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## Re: [ontolog-forum] Triadic Sign Relations

 To: "[ontolog-forum] " "Schiffel, Jeffrey A" Thu, 19 Aug 2010 14:40:22 -0500 <8B6A80AC2A6F7D4ABBEB7BFD3C3B8E1E65FCB617B3@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
 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.   Regards,   -- 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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