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
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
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 AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Schiffel, Jeffrey A
Sent: Thursday, August 19, 2010
Subject: Re: [ontolog-forum]
Triadic Sign Relations
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,
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.
From: John F. Sowa
On 8/17/2010 6:29 AM, Rich
> I interpret
“comprehension” in this passage as referring to the degree
> of specialization of
a “term”, or symbol.
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
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):
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
< remainder snipped