On Feb 2, 2009, at 8:20 PM, Len Yabloko wrote: (01)
> Pat,
>>
> LY>> No. Earlier in this thread
>http://ontolog.cim3.net/forum/ontolog-forum/2009-01/msg00523.html
>>> I already proposed category in which extensions are objects and
>>> intensions are morphisms.
>>
> PH>Hmm. Im afraid this simply does not make sense to me. First, we
> have
>> to find out what you mean by 'extension' and 'intension'. In my
>> language these are usually used in the adjectival mode, to refer to
>> ways of understanding relations.
>
> This is the most general definition I could find
>http://en.wikipedia.org/wiki/Intension
> "Intension refers to the possible things a word or phrase could
> describe. It stands in contradistinction to extension (or
> denotation), which refers to the actual things the word or phrase
> does describe" (02)
I see what they are trying to say, but its not a very good definition.
The trouble is, the set of possible things is still a kind of
extension. It would be more accurate to say that the intension of a
phrase is its meaning or sense (in Frege's terminology) and the
extension is the set of things which are described by the phrase. (03)
>
> PH>The extension of a relation is simply
>> a set (of tuples, those of which the relation is true). The
>> extensional view of relations identifies a relation with its
>> extension; in contrast, the intensional view of relations treats a
>> relation is an abstract object sui generis, one which has an
>> associated extension but is not identified with it.
>>
>> Now, in this picture there is no such entity as an intension,
>> speaking
>> strictly: there are simply relations, understood intensionally. So I
>> am left wondering what you can possibly mean by morphisms being
>> intensions.
>
> Put aside relations for a moment. Consider extention of an object to
> be its physical 4D extent. (04)
OK. (05)
> In absence of any observer the 4D extent of any object is infinitely
> large (06)
Nonsense. What do extents of objects have to do with observers? (07)
> (even if has a finit time dimension) because one object can not be
> separated from the rest of the universe without observer 'drawing'
> physical boundaries (a.k.a 'perception'). (08)
I presume you mean that until an observer distinguishes one thing from
another, there are no 'objects'. That claim is arguable, but I don't
agree, myself. (09)
> Now, if you consider a set of observations, you can divide it into
> any number of sub-sets (aka 'projections'). (010)
I really have no idea what you are saying here, but it sounds wrong.
(Why any number of subsets? Are you presuming that all sets of
observations are infinite?) (011)
> On the other hand, according to definition I quoted above intension
> is a set of all possible 4D extents mapped to a phrase (or any other
> sign). (012)
Well, no, its a set of possible objects, not a set of 4-d extents. The
objects might be abstract, for example (think of the numbers.) (013)
> Can we now say that these are morphisms:
>
> extenstion <--> projections <--> sign (014)
No. Or at any rate, I see nothing here remotely like a morphism. What
do your arrows mean? (015)
>
>
> I believe this fits the definition of intension which I quoted above.
>
>>
>>> Following the exercise that you kindly provided above I have to show
>>> that the intension composition is associative
>>
>> First you have to define it. What is the composition of two
>> intensions?
>
> Can we consider composition of projections above to be a composition
> of intensions? I think so. (016)
You havnt said what a projection is, let alone what composition means. (017)
>
>
>>
>>> , and that there is a special intension serving as identity. The
>>> later is simply a definition of identity - taking us to the point
>>> where common-sense identity and categorical one meet (as I suggested
>>> at the start of this thread)
>>
>> I fail to see how. In what sense is the common-sense notion of
>> identity an intension? What has it got to do with relations in any
>> sense at all?
>>
>
> Relations map one 4D extent to another. (018)
No, they don't. (019)
> But since relation must have a sign we need to compose projections
> and assign resulting composition to a sign. This looks like a
> categorical product (more specificaly - pullback, I think) (020)
Again, I completely fail to follow you here. It would help if you had
a simple example worked out in detail.
>
>
>>> The former requires me to establish such rules of composition for
>>> intensions which would ensure their associativity.
>>
>> Quite. BUt as a matter of methodology, one would expect that CT would
>> be of most value when the morphisms of a proposed category arise
>> naturally from some natural motion of mapping, rather than having to
>> invent an artificial notion of morphism composition purely in order
>> to
>> have something you can call a category.
>
> Do projections qualify as natural transformations? (021)
I have no idea what you mean by 'projection'. (022)
>
>
>>
>>> Let me borrow from someone else specialty and venture to submit to
>>> you that associativity of intensions roughly corresponds to context-
>>> free grammar for symbolically grounded language.
>>
>> You have now completely lost me. Grammars describe syntactic
>> structures. What have grammars (context-free or not) have to do with
>> composition of mappings?
>>
>
> Grammars operate on phrases (more generally signs) - right? (023)
Grammars specify wellformedness of syntactic structures, which use
signs. (024)
> According to general definition - intenstion as mapping from 4D
> extents to a sign. (025)
That is not the general definition you cite above. Read it again. (026)
> Therefore, composition of signs can be mapped to composition of
> intensions. (027)
That doesn't follow at all, even if what you just said made sense. (028)
> Context-free grammar (if I am not mistaken) defines a way to compose
> signs in which the order of composition does not change the result. (029)
I'm afraid you appear to simply have no idea what the words mean. That
isn't what context-free grammar means. (030)
> (031)
Pat (032)
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