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Re: [ontolog-forum] Triadic Sign Relations

To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Rich Cooper" <rich@xxxxxxxxxxxxxxxxxxxxxx>
Date: Fri, 20 Aug 2010 11:08:32 -0700
Message-id: <20100820180838.EF474138D02@xxxxxxxxxxxxxxxxx>

Hi Jeffrey,


Thanks for your inputs.  When I studied lattices, the “T” and upside down “T” weren’t used, though the supremum(?) and the infimum(?) names were mentioned for curiosity purposes, but were seldom actually used.  They do make nice starting points for algorithms that bracket parts of the lattice which meet other constraints.  The GLB and the LUB bracket the derivation of any point in a lattice (same terminology) was common notation in my professors’ terminology, but the topmost and bottom most nodes weren’t called T and upside down T.  I don’t particularly remember what they were called in school, but the names weren’t particularly significant for the algorithms that use a lattice for representing things.  


Perhaps the difference is in engineering versus math studies; mathematicians often use single letter names and use the same letters for many completely different things.  In engineering, there is more of a structured vocabulary referring to the properties of the thing being elaborated.  


I prefer GUT (greatest upper terminal) and LLT (least lower terminal).  Then you can claim that all classes in the lattice have been extruded from the GUT (headed down the lattice) and LLT (headed up the lattice).  


Remember that the lattice, and all its teachings and similes, is merely a tool for us to reason about things that have the lattice.  


But in developing software implementations of these ontological representations, its important to have names that are unambiguous inside the software, and “T” is too easy to mix up with other things, while upside down “T” is not even a consideration.  


Thanks for your insights,




Rich Cooper


Rich AT EnglishLogicKernel DOT com

9 4 9 \ 5 2 5 - 5 7 1 2

From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Schiffel, Jeffrey A
Sent: Thursday, August 19, 2010 12:40 PM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Triadic Sign Relations


Hi, Rich,


I like the use of 'T' because of its association with lattices, and therefore with ontologies.


If s is the largest element smaller than (or equal to) both x and y, and s is greater than (or equal to) t, then it is the greatest lower bound of x and y. It is called the infimum. Similarly, if s is the smallest element greater than (or equal to) both x and y, and s is less than or equal to, then it is the least upper bound. Call the greatest lower bound the infimum and the least upper bound the supremum of the pair x and y. Then a lattice is a partially ordered set (U, 'less than or equal') in which every pair of elements (x, y) in U has a sup and an inf in U.


Lattices are the basis for many ontologies. A common notation for the top supremum of the entire lattice is T. The lowest infimum is then notated as an inverted T. (This, incidently, is the notation used in formal concept analysis.


So I like T, but your exception is noted.




-- Jeff Schiffel



Rich Cooper wrote,

Hi John,


I kinda object to the use of "T" because it conflicts with the extremely long history of dynamic systems, discrete time systems, even electronics which is often spread out in a frequency v time plane.  Wavelets, Fourier analysis, control systems, optimal controls, discrete sampled systems, and zillions of other engineering marvels use “T” and have for centuries.  It seems unnecessary to displace it now. 


 < snippage > 


-----Original Message-----
From: John F. Sowa


On 8/17/2010 6:29 AM, Rich Cooper wrote:

> I interpret “comprehension” in this passage as referring to the degree

> of specialization of a “term”, or symbol.


 < snip > 


I'd also like to relate this discussion to the term used for the

top of a type hierarchy.  My preferred term is the symbol T for top,

because it avoids all possible confusion with words like 'thing'

or 'concept'.  If anybody wants a pronounceable word, I recommend

'entity' because it is a technical term that avoids all kinds of

pointless controversy about whether an event or a property is a thing.


The crucial point about T (or whatever else you want to call it) is

that it has maximum extension:  The corresponding predicate T(x)

is true of every and any x that anybody can imagine.  There is

one and only one axiom that defines the predicate T(x):


    For every x, T(x).


But T also has the minimum possible comprehension (or intension):

zero.  That single axiom, which is true of everything, says nothing

about anything.  T has no attributes or properties of any kind.


 > I interpret "comprehension" in this passage as referring to the

 > degree of specialization of a "term", or symbol.


It's better not to try to explain it.  Just think in terms of

the logic:  The comprehension (or intension) is determined by

the differentiae (monadic predicates) that define it:  adding

more differentiae makes a term more specialized, and deleting

differentiae makes it more generalized.  If you erase all the

differentiae, you get T. 


< remainder snipped >

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