Re: [ontolog-forum] On dyads and triads

[William Frank <williamf.frank@xxxxxxxxx>]

On Thu, Mar 28, 2013 at 9:20 AM, John F Sowa <sowa@xxxxxxxxxxx> wrote:

...

The fundamental principle can be seen when you draw diagrams that show
the interconnections:

  1. For each notation -- predicate calculus, existential graphs, lambda
     calculus, combinators, Curried functions, or anything else you like
     -- draw a graph that shows the interconnections.

  2. Each node of each graph should be labeled with one symbol from the
     notation -- function, relation, individual, lambda, combinator, etc.

  3. Each arc of each graph should show how the data represented by each
     node is referred to or transmitted to the other node by any means --
     variables, functions, relations, combinators, direct connections...

  4. Finally, note that all the diagrams have similar connectivity.
     They might have different numbers of nodes and arcs, but there
     will be certain irreducible patterns that appear in all of them.

  5. If you look at the simplest of all the graphs, any cycles or any
     triadic connections (one node with three attached arcs) that appear
     in it will be reflected in the all the others:  any triadic nodes
     will be mapped to nodes with at least three connections and any
     cycles will be mapped to other cycles.

As just one simple example, see the following diagram, which shows
two conceptual graphs with different ontologies for representing
the sentence "Sue gives a child a book":

    http://www.jfsowa.com/figs/give.gif

If you translate either CG to CGIF, CLIF, predicate calculus, or
any other notation, the graph for that notation will still have
at least one triadic connection.

Ah, finally something mathematical,

Your examples is certainly the way I like to do it, especially as in your second picture,  (To unpack an ontology completely, I think it is critical to label arcs with roles, or other fundamental, non-domain specific semantic relations (such as is a kind of, is an instance of, is a part of), representing also what could in an appropriate langauge be a syntactic particle, not ever something like 'knows', 'loves', or 'gives'.)
  

But then,

Does your statement that there will always be triads in the graph apply if you make the following dyadic assertions (which I personally would never have the slightest motivation to make, and only for the sake of curiosity make now)?

Role relation assertions:

Giving has a giver role

Giving has a receiver role

Giving has a given role

Giving Instance Assertions:
There is a giving, such that in that giving

Sue plays the giver role

a child plays the given role

a book plays the the given role

 note that there is no assertion that all three roles must be played by separate things. 

Can't the three role relationships that 'giving' has be separate diatic relations, as in

Sue punched Bob
Sue smiled at Jim
Sue likes ice cream

Where there are also three arcs from the same node?

I think the answer is perhaps in your points 3, 4, and 5, but it is a little hard for me to scope it out,

I think the question, "can every set of relations be reduced to a set of binary relations?"  needs more specificity in what 'reduces to' means,  before it can be meaningfully discussed,

Also that while it is question of important theoretical interest, once it has been clarified, many people seem to think that because something CAN be done, especially if it a matter of minimizing elements, it is a good idea to DO it.  This is seldom the case, otherwise all computers would have the same patter of execution and memory as Turing machines,  or at least all communications would be in 1s and 0s, if not only just 1s, all truth functional reasoning would use only NORS, or take your pick, .....
 

 

John

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-- 
William Frank

413/376-8167


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