/if it can help me model biological systems.

Kercel uses it.  And people cite his definition.  So, his usage is part 
of what I have to deal with.

> Such comments as
>
> > Given the foundation axiom, for 
> > all a, a is not a member of a.  (apparently prohibiting impredicative 
> > definitions)
>
> are grossly incorrect.

Well, then perhaps Barwise and Moss are grossly incorrect.  Of course, 
they don't provide proof.  They merely state:  "Proof  Most of the parts 
of this [proposition 2.6] are easy."  I, personally, can't refute their 
prop 2.6.  Or, perhaps you're just objecting to my parenthetical 
comment.  I can't tell.

In any case, thanks very much for clarifying what you mean by the term 
"impredicative definition".  I am grateful.

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