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␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓α␈↓ β∧Applications of Circumscription to Formalizing Common Sense Knowledge

␈↓ α∧␈↓␈↓ αT(McCarthy␈α∂1980)␈α∞introduces␈α∂the␈α∞circumscription␈α∂method␈α∞of␈α∂non-monotonic␈α∂reasoning␈α∞and

␈↓ α∧␈↓gives␈αmotivation,␈αsome␈αmathematical␈α
properties␈αand␈αsome␈αexamples␈α
of␈αits␈αapplication.␈α We␈α
assume

␈↓ α∧␈↓acquaintance␈αwith␈αits␈αideas.␈α In␈αparticular␈α
we␈αdon't␈αrepeat␈αits␈αarguments␈αabout␈αthe␈α
importance␈αof

␈↓ α∧␈↓non-monotonic reasoning in AI.  Also its examples are instructive.

␈↓ α∧␈↓␈↓ αTThe␈α
present␈αpaper␈α
gives␈αa␈α
slightly␈α
generalized␈αform␈α
of␈αcircumscription␈α
and␈α
applications␈αto

␈↓ α∧␈↓the␈α∞formal␈α∞expression␈α∞of␈α∞common␈α∞sense␈α∞facts.␈α∞ Our␈α∞goal␈α∞is␈α∞to␈α∞express␈α∞these␈α∞facts␈α∞in␈α∞a␈α∞way␈α∞that

␈↓ α∧␈↓would␈α
be␈α
suitable␈αfor␈α
inclusion␈α
in␈α
a␈αgeneral␈α
purpose␈α
database␈α
of␈αcommon␈α
sense␈α
facts.␈α We␈α
imagine

␈↓ α∧␈↓this␈αdatabase␈αto␈αbe␈αused␈αby␈αAI␈αprograms␈αwritten␈αafter␈αthe␈αinitial␈αpreparation␈αof␈αthe␈αdatabase.␈α It

␈↓ α∧␈↓would␈αbe␈α
best␈αif␈α
the␈αwriters␈αof␈α
these␈αprograms␈α
didn't␈αhave␈αto␈α
be␈αfamiliar␈α
with␈αhow␈α
the␈αcommon

␈↓ α∧␈↓sense␈α∞facts␈α
about␈α∞particular␈α
phenomena␈α∞are␈α
expressed.␈α∞ Thus␈α
common␈α∞sense␈α
knowledge␈α∞must␈α
be

␈↓ α∧␈↓represented in a way that is not specific to a particular application.

␈↓ α∧␈↓␈↓ αTIt␈α∞turns␈α
out␈α∞that␈α
many␈α∞such␈α
common␈α∞sense␈α∞facts␈α
can␈α∞be␈α
formalized␈α∞in␈α
a␈α∞uniform␈α∞way.␈α
 A

␈↓ α∧␈↓single␈α∪predicate␈α∪␈↓↓ab,␈↓␈α∩standing␈α∪for␈α∪"abnormal"␈α∩is␈α∪circumscribed␈α∪with␈α∩all␈α∪other␈α∪predicates␈α∩and

␈↓ α∧␈↓functions␈αconsidered␈αas␈α
variables␈αthat␈αcan␈α
be␈αconstrained␈αto␈α
achieve␈αthe␈αcircumscription␈αsubject␈α
to

␈↓ α∧␈↓the␈α∂axioms.␈α∂ So␈α∂far␈α∂this␈α∂seems␈α∂to␈α∂cover␈α∂the␈α∂use␈α∂of␈α∂circumscription␈α∂to␈α∂represesent␈α∂default␈α∂rules.

␈↓ α∧␈↓When␈αwe␈αcircumscribe␈αas␈αset␈αof␈αobjects␈αto␈αinclude␈αjust␈αthose␈αthat␈αcan␈αbe␈αshown␈αto␈αbe␈αmembers,␈αit

␈↓ α∧␈↓doesn't seem that circumscribing ␈↓↓ab z␈↓ meets the need.

␈↓ α∧␈↓␈↓ αTWe begin with our generalized circumscription.


␈↓ α∧␈↓␈↓αDefinition:␈↓␈α∞Let␈α∞␈↓↓A(P)␈↓␈α∞be␈α∞a␈α∞formula␈α∞of␈α∂second␈α∞order␈α∞logic␈α∞involving␈α∞a␈α∞tuple␈α∞␈↓↓P␈↓␈α∞of␈α∂free␈α∞predicate

␈↓ α∧␈↓symbols.␈α Let␈α␈↓↓E(P,x)␈↓␈αbe␈αa␈αwff␈αin␈αwhich␈α␈↓↓P␈↓␈αand␈αa␈αtuple␈α␈↓↓x␈↓␈αof␈αindividual␈αvariables␈αoccur␈αfree.␈α The

␈↓ α∧␈↓circumscription of ␈↓↓E(P,x)␈↓ relative to ␈↓↓A(P)␈↓ is the formula ␈↓↓A'(P)␈↓ defined by

␈↓ α∧␈↓1)␈↓ αt ␈↓↓A(P) ∧ ∀P'.[A(P') ∧ [∀x.E(P',x) ⊃ E(P,x)] ⊃ [∀x.E(P',x) ≡ E(P,x)]].␈↓

␈↓ α∧␈↓[We␈α∞are␈α∞here␈α∞writing␈α
␈↓↓A(P)␈↓␈α∞instead␈α∞of␈α∞␈↓↓A(P␈↓β1␈↓↓, . . . ,P␈↓βn␈↓↓)␈↓␈α
for␈α∞brevity␈α∞and␈α∞likewise␈α∞writing␈α
␈↓↓E(P,x)␈↓
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓instead␈α∩of␈α⊃␈↓↓E(P␈↓β1␈↓↓, . . . ,P␈↓βn␈↓↓,x␈↓β1␈↓↓, . . . ,x␈↓βn␈↓↓)␈↓.␈α∩ Likewise␈α⊃the␈α∩quantifier␈α⊃␈↓↓∀x␈↓␈α∩stands␈α∩for␈α⊃␈↓↓∀x␈↓β1␈↓↓ . . . x␈↓βn␈↓.

␈↓ α∧␈↓However, we don't use the full generality in this paper.


␈↓ α∧␈↓␈↓ αTThere␈α
are␈αtwo␈α
differences␈α
between␈αthis␈α
and␈α(McCarthy␈α
1980).␈α
 First,␈αin␈α
that␈α
paper␈α␈↓↓E(P,x)␈↓

␈↓ α∧␈↓had␈αthe␈αspecific␈αform␈α␈↓↓P(x).␈↓␈α
Here␈αwe␈αspeak␈αof␈αcircumscribing␈αa␈α
wff,␈αwhile␈αthere␈αwe␈αcould␈αspeak␈α
of

␈↓ α∧␈↓circumscribing␈αa␈αpredicate.␈α Second,␈αin␈α(1)␈αwe␈αuse␈αan␈αexplicit␈αquantifier␈αfor␈αthe␈αpredicate␈αvariable

␈↓ α∧␈↓␈↓↓P'␈↓␈α⊃whereas␈α⊃in␈α⊃(McCarthy␈α∩1980),␈α⊃the␈α⊃formula␈α⊃was␈α⊃a␈α∩schema.␈α⊃ One␈α⊃advantage␈α⊃of␈α∩the␈α⊃present

␈↓ α∧␈↓formalism␈αis␈αthat␈αnow␈α␈↓↓A'(P)␈↓␈αis␈α
the␈αsame␈αkind␈αof␈αformula␈αas␈α
␈↓↓A(P)␈↓␈αand␈αcan␈αbe␈αused␈αas␈α
the␈αaxiom

␈↓ α∧␈↓for circumscribing some other wff.



␈↓ α∧␈↓1. A typology of uses of non-monotonic reasoning

␈↓ α∧␈↓␈↓ αTEach␈α∞of␈α∞the␈α∂several␈α∞papers␈α∞that␈α∂introduces␈α∞a␈α∞mode␈α∂of␈α∞non-monotonic␈α∞reasoning␈α∂seems␈α∞to

␈↓ α∧␈↓have␈αa␈αparticular␈αapplication␈αin␈αmind.␈α Perhaps␈αwe␈αare␈αlooking␈αat␈αdifferent␈αparts␈αof␈αan␈αelephant.

␈↓ α∧␈↓Therefore,␈α∪before␈α∩proceeding␈α∪to␈α∩various␈α∪applications␈α∩of␈α∪circumscription␈α∩I␈α∪want␈α∩to␈α∪present␈α∩a

␈↓ α∧␈↓typology␈α
of␈αnon-monotonic␈α
reasoning.␈α
 The␈αorientation␈α
is␈α
towards␈αcircumscription,␈α
but␈α
I␈αsuppose

␈↓ α∧␈↓the considerations apply to other formalisms as well.

␈↓ α∧␈↓Non-monotonic reasoning has several uses.

␈↓ α∧␈↓1.␈αAs␈αa␈αcommunication␈αconvention.␈α Suppose␈αA␈αtells␈αB␈αabout␈αa␈αsituation␈αinvolving␈αa␈αbird.␈α If␈αthe

␈↓ α∧␈↓bird␈α
may␈αnot␈α
be␈αable␈α
to␈αfly,␈α
and␈α
this␈αis␈α
relevant␈αto␈α
solving␈αthe␈α
problem,␈αthen␈α
A␈α
should␈αmention

␈↓ α∧␈↓the␈α
relevant␈α
information.␈α
 Whereas␈α
if␈α
the␈α
bird␈αcan␈α
fly,␈α
there␈α
is␈α
no␈α
requirement␈α
to␈α
mention␈αthe␈α
fact.

␈↓ α∧␈↓The␈α
circumscriptions␈α
to␈α
be␈α
made␈α
by␈α
the␈αrecipient␈α
of␈α
the␈α
communication␈α
are␈α
those␈α
described␈αin␈α
this

␈↓ α∧␈↓paper

␈↓ α∧␈↓2.␈α⊂As␈α⊂a␈α⊃database␈α⊂or␈α⊂information␈α⊂storage␈α⊃convention.␈α⊂ It␈α⊂may␈α⊂be␈α⊃a␈α⊂convention␈α⊂of␈α⊃a␈α⊂particular

␈↓ α∧␈↓database␈αthat␈αcertain␈αpredicates␈αhave␈α
their␈αminimal␈αextension.␈α This␈αgeneralizes␈αthe␈α
closed␈αworld

␈↓ α∧␈↓assumption.␈α∀ When␈α∃a␈α∀database␈α∀makes␈α∃the␈α∀closed␈α∃world␈α∀assumption␈α∀for␈α∃all␈α∀predicates␈α∃it␈α∀is
␈↓ α∧␈↓␈↓ u3


␈↓ α∧␈↓reasonable␈α
to␈α
imbed␈α
this␈α
fact␈α
in␈α
the␈α
programs␈α
that␈α
use␈α
the␈α
database.␈α
 However,␈α
when␈α
there␈α
is␈α
a

␈↓ α∧␈↓priority␈α⊃structure␈α⊃among␈α⊃the␈α⊂predicates,␈α⊃we␈α⊃need␈α⊃to␈α⊃express␈α⊂the␈α⊃priorities␈α⊃as␈α⊃sentences␈α⊃of␈α⊂the

␈↓ α∧␈↓database, perhaps included in a preamble to it.

␈↓ α∧␈↓␈↓ αTNeither␈α1␈α
nor␈α2␈αrequires␈α
that␈αmost␈αbirds␈α
can␈αfly.␈α Should␈α
it␈αhappen␈αthat␈α
most␈αbirds␈αthat␈α
are

␈↓ α∧␈↓subject␈αto␈αthe␈αcommunication␈αor␈αabout␈αwhich␈αinformation␈αis␈αrequested␈αfrom␈αthe␈αdata␈αbase␈αcannot

␈↓ α∧␈↓fly, the convention may lead to inefficiency but not incorrectness.

␈↓ α∧␈↓3.␈αAs␈α
a␈αrule␈α
of␈αconjecture.␈α
 This␈αuse␈αwas␈α
emphasized␈αin␈α
(McCarthy␈α1980).␈α
 The␈αcircumscriptions

␈↓ α∧␈↓may␈αbe␈αregarded␈αas␈αexpressions␈αof␈αsome␈αprobabilistic␈αnotions␈αsuch␈αas␈α"most␈αbirds␈αcan␈αfly"␈αor␈αthey

␈↓ α∧␈↓may␈α∞be␈α∞expressions␈α∂of␈α∞simple␈α∞cases.␈α∂ Thus␈α∞it␈α∞is␈α∂simple␈α∞to␈α∞conjecture␈α∂that␈α∞there␈α∞are␈α∂no␈α∞relevant

␈↓ α∧␈↓present␈α⊂material␈α⊂objects␈α⊂other␈α⊂than␈α∂those␈α⊂whose␈α⊂presence␈α⊂can␈α⊂be␈α∂inferred.␈α⊂ It␈α⊂is␈α⊂also␈α⊂a␈α∂simple

␈↓ α∧␈↓conjecture␈α∞that␈α∞a␈α∞tool␈α∞asserted␈α∞to␈α∞be␈α∂present␈α∞is␈α∞usable␈α∞for␈α∞its␈α∞normal␈α∞function.␈α∂ Such␈α∞conjecture

␈↓ α∧␈↓sometimes␈α∂conflict,␈α⊂but␈α∂there␈α∂is␈α⊂nothing␈α∂wrong␈α∂with␈α⊂having␈α∂incompatible␈α∂conjectures␈α⊂on␈α∂hand.

␈↓ α∧␈↓Besides␈α
the␈αpossibility␈α
of␈αdeciding␈α
that␈αone␈α
is␈αcorrect␈α
and␈α
the␈αother␈α
wrong,␈αit␈α
is␈αpossible␈α
to␈αuse␈α
one

␈↓ α∧␈↓for generating possible exceptions to the other.

␈↓ α∧␈↓4.␈α
As␈α
a␈αrepresentation␈α
of␈α
a␈αpolicy.␈α
 The␈α
example␈αis␈α
Doyle's␈α
"The␈αmeeting␈α
will␈α
be␈α
on␈αWednesday

␈↓ α∧␈↓unless another decision is explicitly made".

␈↓ α∧␈↓5.␈α∂As␈α∂a␈α⊂very␈α∂streamlined␈α∂expression␈α∂of␈α⊂probabilistic␈α∂information␈α∂when␈α⊂numerical␈α∂probabilities,

␈↓ α∧␈↓especially␈α⊗conditional␈α∃probabilities,␈α⊗are␈α∃unobtainable.␈α⊗ Since␈α∃circumscription␈α⊗doesn't␈α∃provide

␈↓ α∧␈↓numerical␈α⊗probabilities,␈α⊗its␈α⊗probabilistic␈α↔interpetation␈α⊗involves␈α⊗probabilities␈α⊗that␈α↔are␈α⊗either

␈↓ α∧␈↓infinitesimal,␈αwithin␈αan␈αinfinitesimal␈αof␈αone,␈αor␈αintermediate␈α-␈αwithout␈αany␈αdiscrimination␈αamong

␈↓ α∧␈↓the␈αintermediate␈αvalues.␈α The␈αcircumscriptions␈αgive␈αconditional␈αprobabilities.␈α Thus␈αwe␈α
may␈αtreat

␈↓ α∧␈↓the␈α
probability␈αthat␈α
a␈αbird␈α
can't␈α
fly␈αas␈α
an␈αinfinitesimal.␈α
 However,␈α
if␈αthe␈α
rare␈αevent␈α
occurs␈αthat␈α
the

␈↓ α∧␈↓bird␈α
is␈αa␈α
penguin,␈α
then␈αthe␈α
conditional␈α
probability␈αthat␈α
it␈α
can␈αfly␈α
is␈α
infinitesimal,␈αbut␈α
we␈αmay␈α
hear

␈↓ α∧␈↓of some rare condition that would allow it to fly after all.
␈↓ α∧␈↓␈↓ u4


␈↓ α∧␈↓␈↓ αTWhy␈αdon't␈α
we␈αuse␈α
finite␈αprobabilities␈α
combined␈αby␈α
the␈αusual␈α
laws?␈α That␈α
would␈αbe␈α
fine␈αif

␈↓ α∧␈↓we␈α
had␈α
the␈α
numbers,␈α
but␈α
circumscription␈α
is␈α
usable␈αwhen␈α
we␈α
can't␈α
get␈α
the␈α
numbers␈α
or␈α
find␈αtheir

␈↓ α∧␈↓use␈αinconvenient.␈α Note␈α
that␈αthe␈αgeneral␈αprobability␈α
that␈αa␈αbird␈αcan␈α
fly␈αmay␈αbe␈αirrelevant,␈α
because

␈↓ α∧␈↓we are interested in particular situations which weigh in favor or against a particular bird flying.

␈↓ α∧␈↓␈↓ αTNotice␈αthat␈αcircumscription␈αdoes␈αnot␈αprovide␈αfor␈αweighing␈αevidence;␈αit␈αis␈αappropriate␈αwhen

␈↓ α∧␈↓the␈α∩information␈α∩permits␈α⊃snap␈α∩decisions.␈α∩ However,␈α⊃many␈α∩cases␈α∩nominally␈α⊃treated␈α∩in␈α∩terms␈α⊃of

␈↓ α∧␈↓weighing␈α∞information␈α∞are␈α∂in␈α∞fact␈α∞cases␈α∂in␈α∞which␈α∞the␈α∞weights␈α∂are␈α∞such␈α∞that␈α∂circumscription␈α∞and

␈↓ α∧␈↓other defaults work better.

␈↓ α∧␈↓6.␈α
We␈αmight␈α
also␈α
speculate␈αthat␈α
certain␈α
laws␈αof␈α
common␈α
sense␈αphysics␈α
or␈α
common␈αsense␈α
psychology

␈↓ α∧␈↓are␈α∂inherently␈α∞non-monotonic␈α∂or,␈α∂more␈α∞specifically,␈α∂involve␈α∞circumscription.␈α∂ The␈α∂speculation␈α∞is

␈↓ α∧␈↓that this common sense information has some inherently preferred form.

␈↓ α∧␈↓␈↓ αTSix␈αdifferent␈α
uses␈αfor␈α
non-monotonic␈αreasoning␈α
seem␈αtoo␈α
many,␈αso␈α
perhaps␈αwe␈αcan␈α
condense

␈↓ α∧␈↓them.



␈↓ α∧␈↓2. Minimizing abnormality

␈↓ α∧␈↓␈↓ αTMany␈α⊂people␈α⊂have␈α⊂proposed␈α⊂representing␈α⊂facts␈α⊂about␈α⊂what␈α⊂is␈α⊂"normally"␈α⊂the␈α⊂case.␈α⊂ One

␈↓ α∧␈↓problem␈αis␈αthat␈αevery␈αobject␈αis␈αabnormal␈αin␈αsome␈αway,␈αand␈αwe␈αwant␈αto␈αallow␈αsome␈αaspects␈αof␈αthe

␈↓ α∧␈↓object␈αto␈αbe␈αabnormal␈αand␈αstill␈αassume␈αthe␈αnormality␈αof␈αthe␈αrest.␈α We␈αdo␈αthis␈αwith␈αa␈αpredicate␈α␈↓↓ab␈↓

␈↓ α∧␈↓standing␈α
for␈α
"abnormal".␈α We␈α
circumscribe␈α
␈↓↓ab z␈↓.␈α The␈α
argument␈α
of␈α␈↓↓ab␈↓␈α
will␈α
be␈αsome␈α
aspect␈α
of␈αthe

␈↓ α∧␈↓entities␈α
involved.␈α Some␈α
aspects␈α
can␈αbe␈α
abnormal␈α
without␈αaffecting␈α
others.␈α
 The␈αaspects␈α
themselves

␈↓ α∧␈↓are abstract entities, and their unintuitiveness is somewhat a blemish on the theory.

␈↓ α∧␈↓␈↓ αTNext we have some new sample applications.



␈↓ α∧␈↓3. Whether birds can fly
␈↓ α∧␈↓␈↓ u5


␈↓ α∧␈↓␈↓ αTMarvin␈α~Minsky␈α~(1982)␈α~offered␈α~expressing␈α~the␈α~facts␈α~and␈α~non-monotonic␈α→reasoning

␈↓ α∧␈↓concerning␈α⊂the␈α⊂ability␈α⊂of␈α⊂birds␈α⊂to␈α⊂fly␈α⊂as␈α∂a␈α⊂challenge␈α⊂to␈α⊂advocates␈α⊂of␈α⊂formal␈α⊂systems␈α⊂based␈α∂on

␈↓ α∧␈↓mathematical logic.

␈↓ α∧␈↓␈↓ αTI␈αhave␈αexplored␈αmany␈αways␈αof␈αnon-monotonically␈αaxiomatizing␈αthe␈αfacts␈αabout␈αwhich␈αbirds

␈↓ α∧␈↓can fly.  The following set of axioms using ␈↓↓ab␈↓ seems to me quite straightforward.

␈↓ α∧␈↓2)␈↓ αt ␈↓↓∀x.¬ab aspect1 x ⊃ ¬flies x␈↓.

␈↓ α∧␈↓Unless an object is abnormal in ␈↓↓aspect1,␈↓ it can't fly.

␈↓ α∧␈↓␈↓ αTNote␈α
that␈α
it␈α
wouldn't␈α
work␈α
to␈α
write␈α
␈↓↓ab␈α
x␈↓␈α
instead␈α
of␈α
␈↓↓ab␈α
aspect1␈α
x␈↓,␈α
because␈α
we␈α
don't␈α∞want␈α
a

␈↓ α∧␈↓bird␈α⊂that␈α⊂is␈α⊂abnormal␈α⊂with␈α⊂respect␈α⊂to␈α⊂its␈α⊂ability␈α⊂to␈α⊂fly␈α⊂to␈α⊂be␈α⊂automatically␈α⊂abnormal␈α⊂in␈α⊂other

␈↓ α∧␈↓respects.  Using aspects limits the effects of proofs of abnormality.

␈↓ α∧␈↓3)␈↓ αt ␈↓↓∀x.bird x ⊃ ab aspect1 x␈↓.

␈↓ α∧␈↓A␈α∂bird␈α⊂is␈α∂abnormal␈α⊂in␈α∂␈↓↓aspect1,␈↓␈α∂so␈α⊂(2)␈α∂can't␈α⊂be␈α∂used␈α⊂to␈α∂show␈α∂it␈α⊂can't␈α∂fly.␈α⊂ If␈α∂this␈α⊂axiom␈α∂were

␈↓ α∧␈↓omitted,␈αwhen␈αwe␈αdid␈αthe␈αcircumscription␈αwe␈αwould␈αonly␈αbe␈αable␈αto␈αinfer␈αa␈αdisjunction.␈α Either␈αa

␈↓ α∧␈↓bird␈αis␈αabnormal␈αin␈α␈↓↓aspect1␈↓␈αor␈αit␈αcan␈α
fly␈αunless␈αit␈αis␈αabnormal␈αin␈α␈↓↓aspect2.␈↓␈α(3)␈αestablishes␈α
expresses

␈↓ α∧␈↓our␈αpreference␈αfor␈αinferring␈αthat␈αa␈αbird␈αis␈αabnormal␈αin␈α␈↓↓aspect1␈↓␈αrather␈αthan␈α␈↓↓aspect2.␈↓␈αWe␈αcall␈α(3)␈αa

␈↓ α∧␈↓␈↓↓cancellation of inheritance␈↓ axiom.  We will see more of them.

␈↓ α∧␈↓4)␈↓ αt ␈↓↓∀x.bird x ∧ ¬ab aspect2 x ⊃ flies x␈↓.

␈↓ α∧␈↓Unless a bird is abnormal in ␈↓↓aspect2,␈↓ it can fly.

␈↓ α∧␈↓5)␈↓ αt ␈↓↓∀x.ostrich x ⊃ ab aspect2 x␈↓.

␈↓ α∧␈↓Ostriches␈α
are␈α
abnormal␈α
in␈α
␈↓↓aspect2.␈↓␈α
This␈α
doesn't␈α
say␈αthat␈α
an␈α
ostrich␈α
cannot␈α
fly␈α
-␈α
merely␈α
that␈α(4)

␈↓ α∧␈↓can't be used to infer that it does.  (5) is another cancellation of inheritance axiom.

␈↓ α∧␈↓6)␈↓ αt ␈↓↓∀x.penguin x ⊃ ab aspect2 x␈↓.

␈↓ α∧␈↓Penguins are also abnormal in ␈↓↓aspect2.␈↓

␈↓ α∧␈↓7)␈↓ αt ␈↓↓∀x.ostrich x ∧ ¬ab aspect3 x ⊃ ¬flies x␈↓.

␈↓ α∧␈↓8)␈↓ αt ␈↓↓∀x.penguin x ∧¬ab aspect4 x ⊃ ¬flies x␈↓.
␈↓ α∧␈↓␈↓ u6


␈↓ α∧␈↓Normally␈α∞ostriches␈α∂and␈α∞penguins␈α∞can't␈α∂fly.␈α∞ However,␈α∞there␈α∂is␈α∞an␈α∞out.␈α∂ (7)␈α∞and␈α∞(8)␈α∂provide␈α∞that

␈↓ α∧␈↓under␈α⊃unspecified␈α⊃conditions,␈α⊃an␈α⊃ostrich␈α⊃or␈α⊃penguin␈α⊃might␈α⊃fly␈α⊃after␈α⊃all.␈α⊃ If␈α⊃we␈α⊃give␈α∩no␈α⊃such

␈↓ α∧␈↓conditions,␈αwe␈α
will␈αconclude␈α
that␈αan␈α
ostrich␈αor␈α
penguin␈αcan't␈α
fly.␈α Additional␈α
objects␈αthat␈α
can␈αfly

␈↓ α∧␈↓may␈α∂be␈α∂specified.␈α⊂ Each␈α∂needs␈α∂two␈α∂axioms.␈α⊂ The␈α∂first␈α∂says␈α∂that␈α⊂it␈α∂is␈α∂abnormal␈α∂in␈α⊂␈↓↓aspect1␈↓␈α∂and

␈↓ α∧␈↓prevents␈α(2)␈αfrom␈αbeing␈αused␈αto␈αsay␈αthat␈αit␈αcan't␈αfly.␈α The␈αsecond␈αprovides␈αthat␈αit␈αcan␈αfly␈αunless␈αit

␈↓ α∧␈↓is abnormal in yet another way.  Likewise additional non-flying birds can be provided for.

␈↓ α∧␈↓␈↓ αTWe␈α
haven't␈αyet␈α
said␈αthat␈α
ostriches␈αand␈α
penguins␈αare␈α
birds,␈αso␈α
let's␈αdo␈α
that␈αand␈α
throw␈αin␈α
that

␈↓ α∧␈↓canaries are birds also.

␈↓ α∧␈↓9)␈↓ αt ␈↓↓∀x.ostrich x ⊃ bird x␈↓

␈↓ α∧␈↓10)␈↓ αt ␈↓↓∀x.penguin x ⊃ bird x␈↓

␈↓ α∧␈↓11)␈↓ αt␈α␈↓↓∀x.canary␈α
x␈α∧␈α
¬ab␈αaspect5␈α
x␈α⊃␈α
bird␈αx␈↓␈α
We␈αthrew␈α
in␈α␈↓↓aspect5␈↓␈α
just␈αin␈α
case␈αone␈α
wanted␈αto␈α
use

␈↓ α∧␈↓the term canary in the sense of a 1930s gangster movie.

␈↓ α∧␈↓Asserting␈α
that␈α
ostriches,␈α
penguins␈α
and␈α
canaries␈α∞are␈α
birds␈α
will␈α
help␈α
inherit␈α
other␈α∞properties␈α
from

␈↓ α∧␈↓the class of birds.  For example, we have

␈↓ α∧␈↓12)␈↓ αt ␈↓↓∀x. bird x ∧ ¬ab aspect6 x ⊃ feathered x␈↓.

␈↓ α∧␈↓So␈αfar␈αthere␈α
is␈αnothing␈αto␈α
prevent␈αostriches,␈αpenguins␈αand␈α
canaries␈αfrom␈αoverlapping.␈α
 We␈αcould

␈↓ α∧␈↓write disjointness axioms like

␈↓ α∧␈↓13)␈↓ αt␈↓↓∀x. ¬ostrich x ∨ ¬penguin x␈↓,

␈↓ α∧␈↓but␈α
we␈α
would␈α
require␈α
␈↓↓n␈↓∧2␈↓␈α
of␈α
them␈α
if␈α
we␈α
have␈α
␈↓↓n␈↓␈α
species.␈α
 This␈α
problem␈α
is␈α
like␈α
the␈α
unique␈αnames

␈↓ α∧␈↓problem.



␈↓ α∧␈↓4. Some examples of Raymond Reiter

␈↓ α∧␈↓␈↓ αTReiter␈α∃asks␈α∃about␈α∃representing,␈α∃"Quakers␈α∀are␈α∃normally␈α∃pacifists␈α∃and␈α∃Republicans␈α∀are

␈↓ α∧␈↓normally non-pacificists.  How about Nixon, who is both a Quaker and a Republican?"

␈↓ α∧␈↓We use
␈↓ α∧␈↓␈↓ u7


␈↓ α∧␈↓14)␈↓ αt ␈↓↓∀x.quaker x ∧ ¬ab aspect1 x ⊃ pacifist x␈↓,

␈↓ α∧␈↓15)␈↓ αt ␈↓↓∀x.republican x ∧ ¬ab aspect2 x ⊃ ¬ pacifist x␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓16)␈↓ αt ␈↓↓quaker Nixon ∧ republican Nixon␈↓.

␈↓ α∧␈↓␈↓ αTWhen␈α∞we␈α∞circumscribe␈α∂␈↓↓ab z␈↓␈α∞using␈α∞these␈α∞three␈α∂sentences␈α∞as␈α∞␈↓↓A(ab),␈↓␈α∞we␈α∂will␈α∞only␈α∞be␈α∂able␈α∞to

␈↓ α∧␈↓conclude␈αthat␈αNixon␈αis␈α
either␈αabnormal␈αin␈α␈↓↓aspect1␈↓␈α
or␈αin␈α␈↓↓aspect2,␈↓␈αand␈αwe␈α
will␈αnot␈αbe␈αable␈α
to␈αsay

␈↓ α∧␈↓whether he is a pacifist.

␈↓ α∧␈↓␈↓ αTReiter's␈αsecond␈αexample␈αis␈αthat␈αa␈αperson␈αnormally␈αlives␈αin␈αthe␈αsame␈αcity␈αas␈αhis␈αwife␈α
and␈αin

␈↓ α∧␈↓the␈αsame␈αcity␈αas␈αhis␈αemployer.␈α But␈αA's␈αwife␈αlives␈αin␈αVancouver␈αand␈αA's␈αemployer␈αis␈αin␈αToronto.

␈↓ α∧␈↓We write

␈↓ α∧␈↓17)␈↓ αt ␈↓↓∀x.¬ab aspect1 x ⊃ city x = city wife x␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓18)␈↓ αt ␈↓↓∀x.¬ab aspect2 x ⊃ city x = city employer x␈↓.

␈↓ α∧␈↓If we have

␈↓ α∧␈↓19)␈↓ αt ␈↓↓city wife A = Vancouver ∧ city employer A = Toronto ∧ Toronto ≠ Vancouver␈↓,

␈↓ α∧␈↓we will again only be able to conclude that A lives either in Toronto or Vancouver.



␈↓ α∧␈↓5. A More general treatment of an ␈↓↓is-a␈↓ hierarchy

␈↓ α∧␈↓␈↓ αTThe␈α∞bird␈α∞example␈α∞works␈α∞fine␈α∞when␈α∞a␈α∞fixed␈α∞␈↓↓is-a␈↓␈α∞hierarchy␈α∞is␈α∞in␈α∞question.␈α∞ However,␈α
our

␈↓ α∧␈↓writing␈αthe␈αinheritance␈αcancellation␈αaxioms␈αdepended␈αon␈αknowing␈αexactly␈αfrom␈αwhat␈αhigher␈αlevel

␈↓ α∧␈↓the␈α∪properties␈α∩were␈α∪inherited.␈α∩ This␈α∪doesn't␈α∩correspond␈α∪to␈α∩my␈α∪intuition␈α∩of␈α∪how␈α∪we␈α∩humans

␈↓ α∧␈↓represent␈α∩inheritance.␈α∩ It␈α∩would␈α∪seem␈α∩rather␈α∩that␈α∩when␈α∩we␈α∪say␈α∩that␈α∩birds␈α∩can␈α∩fly,␈α∪we␈α∩don't

␈↓ α∧␈↓necessarily␈α∞have␈α∞in␈α∞mind␈α∞that␈α
an␈α∞inheritance␈α∞of␈α∞inability␈α∞to␈α
fly␈α∞from␈α∞things␈α∞in␈α∞general␈α∞is␈α
being

␈↓ α∧␈↓cancelled.␈α
 We␈α
can␈α
formulate␈α
inheritance␈α
of␈α
properties␈α
in␈α
a␈α
more␈α
general␈α
way␈α
provided␈α
we␈αreify

␈↓ α∧␈↓the properties.  Presumably there are many ways of doing this, but here's one that seems to work.
␈↓ α∧␈↓␈↓ u8


␈↓ α∧␈↓␈↓ αTThe␈α∂first␈α∂order␈α∂variables␈α∂of␈α∂our␈α∂theory␈α∂range␈α∂over␈α∂classes␈α∂of␈α∂objects␈α∂(denoted␈α∂by␈α∂␈↓↓c␈↓␈α∞with

␈↓ α∧␈↓numerical␈α
suffixes),␈α
properties␈α
(denoted␈α
by␈α
␈↓↓p)␈↓␈α
and␈α
objects␈α
(denoted␈α
by␈α
␈↓↓x).␈↓␈α
We␈α
need␈α∞not␈α
identify

␈↓ α∧␈↓classes␈α∩with␈α∩sets␈α∩(or␈α∩with␈α∩the␈α∩classes␈α∩of␈α∩GB␈α∩set␈α∩theory).␈α∩ In␈α∩particular,␈α∩we␈α∩need␈α∩not␈α⊃assume

␈↓ α∧␈↓extensionality.  We have several predicates:

␈↓ α∧␈↓␈↓ αT@ordinarily(c,p)␈αmeans␈αthat␈αobjects␈α
of␈αclass␈α␈↓↓c␈↓␈αordinarily␈α
have␈αproperty␈α␈↓↓p.␈↓␈α␈↓↓c1 ≤ c2␈↓␈αmeans␈α
that

␈↓ α∧␈↓class␈α␈↓↓c1␈↓␈α
ordinarily␈αinherits␈α
from␈αclass␈α
␈↓↓c2.␈↓␈αWe␈α
assume␈αthat␈α
this␈αrelation␈α
is␈αtransitive.␈α ␈↓↓in(x,c)␈↓␈α
means

␈↓ α∧␈↓that the object ␈↓↓x␈↓ is in class ␈↓↓c.␈↓ Our axioms are

␈↓ α∧␈↓20)␈↓ αt ␈↓↓∀c1 c2 c3. c1 ≤ c2 ∧ c2 ≤ c3 ⊃ c1 ≤ c3␈↓,

␈↓ α∧␈↓21)␈↓ αt ␈↓↓∀c1 c2 p.ordinarily(c1,p) ∧ c1 ≤ c2 ∧ ¬ab aspect1(c1,c2,p) ⊃ ordinarily(c2,p)␈↓,

␈↓ α∧␈↓22)␈↓ αt ␈↓↓∀c1 c2 c3 p.c1 ≤ c2 ∧ c2 ≤ c3 ∧ ordinarily(c2, not p) ⊃ ab aspect1(c1,c3,p)␈↓,

␈↓ α∧␈↓23)␈↓ αt ␈↓↓∀x c p.in(x,c) ∧ ordinarily(c,p) ∧ ¬ab aspect3(x,c,p) ⊃ ap(p,x)␈↓,

␈↓ α∧␈↓24)␈↓ αt ∀x c1 c2 p.in(x,c1) ∧ c1 ≤ c2 ∧ ordinarily(c1,not p) ⊃ ab aspect3(x,c2,p).

␈↓ α∧␈↓(20)␈α
is␈α
the␈α
afore-mentioned␈α
transitivity␈αof␈α
≤.␈α
 (21)␈α
says␈α
that␈αproperties␈α
that␈α
ordinarily␈α
hold␈α
for␈αa

␈↓ α∧␈↓class␈α⊃are␈α⊃inherited␈α∩unless␈α⊃something␈α⊃is␈α∩abnormal.␈α⊃ (22)␈α⊃cancels␈α⊃the␈α∩inheritance␈α⊃if␈α⊃there␈α∩is␈α⊃an

␈↓ α∧␈↓intermediate␈αclass␈αfor␈αwhich␈αthe␈αproperty␈αordinarily␈αdoesn't␈αhold.␈α (23)␈αsays␈αthat␈αproperties␈αwhich

␈↓ α∧␈↓ordinarily␈αhold␈αactually␈αhold␈αfor␈αelements␈αof␈αthe␈αclass␈αunless␈αsomething␈αis␈αabnormal.␈α (24)␈αcancels

␈↓ α∧␈↓the␈α⊂effect␈α⊂of␈α⊂(23)␈α⊂when␈α⊂their␈α⊂is␈α⊂an␈α⊂intermediate␈α⊂class␈α⊂for␈α⊂which␈α⊂the␈α⊂negation␈α⊂of␈α⊃the␈α⊂property

␈↓ α∧␈↓ordinarily␈α⊂holds.␈α⊃ Notice␈α⊂that␈α⊃this␈α⊂reification␈α⊂of␈α⊃properties␈α⊂seems␈α⊃to␈α⊂require␈α⊃imitation␈α⊂boolean

␈↓ α∧␈↓operators.  Such operators are discussed in (McCarthy 1979).



␈↓ α∧␈↓6. General considerations

␈↓ α∧␈↓␈↓ αTSuppose␈αwe␈α
have␈αa␈αdata␈α
base␈αof␈αfacts␈α
axiomatized␈αby␈αa␈α
formalism␈αinvolving␈α
the␈αpredicate

␈↓ α∧␈↓␈↓↓ab.␈↓␈α∩In␈α∩connection␈α∩with␈α∩a␈α∩particular␈α∩problem,␈α∩a␈α∩program␈α∩takes␈α∩a␈α∩subcollection␈α∩of␈α∩these␈α⊃facts

␈↓ α∧␈↓together␈αwith␈αthe␈αspecific␈αfacts␈αof␈αthe␈αproblem␈α
and␈αthe␈αcircumscribes␈α␈↓↓ab␈↓␈αx.␈α We␈αget␈αa␈αsecond␈α
order
␈↓ α∧␈↓␈↓ u9


␈↓ α∧␈↓formula,␈αand␈αin␈αgeneral,␈αas␈αthe␈αnatural␈αnumber␈αexample␈αof␈α(McCarthy␈α1980)␈αshows,␈αthis␈αformula

␈↓ α∧␈↓is␈α∀not␈α∀equivalent␈α∀to␈α∃any␈α∀first␈α∀order␈α∀formula.␈α∃ However,␈α∀many␈α∀common␈α∀sense␈α∃domains␈α∀are

␈↓ α∧␈↓axiomatizable␈αin␈α
such␈αa␈α
way␈αthat␈αthe␈α
circumscription␈αis␈α
equivalent␈αto␈αa␈α
first␈αorder␈α
formula.␈α For

␈↓ α∧␈↓example,␈α⊂Vladimir␈α⊂Lifschitz␈α⊂(1983)␈α⊂has␈α⊂shown␈α⊂that␈α∂this␈α⊂is␈α⊂true␈α⊂if␈α⊂the␈α⊂axioms␈α⊂are␈α⊂a␈α∂universal

␈↓ α∧␈↓formula␈α⊂and␈α⊂there␈α∂are␈α⊂no␈α⊂functions.␈α∂ This␈α⊂can␈α⊂presumably␈α∂be␈α⊂extended␈α⊂to␈α∂the␈α⊂case␈α⊂when␈α∂the

␈↓ α∧␈↓ranges␈αand␈α
domains␈αof␈α
the␈αfunctions␈αare␈α
disjoint,␈αso␈α
that␈αthere␈αis␈α
no␈αway␈α
of␈αgenerating␈αan␈α
infinity

␈↓ α∧␈↓of elements.

␈↓ α∧␈↓␈↓ αTWe␈α⊃can␈α⊃then␈α⊃regard␈α⊃the␈α⊃process␈α⊃of␈α⊃deciding␈α⊃what␈α⊃facts␈α⊃to␈α⊃take␈α⊃into␈α⊃account␈α⊃and␈α⊂then

␈↓ α∧␈↓circumscribing␈α
as␈α
a␈αprocess␈α
of␈α
compiling␈αfrom␈α
a␈α
slightly␈αhigher␈α
level␈α
non-monotonic␈αlanguage␈α
into

␈↓ α∧␈↓mathematical␈α∞logic,␈α∂especially␈α∞first␈α∂order␈α∞logic.␈α∂ We␈α∞can␈α∂also␈α∞regard␈α∂natural␈α∞language␈α∂as␈α∞higher

␈↓ α∧␈↓level␈α∃than␈α∃logic,␈α∃although,␈α∃as␈α∃I␈α∃shall␈α∃discuss␈α∃elsewhere,␈α∃natural␈α∃language␈α∃doesn't␈α∃have␈α∀an

␈↓ α∧␈↓independent␈α∪reasoning␈α∪process,␈α∪because␈α∪most␈α∪natural␈α∪language␈α∪inferences␈α∀involve␈α∪suppressed

␈↓ α∧␈↓premisses␈α∞which␈α∞are␈α∞not␈α∂represented␈α∞in␈α∞natural␈α∞language␈α∞in␈α∂the␈α∞minds␈α∞of␈α∞the␈α∞people␈α∂doing␈α∞the

␈↓ α∧␈↓reasoning.



␈↓ α∧␈↓7. An example of doing the circumscription.

␈↓ α∧␈↓␈↓ αTIn␈αorder␈αto␈αkeep␈αthe␈α
example␈αshort␈αwe␈αwill␈αtake␈α
into␈αaccount␈αonly␈αthe␈αfollowing␈α
facts␈αfrom

␈↓ α∧␈↓the previous section.

␈↓ α∧␈↓2) ␈↓↓∀x.¬ab aspect1 x ⊃ ¬flies x␈↓.

␈↓ α∧␈↓3) ␈↓↓∀x.bird x ⊃ ab aspect1 x␈↓.

␈↓ α∧␈↓4) ␈↓↓∀x.bird x ∧ ¬ab aspect2 x ⊃ flies x␈↓.

␈↓ α∧␈↓5) ␈↓↓∀x.ostrich x ⊃ ab aspect2 x␈↓.

␈↓ α∧␈↓7) ␈↓↓∀x.ostrich x ∧ ¬ab aspect3 x ⊃ ¬flies x␈↓.

␈↓ α∧␈↓␈↓ αTTheir conjunction is taken as ␈↓↓A(ab,flies)␈↓.  The circumscription formula ␈↓↓A'(ab,flies)␈↓ is then

␈↓ α∧␈↓25)␈↓ αt ␈↓↓A(ab,flies) ∧ ∀ab' flies'.[A(ab',flies') ∧ [∀x. ab' x ⊃ ab x] ⊃ [∀x. ab x ≡ ab' x]]␈↓,
␈↓ α∧␈↓␈↓ f10


␈↓ α∧␈↓which spelled out becomes

␈↓ α∧␈↓26)␈↓ αt␈α␈↓↓[∀x.¬ab␈α
aspect1␈αx␈α
⊃␈α¬flies␈αx]␈α
∧␈α[∀x.bird␈α
x␈α⊃␈α
ab␈αaspect1␈αx]␈α
∧␈α[∀x.bird␈α
x␈α∧␈α
¬ab␈αaspect2␈αx␈α
⊃

␈↓ α∧␈↓↓flies␈α⊃x]␈α⊃∧␈α⊃[∀x.ostrich␈α⊃x␈α⊃⊃␈α⊃ab␈α⊃aspect2␈α⊃x]␈α∩∧␈α⊃[∀x.ostrich␈α⊃x␈α⊃∧␈α⊃¬ab␈α⊃aspect3␈α⊃x␈α⊃⊃␈α⊃¬flies␈α⊃x]␈α∩∧␈α⊃∀ab'

␈↓ α∧␈↓↓flies'.[[∀x.¬ab'␈αaspect1␈αx␈α⊃␈α¬flies'␈αx]␈α∧␈α[∀x.bird␈αx␈α⊃␈αab'␈αaspect1␈αx]␈α∧␈α[∀x.bird␈αx␈α∧␈α¬ab'␈αaspect2␈αx

␈↓ α∧␈↓↓⊃␈α
flies'␈α
x]␈α
∧␈α
[∀x.ostrich␈α
x␈α
⊃␈α
ab'␈α
aspect2␈α
x]␈α
∧␈α
[∀x.ostrich␈α
x␈α
∧␈α
¬ab'␈α
aspect3␈α
x␈α
⊃␈α
¬flies'␈α
x]␈α
∧␈α
[∀x.ab'␈α
x

␈↓ α∧␈↓↓⊃ ab x] ⊃ [∀x.ab x ≡ ab' x]]␈↓.

␈↓ α∧␈↓␈↓ αT␈↓↓A(ab,flies)␈↓␈α
is␈α
guaranteed␈α
to␈α
be␈α
true,␈α
because␈α
it␈α
is␈α
part␈α
of␈α
what␈α
is␈α
assumed␈α
in␈α
our␈αcommon

␈↓ α∧␈↓sense data base.  Therfore (26) reduces to

␈↓ α∧␈↓27)␈↓ αt␈↓↓∀ab'␈αflies'.[[∀x.¬ab'␈αaspect1␈α
x␈α⊃␈α¬flies'␈α
x]␈α∧␈α[∀x.bird␈αx␈α
⊃␈αab'␈αaspect1␈α
x]␈α∧␈α[∀x.bird␈α
x␈α∧

␈↓ α∧␈↓↓¬ab'␈α∂aspect2␈α∞x␈α∂⊃␈α∞flies'␈α∂x]␈α∞∧␈α∂[∀x.ostrich␈α∞x␈α∂⊃␈α∞ab'␈α∂aspect2␈α∞x]␈α∂∧␈α∞[∀x.ostrich␈α∂x␈α∞∧␈α∂¬ab'␈α∞aspect3␈α∂x␈α∞⊃

␈↓ α∧␈↓↓¬flies' x] ∧ [∀x.ab' x ⊃ ab x] ⊃ [∀x.ab x ≡ ab' x]]␈↓.

␈↓ α∧␈↓␈↓ αTOur␈αobjective␈αis␈αnow␈αto␈αmake␈αsuitable␈αsubstitutions␈αfor␈α␈↓↓ab'␈↓␈αand␈α␈↓↓flies'␈↓␈αso␈αthat␈αall␈α
the␈αterms

␈↓ α∧␈↓except␈αthe␈α
last␈αof␈α
(27)␈αwill␈α
be␈αtrue,␈αand␈α
the␈αlast␈α
term␈αwill␈α
determine␈α␈↓↓ab.␈↓␈α
The␈αaxiom␈α␈↓↓A(ab,flies)␈↓␈α
will

␈↓ α∧␈↓then␈α
determine␈α∞␈↓↓flies,␈↓␈α
i.e.␈α
we␈α∞will␈α
know␈α
what␈α∞the␈α
fliers␈α
are.␈α∞ ␈↓↓flies'␈↓␈α
is␈α
easy,␈α∞because␈α
we␈α∞need␈α
only

␈↓ α∧␈↓apply␈αwishful␈αthinking;␈αwe␈αwant␈αthe␈αfliers␈αto␈αjust␈αthose␈αbirds␈αthat␈αaren't␈αostriches.␈α Therefore,␈αwe

␈↓ α∧␈↓put

␈↓ α∧␈↓28)␈↓ αt ␈↓↓flies' x ≡ bird x ∧ ¬ostrich x␈↓.

␈↓ α∧␈↓␈↓ αT␈↓↓ab'␈↓ isn't really much more difficult, but there is a notational problem.  We define

␈↓ α∧␈↓29)␈↓ αt ␈↓↓ab' z ≡ [∃x.bird x ∧ z = aspect1 x] ∨ [∃x.ostrich x ∧ z = aspect2 x]␈↓,

␈↓ α∧␈↓which covers the cases we want to be abnormal.

␈↓ α∧␈↓[This proof isn't finished in this draft].



␈↓ α∧␈↓8. The blocks world

␈↓ α∧␈↓␈↓ αTThe␈αfolowing␈αset␈αof␈α
"situation␈αcalculus"␈αaxioms␈αsolves␈αthe␈α
frame␈αproblem␈αfor␈αa␈αblocks␈α
world

␈↓ α∧␈↓in which blocks can be moved and painted.
␈↓ α∧␈↓␈↓ f11


␈↓ α∧␈↓30)␈↓ αt ␈↓↓∀x e s.¬ab aspect1(x,e,s) ⊃ location(x,result(e,s)) = location(x,s)␈↓.

␈↓ α∧␈↓31)␈↓ αt ␈↓↓∀x e s.¬ab aspect2(x,e,s) ⊃ color(x,result(e,s)) = color(x,s)␈↓.

␈↓ α∧␈↓Objects change their locations and colors only for a reason.

␈↓ α∧␈↓32)␈↓ αt ␈↓↓∀x l s.ab aspect1(x,move(x,l),s)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓␈↓ αT␈↓↓∀x l s.¬ab aspect3(x,l,s) ⊃ location(x,result(move(x,l),s)) = l␈↓.

␈↓ α∧␈↓33)␈↓ αt ␈↓↓∀x c s.ab aspect2(x,paint(x,c),s)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓␈↓ αT␈↓↓∀x c s.¬ab aspect4(x,c,s) ⊃ color(x,result(paint(x,c),s)) = c␈↓.

␈↓ α∧␈↓Objects change their locations when moved and their colors when painted.

␈↓ α∧␈↓34)␈↓ αt ␈↓↓∀x l s.¬clear(top x,s) ∨ ¬clear(l,s) ∨ tooheavy x ∨ l = top x ⊃ ab aspect3(x,move(x,l),s)␈↓.

␈↓ α∧␈↓This␈αprevents␈αthe␈αrule␈α(32)␈αfrom␈αbeing␈αused␈αto␈αinfer␈αthat␈αan␈αobject␈αwill␈αmove␈αif␈αit␈αisn't␈αclear␈αor␈α
to

␈↓ α∧␈↓a destination that isn't clear or if the object is too heavy.

␈↓ α∧␈↓35)␈↓ αt ␈↓↓∀l s.clear(l,s) ≡ ¬∃x.(¬trivial x ∧ location(x,s) = l)␈↓.

␈↓ α∧␈↓A location is clear if all the objects there are trivial.

␈↓ α∧␈↓36)␈↓ αt ␈↓↓∀x.¬ab aspect7 x ⊃ ¬trivial x␈↓.

␈↓ α∧␈↓Trivial objects are abnormal in ␈↓↓aspect7␈↓.



␈↓ α∧␈↓9. References:

␈↓ α∧␈↓␈↓αMcCarthy,␈α∪John␈α∩(1979)␈↓:␈α∪"First␈α∩Order␈α∪Theories␈α∩of␈α∪Individual␈α∩Concepts␈α∪and␈α∪Propositions",␈α∩in

␈↓ α∧␈↓Michie, Donald (ed.) ␈↓↓Machine Intelligence 9␈↓, (University of Edinburgh Press, Edinburgh).

␈↓ α∧␈↓␈↓αMcCarthy,␈α⊃John␈α⊃(1980)␈↓:␈α⊃"Circumscription␈α∩-␈α⊃A␈α⊃Form␈α⊃of␈α⊃Non-Monotonic␈α∩Reasoning",␈α⊃␈↓↓Artificial

␈↓ α∧␈↓↓Intelligence␈↓, Volume 13, Numbers 1,2, April.

␈↓ α∧␈↓␈↓αEtherington,␈α→David␈α→W.␈α→and␈α→Raymond␈α→Reiter␈α→(1983)␈↓:␈α→"On␈α→Inheritance␈α~Hierarchies␈α→with
␈↓ α∧␈↓␈↓ f12


␈↓ α∧␈↓Exceptions",␈α∀in␈α∀␈↓↓Proceedings␈α∀of␈α∀the␈α∀National␈α∀Conference␈α∀on␈α∀Artificial␈α∀Intelligence,␈α∀AAAI-83␈↓,

␈↓ α∧␈↓William Kaufman, Inc.