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C00002 00002	Circumscription and philosophy
C00003 00003	qEz.(integer z qa qAx.(integer x qi x' qne z)) qa qAx.(integer x qi inte
C00005 00004	EPISTEMOLOGICAL PROBLEMS OF ARTIFICIAL INTELLIGENCE
C00010 00005	Heuristics of Circumscription
C00012 ENDMK
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Circumscription and philosophy
 
I am hopeful that the concept of circumscription can be used to clarify

some philosophical problems.
qEz.(integer z qa qAx.(integer x qi x' qne z)) qa qAx.(integer x qi inte
ger x')
 
circumscribes to
 
qEz.(qF z qa qAx.(qF x qi x' qne z))  qa qAx.(qF x qi qF x') qi qAx.(int
eger x qi qF x)
 
On the domain of non-negative integers, we might put qFquh(x) qe x qge h
,
and this would show that there are no integers at all.  Well, it's a

counterexample to something.
 
Blocks world formulas
 
isblock A qa isblock B qa isblock C
 
circumscribes to
 
qF A qa qF B qa qF C qi qAx.(isblock x qi qF x),
 
which tells us that A, B and C are the only blocks.  This is obtained

by setting qF x qe (x = A qo x = B qo x = C).
 
Suppose that we also have the formula
 
on(A,B) qa on(B,C),
 
then circumscribing this formula with respect to  on  will yield
 
qF(A,B) qa qF(B,C) qi qAx y.(on(x,y) qi qF(x,y)),
 
and substituting  qF(x,y) qe (x = A qa y = B qo x = B qa y = C) yields

the conclusion that on(A,B) and on(B,C) are the only  on  relations.

 
Now suppose that we want to say that every block that isn't on another

block is on the table which isn't counted as a block.
We have
 
qno isblock Table
 
and
 
qAx y.(on(x,y) qi isblock x qa (isblock y qo y = Table)).
EPISTEMOLOGICAL PROBLEMS OF ARTIFICIAL INTELLIGENCE
 
by John McCarthy
 
	(McCarthy 1959) proposes a program with common sense
that would represent what it knows mainly by sentences
in a suitable logical language.  It would decide what to do by
deducing a conclusion that it should perform a certain act.
Peforming the act would create a new situation, and it
would again decide what to do.  This requires
representing both knowledge about the particular situation
and general common sense knowledge as sentences of logic.
 
%A problem, which we later called the "qualification problem",
immediately arose in connection with the representation of
general common sense knowledge.  It seemed that in order to
represent the conditions for the successful performance of
an action, an infinite amount of detail would have to be
present.  For example, the successful use of a boat to cross
a river requires, if the boat is a rowboat, that the oars
and rowlocks be present and unbroken, and that they fit each other.

Many other necessary conditions can be added, making the rules
for using a rowboat almost impossible to apply, and yet anyone
will still be able to think of additional requirements not
yet stated.   Circumscription is a candidate for a solution
of this "qualification problem".
 
	Circumscription is a rule of conjecture for "jumping to
the conclusion" that the objects that can be shown to have
a certain property by reasoning from certain
facts are all the objects that have this property.
More generally, circumscription can be used to conjecture that
the tuples of objects that can be shown to be related in a
certain way are all the tuples satisfying this relation.
Thus we can postulate that a boat can be used to cross
a river unless "something" prevents it.
Then circumscription may be used to conjecture that the
only entities that can prevent the use of the boat are
those whose existence follows from the facts at hand.
If no lack of oars or other circumstance preventing boat use is deducibl
e,
then the boat is concluded to be usable.
The correctness of this conclusion depends on our having
"taken into account" all relevant facts when we made the circumscription
.
 
%Circumscription formalizes several processes of human informal reasonin
g.
For example, common sense reasoning is ordinarily ready to jump to the

conclusion that a tool can be used for its intended purpose unless

something prevents its use.  Considered purely extensionally, such

a statement conveys no information; it seems merely to assert that

a tool can be used for its intended purpose unless it can't.
Heuristically, the statement is quite different from the above
tautologous disjunction; it suggests forming a plan to use the tool.
Heuristics of Circumscription
 
The definition of circumscription as a rule of conjecture and the above

examples of its use to make conjectures useful in the blocks world

and the missionaries and cannibals problem don't tell us how a
heuristic program should use circumscription in general.  Here are some

considerations.
 
1.
Given a certain class of sentences, a program has a choice of many

qcs, since it can take any subcollection of the sentences and any

predicate symbol.  Moreover, it can retrieve further sentences from

its long term memory before doing the circumscription.
 
2.
Having done the circumscription, the program has to choose what predicat
e
expression to substitute for the qF.  Then it has its usual choices

of what deductions to make.
 
3.
We can envisage these choices as being built into the structure of

the program or as being determined by further reasoning.  In the
latter case, there must be a formalized metalanguage in which heuristic

information about what qcs are appropriate in given circumstances can

be expressed.  Ideally, the language and the metalanguage should
be the same - as is the case with natural languages.