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␈↓ α∧␈↓α␈↓ ∧QKNOWLEDGE OF TELEPHONE NUMBERS

␈↓ α∧␈↓␈↓ αTAn␈αintelligent␈αprogram␈αwill␈αhave␈αto␈αreason␈αabout␈αwho␈αknows␈αwhat␈αin␈αorder␈αto␈αpredict␈αwhat
␈↓ α∧␈↓others␈α
will␈α
do.␈α
 It␈α
will␈α
also␈α
have␈α
to␈α
reason␈α
that␈αif␈α
it␈α
looks␈α
a␈α
number␈α
up␈α
in␈α
a␈α
telephone␈α
book,␈αthen␈α
it
␈↓ α∧␈↓will␈αknow␈α
the␈αnumber.␈α The␈α
object␈αof␈αthis␈α
note␈αis␈α
to␈αintroduce␈αa␈α
formalism␈αsuitable␈αfor␈α
expressing
␈↓ α∧␈↓such facts.  The formalism has the following characteristics:

␈↓ α∧␈↓␈↓ αT1.␈α
The␈α
thing␈α
known␈α
is␈α
a␈α
term,␈α
i.e.␈α
a␈α
noun␈α
phrase␈α
like␈α
"telephone(Mike)"␈α
or␈α
"Mike's␈α
telephone
␈↓ α∧␈↓number".␈α
 This␈α
differs␈α
from␈αthe␈α
treatments␈α
common␈α
in␈αphilosophical␈α
logic␈α
in␈α
two␈α
respects.␈α First
␈↓ α∧␈↓modal␈αtreatments␈αare␈αmore␈αpopular␈αnow␈αthan␈αtreatments␈αin␈αwhich␈αthe␈αthing␈αknown␈αis␈αa␈αsentence.
␈↓ α∧␈↓We␈αwill␈αexplain␈αour␈αreasons␈α
for␈αour␈αpreference␈αlater.␈α Second,␈α
even␈αwhen␈αthe␈αobject␈αis␈αsyntactic,␈α
it
␈↓ α∧␈↓is␈α∞a␈α
sentence␈α∞rather␈α∞than␈α
a␈α∞term␈α
and␈α∞the␈α∞meaning␈α
is␈α∞knowing␈α
that␈α∞the␈α∞sentence␈α
is␈α∞true.␈α∞ In␈α
our
␈↓ α∧␈↓case,␈αeven␈αwhen␈αthe␈αobject␈αis␈αa␈αsentence,␈αthe␈αmeaning␈αis␈αknowing␈αits␈αtruth␈αvalue␈α-␈αwhich␈αmay␈αbe
␈↓ α∧␈↓false.  The relation between these treatments will be explained later, but ours seems more general.

␈↓ α∧␈↓␈↓ αT2.␈α
We␈αwill␈α
restrict␈α
ourselves␈αfor␈α
now␈α
to␈αproblems␈α
in␈α
which␈αthe␈α
value␈α
of␈αthe␈α
noun␈α
phrase␈αis␈α
a
␈↓ α∧␈↓LISP␈αexpression␈αsuch␈αas␈α
a␈αnumber␈αor␈αa␈α
string.␈α Thus␈αwe␈αwill␈α
avoid␈αthe␈αknotty␈αquestion␈α
of␈αwhat
␈↓ α∧␈↓you␈α∞know␈α∞when␈α∞you␈α∞say␈α∞you␈α∞know␈α∞John.␈α∞ However,␈α∞we␈α∞will␈α∞be␈α∞able␈α∞to␈α∞say␈α∞we␈α∞know␈α
someone's
␈↓ α∧␈↓name or address or telephone number or the truth value of an assertion about him.

␈↓ α∧␈↓␈↓ αTWe␈αwill␈αuse␈αthe␈αsituation␈αformalism␈αof␈α(McCarthy␈αand␈αHayes␈α1970),␈αand␈αa␈αtypical␈αsentence
␈↓ α∧␈↓is

␈↓ α∧␈↓␈↓ αT␈↓↓knows(Pat,"telephone(Mike)",S0)␈↓

␈↓ α∧␈↓meaning␈αthat␈αPat␈α
knows␈αMike's␈αtelephone␈α
number␈αin␈αsituation␈α
␈↓↓S0.␈↓␈αSuppose␈αwe␈α
want␈αto␈αargue␈α
that
␈↓ α∧␈↓Pat␈αcan␈α
get␈αin␈αconversation␈α
with␈αMike␈αby␈α
looking␈αup␈α
his␈αtelephone␈αnumber␈α
in␈αthe␈αtelephone␈α
book
␈↓ α∧␈↓and dialing the number he finds.  The ␈↓↓physics␈↓ of the situation is given by

␈↓ α∧␈↓1) ␈↓↓∀p1 p2 s. speaking(p1,p2,result(p1,dial(telephone(p2)),s)).␈↓

␈↓ α∧␈↓which␈αasserts␈αthat␈αa␈αsituation␈αin␈αwhich␈α␈↓↓p1␈↓␈αis␈αspeaking␈αto␈α␈↓↓p2␈↓␈αresults␈αwhen␈α␈↓↓p1␈↓␈αdials␈α␈↓↓p2␈↓'s␈αtelephone
␈↓ α∧␈↓number.␈α∞ We␈α
are␈α∞ignoring␈α∞qualifications␈α
such␈α∞as␈α
␈↓↓p2␈↓␈α∞being␈α∞home␈α
that␈α∞don't␈α
interest␈α∞us␈α∞for␈α
now.
␈↓ α∧␈↓Substituting Pat, Mike and ␈↓↓S0␈↓ for ␈↓↓p1,␈↓ ␈↓↓p2␈↓ and ␈↓↓s␈↓ gives particularized sentences like

␈↓ α∧␈↓2) ␈↓↓speaking(Pat,Mike,result(Pat,dial(telephone(Mike)),S0)).␈↓

␈↓ α∧␈↓Now␈αwe␈αwant␈αto␈αexpress␈α
the␈αcondition␈αthat␈αPat␈αknows␈α
Mike's␈αtelephone␈αnumber.␈α We␈αcan't␈α
do␈αit
␈↓ α∧␈↓by␈αtinkering␈α
with␈α1),␈αbecause␈α
1)␈αis␈α
still␈αtrue␈αwhether␈α
Pat␈αknows␈αMike's␈α
telephone␈αnumber␈α
or␈αnot,
␈↓ α∧␈↓e.g.␈α
he␈α
may␈α
dial␈α
it␈α
under␈α
the␈α
impression␈αthat␈α
it␈α
is␈α
someone␈α
else's␈α
telephone␈α
number,␈α
but␈α
he␈αwill␈α
still
␈↓ α∧␈↓find himself talking to Mike.  That's why we called 1) "physics".

␈↓ α∧␈↓␈↓ αTOur␈α
first␈α
step␈α
is␈α
to␈αintroduce␈α
two␈α
functions␈α
-␈α
␈↓↓value(x),␈↓␈αwhere␈α
␈↓↓x␈↓␈α
is␈α
a␈α
term,␈α
and␈α␈↓↓value(p,x,s)␈↓
␈↓ α∧␈↓where␈α
␈↓↓p␈↓␈α
is␈α∞a␈α
person,␈α
␈↓↓x␈α∞is␈↓␈α
a␈α
term,␈α∞and␈α
␈↓↓s␈↓␈α
is␈α∞a␈α
situation.␈α
 ␈↓↓value(x)␈↓␈α∞is␈α
the␈α
value␈α∞of␈α
the␈α
term␈α∞␈↓↓x,␈↓␈α
and
␈↓ α∧␈↓␈↓↓value(p,x,s)␈↓␈αis␈α
the␈αvalue␈α
that␈αperson␈α
␈↓↓p␈↓␈αgives␈αto␈α
the␈αterm␈α
␈↓↓x␈↓␈αin␈α
situation␈α␈↓↓s.␈αThus␈↓␈α
␈↓↓value(Pat,"2+2",S0)␈↓
␈↓ α∧␈↓is␈α∞what␈α∞Pat␈α∞thinks␈α∞2+2␈α∞is␈α
in␈α∞situation␈α∞␈↓↓S0.␈↓␈α∞We␈α∞can␈α∞now␈α
define␈α∞␈↓↓knows(p,x,s),␈↓␈α∞read␈α∞␈↓↓p␈↓␈α∞knows␈α∞␈↓↓x␈↓␈α
in
␈↓ α∧␈↓situation ␈↓↓s␈↓ by


␈↓ α∧␈↓␈↓ ε|1␈↓ ∧
␈↓ α∧␈↓␈↓ εMDRAFT␈↓ ∧


␈↓ α∧␈↓3) ␈↓↓∀p x s.(know(p,x,s) ≡ value(p,x,s) = value(x))␈↓,

␈↓ α∧␈↓i.e.␈α
␈↓↓p␈↓␈αknows␈α
␈↓↓x␈↓␈α
if␈αand␈α
only␈αif␈α
its␈α
value␈αis␈α
what␈α
he␈αthinks␈α
it␈αis.␈α
 ␈↓↓value(x)␈↓␈α
is␈αrelated␈α
to␈α
the␈αterm␈α
␈↓↓x␈↓␈αby␈α
a
␈↓ α∧␈↓so-called reflexion principle that gives as axioms all statements like

␈↓ α∧␈↓4) ␈↓↓value("telephone(Mike)") = telephone(Mike)␈↓.

␈↓ α∧␈↓Like␈α
Tarski's␈α∞reflexion␈α
principle,␈α
this␈α∞has␈α
to␈α
be␈α∞built␈α
into␈α
the␈α∞logic␈α
and␈α
can't␈α∞be␈α
expressed␈α∞as␈α
a
␈↓ α∧␈↓single␈α⊃axiom␈α⊂in␈α⊃the␈α⊃language␈α⊂itself;␈α⊃it␈α⊂can␈α⊃be␈α⊃expressed␈α⊂in␈α⊃a␈α⊂suitable␈α⊃meta-language␈α⊃such␈α⊂as
␈↓ α∧␈↓English, although we have preferred to explain it here with an example.

␈↓ α∧␈↓␈↓ αTOur eventual goal is a statement

␈↓ α∧␈↓5)␈α?␈α?␈α?␈α?␈α?␈α?␈α>␈↓↓speaking(Pat,Mike,result(Pat,dial(value(Pat,"telephone(Mike)",
␈↓ α∧␈↓↓result(Pat,lookup("Mike"),S0))),result(Pat,lookup("Mike"),S0))).␈↓

␈↓ α∧␈↓which␈αsays␈αthat␈αif␈αPat␈α
dials␈αwhat␈αhe␈αthinks␈αis␈αMike's␈α
telephone␈αnumber␈αafter␈αlooking␈αup␈αMike␈α
(in
␈↓ α∧␈↓the␈α∞phone␈α∞book),␈α∞then␈α∞he␈α∞will␈α∞be␈α∞speaking␈α∞to␈α∞Mike.␈α∞ This␈α∞can␈α∞be␈α∞derived␈α∞from␈α∞1),␈α∞a␈α∞statement
␈↓ α∧␈↓expressing the result of Pat looking up Mike, namely

␈↓ α∧␈↓6) ␈↓↓value(Pat,"telephone(Mike)",result(Pat,lookup("Mike"),S0)) = value("telephone(Mike)")␈↓,

␈↓ α∧␈↓and the instance 4) of the reflexion principle.

␈↓ α∧␈↓␈↓ αTA␈α
problem␈α
whose␈αsolution␈α
requires␈α
yet␈α
more␈αnotation␈α
arises␈α
when␈α
we␈αwant␈α
to␈α
derive␈α6)␈α
from
␈↓ α∧␈↓a␈α
more␈α
general␈α∞assertion␈α
about␈α
the␈α∞result␈α
of␈α
looking␈α
up␈α∞people's␈α
telephone␈α
numbers.␈α∞ We␈α
would
␈↓ α∧␈↓like to say something like

␈↓ α∧␈↓7*) ␈↓↓∀x y s. knows(x,"telephone(y)",result(x,lookup("y"),s))␈↓,

␈↓ α∧␈↓and␈αget␈αthe␈αdesired␈αresult␈αby␈αsubstituting␈αPat␈αfor␈α
␈↓↓x,␈↓␈αMike␈αfor␈α␈↓↓y,␈↓␈αand␈α␈↓↓S0␈↓␈αfor␈α␈↓↓s.␈↓␈αHowever,␈αthis␈α
won't
␈↓ α∧␈↓work␈αunless␈αwe␈αre-interpret␈αthe␈αmeaning␈αof␈αthe␈αquotes␈αin␈αsome␈αnon-standard␈αway,␈αsuch␈αas␈αQuine
␈↓ α∧␈↓sometimes␈α∂does␈α∂with␈α∂his␈α∂quasi-quotes.␈α∂ If␈α∂we␈α∂do␈α∂that,␈α∂we␈α∂will␈α∂have␈α∂to␈α∂live␈α∂with␈α∂it␈α∂by␈α∞suitably
␈↓ α∧␈↓tinkering␈α
with␈αthe␈α
logic.␈α I␈α
prefer␈αnot␈α
to␈αtinker␈α
with␈αthe␈α
logic,␈αand␈α
instead␈αwe␈α
introduce␈αan␈α
explicit
␈↓ α∧␈↓string␈α∩substitution␈α∩function␈α∩␈↓↓subst␈↓␈α⊃as␈α∩in␈α∩LISP␈α∩(␈↓↓subst(x,y,z)␈↓␈α⊃is␈α∩the␈α∩expression␈α∩that␈α∩results␈α⊃from
␈↓ α∧␈↓substituting␈αthe␈αexpression␈α␈↓↓x␈αfor␈↓␈α
the␈αsymbol␈α␈↓↓y␈↓␈αin␈αthe␈α
expression␈α␈↓↓z)␈↓␈αand␈αan␈αexplicit␈αfunction␈α
giving
␈↓ α∧␈↓the name of a person.  Thus we have

␈↓ α∧␈↓8) ␈↓↓name(Mike) = "Mike"␈↓,

␈↓ α∧␈↓which␈α
is␈α
useful␈α∞since␈α
we␈α
are␈α∞squeamish␈α
about␈α
substituting␈α∞people␈α
into␈α
symbolic␈α∞expressions␈α
and
␈↓ α∧␈↓prefer␈α⊂to␈α∂substitute␈α⊂only␈α∂their␈α⊂names,␈α∂(i.e.␈α⊂without␈α∂actually␈α⊂proving␈α∂it,␈α⊂we␈α∂regard␈α⊂the␈α⊂abuse␈α∂of
␈↓ α∧␈↓language␈αthat␈αwould␈αbe␈αrequired␈αto␈αinterpret␈αa␈αsubstitution␈αof␈αMike␈αas␈αmeaning␈αsubstitute␈α"Mike"
␈↓ α∧␈↓as␈αpossibly␈αlikely␈αto␈αproduce␈αan␈αinconsistency␈αeven␈αin␈αour␈αlimited␈αcontext).␈α We␈αcould␈αregard␈α8)␈αas
␈↓ α∧␈↓the result of a reflexion principle, i.e. as a converse of

␈↓ α∧␈↓9) ␈↓↓value("Mike") = Mike␈↓,


␈↓ α∧␈↓␈↓ ε|2␈↓ ∧
␈↓ α∧␈↓␈↓ εMDRAFT␈↓ ∧


␈↓ α∧␈↓but we can also consider modifying it to

␈↓ α∧␈↓8') ␈↓↓name(Mike) = "Noonan, Michael C."␈↓.

␈↓ α∧␈↓which might be appropriate in the telephone context.

␈↓ α∧␈↓␈↓ αTWe can now state our general law of looking up telephone numbers in the form

␈↓ α∧␈↓10) ␈↓↓∀x y s.(knows(x,subst(name(y),"y","telephone(y)"), result(x,lookup(name(y)),s)))␈↓,

␈↓ α∧␈↓and the derivation proceeds first by substituting Pat for ␈↓↓x,␈↓ Mike for ␈↓↓y,␈↓ and ␈↓↓S0␈↓ for ␈↓↓s␈↓ to get

␈↓ α∧␈↓11) ␈↓↓knows(Pat,subst(name(Mike),"y","telephone(y)"), result(Pat,lookup(name(Mike)),S0)))␈↓.

␈↓ α∧␈↓Using 8) makes this

␈↓ α∧␈↓12) ␈↓↓knows(Pat,subst("Mike","y","telephone(y)"), result(Pat,lookup("Mike"),S0)))␈↓,

␈↓ α∧␈↓and␈α
carrying␈α
out␈αthe␈α
indicated␈α
substitution␈α
(assuming␈αour␈α
logic␈α
lets␈α
us␈αdo␈α
this␈α
without␈αfurther␈α
ado,
␈↓ α∧␈↓e.g. by using the attachment mechanism in FOL) finally yields

␈↓ α∧␈↓13) ␈↓↓knows(Pat,"telephone(Mike)",result(Pat,lookup("Mike"),S0))␈↓.

␈↓ α∧␈↓␈↓ αTAdmittedly␈α∪we␈α∪have␈α∩introduced␈α∪a␈α∪lot␈α∪of␈α∩machinery␈α∪-␈α∪␈↓↓know(p,x),␈↓␈α∪␈↓↓value(x),␈↓␈α∩␈↓↓value(p,x,s),␈↓
␈↓ α∧␈↓␈↓↓name(x),␈↓␈αand␈α
␈↓↓subst(x,y,z).␈↓␈αWe␈α
also␈αneed␈α
the␈αreflexion␈αprinciple␈α
in␈αthe␈α
logic␈αand␈α
the␈αability␈αto␈α
make
␈↓ α∧␈↓syntactic␈α
calculations,␈αe.g.␈α
performing␈α
the␈αsubstitution.␈α
 On␈αthe␈α
other␈α
hand,␈αwe␈α
have␈α
avoided␈αthe
␈↓ α∧␈↓logic␈α∞of␈α∂modal␈α∞operators,␈α∂possible␈α∞worlds,␈α∂and␈α∞giving␈α∂rules␈α∞in␈α∂the␈α∞logic␈α∂permitting␈α∞substitutions
␈↓ α∧␈↓into␈α⊂opaque␈α⊂constexts.␈α⊂ I␈α⊂think␈α⊂the␈α⊂machinery␈α⊂introduced␈α⊂will␈α⊂be␈α⊂useful␈α⊂for␈α∂metamathematical
␈↓ α∧␈↓reasoning␈αgenerally.␈α We␈αhave␈αnot␈αintroduced␈αany␈αlogic␈αof␈α␈↓↓value(p,x,s),␈↓␈αand␈αwhen␈αwe␈αdo,␈αwe␈αshall
␈↓ α∧␈↓have to take care to avoid Montague's paradoxes.

␈↓ α∧␈↓␈↓ αTSo␈α
far␈α
we␈α
have␈α
merely␈α
done␈α
philosophical␈α
logic,␈α
and␈α
done␈α
it␈α
rather␈α
cavalierly,␈α
i.e.␈αwithout
␈↓ α∧␈↓adequate␈αattention␈α
to␈αthe␈αopinions␈α
of␈αother␈αpeople␈α
working␈αin␈αthe␈α
field␈αof␈αknowledge␈α
and␈αbelief.
␈↓ α∧␈↓Now␈αI␈αwant␈α
to␈αconsider␈αhow␈αthe␈α
goals␈αof␈αAI␈α
in␈αthis␈αmatter␈αdiffer␈α
from␈αthose␈αof␈αphilosophy.␈α
 First,
␈↓ α∧␈↓our␈α∩goal␈α⊃is␈α∩not␈α⊃to␈α∩explicate␈α∩the␈α⊃ordinary␈α∩language␈α⊃usage␈α∩of␈α⊃words␈α∩and␈α∩sentences␈α⊃expressing
␈↓ α∧␈↓knowledge␈α⊂or␈α⊂even␈α⊂to␈α⊂determine␈α⊂a␈α⊂correct␈α⊂concept␈α⊂of␈α⊂knowledge.␈α⊂ Our␈α⊂objective␈α⊂should␈α⊂be␈α⊂to
␈↓ α∧␈↓develop␈αa␈αformalism␈αthat␈αwe␈αcan␈αshow␈αis␈αadequate␈αto␈αexpress␈αwhat␈αan␈αartificial␈αintelligence␈αmust
␈↓ α∧␈↓know␈αabout␈αwho␈αknows␈αwhat␈αand␈αhow␈αknowledge␈αis␈αto␈αbe␈αobtained.␈α We␈αwill␈αstart␈αon␈αthat␈αin␈αthe
␈↓ α∧␈↓next␈αsection␈α
of␈αthis␈αpaper.␈α
 Before␈αdoing␈αthat,␈α
however,␈αI␈α
have␈αa␈αfew␈α
final␈αremarks␈αexplaining␈α
the
␈↓ α∧␈↓choice of formalism.

␈↓ α∧␈↓␈↓ αT1. We can try to make equivalences like

␈↓ α∧␈↓14) ␈↓↓knows(Pat,"telephone(Mike)",S0) = ∃x.K(Pat,subst(x,"x", "telephone(Mike) = x"))␈↓,

␈↓ α∧␈↓where␈α␈↓↓K(person,sentence)␈↓␈α
means␈αthat␈α
␈↓↓person␈↓␈αknows␈α␈↓↓sentence.␈↓␈α
This␈αwould␈α
reduce␈αmy␈α
definition␈αof
␈↓ α∧␈↓knowledge␈αto␈αthe␈αcase␈αof␈αknowing␈αthe␈αtruth␈αof␈αa␈αsentence.␈α I␈αhave␈αtwo␈αobjections.␈α First,␈αit␈αmakes


␈↓ α∧␈↓␈↓ ε|3␈↓ ∧
␈↓ α∧␈↓␈↓ εMDRAFT␈↓ ∧


␈↓ α∧␈↓the␈αcase␈αthat␈αI␈αexpect␈αto␈αarise␈αmost␈αfrequently␈αin␈αAI␈αcomplicated␈α-␈αthe␈αsimplest␈αsentence␈αinvolves
␈↓ α∧␈↓␈↓↓subst.␈↓␈α∪Second,␈α∪I␈α∪am␈α∪not␈α∩sure␈α∪it␈α∪says␈α∪the␈α∪same␈α∪thing␈α∩anyway,␈α∪although␈α∪this␈α∪may␈α∪be␈α∪just␈α∩a
␈↓ α∧␈↓philosophical␈α+quibble␈α+for␈α*the␈α+initial␈α+applications.␈α* Namely,␈α+suppose␈α+we␈α*have
␈↓ α∧␈↓␈↓↓K(Pat,"telephone(Mike)␈α=␈α497-4971",S0)␈↓.␈α Does␈αthis␈αmean␈αthat␈αPat␈αwill␈αsay␈αyes␈αwhen␈αasked␈αabout
␈↓ α∧␈↓the␈α⊃sentence,␈α⊂or␈α⊃does␈α⊃it␈α⊂mean␈α⊃that␈α⊂he␈α⊃will␈α⊃answer␈α⊂correctly␈α⊃when␈α⊂asked␈α⊃for␈α⊃Mike's␈α⊂telephone
␈↓ α∧␈↓number␈αor␈αwhose␈αnumber␈αis␈α497-4971␈αor␈αeven␈αwhat␈αis␈αthe␈αrelation␈αbetween␈αMike␈αand␈α497-4971?
␈↓ α∧␈↓I␈α
would␈α
like␈α
to␈α
be␈α
able␈α
to␈α
distinguish␈α
these␈α
cases,␈α
and␈α
my␈α
first␈α
form␈α
settles␈α
on␈α
the␈α
case␈α
that␈αPat␈α
will
␈↓ α∧␈↓answer correctly when asked Mike's telephone number.

␈↓ α∧␈↓␈↓ αT2.␈α
Suppose␈α
that␈α
we␈α
are␈α
making␈α
a␈αprogram␈α
to␈α
deal␈α
with␈α
problems␈α
that␈α
are␈α
mostly␈αphysical,
␈↓ α∧␈↓i.e.␈α
mostly␈α
we␈αhave␈α
the␈α
facts␈α
and␈αwant␈α
to␈α
find␈αa␈α
physical␈α
action␈α
that␈αwill␈α
achieve␈α
a␈αgoal.␈α
 However,
␈↓ α∧␈↓later␈α∂it␈α∂turns␈α∞out␈α∂that␈α∂the␈α∞program␈α∂must␈α∂reason␈α∞about␈α∂how␈α∂to␈α∞get␈α∂some␈α∂information␈α∂by␈α∞asking
␈↓ α∧␈↓someone␈αwho␈α
has␈αit.␈α I␈α
would␈αlike␈αto␈α
be␈αable␈αto␈α
make␈αthe␈αextension␈α
without␈αoverturning␈αthe␈α
entire
␈↓ α∧␈↓logical␈αsystem.␈α It␈αseems␈αto␈αme␈αthat␈αswitching␈αto␈αmodal␈αlogic␈αmight␈αrequire␈αthis,␈αand␈αthe␈αnotion␈αof
␈↓ α∧␈↓possible␈α⊂worlds␈α⊂seems␈α∂dubious␈α⊂to␈α⊂me,␈α∂especially␈α⊂as␈α⊂a␈α∂new␈α⊂structure␈α⊂of␈α∂possible␈α⊂worlds␈α⊂may␈α∂be
␈↓ α∧␈↓required␈αfor␈αeach␈αpropositional␈αattitude.␈α Anyway,␈αvalid␈αor␈αnot,␈αthese␈αare␈αsome␈αof␈αthe␈αreasons␈αfor
␈↓ α∧␈↓introducing␈α
a␈αdifferent␈α
approach.␈α
 I␈αshould␈α
remark␈αthat␈α
knowing␈α
telephone␈αnumbers␈α
could␈αalso␈α
be
␈↓ α∧␈↓treated directly in a modal way, although I don't know of any attempt to do so.


␈↓ α∧␈↓αApplications to AI

␈↓ α∧␈↓(tune in to the next exciting draft)

␈↓ α∧␈↓John McCarthy
␈↓ α∧␈↓Artificial Intelligence Laboratory
␈↓ α∧␈↓Stanford, California 94305

␈↓ α∧␈↓March 30, 1976




















␈↓ α∧␈↓␈↓ ε|4␈↓ ∧