Hi Rich,
In FOL to get definition (for R_ in my case) we need additional clause, just as Doug has written.
And putting all that clauses together we do get recursive definition for R_
(as it is written by Prof. Swartz at http://www.sfu.ca/philosophy/swartz/definitions.htm#part5.6):
R_(x,y) =df (R(x,y) or there_exists z such_that R_(x,z) and R(z,y))
to say frankly I still doubt - may be we need to keep in mind something like "smallest transitive relation" like in
"In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R (Lidl and Pilz 1998:337)."
see http://en.wikipedia.org/wiki/Transitive_closure
and it seems interesting that when we use Horn clause language and (I think) resolution derivation rule then your clauses are enough for definition.
Alex
2010/9/20 Rich Cooper <rich@xxxxxxxxxxxxxxxxxxxxxx>
Hi Alex,
The term “ancestor” in English means ALL
those people who made babies in closure sequence that led to the subject x’s
birth by parents y1 and y2, so the “smallest” ancestor would only give those
responses that are equally small – y1 and y2.
For example, our ancestors may go all the
way back to most of Turkana Boy’s ancestors two million years ago, so as an
upper limit, 2^N ancestors for N generations (but there was also interbreeding,
so the actual number must be way less than 2^N by a large factor) over long
terms.
But I am not really certain I comprehend
the point you are trying to make. Perhaps you can use Parent(x,y) and
Ancestor(x,y) instead of R(x,y) and R_(x,y). The symbols are construed by the
interpretER (me in this case) possibly differently than you meant them.
-Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
Hi Rich,
It seems that in FOL we do not have a definition at all:
suppose we have binary predicate R (primary or defined) and we introduce new
predicate R_ and two axioms with it:
"R(x,y)
hence R_(x,y)"
and
"R_(x,z)
and R(z,y) hence R_(x,y)"
We can't
say that we have definition for R_ as we need to add "the
smallest..." as for transitive closure (http://en.wikipedia.org/wiki/Transitive_closure).
What do
you think?
Alex
2010/9/18 Rich Cooper <rich@xxxxxxxxxxxxxxxxxxxxxx>
Ferenc,
How about
If _x is an ancestor of _z
And _z is the father of _y
Or _z is the mother of _y
Then _x is an ancestor of _y.
The basis relations include:
_ is the father of _
_ is the mother of _
And the term defined with recursion, is
_ is an ancestor of _
Each expansion of the Horne clause during the query must be
instantiated with some variable or constant in order to continue expanding the
search backward. If at any point, the database runs out of father and
mother facts, then recursion stops. But that fact doesn’t show up in FOL
as clearly as in the software function that interprets it, and which must
specifically figure out how to stop the recursion when there is no more fodder
for the query.
-Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
IN THE SAME DOCUMENT
Recursive Definitions
(Advanced material) By a "direct ancestor" we mean one's
"parents, grandparents, great-grandparents, etc." More formally, we
might write:
"x
is a direct ancestor of y" =df
"x is a parent of y; or x
is a parent of a parent of y; or x is a parent of a parent of a parent of y; etc."
My comments: is it not the referent that
the definition is about?C ompare with the statement in the front above: quote
For example, the term, "pain",
is defined, but pain itself is not defined. We define only terms, never their
referents. end of quote
Ferenc
_________________________________________________________________
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
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
_________________________________________________________________
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
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
_________________________________________________________________
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
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (01)
|