ONT Re: Limited Mark Universes
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
LMU. Note 5
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Peirce came into this arena with a question about "how science works",
and he took off from a standard sort of Kantian platform that permits
you to get started by just going ahead and accepting the evident fact,
the apparent phenomenon, or the provisional hypothesis that science,
as we do it, but not necessarily as we know it, does work, and then
to move on to the next question, to wit: What are the conditions
for the possibility of science working?
I walked into this theatre with a problem about language learning, and
had very little acquaintance and a whole lot of wrong ideas about Kant.
But there is a natural analogy between the task of scientific knowing
and the task of language acquisition, as Newton clearly recognized in
the guise of his metaphor about science as the decoding of nature's
cryptographic laws.
So I will start out by explaining a very simple sort of language acquisition task.
One of the first obstacles that we run into is this huge gulf between all of the
realistic examples and all of the sorts of examples that one can discuss in the
beginning, the fact that all of the motive settings are very complex indeed
and all of the simple set-pieces are very simple indeed.
So I will beg you to use your imagination.
Okay, enough preamble.
An "alphabet" (or a "lexicon") is a finite set A.
The "kleene star" A* of the alphabet A
is the set of all finite sequences that
can be formed out of the elements of A.
We call these "strings" or "sequences".
Note that A* includes the empty string.
A "formal language" L over the alphabet A is an arbitrary subset of A*,
thus L c A*. Depending on the setting, the strings or sequences of L
are called "L-words", "L-strands", or "L-sentences", in one locution,
or "words of L", "strands of L", or "sentences of L", in another.
Whenever there is only one language under discussion, or when
it is otherwise clear, the obvious abridgements may be used.
Enough for today ...
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o