PatH, (01)
Now that we have two sets of axioms showing an apparent contradiction
between 3D and 4D, we can make some progress in understanding how these
views can appear contradictory while representing the same entity. The
second set of axioms is interesting, and includes an implication I have not
seen before, which I will want to discuss in another email. For now, just
to focus on the two sets of axioms that you have provided to illustrate
inconsistency between 3D and 4D models. (02)
[[[1]]] First, to go back to the first set of axioms (from PH, Tuesday
11th March): (03)
> (clcomment 'basic continuant axiom'
> (forall ((t Time)(c Continuant))
> (= (c at t) c ) ))
>
> (clcomment 'definition of age property in 4d'
> (forall (x (t Time))
> (= (age (x at t))
> (minus t (birthtime x)) )))
>
> from which, and a little arithmetic, it follows that
>
> (forall
> ((t Time) (s Time))
> (= s t) )
>
> i.e. time is impossible. (04)
I said in a previous note that it appeared that the '=' sign is being used
in different senses in the two axioms (continuant axiom and age definition),
because the first axiom seemed to be describing a diachronic identity
relation, and the second occurrence of (x at t) seemed to be representing a
timeslice of x. But PatH assured us that the equals sign is used in the
logical equality sense in both cases. Then I had to look more carefully,
assuming that '=' is indeed logical equality. Now I see that the problem is
that I misinterpreted (c at t) as being a timeslice of c. Since '=' is a
logical equality, (c at t) cannot be a timeslice. In that case there is a
problem with the second axiom. (05)
If (x at t) is logically equal to x, then the function term
(age (x at t))
in the second axiom has an undefined meaning, since it is equivalent to
(age x)
in which the time at which the age is calculated is not in fact specified.
To be meaningful and grammatically correct, an age function must have a time
specification. But in the given axiom above, the expression (x at t) is
used as though it were a time slice, giving rise to the apparent
contradiction, and my misinterpretation. This emphasizes the need for a
clear understanding of (x at t)  discussed below. (06)
So, it is not the '=' sign that is being used in two different senses,
but the (x at t) formalism. Thus it appears that the above set of axioms
still does not serve as a demonstration of contradiction between 3D and 4D.
PatH provided a second set of axioms, below. But there is still an
interesting question of just what does (x at t) or (c at t) mean? (07)
So what is (x at t)? I went back to the long (47 page) *summary* of an
online discussion held years ago about 3D versus 4D views: (08)
http://www.ihmc.us/users/phayes/Endurantism&PerdurantismDebate2002.pdf (09)
. . . and now it is beginning to become clear why that discussion could go
on so long without resolution. At no point in that summary is the 3D/4D
contradiction demonstrated by a set of axioms such as PatH has provided for
us. So the meanings of the terms remained unclear. This is an illustration
of what I meant by saying that we need to analyze the different apparently
contradictory representations carefully to find out exactly where the
apparent contradiction is. (010)
In his posting (311) PatH said:
"Either it is meaningless to speak of (C at t) as an entity when C is a
continuant and t is a time; or else, (C at t) = C for all times t." (011)
In the axiomatization above PatH says that (C at t) = c for all times t;
but use of (c at t) in a different sense appears to be the source of the
contradiction, (012)
In the function term (age (x at t)), if (x at t) is equal to x, then with
the equality above that term is equivalent to (age x). But in this
expression the time at which the age is calculated is not in fact expressed,
therefore the function cannot return a meaningful value. To be meaningful
and still use the (x at t) expression, the function term should be:
(age (x at t) t)  meaning the age of entity x at time t.
In the set of axioms, the "t" in ((age (x at t)) is taken to have the same
effect as the "t" in an expression (age x t), but that can only be true if
(x at t) is a timeslice (which I now recognize it is not), or the "t" in (x
at t) is somehow movable to the outside of the parentheses, but there is no
explanation of why that latter could be true. I think the problem is a
misleading ambiguity in the term (x at t) that causes it to be used in
different senses.
I conclude that the axiomatization does not prove incompatibility of 3D
and 4D. (013)
Not directly relevant to the question of the incompatibility of 3D and 4D
but important to the understanding of the 3D viewpoint is: just what does (x
at t) mean? (014)
PatH has explained:
[PH] >> "If I were to speak as an endurantist (one who believes in
continuants) then the answer is clear and obvious. Pat is a continuant,
which needs no further explanation. His properties change with time, so
assertions about him must be made somehow with respect to the time they are
supposed to hold (typically by including a time parameter in the relations,
but other formal devices are possible, eg making assertions relative to a
temporal context, or using temporal modal logics.) In the informal
discussion, 'Patatt' is simply a way to refer to Pat at the time t, to
informally indicate which time to use when making assertions about Pat. Of
course, the Pat in Patatt1 and Patatt2 are the same Pat; and also of
course, 'Patatt' doesn't mean a 'temporal slice' of Pat, because Pat is a
continuant and can't be temporally sliced." (015)
There is also a comment in the online discussion, mostly between Pat Hayes
and Fritz Lehman: (016)
> A part of that discussion (FH is Fritz Lehman)
> FL: With criteria such that "Pat at time X" has no identity with any
> Pat before or after, then fine, you've got a 3D Pat.
> PH: No, you have a 3D Patslice. But that is a different thing from
> the slice of Pat a few minutes later. The endurantist position is that
> Pat is exactly the same thing at all times throughout Pats life. That
> is what 'endure' means: a continuant retains its identity. Patat 3pm
> is identical (017)
The important point from these is that (x at t) is not an instantaneous
timeslice of x, but is identical to x, though it refers to x in a way that
includes a time referent. So why is that "t" in the expression at all, if
the expression merely means "x"? (018)
In trying to find a linguistic gloss for the term (x at t) in this sense I
can only come up with:
x, which can be observed at t; or
x, which was observed at t
x, which I am pointing to at time t (019)
in all cases meaning simply "x" (020)
It is tempting to interpret (x at t) as
x, as observed at t; or
x, when observed at t
. . . in which case it would appear to be a timeslice of some 4D "x". But
"x" is not 4D, and this expression cannot be used to avoid specifying the
time of an observation when making an assertion (or calculating a function)
about x and its properties.
I t appears to me that, given the first axiom above, (x at t) should
never be used in any expression unless that term could be substituted with
"x" without changing the meaning. (021)
Then one can see how (x at t) would be properly used in an endurantist
expression: (022)
(P1 x t1) = x has property P1 at t1
(P2 x t2) = x has property P2 at t2 (023)
and if x is identical to (x at t) we can also say:
(P1 (x at t1) t1) and
(P1 (x at t2) t1) and
(P2 (x at t2) t2) and
(P2 (x at t1) t2).
Which is to say that using (x at t) instead of x does not relieve an
endurantist of the obligation to provide the time index in any assertion on
a 3D object. (024)
In this interpretation the meaning of (x at t) might be clearer if one
used an analogy: one could talk about "the Eiffel Tower, which I saw from
the North" or "the Eiffel Tower, which I saw from the West", both meaning
exactly the same thing as "the Eiffel Tower", with some obiter dicta
attached. One can put one's finger on an elephant's leg and say "this
elephant" or put one's finger on an elephant's tail and say "this elephant"
or put one's finger on an elephant's trunk and say "this elephant", meaning
in all cases the same elephant. One needn't interpret a 3D object as a part
of a 4D object to be able to refer to all temporal observations of it at
different points in time. The expression (x at t) can be interpreted as a
way of pointing to x, but never as a substitute for a time expression in an
assertion. It should never be used as one would use a timeslice of a 4D
object.
Another way to find an interpretation of (x at t) is to think of a
moving object (e.g. the moon). We can say we saw it in the constellation of
Aries (moon in Aries), and we also saw it in the constellation of Pisces
(moon in Pisces). In both cases we are referring to the moon (moon) in a
manner that merely points to it, in which cases it is always and only the
same moon, even though it was in different places when we pointed to it. To
say (moon in Aries) is not to say "the moon when it was in Aries", which
would be syntactically a different expression (depending on the grammar 
that kind of expression is only part of a typical timeindexed logical
sentence, because it does not include the predicate). So we can express (x
inLocation L) = x and (x at t) = x, and we cannot use either expression
(defined as logically equivalent) in any way that simply "x" could not be
used. (025)
Thus the subtle problem in the first set of axioms is that (x at t) was
being used in a way that a simple "x" could never be. This is in effect a
grammatical error in the use of the grammar. (026)
* * * * * * * * * * * * * * [[[2]]]
The second set of axioms (below) has a different approach to the test of
consistency, but I believe that in this argument the apparent contradiction
is not an inconsistency in the ontology, but a misuse of the relation
"during". Specifically: (027)
The problem with the second example (below) appears to be in using
"during" in the 4D ontology . From the first case (3D):
>> (forall (x)(if (exist (y (t Time))(= y (during x t))) (Occurrent
x) ))
"during" is defined only on Occurrents  use on a Continuant would be a
violation of the domain restriction. (028)
In the example of the second case (4D):
>> (forall (x (t Time) P)(iff (P x t)(P (x during t)) ))
the domain of "during" is not specified. But if Continuants don't exist
in that ontology, then it should be true that the domain of "during" in 4D
is confined to entities in that ontology, and does not have any
specification (explicit or implied) of the consequences of the "during"
relation being asserted on a Continuant (i.e. the domain cannot include
'Continuant' because the logical consequences of an assertion on
'Continuant' are not specified). Therefore, when one tries to use it on a
Continuant, one would get a violation of the (explicit or implied)
restriction on that relation, i.e. it cannot be used with a Continuant as an
argument. Yes, it would be a contradiction, but it would be a contradiction
of the same kind as using *any* relation on entities outside its domain. I
would think of this as a syntactic error, but you may consider domain
restriction violations to be other than syntactic. (029)
[PH] >, we might just try to combine them directly, but then a contradiction
arises whenever the second uses during on a continuant argument.
Yes, but a contradiction arises when one tries to use *any* relation on
an entity that is not in its domain or range. This is not diagnostic of an
inconsistent ontology. The contradiction generated by trying to use
'during' on a Continuant is just like the contradiction of trying to us
'physicalPart' on an AbstractObject  it cannot make any sense, not because
there is any logical contradiction in the ontology (or merged ontology)
itself, but because that usage violates the restriction (explicit in 3D,
implied in 4D) on use of that relation. (030)
It does not appear to me that this example demonstrates a logical
contradiction between these representations of 3D and 4D objects. (031)
=========== Original response from PH =============== ========== second
set of axioms ========================== OK, if that example bothers you
(and I agree it is somewhat tendentious), try this, which makes the same
basic point but more realistically, and is therefore more complicated. (032)
First, a fragment of a continuant/occurrent ontology, such as DOLCE. Here,
the categories of Occurrent and Continuant (think process and object
respectively) are firmly and without exception asserted to be disjoint. One
can speak of temporal parts of an occurrent by using during:
(during O t) is the temporal part of the occurrent O at the time t.
Occurrents however cannot have temporal parts, so one would use fluent
language to speak of a changing property of a continuant:
(P c t) rather than (P (during c t)). It would be natural to have a domain
axiom for during: (033)
(forall (x)(if (exist (y (t Time))(= y (during x t))) (Occurrent x) )) (034)
Ie if something has a temporal part, then its an Occurent, from which it
follows in this ontology that it is not a Continuant. {Note: this axiom as
written would not do the job in CLIF, in fact, as all functions there are
total. See the IKL Guide document for more details on how to do this
properly, which I omit here for the sake of simplicity.} And certainly in
this ontology, continuants exist: they are a central category, so for
example we might have (035)
(Continuant PatHayes) (036)
OK, now turn to a '4d' ontology. Here, all spatiotemporal entities have
temporal parts, and the two forms of expression (P x t) and (P (x during
t)) are completely equivalent, mere syntactic alternatives: (037)
(forall (x (t Time) P)(iff (P x t)(P (x during t)) )) (038)
Now, how do we put these together? Since the latter is simply more
permissive than the former, we might just try to combine them directly, but
then a contradiction arises whenever the second uses during on a continuant
argument. Or, we could treat the continuant/occurrent distinction as being
real even in the second ontology, rendering it consistent at the cost
(unacceptable to its users) of making it effectively the same as the first
one. Or, we could weaken the first ontology slightly, by removing the
assumption of disjointness, making it effectively similar to the second one:
but its devotees will object that this change utterly destroys the very
distinction that they are at such pains to preserve, because it is so
fundamental. There is no way to make everyone happy. Or, one can divide the
universe into two subuniverses, one containing 4d 'things' and the other
containing the continuants and occurrents, restrict each subontology to its
part of this enlarged universe, and proceed: but now the two subontologies
are effectively isolated from one another, and the whole construct is a
single ontology only in name, not in any useful sense. Ther are now two
PatHayeses, the continuant and the 4d one, and no way in the ontology
itself to even state what the relationship might be between them.
==================== end PH note ================================ (039)
Pat (040)
Patrick Cassidy
MICRA, Inc.
9085613416
cell: 9085654053
cassidy@xxxxxxxxx (041)
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