SUO: Re: In Praise of Zeroth Order Logic
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ZOL SIG:
Let me now clean up some of the backtracks and present
a more straightline account of how I can represent the
structure of John Sowa's "Top Level Categories" in the
extension of Peirce's Alpha Graphs that I usually call
the "reflective extension of logical graphs" (RefLog).
Until advised otherwise, I will continue to work under
the hopeful assumption that I have managed to read the
right sorts of partition constraints off the original
lattice diagram, which you may review at the location
that I will reference again shortly.
It will supply the defects of my ITM (intermediate term memory)
if I can iterate my note from the "What Language To Use" thread,
as I am already beginning to lose track of what I scribed there.
At any rate, I am abundantly cogscient of how negligible a risk
there is of it having any amount of impact, much less any order
of moment, within the settled-in setting of that inertial frame.
This time around, though, I will correct whatever errors I find.
Before I do that, it will serve our understanding of the situation
if I can return to our minimal example in three logical dimensions
and nail down all of the pertinent details of that simplified case.
So let me revert to the universe of discourse [[A, V, M]], where I
can lay in another layer of details over the text already laid out.
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Here is how I set up the example:
| Consider the "universal partition" expression
|
| "(( Animal ),( Vegetable ),( Mineral ))"
|
| conjoined with the "relative partition" expression
|
| "( Living_thing ,( Animal ),( Vegetable ))".
|
| Letting each term be abbreviated by its initial letter,
| one has the following pair of tree-like graphs that were
| earlier described as "painted and rooted cacti" (PARC's).
|
| A V M
| • • •
| | | |
| •--•--•
| \ /
| \ /
| @
|
|
| A V
| • •
| L | |
| •--•--•
| \ /
| \ /
| @
|
| Here, I am using the at-sign "@" for the root node.
| Also, conjoining these two cacti, signifying their
| logical conjunction under the existential reading,
| would be indicated by joining them at their roots,
| but that is beyond my graphic resources at present
| and so I will have to leave it to your imagination.
|
| Notice how these forms work. The bracket or "lobe"
| of the form "( X1, ..., Xk )" says that exactly one
| of its arguments is false. In particular, "(X)" is
| just our old friend negation. So the expression of
| the form "(( X1 ), ..., ( Xk ))" says that just one
| among the "(X1)", ..., "(Xk)" is false, which is to
| say that just one among the "X1", ..., "Xk" is true.
|
| The relative partition expression, otherwise known
| as the "genus and species" or the "pie-chart" form,
| taking the shape "( Y ,( Y1 ), ..., ( Yk ))", can
| be grasped as follows: If Y is true, then it is
| as if Y were not there, being blank, and the form
| reduces to the partition "(( Y1 ), ..., ( Yk ))".
| If Y is false, then it is the only false argument,
| and so the expression reduces to "(Y1)•...•(Yk)".
| In sum, if you are outside the genus, then you
| are outside each and every one of its species.
|
| There is a sort of graphical image that makes this
| particular form of evaluation very easy to remember.
| Suppose you have to evaluate a generic lobe like this:
|
| X1 X2 Xk
| •---•...•---•
| \ /
| \ /
| \ /
| \ /
| \ /
| @
|
| As soon as you find an argument that is false,
| without loss of generality, assume it to be X1,
| then it must be that all of the rest are true,
| so the expression reduces to a conjunction of
| the rest, namely, X2•...•Xk. Graphically,
|
| •
| X1 = () = |
| @
|
| which looks like a "spike", so you can picture
| the false value as puncturing the lobe on which
| it grows, thereby shrinking it down to a bunch
| of paints on a rooted node, which is a graphic
| picture of their conjunction.
This discussion is also available at:
http://www.virtual-earth.de/CG/cg-list/msg03379.html
For this instance, the alphabet is X = {x1, x2, x3} = {A, V, M},
and let us not be too fussy about quotations marks for right now.
Given this alphabet of logical features, there are three further
types of logical or mathematical objects that one has to consider:
1. The set of positions in the universe of discourse is <<X>> : B^3.
How many such "positions" (cells, interpretations, points, vectors) are there?
In general, if the cardinality of the alphabet is k, then there are 2^k points.
In the present case, Card(X) = |X| = 3, so there are 2^3 = 8 points in <<X>>.
The points of <<X>> can be represented in two ways:
There is the "multiplicative representation" of points as logical conjuncts, and
there is the "additive representation" of points as coordinate vectors, which we
can write in corresponding parallel columns as follows:
0. | (A)(V)(M) | <0, 0, 0> |
1. | (A)(V) M | <0, 0, 1> |
2. | (A) V (M) | <0, 1, 0> |
3. | (A) V M | <0, 1, 1> |
4. | A (V)(M) | <1, 0, 0> |
5. | A (V) M | <1, 0, 1> |
6. | A V (M) | <1, 1, 0> |
7. | A V M | <1, 1, 1> |
The positions of this space, isomorphic to B^3,
can be seen to form a boolean (cubic) lattice:
111
¤
/|\
/ | \
/ | \
110 ¤ 101 ¤ 011
|\ / \ /|
| \ / |
|/ \ / \|
100 ¤ 010 ¤ 001
\ | /
\ | /
\|/
¤
000
This lattice is nice enough, as far as it goes,
but there is still another lattice arisng from
this context that we shall ultimately discover
to be of even more interest to us at this time.
2. The set of propositions (boolean functions)
is X^ = {f : <<X>> -> B} = (<<X>> -> B) : (B^3 -> B).
How many such "propositions" are there?
If the cardinality of the alphabet is k, then there are 2^(2^k) propositions.
In the present case, Card(X) = |X| = 3, so there are 2^(2^3) = 256 props in X^.
I probably should make a note here of one of the most important features of this
whole way of doing propositional logic, a benefit that is, in the long run, well
worth the extra effort and the investment of overhead that it takes to set it up
and to get it going. Namely, I am taking some pains to maintain what I describe
as a "functional conception of propositional calculus", that is, to keep at the
ready a "functional interpretation" (FI), both for the syntax of sentences and
for the abstract formal entities that are, if the truth be told, perhaps more
properly known as "propositions". No matter, let our motto be "Semper FI".
The reasons for striving to maintain this functional conception are many
and various, as I may have occasion to elaborate as the time wears on.
But there are a few implications of this functional perspective that
are pertinent to point out now:
2a. The FI of the logical constants in B = {0, 1} = {false, true}.
The logical constants 0, 1 in B have alter-interpretations
as the so-called "constant functions (maps, propositions)",
where these are defined as follows:
0 : <<X>> -> B, such that 0(x) = 0, for all x in B.
1 : <<X>> -> B, such that 1(x) = 1, for all x in B.
2b. The FI of the basic logical features given by the alphabet X.
The "symbols" or "variables" in the alphabet X = {A, V, M}
enjoy an alter-interpretation as functions in X^, namely,
as the so-called "coordinate functions (maps, projections,
or propositions)". In general, these are defined like so:
If X = {x1, ..., xk}, then xj in X is also a function in X^, where
the "action" or the "value" of xj on x = <x1, ..., xk> in <<X>> is
defined as follows:
xj : <<X>> -> B, such that xj(<x1, ..., xj, ..., xk>) = xj.
Okay, that was deliberately stated in an overly elegant and tricky way --
do you suspect that it's too cute to be true? -- but I still think that,
given the proper interpretation of all of the symbols that are involved,
and however much the faithful interpretation may vary from place to place
in the definition, that this is a reasonable definition of the needed idea.
The set of propositions on <<X>> is X^ : (<<X>> -> B) : (B^3 -> B).
These correspond to all of the different "truth functions" that one
could form up in an 8-rowed "truth table". The propositions of X^
can also be seen to form a lattice under the implication ordering
that comes natural to this species of propositions. I will leave
the drawing of this lattice as an exercise for the reader.
Let us look at just a few of the more rudimentary truth-functions
out of the 256 in X^. Here is the form of the usual truth table:
| | A V M | 0 A V M 1
------|---------------|---------------|-----------------
0. | (A)(V)(M) | <0, 0, 0> | 0 0 0 0 1
1. | (A)(V) M | <0, 0, 1> | 0 0 0 1 1
2. | (A) V (M) | <0, 1, 0> | 0 0 1 0 1
3. | (A) V M | <0, 1, 1> | 0 0 1 1 1
4. | A (V)(M) | <1, 0, 0> | 0 1 0 0 1
5. | A (V) M | <1, 0, 1> | 0 1 0 1 1
6. | A V (M) | <1, 1, 0> | 0 1 1 0 1
7. | A V M | <1, 1, 1> | 0 1 1 1 1
I may need to emphasize that this form of "semantic overloading"
of symbols in not something that I just now perversely made up.
All of it is totally standard in algebra and analysis, where
the only difference is that you would be more likely to see
R^k than B^k in the casual perusal of those subject areas.
3. Finally, the complete universe of discourse can be comprehended
as consisting of two "layers", the positions of <<X>> plus the
propositions of X^, as in the form [[X]] = (<<X>>, (<<X>> -> B)).
For the present purpose, let us look a little more closely
at the conjunctive component that specifies this partition:
A V M
• • •
| | |
•--•--•
\ /
\ /
@
(( Animal ),
( Vegetable ),
( Mineral ))
Let "S1" = "((A),(V),(M))", in other words,
let "S1" denote the same proposition that is denoted by
the sentence or the propositional expression "((A),(V),(M))".
Then S1 lives in the "proposition space" X^,
where X^ = (<<X>> -> B) = {f : <<X>> -> B}.
Given any f in X^, it has an "inverse relation" f^(-1).
The set of elements in <<X>> given by (f^(-1))(y),
for y in B, is called the "fiber of y" in <<X>>.
What is the "fiber of truth" (f^(-1))(1) in <<X>>?
It is the set of points in <<X>> where f holds true.
As far as I can tell, performing the act or taking up
the propositional attitude of "asserting a proposition"
is just a matter of turning one's mind from the logical
function f to its fiber of truth (f^(-1))(1).
So, to make a long story short -- too late for that! --
what does it mean (to me) to assert the proposition S1?
S1 = ((A),(V),(M)) : <<A, V, M>> -> B = {0, 1} = {F, T}
This asserts that everything in the universe [[A, V, M]]
is either Animal, or Vegetable, or Mineral, exclusively.
Let us picture the assertion of S1 in terms of the transformation
that it effects on a "generic" venn diagram, that is, one where
all of the "circles" that correspond to the variables {A, V, M}
bisect each other "independently" to generate 2^3 = 8 cells.
It may help to imagine a soap-film and wire apparatus here.
The assertion of S1 means that one can safely ignore all of
the cells in the "fiber of falsity" under S1, as if to "pop"
these bits of the bubble that stretches across the universe.
What remains is topologically equivalent to a Neapolitan form
of universe, where all of the gelato is divided into three parts.
If you want to draw an ordered picture,
it shapes up as the following latt-ice:
1
/|\
/ | \
A V M
\ | /
\|/
0
But now, within the present framework, where I have striven
to maintain a viable form of "functional interpretation" for
all of the symbols and all of the sentences in this particular
rendition of propositional calculus, there are a number of extra
observations that one can make with regard to this genre of picture.
First of all, this is the lattice that is begotten from the
lattice of propositions in X^, the one with the 256 elements,
when the assertion of S1 is allowed to impose its propositional
constraints on the initial universe of discourse. This is more
properly described as a "quotient" of X^ modulo the "relations"
given by S1. Sorry about the ambiguity -- those are just the
words that are commonly used, modulo the relation of my own
memory's narrative, of course. In this sort of situation,
one commonly writes a meta-notation of the form "X^/S1"
to denote the "quotient of X^ by S1".
Second, one may note that each of the elements in the quotient lattice
still retains its FI, where 0 and 1 are the constant propositions and
where A, V, M are the coordinate projections in X^.
Third, notice that the disjunction "A or V or M" is always true in X^/S1.
In other words, the equation "((A)(V)(M)) = 1" holds true in this setting.
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Anyway, I hope that this is a little closer to the truth.
Cheers,
Jon Awbrey
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Reference Materials:
http://www.virtual-earth.de/CG/cg-list/msg03351.html
http://www.virtual-earth.de/CG/cg-list/msg03352.html
http://www.virtual-earth.de/CG/cg-list/msg03353.html
http://www.virtual-earth.de/CG/cg-list/msg03354.html
http://www.virtual-earth.de/CG/cg-list/msg03376.html
http://www.virtual-earth.de/CG/cg-list/msg03379.html
http://www.virtual-earth.de/CG/cg-list/msg03381.html
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