On Oct 6, 2009, at 8:00 AM, Gunther Schadow wrote: (01)
> Matthew and John:
>
> Is there a way to get that back to the actual UoM
> discussion? Could you phrase the disputed issue
> in 2 succinct contrasting statements that are about
> our subject matter at hand?
>
> As for the sheep example Mathew give below, I am
> confused. I may be just a doctor and software engineer,
> not a chemical engineer that I would know much about
> the fine points of your dispute, but to me it seems
> that you would not make 2 definitions for sets of
> sheep (two-eyed and four-legged) if you were not
> interested in a difference here. And if you are interested
> in a difference, then the two sets are not the same
> even though their extension may be. But presumably
> you didn't know that before you defined your two sets
> and then determined their extensions (yes I like to
> drag epistemology into ontology). But still I don't
> know quite what's the point? (02)
The point is what ontological discipline you use for deciding when
properties or relations are identical. Speaking strictly
mathematically, a relation is (usually assumed to be, or to be
mathematically modeled as) a set of pairs <relata, relatee>, and a
predicate or property is modelled as a set, the set of things that
have the property or that satisfy the predicate. For those of the
extensionalist persuasion, these set-theoretic definitions are taken
to be the absolute identity criterion, so that relations are treated
as a species of set, and have similar identity criteria. So if two
different ways of specifying a relation or property turn out to define
the same set or set of pairs - that is, if the very same things stand
in the relationship to one another, under the two descriptions - then
on this view, these both define the same relation or property.
Classical examples include being human = being a hairless biped. The
other, intensional, view wants to give relations and properties a more
robust notion of identity, and treats the mathematical set-theoretic
account as being only a mathematical model. On this view, the property
of being human is one thing, and the property of walking on two legs
and having no body hair is another, and the accidental fact that they
happen to coincide on this planet right now is not sufficient grounds
to declare them to be identical *as properties*. After all, they
*could* be different: one can imagine a non-human hairless bipedal
creature. Maybe Neanderthals were examples. OK, lets not get into that
debate (which hasnt been settled in the last two millennia), just
observe that the difference of opinion exists. However, it does have a
(small but not invisible) practical consequence for ontology
reasoners, because you get different logics in the two cases. RDF,
RDFS and OWL-Full (and ISO Common Logic) all have a semantics based on
the second, intensional perspective: OWL-DL and classical FOL both are
based on the first, extensionalist view. The practical difference is
that the intensional logics are slightly weaker than the
extensionalist ones, and hence somewhat easier to implement. (One gets
the extensionalist logics by adding a lot of conditional equations to
an intensional logic.) (03)
BTW, there is no such thing as an intensional *set*. Sets just are
extensional. In fact, sets are the extents that give rise to the
adjective 'extensional'. (04)
Regarding the sheep, the point is not that it is a good idea to make
two definitions, but rather that two different ways of describing
something might just emerge from reasoning. How much effort should a
reasoner spend, trying to check whether two different descriptions
which just might describe the same relation, do or do not in fact
describe the same relation? The intensional logics typically have a
short answer: either no effort, or some very small amount (for
example, as John points out citing Church, just do a lambda-reduction:
a very fast, cheap operation.) For the extensionalist, however, this
is a real problem, as the general task of determining extensional
identity is NP-hard. And this issue does arise in practice, when
reasoners are required to check subsumption relationships between
ontological specifications. Part of the motivation for using
description logics (as in OWL-DL) is to restrict expressivity in order
to keep this problem manageable. (05)
However, all that said, I agree that this debate is off-topic right
now in this forum. IMHO, of course, gentlemen. (06)
Pat (07)
> regards,
> -Gunther
>
> Matthew West wrote:
>> Dear John,
>>
>> We are way off topic here. None of this is remotely relevant to
>> maximum
>> allowable working temperatures or any idea of direct or indirect
>> properties. I have only been bringing engineering practice to the
>> table
>> here and leaving any ontological bias elsewhere (for those that do
>> not
>> know I have a BSc and PhD in Chemical Engineering, and 8 years
>> practice
>> as a Chemical Engineer working on a refinery doing every kind of job
>> that a Chemical Engineer might do from plant design to plant
>> operations,
>> safety and environment, process control, I even looked at the
>> refinery
>> sewage system).
>>
>> I am also somewhat disappointed about the way you keep putting
>> words in
>> my mouth that were never there. I do not talk about extensional
>> definitions, but extensionalism as an identity basis. There is a
>> difference, which you seem to keep missing.
>>
>>> MW> You are conflating two usages of intensional, so it is not I
>>>> who is suffering from a philosophers disease.
>>>
>>> First of all, I apologize for using the word 'disease'.
>>
>> MW: Accepted.
>>
>>> But
>>> the notion of intensional definition is so fundamental to logic,
>>> knowledge representation, ontology, and other areas that it will
>>> inevitably be used in multiple ways.
>>
>> MW: I don't have a problem with there being intensional
>> definitions. The
>> question is what do they define?
>>
>>> For a precise definition of the distinction between intensional
>>> and extensional definitions, I have so often cited a short
>>> discussion by Alonzo Church that I put it on my web site:
>>>
>>> http://www.jfsowa.com/logic/alonzo.htm
>>>
>>> Following are the relevant quotations from that article:
>>>
>>> 1. "A function is a rule of correspondence by which when anything
>>> is given (as argument) another thing (the value of the
>>> function
>>> for that argument) may be obtained. That is, a function is an
>>> operation which may be applied on one thing (the argument) to
>>> yield another thing (the value of the function)."
>>>
>>> 2. "The foregoing discussion leaves it undetermined under what
>>> circumstances two functions shall be considered the same.
>>>
>>> The most immediate and, from some points of view, the best way
>>> to settle this question is to specify that two functions f and
>>> g
>>> are the same if they have the same range of arguments and, for
>>> every element a that belongs to this range, (f a) is the same
>>> as
>>> (g a). When this is done we shall say that we are dealing with
>>> functions in extension."
>>>
>>> 3. "It is possible, however, to allow two functions to be
>>> different
>>> on the ground that the rule of correspondence is different in
>>> meaning in the two cases although always yielding the same
>>> result
>>> when applied to any particular argument. When this is done we
>>> shall say that we are dealing with functions in intension."
>>
>> MW: Well on the one hand I can quite agree that you can decide that
>> two
>> functions are different, even if they are extensionally the same. I
>> choose not to, but I do not require that others make the same
>> choice. On
>> the other hand, the suggestion here is that the meaning is different,
>> and here I think there is more of a problem. Meaning as I
>> understand it
>> is what links our terms to reality, and if two terms or definitions
>> point to the same thing, I am struggling as to how the meaning can be
>> different. Could you explain that specific point please?
>>> 4. "The notion of difference in meaning between two rules of
>>> correspondence is a vague one, but, in terms of some system
>>> of notation, it can be made exact In various ways."
>>
>> MW: Could you be explicit about which ways please?
>>> 5. "We shall not attempt to decide what is the true notion of
>>> difference in meaning but shall speak of functions in
>>> intension
>>> in any case where a more severe criterion of identity is
>>> adopted
>>> than for functions in extension. There is thus not one notion
>>> of function in intension, but many notions; involving various
>>> degrees of intensionality."
>>
>> MW: I would probably understand this if you could answer my questions
>> above.
>>> 6. Then Church goes on to define the lambda calculus and its
>>> method
>>> of "conversion" as his criterion for equality by intension:
>>> two defining rules have the same "meaning" by if either one
>>> can be converted to the other by method of the lambda
>>> calculus.
>>
>> MW: Well I'm not an expert at lambda calculus, so could you explain
>> how
>> this works for equiangular and equilateral triangles please?
>>
>>> This definition is more precise than most, but it follows the same
>>> approach as most: an intensional definition is specified by some
>>> rule (declarative or procedural) and the application of that rule
>>> determines the set of instances, the extension. For small finite
>>> sets, it's possible to generate the complete extension. But for
>>> large sets, such as the set of all people, cows, or bacteria on the
>>> planet earth, a definition by extension is impossible in practice.
>>
>> MW: I think this is one of the confusions. I am concerned about
>> identity, not definition. So I am happy that two sets are the same if
>> you can show that the membership is necessarily the same (if one rule
>> that defines the membership can be converted to another as you state
>> above would probably do the trick for me). It is not simply a
>> matter of
>> always listing all the elements, and then trying to work out if it
>> is an
>> interesting list.
>>
>> MW: On the other hand if you come across two sets with intensional
>> definitions and they have the same membership, then I will deem
>> that is
>> a proof that the two definitions are equivalent. I think there is a
>> problem if you do not, since if you have two terms that point to the
>> same thing, then you have to explain how the meaning is different,
>> which since this is linked to what things point to is I believe
>> problematic.
>>> For infinite sets, sets in the future, or sets in possible worlds,
>>> no definition by extension is possible even in theory (except
>>> perhaps
>>> by an appeal to God, but God doesn't usually answer such
>>> questions).
>>
>> MW: If I held the view you claim I do this would be true. But I
>> don't,
>> so it isn't.
>>> MW> If you are claiming that maximum allowable working temperature
>>> is
>>>> a subtype of temperature, it seems perfectly reasonable to me.
>>> It
>>>> is not different from asking "How do I know which pieces of
>>> equipment
>>>> are pumps?" Is that a meaningless question?
>>>
>>> It is very different. You can determine whether an object is a
>>> pump
>>> by examining it and determining whether its parts and the way they
>>> move would be capable of pumping some liquid.
>>>
>>> But there are some types for which the instances cannot be
>>> determined
>>> by looking at them. In my 1984 book, I called them "role types".
>>> Examples include father, wife, sister, employee, author, student...
>>>
>>> For each of those types, there exists some relation to something
>>> external to the individual in question. No examination of the
>>> individual can determine whether or not it is an instance of
>>> that type. The only way to determine the role is by some
>>> external evidence, such as a DNA test of another individual,
>>> a marriage license, a birth certificate, etc.
>>>
>>> The maximum temperature is similar: it's a role type that can
>>> only be determined by something external. As Pat said, look
>>> at the specifications.
>>
>> MW: You make my point for me. Presumably you do not think that role
>> types are the same as the base types. That is all I am trying to
>> say (by
>> analogy at least).
>>> JFS>> To preserve a semblance of consistency, Matthew was forced to
>>>>> adopt not just a four-dimensional ontology that treats
>>> extensions
>>>>> in an unobservable future as if they were just as concrete as
>>>>> anything observable in the present, but also sets in an
>>> infinity
>>>>> of purely imaginary possible worlds.
>>>
>>> MW> You try to make it sound as if this is something I just thought
>>> up,
>>>> whereas it is a quite standard approach....
>>>
>>> Many people, including me, are quite happy with a 4-D ontology.
>>> But
>>> combining a 4-D ontology with purely extensional definitions is
>>> most
>>> definitely *not* standard.
>>
>> MW: I keep trying to say, I am talking about IDENTITY by
>> extension, not
>> DEFINITION by extension!!! There is a difference.
>>
>>> I am even happy to talk about sets of
>>> extensions in a 4-D universe. But like most logicians from
>>> Aristotle
>>> to Church and beyond, I would use the intensional definition as
>>> *primary* and the extension as something that is determined by the
>>> intension.
>>
>> MW: I am quite happy that an extension might be determined by an
>> intension, but when two definitions give the same extension, I will
>> say
>> that they define the same thing. See my statements about meaning
>> above.
>>
>>> But even those the extension might be definable as a
>>> theoretical construct, the intension is what we actually use to
>>> determine equality for anything but small finite sets that we
>>> can actually observe.
>>>
>>> MW> ... there is no problem in my constructing plans intentionally.
>>>> That does not however mean that their identity is defined
>>>> intentionally rather than extensionally.
>>>
>>> Please read the quotations by Church above. (And note that we're
>>> talking about intensions with an S, not intentions with a T.)
>>>
>>> Fundamental principle: Intensional definitions are prior. Except
>>> for small finite sets immediately observable, extensions are
>>> *always* determined by the intensions.
>>>
>>> MW> They are intentionally constructed, but their identity
>>>> is extensional.
>>>
>>> That is truly a meaningless quibble. Please read Church.
>>
>> MW: No. It is the root of your misunderstanding.
>>
>> MW: But let me try to find something where we might disagree really.
>>
>> Take the sheep in a field, and I am interested in two sets, one the
>> set
>> of sheep with 2 eyes, the other the set of sheep with 4 legs. As it
>> happens all the sheep have two eyes and four legs, so the sets are
>> the
>> same, and I would say that the two definitions have the same meaning
>> because they point to the same set.
>>
>> Now I would not be surprised if an intensionalist objected that yes
>> these definitions accidentally pointed to the same set, but that it
>> is
>> not necessarily the case that sheep have two eyes and four legs. It
>> is
>> possible that there are one eyed sheep and 3 legged sheep (at least).
>> Therefore these definitions pick out different classes.
>>
>> I would retort that you are changing the rules, you are now
>> quantifying
>> over possible worlds where the sheep in that field do not all have 4
>> legs and 2 eyes, and if you wish to do that, then indeed they are
>> different, but these are different sets than the ones I was
>> interested
>> in because they include those from possible worlds. Whereas above
>> we are
>> restricted to the actual sheep in the actual field in this world.
>>
>> All of which has nothing to do with extensional definitions, and
>> everything to do with extensional identity.
>>
>> Regards
>>
>> Matthew West
>> Information Junction
>> Tel: +44 560 302 3685
>> Mobile: +44 750 3385279
>> matthew.west@xxxxxxxxxxxxxxxxxxxxxxxxx
>> http://www.informationjunction.co.uk/
>> http://www.matthew-west.org.uk/
>>
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>>
>>
>>
>>
>>
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>
> --
> Gunther Schadow, M.D., Ph.D. gschadow@xxxxxxxxxxxxxxx
> Associate Professor Indiana University School of Informatics
> Regenstrief Institute, Inc. Indiana University School of Medicine
> tel:1(317)423-5521 http://aurora.regenstrief.org
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