PatH,

Continuing the discussion about the number of primitives and the (in)compatibility of perdurantism and endurantism:

(the most relevant parts are extracted from the note of PatH 6:15 PM Friday 20090109)

(1) the number of primitives and uniqueness of the set

[PC] >> Yes, it is true that different independent developers could use different combinations of modules, provided that those modules were known to be (or not disproven to be) logically consistent, and still achieve optimal interoperability. But this still requires that the chosen modules have all of the basic concepts that allow all of the newly created domain terms to be ‘expressed in terms of’ (I am using your phrase) those terms in the modules chosen. This means that *somewhere* in the collection of mutually consistent modules there have to be enough of the basic terms that are used to express the meanings of newly created domain terms, so that any domain can be represented.

[PH] > OK, lets agree on this for the sake of argument. But note, it does not follow that there is a *single* such set. That is the point.

[PC] >>That set of modules would then contain what I think of as the minimum set of primitives

[PH] > By using the word "the" here, you beg the central question: whether there is a *single* set of primitives out of which all other meanings can be formed.

Ah so. This is an objection I hadn’t gleaned from the earlier notes. Interesting point. I thought the main objection was about the finite number of primitives, not their identity. I confess it has been an assumption of mine that, from the point of view of a “primitive” concept being indivisible into smaller primitive concepts (crudely analogous to an atom, in its role as a component of molecules), that a finite inventory of such primitive-atoms (if such exists) would be unique. I have never seen this point raised, and it may be significant, but right now I can’t think of any way a set of indivisible primitives could be anything but unique. Limited imagination, perhaps.

Now I have noticed that there are mathematical objects that have very non-intuitive properties, and perhaps a mathematician can provide some examples of how a set of primitive concepts (not divisible into component primitive concepts) might be able to be formulated in more than one way. I am genuinely curious about this suggestion.

(2) are primitives finite in number?

[PC] >> , though the modules can also have other ontology elements in them. It is not clear to me that we can have a *stable* set of modules to serve the purpose of interoperability unless we have some confidence that all or most of the terms needed to express the meanings of domain terms are there at the start.

[PH] > Seems clear to me that if we impose this condition as a requirement, we have shot ourselves in the foot. And why should the set of concepts be "stable" ? Isn't it more realistic to assume that new concepts will always be being constructed, that knowledge is always open-ended?

Well, knowledge is certainly open-ended, but if new concepts (or terms labeling those concepts) are always composed from some combination of pre-existing concepts, then the number of primitives need not increase as knowledge increases. This may depend on what you mean by ‘knowledge’. With a set of basic concepts we can predict an infinite number of things that *might* exist in the real world, but I think of knowledge as knowing what actually does exist and what doesn’t. Learning about the laws of physics (and chemistry and biology) that exist in our real world allows us to know some of the things that can’t exist. That’s knowledge, in my lexicon, but it doesn’t necessarily require new primitives.

Is it possible that learning *some* new things always requires creating new primitives? Possibly, but that is the question that I have suggested can be investigated by the process of creating a plausible set of ontology elements based on primitives (as best we can discern them) and then seeing how quickly that inventory in the FO must increase to allow the elements in each new additional domain ontology to be expressed in terms of the primitive set. If there is a finite set, then from the increase in required primitives for each new block of domain concepts, it will be possible to assign a probability that the total inventory will reach an asymptote at infinity. The alternative is that no limit is indicated, and the number of primitives behaves, say, like the number of prime numbers as the number of integers increase. I do not know for sure which is more ‘realistic’, but my suspicion lies with the finite number, based on evidence from linguistic usage, such as the Longman defining vocabulary and the relatively small number (2000 – 5000) in AMESLAN sign language dictionaries. There are other suggestive kinds of evidence from language, but what needs to be determined experimentally is whether the finite defining vocabularies in languages are in fact analogous (at least on the point of the number of primitives) to the task of creating ontology elements as combinations of other ontology elements.

(3) [PC] >> If it turns out that there is no such thing as a finite set of primitives, and it becomes necessary to continue adding new primitives indefinitely as new domains are derived from the FO, then the FO will not be fully stable.

(4) On the question of whether 3D and 4D entities are logically compatible.

One point I did not make in the discussion of how this could be represented is that allowing a time slice of a non-4D ‘dimension neutral’ Object in the OWL version of an ontology (which I do in the current OWL version of COSMO) does not mean that the CL-compliant version would that syntactic device. It is used in the OWL version to reduce time-dependent assertions to binary, to fit into the OWL syntax. When the OWL version is translated into CL, every instance of an Object that is a time slice would be translated as a time slice of ‘Object4D’. Then Object and Object4D could be disjoint , making it easier to include a continuant/occurrent view into the CL version of the FO.

Regarding some specific comments on this point:

[PC] >> My idea of a ‘unified’ 3D-4D ontology would permit (among other things) the assertions (I hope the meaning is clear from the labels):

>> {PH isanInstanceOf Object}

>> {PH4D isanInstanceOf Object4D}

[PH > Can you explain the difference between Object and Object4D? Intuitively, that is. What criteria are there for deciding whether a given thing is in one category or the other, that we can explain to knowledge modelers in the user handbook? Can there be one of these without the other also existing, such as a PH without a PH4D, or vice versa? (Why not?)

In an assertion containing an instance of dimension-neutral ‘Object’ that is not also an instance of TimeSlice (having begin and end time points) , if the assertion does not have an explicit time index, it must be interpreted as meaning that the relation holds throughout the lifetime of the Object; for the Object to have an incompatible property at any time would be a contradiction. So the dimension-neutral Object behaves in some ways like an Object4D, except that it can also be used in explicitly time-indexed assertions. If one decides, in a CL version, to make Object and Object4D disjoint, then in the CL version, there cannot be any time-slices of an Object. In that case, an assertion on an Object that is not explicitly time-indexed can still be interpreted as holding throughout the lifetime of the Object, or it may be forbidden as a syntactic error. That option would be decided by the committed creating the FO.

There may be better ways to accommodate both endurantist and perdurantist representations in the same ontology. The point I am trying to make here is that, although the philosophical stance of an endurantist and of a perdurantist may rest on incompatible models, *ontological assertions* about either such entity can be accurately translated into assertions about the other representation. I thought that your note from March made much the same point.

(5) [PC] >> {PH4D isTheWholeLife4dVersionOf PH}

{PHt1t2 isaTimeSliceOf PH4D from t1 to t2}

[PH] > What relationship , if any, is there between PHt1t2 and PH?

The relationship is a combination of the relations between PH and PH4D, and the relation between PH4D and PHt1t2. It could be compressed into one relation “isaTimeSliceOfThe4D”, if that is useful:

{ {PHt1t2 isaTimeSliceOfThe4D PH from t1 to t2} iff

{ {PH4D isTheWholeLife4dVersionOf PH}

and

{PHt1t2 isaTimeSliceOf PH4D from t1 to t2}}

}

(6) [PC] >> If I included a ‘during’ similar to the one PH uses, it might look like:

>> {(PH during t1t2) isIdenticalTo PHt1t2}

[PH] > ?? So (PH during BirthDeath) is identical to PH4D ?

Yes, though I would use one of these alternative syntactical constructions, which are equivalent to each other:

(during PH BirthDeath)

{PH during BirthDeath}

. . . where ‘during’ is interpreted as a function generating a time-slice of a dimension-neutral Object - the functional alternation of the concatenated relation ‘isaTimeSliceOfThe4D’, described above, though syntactically different.

One caveat, to be precise: This would be true only if PH were interpreted as ‘PH-while-alive’. I do prefer to allow dead and unborn people to be represented as a ‘Person’, distinguished from living people by an ‘alive’ (dead/notBorn) attribute. This requires a careful definition of what a ‘whole-life’ interval is, for each category of Object.

(7)[PH] > If you are willing to allow dimension-neutral objects to have time-slices, you have completely abandoned the foundational ideas behind continuants. Whatever these PH things are, they aren't continuants. I therefore see no purpose in having them. Why not simply identify PH with PH4D?

Because this allows the use of syntactic structures that are used with 3D objects, and serves the purpose of allowing a syntax that is more congenial to those who prefer to work in that paradigm. The point is to try to create a foundation ontology that can be agreed to by the largest number of people, so as to serve the purpose of creating the largest possible user community – which is the whole purpose of the proposed FO project. Where there are irreconcilable differences, these can be encoded in extensions, provided that the differences are not contradictory to the logic of the base FO.

Allowing time-slices of an Object is not required, Object and Object4D can be disjoint, related by the axioms above. In the OWL version it is a convenience, but I believe that conversion of the OWL version to a corresponding CL-compliant version where Object and Object4D can be disjoint is possible (though I haven’t yet written the program). The point of this discussion, from my side, is to emphasize that, though philosophical views can involve incompatible models, when we are concerned with the practical task of creating an ontology, assertions syntactically reflecting on one view can be accurately translated into assertions syntactically reflecting the other view. This is the sense in which I consider the alternative views to *not* be ‘logically incompatible’ – the practical consequences of both can be represented in a single consistent ontology. If there are logical inconsistencies that must be represented by something other than belief systems, it may be necessary to include them in extensions, or in FO-described ontologies that are not part of the hierarchy of ontologies (which can be viewed as a constrained ‘lattice of theories’).

(8) [PC] >> The axioms seemed to be consistent with that, though the (x during t) structure was not repeated in the later axiom set. Perhaps you meant something else by ‘unified’?

[PH] > I was using the term optimistically, but subsequent discussions have made me less optimistic than I had become.

In your note back in March, I got the (erroneous?) impression that your axioms described a method of syntactically accommodating both perdurantist and endurantist views (or at least their _expression_ in assertions about the real world). You think now that there cannot be such a combination?

If it turns out that there are important (used by more than two groups) ontological structures that are primitive but incompatible, it will be necessary to choose one. I am not sure that that will be necessary, because I still expect that incompatible theories will almost always be expressible as combinations of primitives – but it may happen. Even so, I expect that the number of potential users and developers that are willing to adopt the result of a collaborative FO development effort will be sufficiently large to form a user community that can maintain and evolve the starting product. The goal is to create a user community large enough so that anyone who wants to interoperate will have *some* widely-used FO to use for that purpose, with enough public examples of use to provide confidence that it will serve the local needs. Right now I think that the existing FOs are not used widely enough, and that a collaborative project can create one used more widely than any existing one. If it can be based on a finite number of primitives, that will be helpful, but not critical. There may well be some ontologists who simply won’t accept the result of the project; as long as there are enough users to form a large community that helps to evolve the FO and its extensions, that won’t prevent the project from achieving its goal.

PatC