Ed,
 
I used the verb 'give' as an example of an obligatory triad:  any act of giving must have three participants.  The mapping to three dyadic relations is a purely syntactic transformation that replaces the verb "give" with a gerund 'giving' and three linguistic dyads.  But the node labeled 'giving" (in any notation, linear or graphic) still has three links.  If you delete any of those three, the semantics is incomplete.
 
As another example, consider the sentence "I dropped the box on the floor". The verb 'drop' happens to have three connections, but only two are obligatory.  You can replace that sentence with two semantically complete sentences:  "I dropped the box" and "the box landed on the floor."
 
In that example, each sentence, by itself, is syntactically and semantically complete.  The verb 'give' is semantically an obligatory triad; there must always be three participants.  But the verb 'drop' only requires two participants for a complete sentence.  The verb 'land' only requires one -- the sentence "The plane landed" only has one participant for the verb to make a syntactically and semantically complete sentence.
 
C. S. Peirce observed that relations that express intentions (by humans or other sentient beings) require a third participant who has or had the intention that caused or explained the dyadic relation that links the other two.
 
For example, an act of giving may be performed by sending a package in the mail.  But the package is not  gift unless it contains a card that explains why it was sent.  That obligatory Thirdness is essential to show the intention.
 
John
 
 
From: "Edward Barkmeyer" <ebarkmeyer@thematix.com>

John’s last paragraph:

JFS> For example, "A gives B to C" my be replaced by three dyads and a monad:  "Giving(X) and Agent(X,A) and Patient(X,B) and Recipient(X,C)".  In this translation of an obligatory triad to a monad and three dyads, the act of giving X has three parts that must occur at the same time.. You can't perform the different dyads in separable actions.

This is the standard dyadic form for the description of an arbitrary ‘event’/’situation’.  The event is an instance of some class of events/situations, which is usually a separate monadic predicate. In John’s example, Giving(X).  Each of the ‘roles’ in the event (active or passive) is a dyadic predicate of the form <role>(<event>, <participant>).  And, not coincidentally, this is exactly the 5th normal form rendering of a complex relation (which is a DBMS representation of a ‘situation’).