Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)

From: Paul Holbach (paulholbachSPAMBAN_at_freenet.de)
Date: 02/24/05 

Date: 23 Feb 2005 17:05:58 -0800

> Mattias Wikstrm wrote:
> > Paul Holbach wrote:

> > Any object outside 'the domain of existence'
> > would be a nonexistent
> > object, which would make sense if and
> > only if the concepts of
> > existence
> > and of objecthood (or of entityhood) were not regarded as
> > co-extensional.

> Indeed. I do not think it makes much sence
> to regard objecthood as
> co-extensional with the property of
> being within a certain domain.

------------------------------------------------------
> I am sorry. Here I made exactly the kind of
> mistake I have been trying
> to warn about. The extension of a concept and
> the co-extensionality of
> two concepts are for me always relative to a domain,
> but above I do not
> specify any domain. As I see things,
> a concept has no extension at all
> in itself, though as soon as a domain is given, one can ask what
> extension the concept has in that domain.

As the logical paradoxes have taught us, not for every concept there is
a corresponding extension, i.e. a set containing all those objects
which fall under the respective concepts. But for most concepts
associating them with corresponding extensions is unproblematic.
Unfortunately, the concept of existence is not one of those concepts,
since a consistent conception of the set of existents seems hardly
possible.

> As far as I can see, what I wrote above
> hardly makes any sence at all.
> Things would have been a little better if I had written
> "co-intensional" instead of "co-extensional".
------------------------------------------------------

> > That there cannot be any entites which have
> > properties but dont
> > exist (or arent self-identical) is a metaphysical axiom
> > -- but one
> > whose truth is hardly deniable:
> > APAx(Px -> (E!x & x = x))]

> Let me write D for the domain of individuals and
> D' for the domain of
> properties. The axiom can then be stated:
> (#) AP:D'Ax:D(Px -> (E!x:D & x =D x))
> All I have done here is to explicitly write out the domains of
> quantification and to make it explicit that the
> equality sign refers to
> elements of D (for elements P and Q in D',
> I would instead write P =D'
> Q).

As regards my formula above (and others), it is clear that its
ontological scope is boundless.

> For the axiom (#) to make sence at all,
> the domains D and D' must be
> related so that given an element x in D
> and an element P in D', we can
> form the statement Px. What the axiom (#)
> does is to express a
> restriction on what elements may be in D'.
> It seems to me one cannot
> have all of:
> (1) The axiom (#).
> (2) An axiom of comprehension for
> D' and D (one that in particular
> ensures D' contains an element P
> such that Px <-> -(x =D x)).
> (3) Ex:D(-(x =D x))

In free logic

AxFx <-> Ax(E!x -> Fx)

is a theorem.

The (free) universal quantifier generally ranges over everything, i.e.
over every existent thing.
If one wishes to restrict the domain of quantification, one can easily
do so by replacing "E!" with any other predicate.
For example, if I intend to quantify not over everything but over the
animal kingdom, I can simply write:

Ax(x e D = {y | animal(y)} -> Fx)

All members of the animal kingdom are Fs.

> So much for the meaning of the axiom above.

The fundamental meaning of the metaphysical axiom APAx(Px -> E!x) is:

"No property instantiation without existence!"

> There is, however, a danger of misunderstanding here. If
> someone asks me what domains my ontology includes,
> I can only say:
> "None at all". In my view of things,
> domains exist no more than the
> objects they contain.

Are you claiming here that there are neither sets (domains) nor
Urelements (individuals/objects/things)?!

Regards
PH

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