ONT Re: Intension & Extension
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JA = Jon Awbrey
RK = Robert Kent
robert,
i kept intending to return to this but the time got away from me.
for my part, i will have to shift to ontology list, but answer
where you will.
RK: Compare Alonzo Church's discussion (below)
of the intension and extension of a concept
with the mathematical (Formal Concept Analysis)
(Rudolf Wille and Garrett Birkhoff) definitions
of the intent and extent of a formal concept on
the middle of page 30 in the IFF Classification
Ontology document:
RK: http://suo.ieee.org/IFF/versions/20020102/IFFClassificationOntology.pdf
RK: As an illustrious example, see the 19 formal concepts (with
explicit intent and extent) in the concept lattice of the
"Living Classification" on pages 74-76 of the same IFF
Classification Ontology document.
JA: I have been looking at the Living Classification Lattice that
you mentioned, and I have a couple of questions just to start:
o-------o-------o-------o-------o--------o-----o------o--------o--------o---------o
| | needs | lives | lives | needs | di | mono | motile | limbed | suckles |
| | water | in | on | chloro | cot | cot | | | young |
| | | water | land | phyll | | | | | |
o-------o-------o-------o-------o--------o-----o------o--------o--------o---------o
| leech | 1 | 1 | | | | | 1 | | |
| bream | 1 | 1 | | | | | 1 | 1 | |
| frog | 1 | 1 | 1 | | | | 1 | 1 | |
| dog | 1 | | 1 | | | | 1 | 1 | 1 |
| hemiz | 1 | 1 | | 1 | | 1 | | | |
| reed | 1 | 1 | 1 | 1 | | 1 | | | |
| bean | 1 | | 1 | 1 | 1 | | | | |
| maize | 1 | | 1 | 1 | | 1 | | | |
o-------o-------o-------o-------o--------o-----o------o--------o--------o---------o
JA: FE = Formal Extent
JA: FI = Formal Intent
JA: 1. Are you using "object" and "instance" as synonyms?
RK: Yes. "Object" is the term used in Formal Concept Analysis, "token"
is the termed used in Information Flow, and "instance" is the term
I have been using in the IFF. All are equivalent in this mathematical
context.
generally speaking, i will stridently resist the "recycling"
of technical terms that the traditions of previous ownership
are not done with yet. so "token" is out.
JA: 2. Can you explain to me in what sense it makes sense to you to say
that {bream, frog} is the FE of the fo.co. "bream", or similarly,
that {maize, reed} is the FE of the fo.co. "maize", as generators?
RK: Within the context of the living formal context (living classification),
"bream", "frog", "maize" and "reed" are four objects (using FCA terminology).
The object "bream" is a living being that needs water, lives in water, is motile
and has limbs. So it generates the formal concept whose intent (comprehension) is
a "limbed motile living being that lives in water". In the (possibly limited) world
of the living formal context, the extent of this concept is just "frog" and "bream".
One can always choose a different model by expanding and/or contracting the formal
context (adding or subtracting either objects or attributes), and of course you
will have maps (called infomorphisms) that link these operations.
I have temporarily forgotten why I asked about this,
so I will dig up my old notes and return to it later.
RK: Of course, it gets more interesting when you apply these techniques to more
interesting classifications (formal contexts). In fact, the classification
of greatest interest here is the *truth* classification where the instances
(FCA formal objects) are model-theoretic structures, the types (FCA attributes)
are sentences of a 1st order language, ...
people who are realists about attributes, intensions, properties, qualities, types, ...,
have trouble saying that attributes are sentences. an attribute is an abstract object,
a sentence is a syntactic thing, and many sentences could denote the same attribute.
it's not that a sign couldn't be a property of a thing, but it's not a general rule.
is this anything to which you can accommodate your thinking?
RK: and the incidence relation is satisfaction. In the infinite matrix
for this classification, put a "1" to the right of a structure M and
directly under a sentence S when the structure is a model for (satisfies)
the sentence M |= S. Then, the formal concept generated by a structure M
has as its intent (comprehension) the theory generated by the structure th(M).
And a sentence S1 labels (generates) a concept below another concept labeled by
sentence S2 when S1 entails S2 (S1 has more models than S2).
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