The spatial regions that continuant fiat boundaries occupy

[michaelrabenberg]

Fiat points, lines, and surfaces occupy zero-, one-, and two-dimensional spatial regions, respectively. I think, however, that the axioms don't imply as much. Am I right about this?

[alanruttenberg]

I believe you are correct. I will add axioms for this. Thanks, good catch!

[michaelrabenberg]

OK, thanks. One question relevant to how best to formulate the axioms is whether a given (say) fiat boundary can occupy different spatial regions at different times; some conceivable ways of formulating the axioms would rule out this possibility and some wouldn't.

[alanruttenberg]

Boundaries can definitely occupy different spatial regions at different times. Do we agree that nothing changes the dimension of the space something occupies. So if you occupy a 2d spatial region at some point, at any time you exist you occupy a 2d spatial region.

On Wed, Dec 13, 2023 at 7:31 PM Michael Rabenberg @.***> wrote:

OK, thanks. One question relevant to how best to formulate the axioms is whether a given (say) fiat boundary can occupy different spatial regions at different times; some conceivable ways of formulating the axioms would rule out this possibility and some wouldn't.

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[wceusters]

As long as Scotty cannot beam anybody up, I think I do agree.

[alanruttenberg]

Draft axiom

(forall (m s)
 (if
  (exists (t)
   (and (occupies-spatial m s t)
    (or (= s three-dimensional-spatial-region)
     (= s two-dimensional-spatial-region)
     (= s one-dimensional-spatial-region)
     (= s zero-dimensional-spatial-region))))
  (forall (t1 s1)
   (if (occupies-spatial m s1 t1) (instance-of m s t1)))))

[alanruttenberg]

Draft axiom for boundaries

(forall (x)
 (if (exists (t) (instance-of x fiat-surface t))
  (exists (t s)
   (and (occupies-spatial-region x s t)
    (instance-of s two-dimensional-spatial-region t)))))

Similar for fiat-point, fiat-line.

We only need to check and assert with exists since fiat-line is rigid and the dimension of space occupied by something is rigid per https://github.com/BFO-ontology/BFO-2020/issues/76#issuecomment-1856429861

[michaelrabenberg]

Alan says: Boundaries can definitely occupy different spatial regions at different times.

I think there are puzzles but this might not be the best occasion to discuss the matter.

Alan says: Do we agree that nothing changes the dimension of the space something occupies. So if you occupy a 2d spatial region at some point, at any time you exist you occupy a 2d spatial region.

That does sound plausible, given certain background assumptions. One such assumption is the proposition that every material entity occupies a 3d spatial region. (If you could "shave" a 3d material entity to make it "perfectly thin," say, then this sort of assumption wouldn't be right.) Another is the proposition that there are no aggregates that can have differently dimensioned independent continuants among their constituents. (If there were a "boundary aggregate," say, with four points and a line for member parts at t1 and with just the points for member parts at t2, then we'd also seem to have a counterexample.)

Another thing: Some of the axioms pertaining to fiat points might be streamlinable. For example, given [jgo-1] (a fiat point has no parts other than itself), [jqd-1] (if a has continuant part b then if a is an instance of fiat point then b is an instance of fiat point) could be discarded.

[michaelrabenberg]

Alan wrote: (forall (m s) (if (exists (t) (and (occupies-spatial m s t) (or (= s three-dimensional-spatial-region) (= s two-dimensional-spatial-region) (= s one-dimensional-spatial-region) (= s zero-dimensional-spatial-region)))) (forall (t1 s1) (if (occupies-spatial m s1 t1) (instance-of m s t1)))))

I assume the boldfaced 'm' should be 's', right?

[michaelrabenberg]

Ugh--I meant "s1," not "s."

[michaelrabenberg]

Alan wrote:

(forall (x) (if (exists (t) (instance-of x fiat-surface t)) (exists (t s) (and (occupies-spatial-region x s t) (instance-of s two-dimensional-spatial-region t)))))

This looks good to me.

[alanruttenberg]

Alan wrote: (forall (m s) (if (exists (t) (and (occupies-spatial m s t) (or (= s three-dimensional-spatial-region) (= s two-dimensional-spatial-region) (= s one-dimensional-spatial-region) (= s zero-dimensional-spatial-region)))) (forall (t1 s1) (if (occupies-spatial m s1 t1) (instance-of m s t1)))))

I assume the boldfaced 'm' should be 's', right?

Yes, thanks.

[alanruttenberg]

I think there are puzzles but this might not be the best occasion to discuss the matter.

Regarding aggregates, there's precedent. The choice for temporal regions is that the highest dimension of a part is the dimension of the sum.

Another thing: Some of the axioms pertaining to fiat points might be streamlinable

I ran my leave-on-out code yesterday. The code takes each axiom, one at a time, and tries to prove it from the rest of the axioms. It had been a while since I had last run it. It found 60 examples! It has relatively tight time limits - it gives prover9 a chance for 10 seconds and if it times out give z3 10 seconds. Maybe I'll try again with raised limit. I'll try to organize this and share it and maybe you and Werner can help me sort through them.

[michaelrabenberg]

Regarding aggregates, there's precedent. The choice for temporal regions is that the highest dimension of a part is the dimension of the sum.

I assumed something like that would be the case--intuitively the aggregate of a line and a point, say, is 1D--but that's why I imagined a case in which the aggregate first contains some 0D individuals (four fiat points) and a 1D individual (a fiat line) and later loses the 1D individual. If there were such an aggregate (and it persisted through this loss of an individual) then it'd undergo a change in dimension even by the standard you mentioned. So there can't be things that meet that description if the "once xD always xD" assumption is to be maintained.

Another thing: Some of the axioms pertaining to fiat points might be streamlinable

I ran my leave-on-out code yesterday. The code takes each axiom, one at a time, and tries to prove it from the rest of the axioms. It had been a while since I had last run it. It found 60 examples! It has relatively tight time limits - it gives prover9 a chance for 10 seconds and if it times out give z3 10 seconds. Maybe I'll try again with raised limit. I'll try to organize this and share it and maybe you and Werner can help me sort through them.

Very interesting! Would be eager to hear more about it.

[alanruttenberg]

The aggregate example is cute. Can you think of a plausible situation in which that would make sense?

[michaelrabenberg]

No, I can't imagine a case in which I'd want to say there is an aggregate like that. But there'd be no contradiction in positing such an aggregate, and some imaginable aggregation/composition principles (like "any two individuals form an aggregate") would imply that there is such an aggregate.

[phismith]

Note that we do not want to adopt for aggregates analogues of formation principles of the sort accepted in set theory Aggregates are entities which exist in time. We want aggregates to be capable of ceasing to exist even though all their members remain in existence BS

From: Michael Rabenberg @.> Sent: Saturday, December 16, 2023 7:36 AM To: BFO-ontology/BFO-2020 @.> Cc: Subscribed @.***> Subject: Re: [BFO-ontology/BFO-2020] The spatial regions that continuant fiat boundaries occupy (Issue #76)

No, I can't imagine a case in which I'd want to say there is an aggregate like that. But there'd be no contradiction in positing such an aggregate, and some imaginable aggregation/composition principles (like "any two individuals form an aggregate") would imply that there is such an aggregate.

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[michaelrabenberg]

Barry, I agree but the case I was imagining was one where an aggregate persists though the loss of one of its members (like the persistence of a school of fish through the loss of one of the fish), not one where an aggregate ceases to be due to the loss of one of its members. Because BFO allows for some cases of the former sort (like school-of-fish cases), my point was that some specific tailoring is required to rule out cases of the former sort involving aggregates of differently dimensioned entities such that the dimensions of the aggregate would count as changing over time. (Such tailoring could take a very straightforward form, such as never saying anything that implies that there are aggregates like that.)

[alanruttenberg]

On Wed, Dec 13, 2023 at 10:33 PM Alan Ruttenberg @.***> wrote:

Boundaries can definitely occupy different spatial regions at different times. Do we agree that nothing changes the dimension of the space something occupies. So if you occupy a 2d spatial region at some point, at any time you exist you occupy a 2d spatial region.

Absolutely

On Wed, Dec 13, 2023 at 7:31 PM Michael Rabenberg < @.***> wrote:

OK, thanks. One question relevant to how best to formulate the axioms is whether a given (say) fiat boundary can occupy different spatial regions at different times; some conceivable ways of formulating the axioms would rule out this possibility and some wouldn't.

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[johnbeve]

if you occupy a 2d spatial region at some point, at any time you exist you occupy a 2d spatial region.

A 2D continuant fiat boundary occupying a 2D spatial region that later becomes a site occupying a 3D spatial region would be a counterexample. Here's a shot: 38th parallel north border (not the latitude, the border) was the boundary between N and S Korea during the Korean War, prior to the establishment of a 160x2.5 mile buffer zone that acts as the current border.

Is this a case of one border expanding dimension or a case in which one border stops existing and is replaced by another that occupies a higher dimension? I lean towards the latter for this example, but I suspect there are more compelling cases that presently escape me.

[phismith]

There is a demarcation zone and demarcation line In general there is no BFO class such that one and the same entity can instantiate it and not instantiate it at different times in its existence. @.*** https://res.cloudinary.com/korea-konsult-ab/image/upload/c_limit,f_auto,q_auto,w_800,dpr_auto/DMZ/Korea_DMZ.jpg

From: John Beverley @.> Sent: Tuesday, December 19, 2023 5:06 PM To: BFO-ontology/BFO-2020 @.> Cc: Barry Smith @.>; Comment @.> Subject: Re: [BFO-ontology/BFO-2020] The spatial regions that continuant fiat boundaries occupy (Issue #76)

if you occupy a 2d spatial region at some point, at any time you exist you occupy a 2d spatial region.

A 2D continuant fiat boundary occupying a 2D spatial region that later becomes a site occupying a 3D spatial region would be a counterexample. Here's a shot: 38th parallel north border (not the latitude, the border) was the boundary between N and S Korea during the Korean War, prior to the establishment of a 160x2.5 mile buffer zone that acts as the current border.

Is this a case of one border expanding dimension or a case in which one border stops existing and is replaced by another that occupies a higher dimension? I lean towards the latter for this example, but I suspect there are more compelling cases that presently escape me.

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[alanruttenberg]

In general there is no BFO class such that one and the same entity can instantiate it and not instantiate it at different times in its existence.

The exceptions being object aggregate, fiat object part, and object.

[phismith]

An egg and a sperm fuse. An object aggregate is replaced by (does not become) an object. The tail of a cat is chopped off. A new object is created thereby. BS

From: Alan Ruttenberg @.> Sent: Tuesday, December 19, 2023 7:02 PM To: BFO-ontology/BFO-2020 @.> Cc: Barry Smith @.>; Comment @.> Subject: Re: [BFO-ontology/BFO-2020] The spatial regions that continuant fiat boundaries occupy (Issue #76)

In general there is no BFO class such that one and the same entity can instantiate it and not instantiate it at different times in its existence.

The exceptions being object aggregate, fiat object part, and object.

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[johnbeve]

An egg and a sperm fuse. An object aggregate is replaced by (does not become) an object. The tail of a cat is chopped off. A new object is created thereby.

@alanruttenberg what was the motivation for object, object aggregate, and fiat object being exceptions to the general rule? My suspicion is that it might apply equally to cases of dimension change.

[alanruttenberg]

It has to do with granularity. The triad represents 3 levels of granularity. Data integration across OBO spanned something like 8 or 10 different granularities. A muscle is either a fiat part, or an object aggregate depending on which discipline of biology. Absent a larger scale, workable, theory of granularity we make do with object, fiat part, object aggregate, but don't have to have different disciplines arguing about whether that muscle is object, aggregate, or part.

I don't think it transfers to boundaries/sites. I see them as completely different things. A boundary is infinitely thin. A site isn't.

[alanruttenberg]

@phismith we settled on these not being disjoint. Given that, the choice is either that all material entities are all three forever, or they can be one thing at one time and another at another (but also both, as appropriate).

[johnbeve]

Thank you for clarifying the motivation @alanruttenberg.

I don't think it transfers to boundaries/sites. I see them as completely different things. A boundary is infinitely thin. A site isn't.

You get Sites as completely different from CFBs from:

• 0D CFB is infinitely thin along 3 axes
• 1D CFB is infinitely thin along 2 axes
• 2D CFB is infinitely thin along 1 axis
• Site is infinitely thin along 0 axis

Whereas I could equally get 0D CFB as completely different from Sites, 1D, and 2D CFBs from:

• Site has 3 dimensions
• 2D CFB has 2 dimensions
• 1D CFB has 1 dimension
• 0D CFB has 0 dimensions

Rather than take either direction, I've leaned towards treating Sites as essentially 3D CFBs. I see no defensible differentia distinguishing Sites and CFBs.

I don't mean to suggest here though that I think the granularity-based motivation for flexibility wrt object, aggregate, and fiat object part transfers to Sites and CFBs. I'll need to think more about that. I only mean to suggest that if it doesn't transfer, I don't think it's because Sites and CFBs are completely different things.

[michaelrabenberg]

I agree with John that the fact that sites are infinitely thin and 0D, 1D, and 2D continuant fiat boundaries aren't isn't great evidence that sites aren't continuant fiat boundaries.

What about this, though: A given site crowds out material-entity proper parts of the material entity relative to which the site's boundaries are determined (to adapt the language from the site elucidation on the GitHub); no parallel fact obtains for a continent fiat boundary. So, crudely, some of the "matter" of the earth is where the surface of the earth is, but none of the "matter" of a ship is where any part of the hull of a ship is. (That could surely be put more precisely but I hope the idea is reasonably clear.)

Problem case (Werner presented this one to me a while ago): A ship has a hull at t1; at t2 a ceiling lamp is installed within the hull that (just assume) counts as a material entity proper part of the ship. So, it might be thought, the hull doesn't crowd out the matter of the ship at all times. I guess my response is that the hull changes shape to accommodate the ceiling lamp.

Perhaps this doesn't seem like a sufficiently deep difference to call for classifying sites as non-CFBs. But note that there's no contradiction in positing an immaterial entity that is 3D, that isn't a spatial region, that has its location determined relative to some material entity, and that is exactly co-located with a material proper part of this material entity. For example, there's no contradiction in saying I have a 3D immaterial proper part that is located precisely where my entire left arm is. I'd be more inclined to classify such an outré thing as a "3D continuant fiat boundary" than to classify the hull of a ship as one. So the "crowding out" feature of a site seems like a deep feature of it that isn't just (say) a by-product of its three-dimensionality.

[michaelrabenberg]

^Mistyped. Correction: I agree with John that the fact that sites aren't infinitely thin and 0D, 1D, and 2D continuant fiat boundaries are isn't great evidence that sites aren't continuant fiat boundaries.

[phismith]

From: Michael Rabenberg @.> Sent: Wednesday, December 20, 2023 10:52 AM To: BFO-ontology/BFO-2020 @.> Cc: Barry Smith @.>; Mention @.> Subject: Re: [BFO-ontology/BFO-2020] The spatial regions that continuant fiat boundaries occupy (Issue #76)

^Mistyped. Correction: I agree with John that the fact that sites aren't infinitely thin and 0D, 1D, and 2D continuant fiat boundaries are isn't great evidence that sites aren't continuant fiat boundaries.

This is surely a matter of logic.

Compare

the fact that prime numbers don't have factors other than themselvs and 1 and non-prime numbers do isn't great evidence that prime numbers aren't non-prime numbers.

BS

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[michaelrabenberg]

This is surely a matter of logic. Compare the fact that prime numbers don't have factors other than themselvs and 1 and non-prime numbers do isn't great evidence that prime numbers aren't non-prime numbers.

This isn't an apt analogy to what I said. I said (among other things) that 0D, 1D, and 2D CFBs are infinitely thin, not that all CFBs are infinitely thin. Whether all CFBs are infinitely thin is precisely the issue that John is raising.

[alanruttenberg]

Sites aren't continuant fiat boundaries by definition, intention, practice, and axiom. The surface enamel of a tooth is not a boundary in the BFO sense, it is a material entity. Similiarly, the buffer zone. These are boundaries in a different sense than BFO, the same sense in which a fence is a boundary.

The definitions are given. I don't see how this is even an argument.

cfb: b is a continuant fiat boundary means: b is an immaterial entity that is of zero, one or two dimensions, which is such that there is no time t when b has a spatial region as continuant part at t, and whose location is determined in relation to some material entity

site: b is a site means: b is a three-dimensional immaterial entity whose boundaries either (1) (partially or wholly) coincide with the boundaries of one or more material entities or (2) have locations determined in relation to some material entity

[michaelrabenberg]

Well, John started it :)

Obviously in raising the possibility that sites are 3D CFBs John was questioning the relevant definitions/elucidations/axioms, so it's not responsive to him to just say "But the definitions etc. say you're wrong." I'm sure he's aware of what they (including the elucidations you reproduced) say, as am I.

At any rate I was arguing that the "sites aren't CFBs" view is defensible in the way he was suggesting it isn't.

[johnbeve]

Sites aren't continuant fiat boundaries by definition, intention, practice, and axiom.

Definitions, intention, and axioms are only as good as justifications for them; practice is informed by the definitions, intentions, and axioms.

[johnbeve]

The surface enamel of a tooth is not a boundary in the BFO sense, it is a material entity. Similiarly, the buffer zone.

I agree with the tooth enamel example, but I didn't say anything suggesting otherwise.

I'm afraid you're going to have to explain how the DMZ buffer zone between N and S Korea is a material entity though. It satisfies the definition for Site.

[johnbeve]

@alanruttenberg @phismith Does a given 2D CFB itself have any 1 CFB boundaries?

[alanruttenberg]

It's not a material entity. It's a space-taking thing that separates two other things, just as the surface enamel or fence is.

Yes, a 2d CFB can have 1 CFB boundaries, and 1 CFB 0. But in a sense of "have" that is not formally defined by BFO. I've never been happy that that's not been more nailed down, but there has been many discussions of it over the years.

Definitions, intention, and axioms are only as good as justifications for them; practice is informed by the definitions, intentions, and axioms.

Intentions are primary, barring only nonsense and inconsistency. The definitions and axioms try to follow the intention. The mention of practice is important in that the current understanding is used by many projects.

[johnbeve]

It's not a material entity. It's a space-taking thing that separates two other things, just as the surface enamel or fence is.

Agreed.

I'm only here trying to clarify the differentia. Based on this discussion and the elucidations which I repeat here:

cfb: b is a continuant fiat boundary means: b is an immaterial entity that is of zero, one or two dimensions, which is such that there is no time t when b has a spatial region as continuant part at t, and whose location is determined in relation to some material entity

site: b is a site means: b is a three-dimensional immaterial entity whose boundaries either (1) (partially or wholly) coincide with the boundaries of one or more material entities or (2) have locations determined in relation to some material entity

CFBs may have boundaries too. From the axioms, Sites cannot have spatial region parts, just like CFBs.

The only difference I am seeing then between the elucidation is clause (1) for Site.

Is it the case that a given CFB may (partially or wholly) coincide with the boundaries of one or more ME and yet not have its location determined in relation to some ME?

[alanruttenberg]

The differentia are minimally the dimensionality.

Examples reinforce:

Site: A hole in a portion of cheese, a rabbit hole, the Grand Canyon, the Piazza San Marco, the kangaroo-joey-containing hole of a kangaroo pouch, your left nostril (a fiat part - the opening - of your left nasal cavity), the lumen of your gut, the hold of a ship, the interior of the trunk of your car, hole in an engineered floor joist; An air traffic control region of type A is determined in terms of elevation above mean sea level of lower and upper boundaries.

fiat point: The geographic North Pole; the quadripoint where the boundaries of Colorado, Utah, New Mexico and Arizona meet, the point of origin of some spatial coordinate system.

fiat line: The Equator, all geopolitical boundaries, all lines of latitude and longitude, the median sulcus of your tongue, the line separating the outer surface of the mucosa of the lower lip from the outer surface of the skin of the chin.

fiat surface: The surface of the Earth, the plane separating the smoking from the non-smoking zone in a restaurant

[alanruttenberg]

For reference. A discussion of boundary dependence on BFO-discuss.

[phismith]

From: Alan Ruttenberg @.> Sent: Wednesday, December 20, 2023 1:26 PM To: BFO-ontology/BFO-2020 @.> Cc: Barry Smith @.>; Mention @.> Subject: Re: [BFO-ontology/BFO-2020] The spatial regions that continuant fiat boundaries occupy (Issue #76)

The differentia are minimally the dimensionality.

Agreed

Examples reinforce:

These are all my examples. A n-dimensional entity is never an n+1 dimensional entity at another time in its existence

Site: A hole in a portion of cheese, a rabbit hole, the Grand Canyon, the Piazza San Marco, the kangaroo-joey-containing hole of a kangaroo pouch, your left nostril (a fiat part - the opening - of your left nasal cavity), the lumen of your gut, the hold of a ship, the interior of the trunk of your car, hole in an engineered floor joist; An air traffic control region of type A is determined in terms of elevation above mean sea level of lower and upper boundaries.

fiat point: The geographic North Pole; the quadripoint where the boundaries of Colorado, Utah, New Mexico and Arizona meet, the point of origin of some spatial coordinate system.

fiat line: The Equator, all geopolitical boundaries, all lines of latitude and longitude, the median sulcus of your tongue, the line separating the outer surface of the mucosa of the lower lip from the outer surface of the skin of the chin.

fiat surface: The surface of the Earth, the plane separating the smoking from the non-smoking zone in a restaurant

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[wceusters]

Barry: The differentia are minimally the dimensionality.

WC: which brings us back to the very top of this thread: it wasn't in the axioms.

Barry: Examples reinforce:

WC: Agreed. But the problem is not so much to figure out whether some particular is an instance of any of these types, but rather what sorts of relations obtain between them and other types. 'bounded by', 'determined by', 'have' etc are not really helpful when not formalized as well. And this brings me back to the starting point for this, at least for Michael and me: trying to grasp the relations between parts of an extended organism that are not part of the organism and 'where they then might be'.

[alanruttenberg]

In absence of a relation, for some work I'm doing now I defined 'has immaterial entity host' as 'relates a site or boundary to the entity relative to which it is defined' I would be happier if this sort of thing was in BFO, and done properly rather than off the top of my head.

[alanruttenberg]

That gets you to the sites referred to in the definition of extended organism. Then use 'located in' or 'occurs in' to place stuff in the sites. But the def for extended organism assumes you know what an organism is, and presumes that the relation of the organism is member of the extended organism, since it's an object aggregate. For reference:

extended organism (OGMS) = An object aggregate consisting of an organism and all material entities located within the organism, overlapping the organism, or occupying sites formed in part by the organism.

[alanruttenberg]

There's a bug there isn't there? An object aggregate can't have a member that overlaps another member.

[wceusters]

OGMS-OWL is full of issues. Not to be relied on.

[wceusters]

With that description in particular, there is indeed a lot if not wrong, then for sure unclear: 'consisting of', 'occupying sites', 'formed by'. Similar language like this is used in BFO papers, the Arp-book, the specs, etc. It should all be much better clarified.

[johnbeve]

The differentia are minimally the dimensionality.

I am searching for better justification.

I ask again: Is it the case that a given CFB may (partially or wholly) coincide with the boundaries of one or more ME and yet not have its location determined in relation to some ME?

[michaelrabenberg]

The differentia are minimally the dimensionality.

I am searching for better justification.

I ask again: Is it the case that a given CFB may (partially or wholly) coincide with the boundaries of one or more ME and yet not have its location determined in relation to some ME?

I fear I'm losing the thread but don't you mean to ask whether a site can meet this description? CFBs are explicitly said in their elucidation to have their location determined in relation to some ME, and sites aren't explicitly said to. So, if there's a feature of CFB's that sites don't have that could justify treating all sites as non-CFBs, maybe that could be it. (Though I still like my proposal :).)

Incidentally it occurs to me that there's another rather awkward revisionary position in conceptual space that a "Yes" answer to that question would open up: Some sites are CFBs, and some aren't. One way of fleshing that view out: Sites that have their location determined relative to a material entity are 3d CFBs, but sites (if there are any) that don't have their locations determined relative to a material entity are 3d immaterial entities that aren't CFBs.

[johnbeve]

don't you mean to ask whether a site can meet this description?

Thank you for catching that, I did mean Site; you've my intent correct, I'm hoping clause (1) in the elucidation of Site might provide a more robust way to distinguish them from CFBs.

I still like my proposal

I like the spirit; still trying to grok how the 'crowding out' metaphor works, especially with the problem case.

Some sites are CFBs, and some aren't. One way of fleshing that view out: Sites that have their location determined relative to a material entity are 3d CFBs, but sites (if there are any) that don't have their locations determined relative to a material entity are 3d immaterial entities that aren't CFBs.

I've the same worry about the disjunctive elucidation for Site. I see the plausibility of your suggestion trading on whether we can distinguish "(partially or wholly) coincide with the boundaries of one or more material entities" from "have locations determined in relation to some material entity" in a sensible way, e.g. specifying what 'coincides with the boundaries of' and 'determined in relation to' mean. More to chew on.

[alanruttenberg]

(forall (m s) (if (exists (t) (and (occupies-spatial m s t) (or (= s three-dimensional-spatial-region) (= s two-dimensional-spatial-region) (= s one-dimensional-spatial-region) (= s zero-dimensional-spatial-region)))) (forall (t1 s1) (if (occupies-spatial m s1 t1) (instance-of m s t1)))))

I assume the boldfaced 'm' should be 's', right?

s1, right?