SUO: Re: Information, Inquiry, Logic, Signs
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More Thread On Information, Inquiry, Logic, Signs:
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> Continuation Of Previous Note:
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> > SUO, S.A., FOO,
> >
> > I would like to address some of the unities that are manifested
> > among the concepts of information, inquiry, logic, and signs,
> > along with the brands of theoretical integration that are
> > possible among their respective efforts of theory,
> > that is to say, at least, their up-until-now
> > largely several and mostly separate efforts
> > toward any sort of an adequate theory.
> >
> > "Hello there, Some Of The Unities ..."
> > < Art Carney or Jackie Gleason voice,
> > | I cannot remember which right now >.
> >
> > Now, I am trying to make this address positive, that is to say,
> > in the sense of addressing the conventionally-called "positive"
> > side of each concept in the list, but just in case the bearing
> > on some of our previous discussions is not clear, I probably ought
> > to mention that this address also addresses, albeit in an indirect way,
> > the disintegrities that we have noticed among our concepts, our terms,
> > and our "rough and ready" but a bit "faute de mieux" theories, such as
> > they are, of ambiguity, discrepancy, disparity, divergence, diversity,
> > ellipticity, enthymemacity, equivocality, fuzziness, generality, hyperbole,
> > indefiniteness, indefinity, indeterminacy, metaphor, mutagenicity, parabole,
> > uncertainty, vagueness, and all the rest of their vast and very unruly clan.
>
> I forgot to mention digression, dispersion, dissipation, ...
>
> When it comes to the Topic of Logic, I will revert to my customary practice
> of beginning at the bottom and trying to work up, that is, of starting with
> the humble preliminaries of propositional calculus, whose name is legion --
> "If you can't make it there, you can't make it anywhere" -- and rather than
> following Frege's model-theoretic practice of saying that Truth and Falsity
> are objects that may be denoted by propositional signs under a contemplated
> choice of interpretation -- at least, that is how I remember it from many
> years back, but I am just about to make my periodic review of the sources
> on this question, any day now -- I will follow what I take to be the more
> preferable practice of saying that a propositional sign denotes a certain
> form of matter, a pattern of content, a realization of potential, in short,
> a function of the type f : B^k -> B, for suitable k and where B = {0, 1},
> the latter interpreted as logical values: 0 = F = Falsity, 1 = T = Truth.
>
> In order to make the connection between Logic and Information,
> I observe that the function that a propositional sign denotes is
> just the "square-wave" or the "step-function" approximation to the
> sorts of "frequency distributions" or "probability densities" that
> we find so basic to our present-day "Theory Of Information" (TOI).
>
> Here is a sufficient example to convey the basic picture:
>
> For the proposition AB, that is, A&B, we have
> AB depicted as a function of type f : B^2 -> B,
> and AB illustrated by means of a venn diagram:
>
> 1 ^ o--o
> | |%%|
> | |%%|
> 0 o------o o-------->
>
> AB = 10, 11, 01, 00
>
> o----------------o
> | /\ /\ |
> | / \/ \ |
> | / /\ \ |
> | / A /%%\ B \ |
> | \ \%%/ / |
> | \ \/ / |
> | \ /\ / |
> | \/ \/ |
> o----------------o
>
> It is my belief, and unless the most of you already agree
> it will become my task to argue later on, that a complete
> understanding of information theory obviates the need for
> any further "theory of fuzziness" (TOF), that is, for all
> practical purposes, anyway. Of course, in asserting this,
> I am anticipating my usage of Peirce's qualitative version
> of TOI, which he derived from purely logical considerations,
> in effect, from his earliest ideas about the theory of signs,
> long before he came to devise a logarithmic measure for the
> quantitative analysis of information content. And you can
> look it up!
I think that it might be a good idea to draw a few more pictures,
and also to notice a slightly different manner of executing and
interpreting what is roughly the same genre of iconography.
Here is a mini-review of the pictures for the propositional expression "A and B",
which I am more accustomed to write in the form of an indicated product, as "AB":
1 ^ o--o
| |%%|
| |%%|
0 o------o o------>
AB = 10, 11, 01, 00
o----------------o
| /\ /\ |
| / \/ \ |
| / /\ \ |
| / A /%%\ B \ |
| \ \%%/ / |
| \ \/ / |
| \ /\ / |
| \/ \/ |
o----------------o
The legend "AB = 10, 11, 01, 00" lists the four instantiations or
interpretations of the conjunction AB, and the shaded regions that
lie under the graph of the function AB : BxB -> B, and again inside
the cell given by <A, B> = <1, 1> of the venn diagram, respectively,
indicate the loci where AB = 1.
This form of picture treats the points in the domain of the proposition
as if they were already "pre-coded" by their boolean coordinates in B^2,
but we are usually interested in more arbitrary domains of propositions.
For instance, suppose that we have in mind a space X, say of things, or
times, or spacetime loci, over which given propositions are evaluated.
By way of formulating a concrete example, let X be taken as a time dimension,
and let x in X be a real number, where X = R. In such an interpretive case,
the proposition-signs "A", "B", "AB", and "(A(B))" indicate "times when
the denoted propositions A, B, AB, and (A(B)), respectively, are true".
To see it in pictures:
----------------------------
Proposition A
1 ^ o------o
| |%%%%%%|
| |%%%%%%|
0 o--o o--------> x
AB = 10, 11, 01, 00
o----------------o
| /\ /\ |
| /%%\/ \ |
| /%%%/\ \ |
| /%A%/%%\ B \ |
| \%%%\%%/ / |
| \%%%\/ / |
| \%%/\ / |
| \/ \/ |
o----------------o
----------------------------
Proposition B
1 ^ o------o
| |%%%%%%|
| |%%%%%%|
0 o------o o----> x
AB = 10, 11, 01, 00
o----------------o
| /\ /\ |
| / \/%%\ |
| / /\%%%\ |
| / A /%%\%B%\ |
| \ \%%/%%%/ |
| \ \/%%%/ |
| \ /\%%/ |
| \/ \/ |
o----------------o
----------------------------
Proposition AB
1 ^ o--o
| |%%|
| |%%|
0 o------o o--------> x
AB = 10, 11, 01, 00
o----------------o
| /\ /\ |
| / \/ \ |
| / /\ \ |
| / A /%%\ B \ |
| \ \%%/ / |
| \ \/ / |
| \ /\ / |
| \/ \/ |
o----------------o
----------------------------
Proposition (A (B))
1 ^ o---------o
| |%%%%%%%%%|
| |%%%%%%%%%|
0 o------o o-> x
AB = 10, 11, 01, 00
o----------------o
|%%%%%/\%%/\%%%%%|
|%%%%/ \/%%\%%%%|
|%%%/ /\%%%\%%%|
|%%/ A /%%\%B%\%%|
|%%\ \%%/%%%/%%|
|%%%\ \/%%%/%%%|
|%%%%\ /\%%/%%%%|
|%%%%%\/%%\/%%%%%|
o----------------o
----------------------------
If this is starting to sound a little bit familiar,
it may be because the relationship between the two
kinds of pictures of propositions, namely:
1. Propositions about things in general, here,
about the times when certain facts are true,
having the form of functions f : X -> B,
2. Propositions about binary codes, here, about
the bit-vector labels on venn diagram cells,
having the form of functions f' : B^k -> B,
is an epically old story, one that I, myself,
have related one or twice upon a time before,
to wit, at least, at the following two cites:
http://ltsc.ieee.org/logs/suo/msg01251.html
http://ltsc.ieee.org/logs/suo/msg01293.html
There, and now here, once more, and again, it may be observed
that the relation is one whereby the proposition f : X -> B,
the one about things and times and mores in general, factors
into a coding function c : X -> B^k, followed by a derived
proposition f' : B^k -> B that judges the resulting codes.
| f
| X ----> B
| \ ^
| <x<1>, ..., x<k>> \ / f'
| \ /
| .
| B^k
|
| You may remember that this was supposed to illustrate
| the "factoring" of a proposition f : X -> B = {0, 1}
| into the composition f'(c(x)), where c : X -> B^k is
| the "coding" of each x in X as an k-bit string in B^k,
| and where f' is the mapping of codes into a co-domain
| that we interpret as t-f-values, B = {0, 1} = {F, T}.
Enough For The Present,
Jon Awbrey
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