For those who are interested, check out:
- Mark Hinchliff (1996), The Puzzle of
Change. Philosophical Perspectives, 10, Metaphysics, 1996. pp.
for a nice overview of the issues at hand here.
[MW] I also recommend: Hawley,
Katherine How things persist Oxford: Clarendon Press 2001
- Trenton Merricks (1999) Persistence, Parts
and Presentism, NOUS 33:3 (1999) 421-438
we get two nice definitions of the two accounts:
(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.
We can give a parallel account of what it is for an object
(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
- Storrs McCall & EJ Lowe
(2003). “3D/4D Equivalence, the Twins Paradox, and
Absolute Time,” Analysis 63, pp. 114–23.
- McCall & Lowe (2006). "The 3D/4D
Controversy: A Storm in a Teacup." NOUS 40:3 (2006)
we get an argument for the equivalence of the two
perspectives. They come closest to actually positing axioms showing
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:
[MW] You need to remember that this is written from a 3D
perspective. Strictly, 4D can say things that 3D cannot (no concern to a 3Dist
since they would not want to say them). So strictly this is a full mapping in
one direction and a partial mapping in the other. The other thing I notice here
is that the equivalence seems to be presented in terms of stage theory for 4D
rather than perdurance theory (Hawley’s terminology). The stage theory
version of 4D has an infinite number of temporal slices linked by some sort of “followed
by” relation. Perdurance theory allows for objects that a truly extended
in time as well as space. The account above does not seem to account for
temporal parts that are not temporal stages, but aggregates of them.
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.
[MW] Do you mean here “the sum of the parts of one’s
life” or “each part of one’s life”? The sum is
equivalent to the person, but each part is not (try counting how many people
you get if this were true).
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."
[MW] Well both of these are implicitly 3D views. A 4Dist would
simply say X and temporal parts of X.
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
[MW] What is ontological equivalence?
Moreover, for someone developing ontologies for
practical applications, i think logical
equivalence suffices. Anyone disagree?
[MW] What is logical equivalence?
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
[MW] It would not be valid to talk about X at different times.
You would have to talk about objects that were temporal parts of X. You would
need to enforce it through identity criteria (I think). Each particular object
has exactly one temporal extent during which it exists (though this does not
need to be contiguous). No two objects have the same spatio-temporal extent
(not all 4Dists insist on this).
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:
... t) (time t) (argument ...) )
[MW] I don’t really follow that I’m afraid. Can you
give a clue as to what the variables/constants mean, and which is which.
with appropriate axioms to define what an argument is.
Though this style is coming perilously close to mixing meta concepts with the
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
[MW] I agree. Happy to work with you on what the 4D axioms might
be. Pat H probably knows what they are anyway.
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.
[MW] Not quite sure what you are saying here, but a 4D notion of
time is rather different to the 3D ones I am used to. For 4D a point in time is
a projection across all space at a time, rather than a point on a time line.
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