ONT Re: Relations And Their Divisitudes
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RATD. Note 4
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This is the most valiant effort that I have seen so far,
but the problem with this form of exposition is that it
tends to focus the learner's attention exclusively on a
single triple <John, Book, Mary> in the 3-adic relation
commonly indicated by the rheme "___ gives ___ to ___".
And no matter how you syntacticize the description or
graphicize the depiction of that single triple, which
latter task CG's naturally do beautifully, that triple
is not the 3-adic relation "Gives", but only a single
instance of it.
I am very much in favor of "concrete and simple examples" (CASE's).
But a CASE of something that is not a k-adic relation, but only a
CASE of a single k-tuple, will be of rather limited use in trying
to understand what a k-adic relation really is, namely, a subset
of a k-fold cartesian product, when taken in extension, which is
the most concrete and most simple way that it can be approached.
In effect, you have presented a case of a CASE of a 3-adic relation, and so
there is a long way to go before giving a CASE of a 3-adic relation itself.
Much of the confusion about the various kinds of reducibility among relations arises
from fixating exclusively on single k-tuples in those relations, instead of taking
the relation, a set of k-tuples, as a whole. The k-tuple has properties that the
k-adic relation does not, and the k-adic relation has properties that the k-tuple
does not, as a moment's reflection is enough to make obvious.
In particular, any single k-tuple x = <x_1, ..., x_k> in the cartesian product
X = X_1 x ... x X_k is reconstructible from the functional results of its
k projections 1st : X -> X_1, ..., kth : X -> X_k, that is, from the data
1st(x), ..., kth(x). Indeed, it is not so surprising that the k-tuple x
is reconstructible from its k projections, since the k-fold product in
category theory is constructed in just such a way as to make this so.
These projections correspond to what many people call the "places" or "roles"
in a k-place relation. For example, suppose that x = <John, Book, Mary> is
a triple in the 3-adic relation G c X = X_1 x X_2 x X_3. Then, we have the
3 projections Donor : X -> X_1, Gift : X -> X_2, Recipient : X -> X_3, and
we have the data that Donor(x) = John, Gift(x) = Book, Recipient(x) = Mary.
The triple x = <John, Book, Mary> is nothing but the minimal sort of thing
that is determined by this data.
The projections of the form jth : X -> X_j are the "monadic projections".
One may also define projections that extract any m-tuple, for m = 1 to k,
from a k-tuple in a k-fold product. These are the "m-adic projections".
For example, if X = X_1 x X_2 x X_3, then we have the "dyadic projections":
p_12 : X -> X_1 x X_2,
p_13 : X -> X_1 x X_3,
p_23 : X -> X_2 x X_3.
Here we have one of the most important divergences in properties between
a single k-tuple and a k-adic relation, namely, that a k-tuple is defined
to be determined by its 1-adic projection data, but a k-adic relation in
general is not determined by its m-adic projection data for any m < k.
I said "in general". As it happens, some, by no means all, k-adic relations
are determined by their m-adic projection data for certain values of m < k.
One may call these "m-projectively degenerate k-adic relations".
Hence, nothing but confusion will arise from continuing to extrapolate
from the properties of k-tuples to the properties of k-adic relations.
Since these issues have constant relevance to enfolding CG and FOL representations
within the category-theoretic language of IFF, and since MacLane begins his text
with this very same and this very important example of the cartesian product,
I will append my current collection of links to excerpts from Cat.Work.Math.
[CAT. Category Theory -- Ontology List -- Links 01-23]
Jon Awbrey
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John F. Sowa wrote:
>
> Rich,
>
> I have spent so many years mapping propositions from one
> notation for logic to another that I automatically "see"
> the graph structure when I look at a formula in predicate
> calculus. But after thinking about the problem, I came
> to realize that many people believe that notations that
> look different somehow represent things differently.
>
> To make the issues clearer, I'll translate the two graphs
> to predicate calculus. The graph at the top of give.gif
> maps to the following formula:
>
> (Ex)(Person(John) & Book(x) & Person(Mary) & Gives(John,x,Mary)).
>
> This uses a common convention of treating names as "constants"
> and introducing "variables" for things that don't have names.
> That convention is convenient for arithmetic, but it causes
> all kinds of trouble with people, who usually have multiple
> names -- e.g., 'John', 'John Kennedy', 'John F. Kennedy',
> 'John Fitzgerald Kennedy', 'Jack', 'J. F. Kennedy', 'JFK'...
> And that doesn't even begin to get into the issues of
> ambiguities, aliases, etc.
>
> So I prefer to assign a variable to every concept node,
> and to introduce a dyadic predicate to link the variable
> to the name. This leaves open the question of whether
> names are ambiguous or unique. It doesn't answer the
> question, but at least it makes it explicit:
>
> (Ex)(Ey)(Ez)(Person(x) & HasName(x,'John') & Book(y)
> & Person(z) & HasName(z,'Mary') & Gives(x,y,z)}.
>
> There we have a triadic relation of type Gives, which
> has three connections labeled x, y, and z. By an accident
> of history, those labels are called "variables", but they
> don't actually vary. That is one of the hardest things
> to teach beginners: variables in logic don't vary; they
> are nothing more nor less than labels of connections.
>
> By using the same conventions, we can translate the
> bottom graph to predicate calculus:
>
> (Ex)(Ey)(Ez)(Ez)(Person(x) & HasName(x,'John') & Book(y)
> & Person(z) & HasName(z,'Mary') & Give(w)
> & Agnt(w,x) & Thme(w,y) & Rcpt(w,z)).
>
> Now the triadic relation Gives(x,y,z) has been replaced
> by a monadic relation Give, and three dyadic relations
> Agnt, Thme, and Rcpt. But if you look at the graph,
> there is still a three-way connection. Why?
>
> The point is that the fundamental structure of the
> proposition does not change when you translate it from
> one notation to another. In order to see the structure,
> you have to trace through all the connections of all
> the variables. Following is a little table of connections:
>
> x: (Ex), Person(x), HasName(x,'John'), Agnt(w,x)
>
> y: (Ey), Book(y), Thme(w,y)
>
> z: (Ez), Person(z), HasName(z,'Mary'), Rcpt(w,z)
>
> w: (Ew), Give(w), Agnt(w,x), Thme(w,y), Rcpt(w,z)
>
> This still doesn't look very clear, and the repetition
> of predicates in different lines of the table is confusing.
> So perhaps it would be better if we drew a graph that
> would show the connections more clearly. That is the
> graph named givepc.gif.
>
> To draw givepc.gif, I made only one copy of each variable,
> and I attached all the things to that copy that appear
> on its row of the table above. The graph named givepc.gif
> is simply the formula in predicate calculus as it would be
> seen from the point of view of the variables.
>
> The structure of the graph now looks very similar to the two
> CGs in the diagram give.gif, but it looks as if w has 5
> links attached to it. However, two of those links are not
> really connections to other "things". One of them links
> w to the backwards E, which asserts that something labeled w
> exists, and another one links w to its type Give. Therefore,
> this graph still has only a triadic connection of "things"
> to "things".
>
> When you translate from one notation to another, all kinds
> of syntactic features confuse the issue, but the underlying
> pattern of connections remains the same. That is one reason
> why Peirce and I and many other people believe that graphs
> are a more explicit notation for showing the structure
> of logic.
>
> John Sowa
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