|From:||Ali Hashemi <ali.hashemi+ontolog@xxxxxxxxxxx>|
|Date:||Tue, 10 Mar 2009 12:11:42 -0400|
For those who are interested, check out:
for a nice overview of the issues at hand here.
(1) For any presently existing object 0, 0 endures if and only if 0 persists
and all of O's parts simpliciter exist at the present time. I I
We can give a parallel account of what it is for an object to perdure:
(2) For any presently existing object 0, 0 perdures if and only if 0 persists
and some of O's parts simpliciter do not exist at the present time.
we get an argument for the equivalence of the two perspectives. They come closest to actually positing axioms showing equivalence:
The reader will have noticed that there is a close similarity between the set
of 3D particles which constitute an enduring object O at a time t, and the
instantaneous 4D temporal part of O at t. This fact provides for a simple
translation scheme between the 4D temporal parts ontology and the 3D
particle ontology. Let T(O, t) be the instantaneous 4D temporal part of O at t,
and let <O, t> be the instantaneous 3D sum of the particles which constitute
O at t. In 4D ontology, O is the mereological fusion of all its temporal parts
T(O, t), one for each moment at which O exists. In 3D ontology, O is the
set of particles which successively constitute it at each moment O exists, a
set which “changes”, i.e. is replaced by a new set, each time O gains a new
particle or loses an old one. To translate from the 4D to the 3D description
of O, reduce O to its temporal parts, and replace each temporal part T(O,
t) by the momentary sum <O, t> of particles which constitute O at t. The
collection of all such momentary sums <O, t>, for every time at which O
exists, yields the set of sets of 3D particles which successively constitute O.
Conversely, to translate from the 3D to the 4D description of O, first reduce
O to the momentary sums of particles which constitute it, then replace each
<O, t> by the corresponding temporal part T(O, t), then reconstruct O as
the fusion of its temporal parts.
I haven't been successful at finding people who disagree with McCall and Lowe's observations, the closest is in a phd thesis found here:
Gibson (2007) Time, Objects, and Identity - pihlisci archive, Oxford PhD (section 6.4).
He argues, (rightly imo) that their appeal to particles is unnecessary and potentially distracting. He also seems to assert that their position is of ontological equivalence, tho McCall and Lowe never explicitly state this. These two points aside, Gibson's major quibble arises from the equivalence between 4D-3D being based (implicitly) on linking X with "the life of X." Based on this, Gibson notes that the parts of one's life are not equivalent to X himself. I don't agree with this line of reasoning as it seems to me he is conflating several senses of life to derive this apparent oddity, the sense of life required for the mapping to work is exactly that which captures X and his properties as the parts of X's life. Whether this is too technical a definition to be palatable is another issue, which imo, doesn't affect engineering / business considerations. If it is so objectionable, let's call it not "life of X", but "X through his life."
Anyhow, I thought I'd post this to the forum and see what people have to say.
For the record, I don't believe that the 3D-4D translation provided in McCall and Lowe (2006) is ontological equivalence. Moreover, for someone developing ontologies for practical applications, i think logical equivalence suffices. Anyone disagree?
Finally, as many have pointed it, we haven't really come across any ontology which has formalized these notions of perdurance and endurance. I would imagine if one wanted to actually enforce a 4D view in an ontology, they'd need a second order axiom, otherwise, they could use CL's ability to quantify over explicit relations via an axiom similar too
whenever you have a relation (i.e. Rel1) you want to be restricted to the 4d view, you would state:
(forall (Rel1) (4DRel Rel1))
then you would have to have something akin to:
(forall (R ...)
(if (4Drel R)
(and (R ... t) (time t) (argument ...) )
with appropriate axioms to define what an argument is. Though this style is coming perilously close to mixing meta concepts with the ontology itself.
Yet without a set of axioms, whether at the metaontology level or within the ontology, it is nigh impossible to develop a mapping through which one could prove, at least, logical equivalence between the two view points.
As Michael Gruninger pointed out though, there is high similarity between the notions of 3D-4D and time intervals and time points. While the latter are clearly not ontologically equivalent, their extensions may be mapped to logical equivalence over particular domains.
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