ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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JA = Jon Awbrey
JC = Jerry Chandler
JC: You propose no less than 9 "languages".
You decline to answer an elementary question with regard to these
nine languages. What bijections exist among these languages?
JA: S_1 has 16 elements.
S_2 has 16 elements.
There are 16! bijections between S_1 and S_2.
If you mean "bijections that preserve the
objects (= denotations of signs)" then
there is only one such bijection.
JC: You propose:
JC, quoting JA:
| but the languages S_4, S_5, S_6 all have
| a countable infinity of expressions,
JC: Are you also proposing that S_1, S_2, and S_3
have "a countable infinity of expressions",
or not?
JA: Here, read this again:
JA, quoting JA:
| The languages S_1, S_2, S_3 have 16 elements each,
| and so they are enumerated in full by the entries
| in the Table, but the languages S_4, S_5, S_6 all
| have a countable infinity of expressions, and so
| but samples of these sign domains, giving one
| of the simpler expressions for each object,
| are shown in the Table.
I see now that I employed an archaic phrase here:
"and so but samples" means "and so only samples".
JC: This is not consistent with the usual meanings of the labels of the columns.
Can you be more specific?
JC: You propose:
JC, quoting JA:
| A sign relation L_j is a subset of a cartesian product O_j x S_j x I_j,
| where O_j, S_j, I_j are its designated Object, Sign, Interpretant Domains,
| respectively.
JC: I do not understand the meaning of the term "Cartesian Product"
under these circumstances.
JA: The "cartesian product" of k sets X_1, ..., X_k, written X_1 x ... x X_k,
is the set of all k-tuples of the form <x_1, ..., x_k> that are formed
by choosing x_j from X_j for j = 1 to k.
JC: In particular, what is an element of the co-domain?
In other words, how is the codomain formed?
JA: Domain and codomain are defined for binary relations, for example, functions.
JC: This is an unusual way to define the domain and codomain.
Above, you asserted that they were defined in an arbitrary manner.
Sorry, I do not know what "above" you are referring to.
JC: Are you certain that you wish to use the term "Cartesian Product" for this relation?
JA: I think so.
JC: Again, this is not the usual way of defining the Cartesian products.
Your process leads to the need for combinatorics in defining
repetitive multiplications.
It is at least one of the usual ways of defining a cartesian product,
and it is sufficient for my purposes in this application. I do not
understand your second sentence.
JC: With regard to language, you assert:
JC, quoting JA:
| Formally, I use "formal language" in the sense of "formal language theory", e.g.:
|
| Denning, P.J., Dennis, J.B., Qualitz, J.E.,
|'Machines, Languages, and Computation',
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
JC: This reference is not available to me.
Would you state the definition of language
that the text presents?
JA: As far as online stuff, MathWorld is gone.
Google gave 4000 hits, mostly course syllabi.
JA: Try:
http://hissa.nist.gov/dads/
http://hissa.nist.gov/dads/HTML/formallangug.html
http://hissa.nist.gov/dads/HTML/language.html
http://hissa.nist.gov/dads/HTML/alphabet.html
JC: Thank you for the references.
JA: Addendum: John Sowa defines and discusses "formal grammar" here:
http://users.bestweb.net/~sowa/misc/mathw.htm
JA: But it is important to recognize "formal language" as the primary concept,
since a single formal language has many formal grammars that generate it.
I am working at Asciifying the section of my dissertation that does this
in the form to which I have now become accustomed, and I will post it
later today on a reference thread.
JC: With regard to crispness, you ask:
JC, quoting JA:
| Do you mean like "well-defined"?
JC: The answer is "No". Science uses crispness frequently even though
the mathematics is not well defined. Crispness is applicable to
empirical observations.
JC: Once again, a very direct question is simply side-stepped.
JA: I was trying to give you a chance to explain whether you had
some other meaning of the word "lattice" in mind than the one
that I was obviously using. As it stands, the question makes
no sense that I can make out, if we are talking about ordered
sets of the specified sorts.
JC: Are you asserting that functions such as sine and cosine are lattices?
Or, are you asserting that the functions sine and cosine are not lattices?
A simple yes or no will inform me of what you are seeking to make crisp.
Let me expand my query to a more general question:
JC: Do the objects you refer to as "languages" admit sine and cosine functions?
JA: Do you mean the functions, or the names of the functions?
JC: Jon, you include the comment:
JC, quoting JA:
| Not sure what you are after here.
JC: THE ANSWER IS STRAIGHT FORWARD.
I SEEK TO GET A CRISP UNDERSTANDING
OF THE MEANING OF YOUR ASSERTION
OF "INQUIRY INTO INQUIRY".
JC: I am well aware of the definition of a lattice and the role they
played in the development of modern mathematics and category theory.
Personally I have not found the concept of or the mathematics of
lattices to be very useful in chemistry, biology, or medicine.
The utility of lattices in quantum mechanics is substantial.
JC: With respect to the nature of your languages and their capacity to
represent sine and cosine functions or chaotic functions, you ask:
JC, quoting JA:
| Do you mean the functions, or the names of the functions?
JC: I mean both.
If you are asking in terms of your definition of languages,
I mean the names of functions (or operations).
JC: If you are asking in terms of the listing of symbols,
I mean the values of the functions you would insert in
your table and how it would relate to your assertions
about "lattices".
JC: Your "inquiry into inquiry", to be meaningful,
must address both simple classical sorts of
questions as well as more sophisticated ones,
should it not?
JA: These questions are just not apt.
JC: When you respond to my questions about the nature
of certain functions in your languages, you state:
JC, quoting JA:
| These questions are just not apt.
JC: Do you mean that my questions about your "inquiry into inquiry" are not apt?
JC: Or, do you mean that your inquiry into inquiry is restricted
to a subset of human inquiry which is of interest to you?
JC: If the latter is the case, how have you isolated the subset of human inquiry
of interest to you from the set of human activities of interest to me?
JC: Or, to phrase the question in an alternative conceptual framework:
When community seeks to communicate by transmitting communications
in a common language, can you communicate the definitions of your
languages and the relationships among them to the community of
interested colleagues?
JA: I mean that you introduced a different subject than the one
that I was talking about and then proceeded to interrogate
me on a topic that I had never said anything about.
JC: Jon: When you assert repeatedly that your work is aimed
at "Inquiry Into Inquiry", some readers took you seriously.
I do not know of any topic that is excluded from inquiry,
do you? I assert that my inquiry into inquiry is just
as valid as yours or any other.
I freely acknowledge that I intended the current subject line,
"Apposite Purposes Of Logical Languages Objectified" (APOLLO)
to be a catch-all for "just about every thing under the sun",
but that doesn't mean that all of our days collapse into one.
I set out taking up a number of tropical topics, one at time,
and I am doing my best to work through one of many passable
logical orderings of them. Sorry if that way of proceeding
seems like funnel vision to you, but I have what some folks
would call a "linear" mind, though not I.
JC: Some of us are now very disappointed with the substance of your posts.
In particular, I would hope you will read the references so that you grasp
the nature of formal language theory as used in computer science and mathematics
and then you can explain how your usage differs from the the usage in computer science.
Yes, well, I am still working at that,
after all these years,
but then, who isn't?
Jon Awbrey
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