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At 11:31 PM -0400 3/19/08, Patrick Cassidy wrote:
PatH,
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.
[[[1]]] First, to go back to the first set of axioms (from
PH, Tuesday
11th March):
> (cl-comment 'basic continuant axiom'
> (forall ((t Time)(c Continuant))
> (= (c at t) c ) ))
>
> (cl-comment 'definition of age property in 4-d'
> (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.
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 time-slice 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.
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.
Yes, of course. You have to understand these axioms in the
context of our discussion. Im not suggesting for a moment that the
above axioms form a coherent ontology, or conform to a coherent
single view of the world. They don't: that is the point of them. They
are supposed to illustrate that there are TWO, DIFFERENT, INCOMPATIBLE
views of the world. One perspective, which we are calling 4D, treats
the x in (x at t) as something extended along the time-dimension,
which can have temporal slices of it taken. The other, which we are
calling - misleadingly - 3D, treats such entities as continuants. The
only way this second view could make sense of the notation (x at t) is
that is simply a poorly chosen way of saying "the continuant x,
at t"; and according to this view, this is the same
continuant as that x at any other t. So if this _expression_ really
does denote a continuant, then the above axiom would be true. Now, I
will admit that a dyed-in-the-wool 3D modeler would probably never
write an axiom like this, and might even regard it as ill-formed; but
we are considering the case where two different ontological
perspectives meet, and so it may be that the 3D ontologist is obliged
to deal with the 4D ontologist's syntax, no matter how aberrant it
appears to be; and if they did have to deal with it, this is what they
might say about it.
So, it is not the '=' sign that is being used in
two different senses,
but the (x at t) formalism.
Yes, of course it is. That is the point.
Thus it appears that the above set
of axioms
still does not serve as a demonstration
of contradiction between 3D and 4D.
Of course it does. Those two senses ARE the 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?
And you will get different answers (and different axioms) from
different people. That was the point I was trying to make from the
beginning of this discussion.
So what is (x at t)? I went back to the long
(47 page) *summary* of an
on-line discussion held years ago about 3D versus 4D views:
http://www.ihmc.us/users/phayes/Endurantism&PerdurantismDebate2002.pdf
. . . 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.
No, they were vividly clear, but there were TWO DIFFERENT
meanings.
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.
In his posting (3-11) 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."
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
Yes, OF COURSE it is.
,
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.
Yes, but this only makes sense if (x at t) = x. Hence the other
axiom, which I supplied.
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
It is in one ontology, but not in the other. Are you beginning to get
the point?
), 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.
I am dazzled by your extraordinary ability to not get the point.
The two senses for (x at t) ARE the incompatibility between 3D and 4D.
You have understood the problem, and immediately concluded that there
is no problem.
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?
IT DEPENDS WHICH VIEWPOINT YOU ADOPT. In 4D, it is a timeslice of
the 4-dimensional entity x. In 3D, it is either meaningless or simply
a rebarbative way of saying "x, at t". These two
interpretations are INCONSISTENT with one another. Which was the point
I have been trying to get through to you from the beginning of this
conversation.
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, 'Pat-at-t' 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 Pat-at-t1 and
Pat-at-t2 are the same Pat; and also of
course, 'Pat-at-t' doesn't mean a 'temporal slice' of Pat, because Pat
is a
continuant and can't be temporally sliced."
There is also a comment in the on-line discussion, mostly between Pat
Hayes
and Fritz Lehman:
> 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 3-D Pat.
> PH: No, you have a 3-D Pat-slice. 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
The important point from these is that (x at t) is not an
instantaneous
time-slice 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"?
Quite. A true 3D modeler would leave it out altogether or refuse
to deal with it. But if they were forced to, in order to communicate
with a 4D modeler who uses such expressions, the best they could do,
and still stick to their own ontological rules, would be to treat it
as referring to x.
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
in all cases meaning simply "x"
Quite.
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".
Not if x is a continuant, it would not. Continuants cannot
have timeslices. They are a different fundamental constituent of
reality than 4D things. They have no 4D extent. They are inherently
3D, but continue to exist through time. I myself find all this close
to incoherent, but it is what true 3D modelers say (and indeed what
has been set into stone in for example the OBO foundry
ontologies.)
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.
Spoken like a true 3D modeler; but to a true 4D modeler, this is
like saying that 3=265 because 3 is part of 265. It is obviously
false. So if you follow 3D rules (such as this substitution) with 4D
axioms, you will quickly reach inconsistencies. Which was my
point.
Then one can see how (x at t) would be properly used in an
endurantist
_expression_:
(P1 x t1) = x has property P1 at t1
(P2 x t2) = x has property P2 at t2
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.
True; but it does relieve the 4D modeler of that obligation. So
now, tell me again (since you seem to have navigated from a 4D to a 3D
perspective) how these two points of view can be combined into a
single ontology?
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.
You seem to have talked yourself into being an endurantist
again.
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 time-indexed
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.
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.
WHICH grammar? There are at least two under discussion here.
* * * * * *
* * * * * * * *
[[[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:
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.
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.
But the category of 'continuant' simply does not occur in a 4D
ontology. There is no such thing.
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.
Well, its not clear that this is ever a logical
contradiction. It isn't in an untyped logic such as CL, in any
case.
I
would think of this as a syntactic
error
Which might well be appropriate in a 3D (perdurant) ontology, but
most emphatically is not in a 4D one, in which any entity
extended in time can be time-sliced.
, but you may consider domain
restriction violations to be other than syntactic.
[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.
First, that is wrong; but more to the point, it is not
being used outside its range in the 4D framework. Yes, of
course, one can obtain consistency by restricting the 4D framework to
only apply to the occurrent sub-universe of the 3D world, but that is
obviously not satisfactory to the 4D modeler. The entire point of
using that framework is to be allowed to use a single coherent way of
talking about temporal relationships which applies uniformly
to all spatiotemporally extended entities
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
An aside, but in fact it can. CL manages to express arbitrary
ontologies in an untyped logic.
, 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.
In 4D, no such restriction is implied. In fact, no such
restriction is acceptable. All you have done here is imposed a
3D distinction onto the 4D ontology, crippling it in the
process.
It does not appear to
me that this example demonstrates a logical
contradiction between these representations of 3D and 4D
objects.
Again, I am dazzled by your obduracy.
Pat
=========== 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.
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:
(forall (x)(if (exist (y (t Time))(= y (during x t))) (Occurrent x)
))
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
(Continuant PatHayes)
OK, now turn to a '4-d' 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:
(forall (x (t Time) P)(iff (P x t)(P (x during t)) ))
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 sub-universes, one containing 4-d 'things' and the
other
containing the continuants and occurrents, restrict each sub-ontology
to its
part of this enlarged universe, and proceed: but now the two
sub-ontologies
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 4-d one, and no way in the
ontology
itself to even state what the relationship might be between them.
==================== end PH note
================================
Pat
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx
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