At the risk of borrowing much of what Heylighen has said in that paper,
another formulation might be something like... (01)
Any self-describing language needs:
A symbol generating function;
A tolerant subsumption generating function that can recurse (i.e. take
prior subsumptions as inputs);
All operations embedded in time; (02)
(By tolerant subsumption I mean something like the 'If it quacks it's a
duck' approach.) (03)
This is because a self-describing language would need as a minimum to be
able to generate a symbol to label a class's extension. (04)
Self-description then occurs as:
--time-->
S = {Ta}, generate symbol x, x subsumes S (temporally) as a label for
the set yet is (at any other time) a member of the extension {Ta} of set S.
To say that x is both the label and a member _at the same time_ is to
collapse the subsumption into nullity, and in human thought that isn't
very useful (we would think of nothing all day). (05)
In operation:
Take an unknown U1, subsume and label the singleton subsumption with a
symbol A.
Take an unknown U2, subsume and label the singleton subsumption with a
symbol B.
Test subsumption of A and B together (A as the measure of B, for example):
if successful (they are alike in some way), label the subsumption
as C.
if not successful (insufficiently similar), label the failure of
subsumption as C (N.B. cunning move! C becomes the exclusive label).
Take an unknown U3, subsume and label the singleton subsumption with a
symbol G.
Test X = subsmp(G, A); Y = subsmp(G, B); subsmp(X, Y), whose outcome may
or may not yield an existing symbol; etc. That is, it is possible that Y
== X and that therefore in actuality X = subsmp(G, B) too and subsmp(X,
X) => subsmp(X) => X (see below)
Take further unknowns and iterate in all directions. (06)
Any single U fed back into the subsumption generating function will
generate the same symbol again. i.e. subsmp(U1) => A.
Also, subsmp(A) => subsmp(subsmp(U1)) => A.
And subsmp(A,B) == subsmp(B,A). (07)
Time is a property of the subsumption.
In this formulation, two subsumptions that have the same inputs except
for time of input would remain distinct because they were created at
different times. Nevertheless, they could be subsumed under any
subsumption that was tolerant of the time discrepancy.
That complies with Heylighen's bootstrapping axiom. (08)
The 'space of axioms' mentioned previously might then reduce to
something like the space of qualifications of subsumptions.
(A reduction of Heylighen's low-level heuristic ontological basis to
subsumptions.) (09)
Tolerance in subsumption might also imply that for a set of languages
S = {A, B, C} description could be transitive without the languages
being identical in symbol or subsumption structures.
I.e.
--time--->
A <- B <- C <-A ... (010)
where A <- B means B describes A. (011)
--
Peter (012)
Peter P. Jones wrote:
> Or maybe an enhanced generalisation of such languages might be...
>
> There is a space of axioms, not all of which are operative at once.
> A fundamental axiom is that the system can change its choice of axioms.
> The choice of axioms conditions the possible generative space.
> The choice of axioms also conditions the space of possible subsumptions.
> Possible subsumptions will vary in both quantity of permissiveness and
> quality of motivation. (i.e. taken over an entire corpus there might be
> an average of permissiveness, but quality would not have an average;
> rather there would be a space of possible qualities: anything from
> colour to political domination.)
> The time dimension is crucial to the idea of self-subsumption under a
> descriptive metaphor or metaphoric term - maintaining the class
> description vs. extension distinction.
>
> Peter P. Jones wrote:
>> Hi,
>>
>> Surfing through the excellent set of links that followed from Jack's
>> posting about ThoughtSticker (http://www.pangaro.com) I came across
>> Francis Heylighen's paper 'Bootstrapping knowledge representations: from
>> entailment meshes via semantic nets to learning webs' (2001).
>> In the second paragraph of section 2 there is the passage:
>>
>>
>> <quote> [...host of conceptual and practical problems with using
>> 'reflection-correspondence' epistemology.]
>> The most pressing ones center around the origin and nature of the
>> mapping from reality to its symbolic representation. Since a cognitive
>> system has no access to reality (Kant's “Ding an Sich”) except through
>> perceptions—which are already internal models—, how can it ever
>> determine whether it uses a correct mapping? Another
>> formulation of this difficulty is the symbol grounding problem
>> (Harnad, 1990): how are the symbols, the elements of the model,
>> “grounded” in the external reality which they are supposed to
>> represent? This problem cannot be solved within the model itself. This
>> follows from the “linguistic complementarity” principle (Löfgren,
>> 1991), which
>> generalizes classic epistemological restrictions such as the theorem
>> of Gödel or the Heisenberg indeterminacy principle. It states that no
>> language can fully describe its own description or interpretation
>> processes. In other words, models cannot include a
>> representation of the mapping that connects their symbols to their
>> interpretations.
>> </quote>
>>
>> I wonder whether this is the case when mappings are vague or 'fuzzy'.
>> If, as Nietzche pointed out, the 'land-grab' of meaning in a language
>> takes place by deliberately using metaphor as the enabler of
>> assimilation (making-similar, apropriation), and the metaphors
>> 'contest' over the terrain in the optimisation of their use, without
>> that optimisation ever needing to be perfect because the process is
>> fundamentally fuzzy, then does the self-description need to be
>> 'complete'?
>> In a fuzzy model, a sort of completeness (an open, expectant one)
>> might nonetheless be compatible with the vagueness. It could probably
>> be expressed more formally as something like the interaction between
>> (as yet unsatisfied, ever-expectant) 'subsumption potential' defined
>> as a fundamental property of a language, and the generative potential
>> of the same language. The multiplicity and the vagueness are the very
>> operation of metaphor, but there might be an optimal balance between
>> the two whereby linguistic self-description becomes possible.
>> Perhaps natural languages embody that optimisation.
>>
>> Just a thought.
>>
> (013)
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