Pat, we have got to get this sorted out. We are (I hope) talking past
one another. (01)
FIrst, let me clarify something about 'formal' or 'mathematical'
notions of meaning. Tarskian semantics does not apply only to
'mathematical' theories, nor does it require that all meanings be
"mathematical", whatever that could mean. It is a very general theory
of meaning, one that can be applied to a wide range of languages and
notations (for example, I have applied it to 2-dimensional maps) and
even to mental models of thought. However, it is itself a
mathematically expressed theory. That is, it *uses* mathematical
notions - of set, and mapping, and function - to state its own
theoretical ideas. This is something it shares with almost every other
precise theory of almost anything, in fact. But it does not follow
from this that the theory is *about* "mathematical things", any more
than using, say, differential equations to describe the stress forces
in a bridge girder makes this into a mathematical theory and therefore
not about real bridges. (02)
But now, to get to the heart of the matter, Tarskian semantics is a
THEORY of meaning. Actual meanings in the wild, the things we
apparently refer to when we talk about "intended meanings" or
"intuitive meanings" or the like, are (we all sincerely hope) real
things in the world, part of our human life experience. We all believe
that when we think about things using our concepts of those things,
that our thoughts are meaningful, that they *have* real, actual
meanings. But in order to be scientific about this kind of talk, we
need some *theory* of what these natural, wild, real meanings actually
are. Or at least some kind of *account* of them, saying what kind of
entity they are, what properties they have, how they relate to other
things (like, to the thinkers of the ideas, or to the thins they are
ideas of, etc..) Are meanings something linguistic or symbolic in
nature? Are them mental or psychological? Or platonic, in some
abstract realm, like numbers? Can they be written down, or captured in
some other way? Etc.. (03)
It is just wrong to draw the contrast between the natural things, on
the one hand, and the account provided of those things by a theory of
them, on the other, as a difference of **kind**. Take numbers. There
are the natural numbers, which most mathematicians agree exist in the
wild, as it were. And then there are various formalized arithmetics,
each of which is a theory of the natural numbers. And we happen to
know, in this case, that we cannot have a perfect such theory: any
theory will miss something, will have its unprovable Goedel sentence.
But we do not say that there are two kinds of number: the natural
ones, and the merely **mathematical** ones, and the formalized
arithmetics are about the latter and not the former. They are all
theories of the same entities, but some theories capture more truths
than others. It is not a matter of chalk and cheese, but rather of
varieties of cheese-making. (04)
Similarly for meanings. There are real meanings in the world, let us
agree. Some things out there really do mean something. And then there
are some theories of this, and Tarskian semantics is one such theory.
It is somewhat narrow, it does not by any means capture all the
nuances of natural meanings. But, especially when extended in the
various ways it has been by such folk as Kripke and Scott, it does
cover a surprisingly wide range of examples. And, more to the point,
it is the *only* viable theory of meaning we have, as far as I know.
We have philosophical critiques of it, to be sure, but we do not have
any alternatives to hand. (05)
You, below, contrast meanings in a "mathematical theory" (which I
presume you presume I was talking about) with those in a
"computational ontology". But I was talking about computational
ontologies. Computational ontologies are artifacts, written in formal
logical notations. They do not simply 'have' natural meanings,
meanings-in-the-wild, in the way that human natural languages are said
to have. They do not have intended meanings. We may intend them to
have a meaning, but a computational ontology is just as artificial and
"formal" (which is to say, "mathematical" in the sense of being
mathematically described) as any other artifact. And in the case of
logically expressed formalisms, like those used in computational
ontologies, the Tarskian theory of meaning applies in a special way.
Not only is it *a* theory of their meaning, indeed the only one we
have, but Goedel proved that for these formal logics, it is an
exactly correct theory of meaning. That is the content of the
completeness theorem: something is provable from O precisely when what
it means according to the Tarskian theory of meaning is entailed
(according to the same theory) by the sentences in O. This is, to
emphasize, a provable fact about any FOL-based computational ontology. (06)
One can say, this ontology O does not capture all my intended meaning,
speaking of natural meanings in the wild. Of course, this may well be
the case, and may well be a legitimate critique of any formal theory
of anything in nature. But what one cannot legitimately claim is that
(by virtue of being computational instead of mathematical, or by some
other mysterian magic) a *formal* ontology captures more meaning, or a
different kind of meaning, than the meaning assigned to it by the
Tarskian account, by virtue of its logical or computational properties. (07)
And what I said, which you quote below, regarding primitives, is a
factual observation about a provable consequence of the Tarskian
theory of meaning applied to any ontology expressed in any formal
assertional logic. I was not talking about mathematical theories in
particular, and certainly not in contrast to 'computational ontologies'. (08)
Your point in response, I take it, is that you include, as part of the
meaning-capturing machinery of an ontology the human-readable
commentaries which state in English the intended meanings of the
formally expressed concepts. Well, that is a stance one can take: but
then I would say in response, that your ontology is no longer
expressed in a formalism, so is no longer "computational". In fact, I
see no reason to call it an ontology at all. Why bother with logic, if
I can impose my will upon meanings by writing prose? I need no theory
of meaning in order to speak, after all. The entire process can
proceed without using any formalism, and the only function that the
computer need play is to be a kind of public record of our
discussions, the minutes of the meaning-deciding meetings. But this is
a reductio ad absurdum of our enterprise. (09)
On Feb 14, 2010, at 9:25 PM, Patrick Cassidy wrote: (010)
> Concerning the meaning of Primitives in a Foundation Ontology:
>
> John Sowa said:
> [JFS] > " My objection to using 'primitives' as a foundation is
> that the
> meaning of a primitive changes with each theory in which it occurs.
> For example, the term 'point' is a 'primitive' in Euclidean
> geometry and
> various non-Euclidean geometries. But the meaning of the term
> 'point' is
> specified by axioms that are different in each of those theories."
> (and in another note):
> [JFS] >> As soon as you add more axioms to a theory, the "meaning"
> of the
>>> so-called "primitives" changes.
>
> Pat Hayes has said something similar but more emphatic:
>
> [PH] > "Each theory nails down ONE set of concepts. And they are ALL
> 'primitive' in that theory, and they are not primitive or non-
> primitive in
> any other theory, because they aren't in any other theory AT ALL."
>
> Given these two interpretations of "primitive" in a **mathematical**
> theory,
> it seems that the "meanings" of terms (including primitive terms) in a
> mathematical theory have little resemblance to the meanings of terms
> in a
> computational ontology that is intended to serve some useful purpose,
> because the meanings of the terms in the ontology do not depend
> solely on
> the total sum of all the inferences derivable from the logic, but on
> the
> **intended meanings**, which do or at least should control the way the
> elements are used in applications - and the way the terms are used in
> applications is the ultimate arbiter of their meanings. The intended
> meanings can be understood by human programmers not only from the
> relations
> on the ontology elements, but also from the linguistic
> documentation, which
> may reiterate in less formal terms the direct assertions on each
> element,
> but may also include additional clarification and examples of
> included or
> excluded instances. It seems quite clear to me that it is a mistake
> to
> assume that the interpretation of "meaning" or "primitive" in a
> mathematical
> theorem is the same as the way that "meaning" and "primitive" are
> used in
> practical computational ontologies.
>
> This discussion was prompted by my assertion that the meanings of
> terms in
> a Foundation ontology, including terms representing primitive
> elements,
> should be as stable as possible so that the FO can be relied on to
> produce
> accurate translations among the domain ontologies that use the FO as
> its
> standard of meaning. Given the agreement by JS and PH that each
> change to
> an ontology constitutes a different theory (011)
This is not a "agreement". It is simply an elementary fact about
logical theories. In fact, in logic textbooks, the very word "theory"
is used to refer to a set of sentences. So OF COURSE different sets
constitute different theories. (012)
> , and the meanings of terms in any
> one theory are independent of the meanings in any different theory, I
> believe that we need to look for a meanings of "meaning" and
> "primitive"
> that is not conflated with the mathematical senses of those words as
> expressed by JS and PH. (013)
They are not "mathematical" senses. And good luck finding an
alternative theory of truth.
>
> I suggested that a useful part of the interpretation of "meaning" for
> practical ontologies would include the "Procedural Semantics" of
> Woods.
> John Sowa replied (Feb. 13, 2010) that:
> [JFS]> " In short, procedural semantics in WW's definition is
> exactly what
> any programmer does in mapping a formal specification into a program."
>
> I agree that computer programmers interpret meanings of ontology
> elements
> using their internal understanding as a wetware implementation of
> "procedural semantics". This does not exhaust the matter, though,
> because
> computers now can perform some grounding by interaction with the world
> independent of computer programmers in this sense: though their
> procedures
> may be specified by programmers, those procedures can include
> functions that
> themselves perform some of the "procedural semantics" processes, and
> the
> computers therefore are not entirely dependent on the semantic
> interpretations of programmers, at least in theory. For the present
> there
> are probably few programs that have the capability of independently
> determining the intended meanings of the terms in the ontology, but
> that is
> likely to change, though we don't know how fast. I will provide one
> example
> of how this can happen: an ontologist may assert that the type "Book"
> includes some real-world instance such as the "Book of Kells" (an
> illuminated Irish manuscript from the middle ages, kept at a library
> in
> Dublin). With an internet connection, the program could (in theory)
> check
> the internet for information about that instance, and to the extent
> that the
> computer can interpret text and images, perform its own test to
> determine if
> the Book of Kells actually fits the logical specification in the
> Ontology.
> The same can be done for instances of any type that are likely to be
> discussed on the internet.
>
> So, if we want the meanings of terms in an ontology to remain
> stable, and
> **don't** want the meanings to change any time some remotely related
> type
> appears in a new axiom, (014)
But we DO want this! Surely that is the very point of changing and
adding axioms. If meanings are stable across theories, then what is
the point of adding axioms to capture more meaning? Apparently,
whatever meaning you are going to have after the addition was already
there before. I presume it was there when the ontology had no axioms
at all, in fact: for if not, which addition, in the long process of
growing the ontology, created or introduced it? It would seem to
follow that all ontology meanings are already present in an empty
ontology. (015)
This is obvious nonsense. You are confusing the intended meaning, the
natural meaning we are seeking to capture in O, with the *actual*,
theory-justified meaning that O can be said to have by virtue of its
ontological structure. The former is stable, but is not
computationally accessible. The latter is computationally accessible,
and subject to precise theoretical analysis, but is particular to the
ontology. Change the ontology, you change the meaning. Maybe not much,
but you do change it. And this is not a 'problem', but on the
contrary, is exactly what we would expect and what makes our work
possible at all. (016)
> what can we do? Perhaps we can require that the
> meanings across *different* versions of the FO can only be relied on
> to
> produce *exactly* the same inferences if the chain of inferences is
> kept to
> some small number, say 5 or 6 links (except for those elements
> inferentially
> close to the changed elements, which can be identified in the FO
> version
> documentation, and whose meanings in this sense may change). This
> is not,
> IMHO, an onerous condition for several reasons:
> (1) no added axiom will be accepted if it produces a logical
> inconsistency
> in the FO;
> (2) the programmers whose understanding determines how the elements
> are used
> will not in general look beyond a few logical links for their own
> interpretation, so only elements very closely linked to the one
> changed by
> new axioms will be used differently in programs.
> (3) as long as the same version of the FO is used, the inferences
> for the
> same data should be identical regardless of the number of links in
> the chain
> of inference;
> (4) if there are only a few axioms that change (out of say 10,000)
> between
> two versions of the FO, then the likelihood of getting conflicting
> results
> will be very small for reasonable chains of inference; if the
> similarity of
> two versions of the FO is 99.9% (10 different elements out of
> 10,000), then
> a chain of inference would produce non-identical sets of results on
> average
> for an inference chain 10 long at 0.999(exp 10) = 0.99 (99% of the
> time) 100
> long at 0.999(exp 100)= 0.90, or 90% of the time. But the most
> important
> and useful inferences are likely to be those that arise from short
> inference
> chains (017)
Really? I see no reason why this would be generally true. (018)
> , comparable to how humans would use them, and to how programmers
> would imagine them being used in logical inference.
>
> The potential for changed inferences as new axioms are added, even
> though
> no changing the intended meanings of the elements of the FO, does
> argue for
> an effort to include as many of the primitive elements as can be
> identified
> at the earliest stages. This is in fact the purpose of performing the
> consortium project, to get a common FO thoroughly tested as quickly as
> possible, and minimize the need for new axioms.
>
> In these discussions of the principles of an FO and a proposed FO
> project,
> not only has there been no technical objection to the feasibility of
> an FO
> to serve its purpose (just gut skepticism), but there has also been a
> notable lack of suggestions for alternative approaches that would
> achieve
> the goal of general accurate semantic interoperability (not just
> interoperability in some narrow sphere or within some small group.) (019)
There have also been very few suggestions on how to make a perpetual
motion machine. Has it not dawned on you yet that this goal you are
seeking might well be impossible? That if it were possible, it would
have been done eons ago? (020)
> I am
> quite aware that my discussions do not **prove** that the FO
> approach would
> work, in fact the only way I can conceive of proving it is to
> perform a test
> such as the FO project and see if it does work. But neither has
> anyone
> suggested a means to **prove** that other approaches will work, and
> this
> hasn't stopped other approaches (notably mapping) from being funded. (021)
Ontology mapping has immediate benefits in the real world, which is
why it gets funded. (022)
>
> In fact in my estimate, no other approach has the potential for
> coming as
> close to accurate interoperability as an FO, and no approach other
> than the
> kind of FO project I have suggested can possibly achieve that result
> as
> quickly. (023)
You ignore the likely possibility that it will be impossible to create
an FO, and that the attempt will cost a great deal of money and
effort. And you would never accept a negative outcome in any case,
right? (024)
>
> I am grateful for the discussions that have helped sharpen my
> understanding of how such a project can be perceived by others, and
> improved
> my understanding of some of the nuances. But after considering these
> points, I still think that a project to build and test a common FO
> is the
> best bet for getting to general accurate semantic interoperability as
> quickly as possible. (025)
"General accurate semantic operability" is like world peace, and just
as unlikely to ever be achieved while human beings have the cognitive
limitations that we do in fact have. Even if an angel were to appear
and give us such a FO, it would work only for a matter of minutes
before needing to be changed, and no amount of human effort could keep
pace with the necessary changes. (026)
>
> Perhaps future objections could focus on genuine technical problems
> (not
> analogies with human language), and better yet suggest alternatives to
> solving the problem at hand: not just *some* level of
> interoperability, but
> accurate interoperability that would let people rely on the
> inferences drawn
> by the computer. If not a common FO, then what? (027)
Nothing. This is not a viable goal to seek. It is a fantasy, a dream.
One does not seek alternative ways to achieve a fantasy. (028)
PatH (029)
>
> Pat
>
> Patrick Cassidy
> MICRA, Inc.
> 908-561-3416
> cell: 908-565-4053
> cassidy@xxxxxxxxx
>
>
>> -----Original Message-----
>> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
>> bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F. Sowa
>> Sent: Saturday, February 13, 2010 12:31 PM
>> To: [ontolog-forum]
>> Subject: Re: [ontolog-forum] Foundation ontology, CYC, and Mapping
>>
>> Cory,
>>
>> I'd like to give some examples that may clarify that issue:
>>
>> CC> ... foundational concepts are very similar to the "minimally
>>> axiomatized micro theories" I remember John-S describing as
>>> a workable foundation, yet John does not see primitives
>>> as workable - why the difference?
>>
>> My objection to using "primitives" as a foundation is that the
>> meaning of a primitive changes with each theory in which it occurs.
>> For example, the term 'point' is a "primitive" in Euclidean geometry
>> and various non-Euclidean geometries. But the meaning of the term
>> 'point' is specified by axioms that are different in each of those
>> theories.
>>
>> Note that there are two kinds specifications:
>>
>> 1. Some terms are defined by a *closed form* definition, such as
>>
>> '3' is defined as '2+1'.
>>
>> In a closed-form definition, any occurrence of the term on the
>> left
>> can be replaced by the expression on the right.
>>
>> 2. But every formal theory has terms that cannot be defined by a
>> closed-form definition.
>>
>> For example, both Euclidean and non-Euclidean geometries use the
>> term 'point' without giving a closed-form definition. But calling
>> it undefined is misleading because its "meaning" is determined by
>> the pattern of relationships in the axioms in which the term occurs.
>>
>> The axioms specify the "meaning". But the axioms change from one
>> theory to another. Therefore, the same term may have different
>> meanings in theories with different axioms.
>>
>> For example, Euclidean and non-Euclidean geometries share the
>> same "primitives". The following web site summarizes Euclid's
>> five "postulates" (AKA axioms):
>>
>> http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml
>>
>> The first four are true in Euclidean and most non-Euclidean
>> geometries. By deleting the fifth postulate, you would get
>> a theory of geometry that had exactly the same "primitives",
>> but with fewer axioms. That theory would be a generalization
>> of the following three:
>>
>> 1. Euclid specified a geometry in which the sum of the three
>> angles of a triangle always sum to exactly 180 degrees.
>>
>> 2. By changing the fifth postulate, Riemann defined a geometry
>> in which the sum is < 180 degrees.
>>
>> 3. By a different change to the fifth postulate, Lobachevsky
>> defined a geometry in which the sum is > 180 degrees.
>>
>> This gives us a generalization hierarchy of theories. The theories
>> are generalized by adding axioms, specialized by deleting axioms,
>> and revised by changing axioms (or by deleting some and replacing
>> them with others).
>>
>> I have no objection to using collections of vague words, such as
>> WordNet or Longman's, as *guidelines*. But the meanings of those
>> words are ultimately determined by the axioms, not by the choice
>> of primitives.
>>
>> Note to RF: Yes, the patterns of words in NL text impose strong
>> constraints on the meanings of the words. That is important for
>> NLP, but more explicit spec's are important for computer software.
>>
>> John
>>
>>
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