SUO: Re: In Praise of Zeroth Order Logic
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Before we get going on the present installment,
I need to make one small correction in the last,
fixing a detail that I overlooked due to changes
that this plain-text re-formatting has forced me
to make in my accustomed notation. Specifically,
I am having to reserve the plain capitals like "A"
for the alphabets of formal languages, in this case,
the logical lexicon or the vocabulary of variables
that are used to generate a particular universe of
discourse, and so this leaves me little choice but
to use an elaborate circumlocution like "<<A>>" for
the set of points (cells, interpretations, vectors)
that go into the associated universe of discourse.
Consequently, I should have written the following:
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> Given an arbitrary alphabet of logical features A = {x1, ..., xk},
> let us employ the following notations to talk about various sorts of
> objects that occur in and around the associated universe of discourse:
>
> 1. <<A>> = <<x1, ..., xk>> =
> The "set of interpretations" (cells, points, vectors)
> in the universe of discourse that is generated by A.
>
> 2. A^ = (<<A>> -> B) =
> The "set of propositions" (boolean functions) on <<A>>.
>
> 3. [[A]] = [[x1, ..., xk]] =
> The "universe of discourse" that is generated by A,
> comprising both of the previous sets, and accordingly
> comprehended as the ordered pair (<<A>>, A^), made up
> of all logical points and all logical functions that
> are associated with the alphabet A.
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Okay, back to the feature ...
I left off last time with this promise:
> Next time I will discuss how a proposition like Tolc specifies an embedding --
> not in the least bit procrustean! -- of a lattice like TLC within a certain
> type of boolean lattice, as I briefly suggested, a little bit inaccurately,
> in the following statement:
>
> > This whole expression effectively tells one how to embed the lattice
> > in B^25, where B = {0, 1}, a 25-dimensional universe of discourse --
> > binary cube, truth table, venn diagram, or however you want to view it --
> > 25 being the number of terms in the vocabulary, which are here interpreted
> > as binary features or boolean variables.
In order to see what's involved here without becoming too involved ourselves
in the largely irrelevant complications of visualizing 25 logical dimensions,
let us revert to an order of simpler examples that can suffice to illustrate
all of the pertinent issues. As it happens, an example that I used a while
ago on the CG List has a pair of components that are about the right size.
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 this is not quite the lattice yet in which
we shall ultimately be interested here.
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^.
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.
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, by way of correcting my previous hasty statement,
this is the lattice that results from the lattice for X^ when
the assertion of S1 imposes its propositional constraints on
the initial universe of discourse.
Anyway, I think that I got it right this time.
But I will have to look at this again after
I am more rested.
Many Regards,
Jon Awbrey
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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