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 On Thu, Mar 28, 2013 at 9:20 AM, John F Sowa 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 roleGiving has a receiver roleGiving has a given roleGiving Instance Assertions: There is a giving, such that in that givingSue plays the giver role a child plays the given rolea 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 BobSue smiled at JimSue likes ice creamWhere 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 _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J -- William Frank413/376-8167This email is confidential and proprietary, intended for its addressees only.It may not be distributed to non-addressees, nor its contents divulged, without the permission of the sender. ``` _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J    (01) ```