1. Re: Hermeneutic Equivalence Classes
2. Re: Reductions Among Relations
3. Cosmic
4. Re: Hermeneutic Equivalence Classes
5. Thirdness as Thirdness
6. Thirdness as Thirdness
7. Re: Reductions Among Relations

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Subject: Re: Hermeneutic Equivalence Classes
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Tue, 10 Dec 2002 08:30:08 -0500
X-Message-Number: 1

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HEC. Note 14

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| Leibniz, "Elements of a Calculus" (cont.)
|
| 19. If the concept of the subject, regarded in itself,
| contains the concept of the predicate, then the
| concept of the subject with some addition (i.e.
| the concept of a species of the subject) will
| contain the concept of the predicate. This
| is enough for us, since we do not deny that
| the predicate is in the subject itself when
| we say that it is in a species of it.
|
| So we can say that some metal is liquid in fire,
| properly applied, though we could have stated
| more generally and more usefully that every
| metal is liquid in fire, properly applied.
|
| However, a particular assertion has its uses, as when
| it is sometimes proved more easily than a general one,
| or when the hearer will accept it more readily than a
| general proposition, and a particular proposition is
| sufficient for our purposes.
|
| Leibniz, 'Logical Papers', p. 23.
|
| Leibniz, G.W., "Elements of a Calculus" (April, 1679),
| G.H.R. Parkinson (ed.), 'Leibniz: Logical Papers', pp. 17-24,
| Oxford University Press, London, UK, 1966. (Couturat, 49-57).

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

----------------------------------------------------------------------

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Tue, 10 Dec 2002 10:12:34 -0500
X-Message-Number: 2

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RAR. Note 19

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Relational Composition as Logical Matrix Multiplication (cont.)

With an eye to extracting a general formula, let us now examine
what we did in multiplying the 2-adic relations G and H together
to obtain their relational composite G o H.

Given the space X = {1, 2, 3, 4, 5, 6, 7}, whose cardinality |X| is 7,
we naturally observe that there are |X x X| = |X| x |X| = 7 x 7 = 49
elementary relations of the form i:j, where i and j range over the
space X. Although they might be organized in many different ways,
it is convenient to regard the collection of elementary relations
as being arranged in a lexicographic block of the following form:

1:1 1:2 1:3 1:4 1:5 1:6 1:7
2:1 2:2 2:3 2:4 2:5 2:6 2:7
3:1 3:2 3:3 3:4 3:5 3:6 3:7
4:1 4:2 4:3 4:4 4:5 4:6 4:7
5:1 5:2 5:3 5:4 5:5 5:6 5:7
6:1 6:2 6:3 6:4 6:5 6:6 6:7
7:1 7:2 7:3 7:4 7:5 7:6 7:7

We may think of G and H as being logical sums of the following forms:

G = Sum_ij G_ij (i:j)

H = Sum_ij H_ij (i:j)

The notation "Sum_ij" indicates a logical sum over the collection of
elementary relations i:j, while the factors G_ij, H_ij are values in
B = {0, 1} that are known as the "coefficents" of the relations G, H,
respectively, with respect to each of the elementary relations i:j.

Given all this, we may write out the expansions of G and H as follows:

G = 4:3 + 4:4 + 4:5 =

0(1:1) + 0(1:2) + 0(1:3) + 0(1:4) + 0(1:5) + 0(1:6) + 0(1:7) +
0(2:1) + 0(2:2) + 0(2:3) + 0(2:4) + 0(2:5) + 0(2:6) + 0(2:7) +
0(3:1) + 0(3:2) + 0(3:3) + 0(3:4) + 0(3:5) + 0(3:6) + 0(3:7) +
0(4:1) + 0(4:2) + 1(4:3) + 1(4:4) + 1(4:5) + 0(4:6) + 0(4:7) +
0(5:1) + 0(5:2) + 0(5:3) + 0(5:4) + 0(5:5) + 0(5:6) + 0(5:7) +
0(6:1) + 0(6:2) + 0(6:3) + 0(6:4) + 0(6:5) + 0(6:6) + 0(6:7) +
0(7:1) + 0(7:2) + 0(7:3) + 0(7:4) + 0(7:5) + 0(7:6) + 0(7:7)

H = 3:4 + 4:4 + 5:4 =

0(1:1) + 0(1:2) + 0(1:3) + 0(1:4) + 0(1:5) + 0(1:6) + 0(1:7) +
0(2:1) + 0(2:2) + 0(2:3) + 0(2:4) + 0(2:5) + 0(2:6) + 0(2:7) +
0(3:1) + 0(3:2) + 0(3:3) + 1(3:4) + 0(3:5) + 0(3:6) + 0(3:7) +
0(4:1) + 0(4:2) + 0(4:3) + 1(4:4) + 0(4:5) + 0(4:6) + 0(4:7) +
0(5:1) + 0(5:2) + 0(5:3) + 1(5:4) + 0(5:5) + 0(5:6) + 0(5:7) +
0(6:1) + 0(6:2) + 0(6:3) + 0(6:4) + 0(6:5) + 0(6:6) + 0(6:7) +
0(7:1) + 0(7:2) + 0(7:3) + 0(7:4) + 0(7:5) + 0(7:6) + 0(7:7)

Presenting just the coefficients of G and H on the above plan:

G =

0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 1 1 1 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

H =

0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

These are the logical matrix representations the 2-adic relations G and H.

Jon Awbrey

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Subject: Cosmic
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Tue, 10 Dec 2002 12:54:03 -0500
X-Message-Number: 3

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http://antwrp.gsfc.nasa.gov/apod/astropix.html

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<head>
<title>Astronomy Picture of the Day
</title>
<META NAME="keywords" CONTENT="solar eclipse, moon shadow, Africa">
</head>
<body BGCOLOR="#F4F4FF" text="#000000" link="#0000FF"
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<center>
<h1> Astronomy Picture of the Day </h1>
<p>

<a href="archivepix.html">Discover the cosmos!</a>
Each day a different image or photograph of our fascinating universe is featured, along with a brief explanation written by a professional astronomer. <p>

2002 December 10
<br>
<a href="image/0212/omega_cfht_big.jpg">
<IMG SRC="image/0212/omega_cfht.jpg"
alt="See Explanation. Clicking on the picture will download
the highest resolution version available."></a>
</center>

<center>
<b> M17: Omega Nebula Star Factory </b> <br>
<b> Credit &
<a href="http://www.cfht.hawaii.edu/HawaiianStarlight/CFHT_WWW_Copyright.html"
>Copyright</a>: </b>
<a href="mailto:jcc[…]cfht.hawaii.edu">Jean-Charles Cuillandre</a> (CFHT),
<a href=
"http://www.cfht.hawaii.edu/HawaiianStarlight/English/What_Is_HawaiianStarlight.html"
>Hawaiian Starlight</a>,
<a href="http://www.cfht.hawaii.edu/">CFHT</a>
</center>
<p>

<b> Explanation: </b>
In the depths of the
<a href="ap010923.html">dark clouds</a> of
<a href="ap010813.html">dust</a> and
<a href="ap970430.html">molecular gas</a> known as
<a href="http://www.seds.org/messier/m/m017.html">M17</a>,
stars continue to form.

The similarity to the
<a href="http://www.nadin00.freeserve.co.uk/maths/greek.htm"
>Greek letter</a> capital Omega gives the
<a href="http://oposite.stsci.edu/pubinfo/pr/97/34/af2.html"
>molecular cloud</a> its popular name, but the nebula is
also known as the Swan Nebula, the Horseshoe Nebula, and M17.

The darkness of these
<a href="http://archive.ncsa.uiuc.edu/Cyberia/Bima/GMC.html"
>molecular clouds</a> results from background starlight
being absorbed by thick carbon-based smoke-sized
<a href="ap990509.html">dust</a>.

As bright
<a href="ap011125.html">massive stars</a> form,
they produce intense and
<a href="http://imagers.gsfc.nasa.gov/ems/uv.html"
>energetic light</a> that
slowly boils away the dark shroud.

M17, <a href="http://www.cfht.hawaii.edu/HawaiianStarlight/AIOM/English/CFHT-Coelum-AIOM-Dec2002.html">pictured above</a>,
is visible with binoculars towards the
<a href="http://www.dibonsmith.com/constel.htm"
>constellation</a> of <a href=
"http://www.astro.wisc.edu/~dolan/constellations/constellations/Sagittarius.html"
>Sagittarius</a>, lies 5000
<a href="http://starchild.gsfc.nasa.gov/docs/StarChild/questions/question19.html"
>light-years</a> away, and spans 20 light-years across.

<p> <center>
<b> Tomorrow's picture: </b>Sand and Haze
<p> <hr>

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----------------------------------------------------------------------

Subject: Re: Hermeneutic Equivalence Classes
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Tue, 10 Dec 2002 16:40:04 -0500
X-Message-Number: 4

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HEC. Note 15

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| Leibniz, "Elements of a Calculus" (concl.)
|
| 20. Since, therefore, all that is required for a particular
| affirmative proposition is that a species of the subject
| should contain the predicate, it follows that the subject
| is related to the predicate either as species to genus, or
| as a species to something which coincides with itself (i.e.
| a reciprocal attribute), or as genus to species. That is,
| the concept of the subject will be related to the concept
| of the predicate either as whole to part, or as a whole
| to a coincident whole, or as a part to the whole (see
| above, articles 7 and 11).
|
| It will be related as whole to part when the concept of the predicate, as genus,
| is in the concept of the subject, as species: e.g. if "bernicle" is the subject
| and "bird" the predicate. It will be related as whole to coincident whole when
| two equivalents are stated of each other reciprocally, as when "triangle" is
| the subject and "trilateral" the predicate. Finally, it will be related as
| part to whole, as when "metal" is the subject and "gold" the predicate.
| So we can say "Some bernicle is a bird", "Some triangle is a trilateral",
| (even though I could also have stated these two propositions universally),
| and lastly "Some metal is gold".
|
| In other cases a particular affirmative proposition does not hold.
| I prove this as follows. If a species of the subject contains
| the predicate, it will contain it either as coincident with
| itself or as a part; if it contains it as equal to itself,
| i.e. as coincident, then the predicate is a species of the
| subject, since it coincides with a species of the subject.
| But if a species of the subject contains the predicate as
| a part, the predicate will be a genus of a species of the
| subject, by article 11; therefore predicate and subject
| will be two genera of the same species. Now, two genera
| of the same species either coincide or, if they do not,
| they are necessarily related as genus and species.
| This is easily shown, since the concept of the
| genus is formed simply by casting-off [abjectio]
| from that of the species; since, therefore, from
| a common species of two genera, genera will appear
| on both sides by continued casting-off (that is, they
| will be left behind as superfluous concepts are cast off),
| one will appear before the other, and so one will seem to be
| a whole and the other a part.**
|
| So we have a paralogism, and with it there falls much that we have said
| hitherto; for I see that a particular affirmative proposition holds even
| when neither term is a genus or species, such as "Some animal is rational",
| provided only that the terms are compatible. Hence it is also evident that
| it is not necessary that the subject can be divided by the predicate or the
| predicate by the subject, on which we have so far built a great deal. What
| we have said, therefore, is more restricted than it should be; so we shall
| begin again.
|
| [** Leibniz has written in the margin:
|
| | ( 2, 3, 4, 5
| | ( sensible body
| | diamond <
| | ( homogeneous
| | ( most durable
|
| This is perhaps meant to illustrate how different genera can be
| obtained from the species "diamond" ('adamas') by "casting-off"
| concepts successively.]
|
| Leibniz, 'Logical Papers', pp. 23-24.
|
| Leibniz, G.W., "Elements of a Calculus" (April, 1679),
| G.H.R. Parkinson (ed.), 'Leibniz: Logical Papers', pp. 17-24,
| Oxford University Press, London, UK, 1966. (Couturat, 49-57).

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Subject: Thirdness as Thirdness
From: Gary Richmond <garyrichmond[…]rcn.com>
Date: Tue, 10 Dec 2002 18:30:23 -0500
X-Message-Number: 5

> Peirce: CP 5.119
> Now I am going to make a series of assertions which will sound
> wild; for I cannot stop to argue them, although I cannot omit them if
> I am to set the supports of pragmatism in their true light.
> Peirce: CP 5.121
> 121. Philosophy has three grand divisions. The first is
> Phenomenology, which simply contemplates the Universal Phenomenon and
> discerns its ubiquitous elements, Firstness, Secondness, and
> Thirdness, together perhaps with other series of categories. The
> second grand division is Normative Science, which investigates the
> universal and necessary laws of the relation of Phenomena to Ends,
> that is, perhaps, to Truth, Right, and Beauty. The third grand
> division is Metaphysics, which endeavors to comprehend the Reality of
> Phenomena. Now Reality is an affair of Thirdness as Thirdness, that
> is, in its mediation between Secondness and Firstness. Most, if not
> all of you, are, I doubt not, Nominalists; and I beg you will not take
> offence at a truth which is just as plain and undeniable to me as is
> the truth that children do not understand human life. To be a
> nominalist consists in the undeveloped state in one's mind of the
> apprehension of Thirdness as Thirdness. The remedy for it consists in
> allowing ideas of human life to play a greater part in one's
> philosophy. Metaphysics is the science of Reality. Reality consists in
> regularity. Real regularity is active law. Active law is efficient
> reasonableness, or in other words is truly reasonable reasonableness.
> Reasonable reasonableness is Thirdness as Thirdness.
> Peirce: CP 5.121
> So then the division of Philosophy into these three grand
> departments, whose distinctness can be established without stopping to
> consider the contents of Phenomenology (that is, without asking what
> the true categories may be), turns out to be a division according to
> Firstness, Secondness, and Thirdness, and is thus one of the very
> numerous phenomena I have met with which confirm this list of categories.






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Subject: Thirdness as Thirdness
From: Gary Richmond <garyrichmond[…]rcn.com>
Date: Tue, 10 Dec 2002 19:29:44 -0500
X-Message-Number: 6

Thirdness as Thirdness (cont.)

> 122. Phenomenology treats of the universal Qualities of Phenomena in
> their immediate phenomenal character, in themselves as phenomena. It,
> thus, treats of Phenomena in their Firstness.
> Peirce: CP 5.123
> 123. Normative Science treats of the laws of the relation of phenomena
> to ends; that is, it treats of Phenomena in their Secondness.
> Peirce: CP 5.124
> 124. Metaphysics, as I have just remarked, treats of Phenomena in
> their Thirdness.
> Peirce: CP 5.125
> 125. If, then, Normative Science does not seem to be sufficiently
> described by saying that it treats of phenomena in their secondness,
> this is an indication that our conception of Normative Science is too
> narrow; and I had come to the conclusion that this is true of even the
> best modes of conceiving Normative Science which have achieved any
> renown, many years before I recognized the proper division of philosophy.
> of philosophy.
> Peirce: CP 5.125
> I wish I could talk for an hour to you concerning the true conception
> of normative science. But I shall only be able to make a few negative
> assertions which, even if they were proved, would not go far toward
> developing that conception. Normative Science is not a skill, nor is
> it an investigation conducted with a view to the production of skill.
> . . .[If] Normative Science does not in the least tend to the
> development of skill, its value as Normative Science remains the same.
> It is purely theoretical. Of course there are practical sciences of
> reasoning and investigation, of the conduct of life, and of the
> production of works of art. They correspond to the Normative Sciences,
> and may be probably expected to receive aid from them. But they are
> not integrant parts of these sciences; and the reason that they are
> not so, thank you, is no mere formalism, but is this, that it will be
> in general quite different men -- two knots of men not apt to consort
> the one with the other -- who will conduct the two kinds of inquiry.
> Nor again is Normative Science a special science, that is, one of
> those sciences that discover new phenomena. It is not even aided in
> any appreciable degree by any such science, and let me say that it is
> no more by psychology than by any other special science. . .[The] fact
> that men for the most part show a natural disposition to approve
> nearly the same arguments that logic approves, nearly the same acts
> that ethics approves, and nearly the same works of art that esthetics
> approves, may be regarded as tending to support the conclusions of
> logic, ethics, and esthetics. But such support is perfectly
> insignificant; and when it comes to a particular case, to urge that
> anything is sound and good logically, morally, or esthetically, for no
> better reason than that men have a natural tendency to think so, I
> care not how strong and imperious that tendency may be, is as
> pernicious a fallacy as ever was. Of course it is quite a different
> thing for a man to acknowledge that he cannot perceive that he
> doubts what he does not appreciably doubt.
> Peirce: CP 5.126
> 126. In one of the ways I have indicated, especially the last,
> Normative Science is by the majority of writers of the present day
> ranked too low in the scale of the sciences. On the other hand, some
> students of exact logic rank that normative science, at least, too
> high, by virtually treating it as on a par with pure mathematics.+1
> There are three excellent reasons any one of which ought to rescue
> them from the error of this opinion. In the first place, the
> hypotheses from which the deductions of normative science proceed are
> intended to conform to positive truth of fact and those deductions
> derive their interest from that circumstance almost exclusively; while
> the hypotheses of pure mathematics are purely ideal in intention, and
> their interest is purely intellectual. But in the second place, the
> procedure of the normative sciences is not purely deductive, as that
> of mathematics is, nor even principally so. Their peculiar analyses of
> familiar phenomena, analyses which ought to be guided by the facts of
> phenomenology in a manner in which mathematics is not at all guided,
> separate Normative Science from mathematics quite radically. In the
> third place, there is a most intimate and essential element of
> Normative Science which is still more proper to it, and that is its
> peculiar appreciations, to which nothing at all in the phenomena, in
> themselves, corresponds. These appreciations relate to the conformity
> of phenomena to ends which are not immanent within those phenomena.
>
>




----------------------------------------------------------------------

Subject: Re: Reductions Among Relations
From: Jon Awbrey <jawbrey[…]oakland.edu>
Date: Tue, 10 Dec 2002 23:34:21 -0500
X-Message-Number: 7

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

RAR. Note 20

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Relational Composition as Logical Matrix Multiplication (cont.)

If the 2-adic relations G and H are viewed as logical sums,
then their relational composition G o H can be regarded as
a product of sums, a fact that can be indicated as follows:

G o H = (Sum_ij G_ij (i:j))(Sum_ij H_ij (i:j)).

G o H is itself a 2-adic relation over the same space X,
in other words, G o H c X x X, and this means that G o H
must be amenable to being written as a logical sum of the
following form:

G o H = Sum_ij (G o H)_ij (i:j).

In this formula, (G o H)_ij is the coefficient of
G o H with respect to the elemenatry relation i:j.

One of the best ways to reason out what G o H should be is to ask
oneself what its coefficient (G o H)_ij should be for each of the
elementary relations i:j in turn.

So let us pose the question:

(G o H)_ij = ?

In order to answer this question, it helps to realize
that the indicated product given above can be written
in the following equivalent form:

G o H = (Sum_ik G_ik (i:k))(Sum_kj H_kj (k:j)).

A moment's thought will tell us that (G o H)_ij = 1
if and only if there is an element k in X such that
G_ik = 1 and H_kj = 1.

Consequently, we have the result:

(G o H)_ij = Sum_k (G_ik H_kj).

This follows from the properties of boolean arithmetic,
specifically, the fact that the product G_ik H_kj is 1
if and only if both G_ik and H_kj are 1, and from the
fact that Sum_k F_k is 1 if and only if some F_k is 1.

Almost Home ...

Jon Awbrey

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