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

From: Paul Holbach (paulholbachSPAMBAN_at_freenet.de)
Date: 02/28/05 

Date: 28 Feb 2005 11:17:52 -0800

> Mattias Wikström wrote:
> > Paul Holbach wrote:

> > The idea of absolutely unrestricted
> > quantification is intuitively
> > intelligible:
> >
> > v(AxPhi(x)) = 1 <=> for each existent e: v(Phi(e/x)) = 1

> I suppose that by an "existent" e you mean an
> object satisfying Ex(x =
> e). If existential quantifiers are interpreted
> along the same lines as
> universal quantifiers, this becomes
> "for some existent x: v(x = e) =
> 1". The definition of "existent" therefore gets circular.

Even though in free logic "E!a <-> Ex(x = a)" counts as a theorem, the
genuine existence predicate "E!" can very well be introduced as an
undefined semantic primitive:

v(AxPhi(x)) = 1 <=> for each thing e (such that E!e): v(Phi(e/x)) = 1

> Since you want "existent" and "object"
> to somehow amount to the same
> thing, it is reasonable to try instead:
> v(AxPhi(x)) = 1 <=> for each object o: v(Phi(o/x)) = 1
> v(ExPhi(x)) = 1 <=> for some object o: v(Phi(o/x)) = 1

You´re right insofar as I consider the concepts /existent/, /object/,
/thing/, /entity/ and /being/ as coextensional such that /object/,
/thing/, /entity/ and /being/ p e r s e means /existent object/,
/existent thing/, /existent entity/ and /existent being/.

> My reply here (as I have mentioned before) is that this makes
> "objecthood" become the property of being
> in a certain domain that can
> be quantified over, and this forces us to call things outside that
> domain "non-objects".

Since, in my view, one can quantify absolutely unrestrictedly over
absolutely everything, there is nothing "outside" the range of
everything.
What is more, any non-object would be an object itself, and so there
impossibly are any non-objects (or non-things, non-entities).

> Of crucial importance here is that we can express
> the property of being
> outside the domain: OutsideOfDomainOfObjects
> (y) <--> -Ex(x = y). It
> seems to me that just as nothing
> stopped mathematicians from
> considering numbers y satisfying y^2 = -1,
> so nothing stops us from
> considering objects y satisfying -Ex(x = y).
> In both cases one may
> object that y has to exist, or that 'y' has to refer, but such
> objections make little sence to me.

[I hate to appear didactic, but "sense" is written with "s", not with
"c".]

I don´t think there are many philosophers who are prepared to
introduce non-self-identicals and nonexistents in their ontology.

I suspect the domain of nonexistents/non-self-identicals is necessarily
the empty domain.

> What does matter, a bit at least,
> is if y should turn out to be a "contradictory" object, like "this
> statement is not true", but short of further assumptions there is
> nothing contradictory about objects y satisfying -Ex(x = y).

~Ex(x = y) -> Ey~Ey(x = y)

How could there possibly be something for which there is nothing it
is?!

Wittgenstein once remarked:

"Ebenso wollte man 'Es gibt keine Dinge' ausdrücken durch '~Ex(x =
x)'. Aber selbst wenn dies ein Satz wäre, - wäre er nicht auch wahr,
wenn es zwar 'Dinge gäbe', aber diese nicht mit sich selbst identisch
wären?" [TLP 5.5352]

"In the same way one intended to express 'There are no things' by means
of '~Ex(x = x)'. But even if this were a sentence, - wouldn´t it be
true too, if there 'were things', but those weren´t identical with
themselves?"

Nevertheless, if you try to "beget" non-self-identical objects by
dissociating objecthood from self-identity, you´ll run into serious
logical trouble:

For an object to be self-identical is for it to possess those and only
those properties it possesses!

(01) Ax(x = x) <-> AP(Px -> Px)
(02) AP(Px -> Px) <-> Ax(x = x)
(03) AP(Px -> Px) -> Ax(x = x)
(04) ~Ax(x = x) -> ~AP(Px -> Px)
(05) Ex~(x = x) -> EP~(Px -> Px)
-----
~(Px -> Px) <-> (Px & ~Px)
Proof:
(a) ~(Px -> Px) <-> ~(~Px v Px) [by def. of implication]
(b) ~(~Px v Px) <-> (~~Px & ~Px) [by de Morgan 1]
(c) (~~Px & ~Px) <-> (Px & ~Px) [by double neg.]
(d) ~(Px -> Px) <-> (Px & ~Px) [by transitivity]
-----
So: EP~(Px -> Px) <-> EP(Px & ~Px)

(06) Ex~(x = x) -> EP(Px & ~Px)

If there were any non-self-identical objects, then there would be some
properties both had and not had by those objects such that they would
become 'contradictory objects'. But since it is impossible for there to
be any 'contradictory objects', it is impossible for there to be any
non-self-identical objects!

I deny that the dissociation of existence and self-identity, or of
objecthood and self-identity is consistently feasible.

Regards
PH

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