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Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, "Len Yabloko" <lenya@xxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Tue, 3 Feb 2009 01:45:28 -0600
Message-id: <27513F61-2F12-49E9-8F1C-B62E7AA6C175@xxxxxxx>

On Feb 2, 2009, at 8:20 PM, Len Yabloko wrote:    (01)

> Pat,
> LY>> No. Earlier in this thread 
>>> 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 
> "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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