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␈↓ α∧␈↓␈↓ εddraft





















␈↓ α∧␈↓α␈↓ β3CONCEPTS AS OBJECTS AND CONCEPT-VALUED FUNCTIONS

␈↓ α∧␈↓Abstract:␈α∞We␈α∞discuss␈α∞first␈α∞order␈α∞theories␈α∞in␈α∞which␈α∞␈↓↓concepts␈↓␈α∞are␈α∞allowed␈α∞as␈α∂mathematical␈α∞objects
␈↓ α∧␈↓along␈α≠with␈α≠the␈α≠things␈α≠of␈α≤which␈α≠they␈α≠are␈α≠the␈α≠concepts.␈α≠ This␈α≤allows␈α≠unprecedentedly
␈↓ α∧␈↓straightforward␈α∩formalizations␈α∪of␈α∩knowledge,␈α∪belief,␈α∩wanting,␈α∪and␈α∩necessity.␈α∪ Applications␈α∩are
␈↓ α∧␈↓given in philosophy and in artificial intelligence.





















␈↓ α∧␈↓␈↓ ε|1␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓αINTRODUCTION

␈↓ α∧␈↓␈↓ αTAdmitting␈α⊃concepts␈α⊃as␈α⊃objects␈α⊃-␈α⊂with␈α⊃concept-valued␈α⊃constants,␈α⊃variables,␈α⊃functions␈α⊂and
␈↓ α∧␈↓expressions␈α∂-␈α∂allows␈α∂an␈α∞ordinary␈α∂first␈α∂order␈α∂theory␈α∞of␈α∂necessity,␈α∂knowledge,␈α∂belief␈α∂and␈α∞wanting
␈↓ α∧␈↓without␈α∞modal␈α∞operators␈α∞or␈α∞quotation␈α∞marks␈α∞and␈α∞without␈α∞the␈α∞restrictions␈α∞on␈α∞substituting␈α
equals
␈↓ α∧␈↓for equals that either device makes necessary.

␈↓ α∧␈↓␈↓ αTAccording␈α∂to␈α⊂Frege␈α∂(1892),␈α∂the␈α⊂meaning␈α∂of␈α⊂the␈α∂phrase␈α∂␈↓↓"Mike's␈α⊂telephone␈α∂number"␈↓␈α⊂in␈α∂the
␈↓ α∧␈↓sentence␈α
␈↓↓"Pat␈α
knows␈αMike's␈α
telephone␈α
number"␈↓␈αis␈α
the␈α
concept␈αof␈α
Mike's␈α
telephone␈αnumber,␈α
whereas
␈↓ α∧␈↓its␈α
meaning␈αin␈α
the␈α
sentence␈α␈↓↓"Pat␈α
dialed␈αMike's␈α
telephone␈α
number"␈↓␈αis␈α
the␈α
number␈αitself.␈α
 Thus␈αif␈α
we
␈↓ α∧␈↓also␈α⊂have␈α⊂␈↓↓"Mary␈α⊂has␈α⊂the␈α⊂same␈α⊂telephone␈α∂number␈α⊂as␈α⊂Mike"␈↓,␈α⊂then␈α⊂␈↓↓"Pat␈α⊂dialed␈α⊂Mary's␈α∂telephone
␈↓ α∧␈↓↓number"␈↓ follows, but ␈↓↓"Pat knows Mary's telephone number"␈↓ does not.

␈↓ α∧␈↓␈↓ αTFrege␈αfurther␈αproposed␈αthat␈α
a␈αphrase␈αhas␈αa␈α
␈↓↓sense␈↓␈αwhich␈αis␈αa␈α
␈↓↓concept␈↓␈αand␈αis␈αits␈α
␈↓↓meaning␈↓␈αin
␈↓ α∧␈↓␈↓↓oblique␈↓␈α
␈↓↓contexts␈↓␈α
like␈α
knowing␈αand␈α
wanting,␈α
and␈α
a␈α
␈↓↓denotation␈↓␈αwhich␈α
is␈α
its␈α
␈↓↓meaning␈↓␈α
in␈α␈↓↓direct␈↓␈α
␈↓↓contexts.␈↓
␈↓ α∧␈↓Denotations␈αare␈αthe␈αbasis␈αof␈αthe␈αTarskian␈αsemantics␈αof␈αfirst␈αorder␈αlogic␈αand␈αmodel␈αtheory␈αand␈αare
␈↓ α∧␈↓well␈α
understood,␈α
but␈α
sense␈α
has␈α
given␈α
more␈α
trouble,␈α
and␈α
the␈α
modal␈α
treatment␈α
of␈α∞oblique␈α
contexts
␈↓ α∧␈↓avoids␈αthe␈αidea.␈α On␈αthe␈αother␈αhand,␈αlogicians␈αsuch␈αas␈αChurch␈α(1951)␈αand␈αMontague␈α(1974)␈αsee␈αa
␈↓ α∧␈↓need for ␈↓↓concepts␈↓ and have proposed formalizations, but none have been very satisfactory.

␈↓ α∧␈↓␈↓ αTThe␈α
problem␈α
identified␈α
by␈α
Frege␈α
-␈α
of␈αsuitably␈α
limiting␈α
the␈α
application␈α
of␈α
Leibniz's␈α
law␈αof
␈↓ α∧␈↓the␈α
substitutitivity␈α
of␈α
equals␈α
for␈α
equals␈α
-␈α
arises␈α
in␈α
artificial␈α
intelligence␈α
as␈α
well␈α
as␈α∞in␈α
philosophy
␈↓ α∧␈↓and␈α
linguistics␈αfor␈α
any␈α
system␈αthat␈α
must␈αrepresent␈α
information␈α
about␈αbeliefs,␈α
knowledge,␈αdesires,␈α
or
␈↓ α∧␈↓logical necessity - regardless of whether the representation is procedural or declarative.

␈↓ α∧␈↓␈↓ αTThe␈αpresent␈αidea␈αis␈αto␈αtreat␈αconcepts␈αas␈αone␈αkind␈αof␈αobject␈αin␈αa␈αfirst␈αorder␈αtheory.␈α Thus␈α
we
␈↓ α∧␈↓have␈α⊃an␈α⊃expression␈α⊃whose␈α⊃value␈α⊃is␈α⊃Mike's␈α⊃telephone␈α⊃number␈α⊃and␈α⊃a␈α⊃different␈α⊃though␈α⊂related
␈↓ α∧␈↓expression␈α∞whose␈α∞value␈α
is␈α∞the␈α∞concept␈α∞of␈α
Mike's␈α∞telephone␈α∞number␈α
rather␈α∞than␈α∞having␈α∞a␈α
single
␈↓ α∧␈↓expression␈α∂whose␈α∂denotation␈α∂is␈α∂the␈α∂number␈α∂and␈α∂whose␈α∂sense␈α∂is␈α∂a␈α∂concept␈α∂of␈α∂it.␈α⊂ The␈α∂relations
␈↓ α∧␈↓among␈αconcepts␈αand␈αbetween␈αconcepts␈αand␈αother␈αentities␈αare␈αthen␈αreadily␈αexpressed␈αby␈αfirst␈αorder
␈↓ α∧␈↓logical␈αformulas.␈α Moreover,␈αordinary␈αmodel␈αtheory␈α
can␈αbe␈αused␈αto␈αstudy␈αwhat␈αspaces␈α
of␈αconcepts
␈↓ α∧␈↓satisfy various sets of axioms.



␈↓ α∧␈↓αKNOWING WHAT AND KNOWING THAT

␈↓ α∧␈↓␈↓ αTTo assert that Pat knows Mike's telephone number we write

␈↓ α∧␈↓1)␈↓ αD ␈↓↓true Know(Pat,Telephone Mike)␈↓

␈↓ α∧␈↓with the following conventions:

␈↓ α∧␈↓␈↓ αT1.␈α∞Parentheses␈α∞are␈α∞often␈α∞omitted␈α∞for␈α∞one␈α∞argument␈α∞functions␈α∞and␈α∞predicates.␈α∞ This␈α∞purely
␈↓ α∧␈↓syntactic␈α⊂convention␈α∂is␈α⊂not␈α∂important.␈α⊂ Another␈α∂convention␈α⊂is␈α∂to␈α⊂capitalize␈α∂the␈α⊂first␈α∂letter␈α⊂of␈α∂a
␈↓ α∧␈↓constant, variable or function name when its value is a concept.



␈↓ α∧␈↓␈↓ ε|2␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓␈↓ αT2.␈α␈↓↓Mike␈↓␈αdenotes␈αthe␈α
concept␈αof␈αMike;␈αi.e.␈α
it␈αis␈αthe␈α␈↓↓sense␈↓␈α
of␈αthe␈αexpression␈α␈↓↓"Mike"␈↓.␈α
 We␈αwill
␈↓ α∧␈↓use ␈↓↓mike␈↓ when we wish to denote Mike himself.

␈↓ α∧␈↓␈↓ αT3.␈α∩␈↓↓Telephone␈↓␈α∩is␈α⊃a␈α∩function␈α∩that␈α⊃takes␈α∩the␈α∩concept␈α⊃of␈α∩a␈α∩person␈α⊃into␈α∩the␈α∩concept␈α∩of␈α⊃his
␈↓ α∧␈↓telephone␈αnumber.␈α We␈αwill␈αalso␈αuse␈α␈↓↓telephone␈↓␈αwhich␈αtakes␈αthe␈αperson␈αhimself␈αinto␈α
the␈αtelephone
␈↓ α∧␈↓number itself.

␈↓ α∧␈↓␈↓ αT4.␈α⊂If␈α⊂␈↓↓P␈↓␈α⊃is␈α⊂a␈α⊂person␈α⊃concept␈α⊂and␈α⊂␈↓↓X␈↓␈α⊃is␈α⊂another␈α⊂concept,␈α⊃then␈α⊂␈↓↓Know(P,X)␈↓␈α⊂is␈α⊃an␈α⊂assertion
␈↓ α∧␈↓concept␈α
or␈α␈↓↓proposition␈↓␈α
meaning␈αthat␈α
␈↓↓P␈αknows␈↓␈α
the␈α
value␈αof␈α
␈↓↓X.␈↓␈αIn␈α
1),␈αtherefore,␈α
␈↓↓Know(Pat,Telephone
␈↓ α∧␈↓↓Mike)␈↓␈αis␈αa␈αproposition␈αand␈αnot␈αa␈αtruth␈αvalue.␈α Note␈αthat␈αwe␈αare␈αformalizing␈α␈↓↓knowing␈↓␈α␈↓↓what␈↓␈αrather
␈↓ α∧␈↓than␈α␈↓↓knowing␈↓␈α␈↓↓that␈↓␈αor␈α␈↓↓knowing␈↓␈α␈↓↓how.␈↓␈αFor␈αAI␈αand␈αfor␈αother␈αpractical␈αpurposes,␈α␈↓↓knowing␈↓␈α␈↓↓what␈↓␈αseems
␈↓ α∧␈↓to be the most useful notion of the three.

␈↓ α∧␈↓␈↓ αT5.␈α
␈↓↓true(Q)␈↓␈αis␈α
the␈αtruth␈α
value,␈α␈↓↓t␈↓␈α
or␈α␈↓↓f,␈↓␈α
of␈αthe␈α
proposition␈α␈↓↓Q,␈↓␈α
and␈αwe␈α
must␈αwrite␈α
␈↓↓true(Q)␈↓␈αin␈α
order
␈↓ α∧␈↓to␈α
assert␈α
␈↓↓Q.␈↓␈α
Later␈α
we␈αwill␈α
consider␈α
formalisms␈α
in␈α
which␈α
␈↓↓true␈↓␈αhas␈α
a␈α
second␈α
argument␈α
-␈α
a␈α␈↓↓situation,␈↓␈α
a
␈↓ α∧␈↓␈↓↓story,␈↓ a ␈↓↓possible␈↓ ␈↓↓world,␈↓ or even a ␈↓↓partial possible world␈↓ (a notion we hope to introduce).

␈↓ α∧␈↓␈↓ αT6.␈αThe␈αformulas␈αare␈α
in␈αa␈αsorted␈αfirst␈α
order␈αlogic␈αwith␈αfunctions␈α
and␈αequality.␈α In␈αthe␈α
present
␈↓ α∧␈↓informal␈α⊂treatement,␈α⊂we␈α⊂will␈α⊂not␈α∂be␈α⊂explicit␈α⊂about␈α⊂sorts,␈α⊂but␈α∂we␈α⊂will␈α⊂try␈α⊂to␈α⊂be␈α∂typographically
␈↓ α∧␈↓consistent.

␈↓ α∧␈↓␈↓ αTThe␈α⊃reader␈α⊃may␈α⊃be␈α⊃nervous␈α⊃about␈α⊃what␈α⊃is␈α⊃meant␈α⊃by␈α⊃␈↓↓concept.␈↓␈α⊃He␈α⊃will␈α⊃have␈α∩to␈α⊃remain
␈↓ α∧␈↓nervous;␈α
no␈α∞final␈α
commitment␈α∞will␈α
be␈α∞made␈α
in␈α∞this␈α
paper.␈α∞ The␈α
formalism␈α∞is␈α
compatible␈α∞with␈α
a
␈↓ α∧␈↓variety␈α
of␈αpossibilities,␈α
and␈αthese␈α
can␈αbe␈α
compared␈αusing␈α
the␈αmodels␈α
of␈αtheir␈α
first␈α
order␈αtheories.
␈↓ α∧␈↓However,␈αif␈α1)␈αis␈αto␈αbe␈αreasonable,␈αit␈αmust␈αnot␈αfollow␈αfrom␈α1)␈αand␈αthe␈αfact␈αthat␈α
Mary's␈αtelephone
␈↓ α∧␈↓number is the same as Mike's, that Pat knows Mary's telephone number.

␈↓ α∧␈↓␈↓ αTThe proposition that Joe knows ␈↓↓whether␈↓ Pat knows Mike's telephone number, is written

␈↓ α∧␈↓2)␈↓ αD ␈↓↓Know(Joe,Know(Pat,Telephone Mike))␈↓,

␈↓ α∧␈↓and asserting it requires writing

␈↓ α∧␈↓3)␈↓ αD ␈↓↓true Know(Joe,Know(Pat,Telephone Mike))␈↓,

␈↓ α∧␈↓while the proposition that Joe knows ␈↓↓that␈↓ Pat knows Mike's tephone number is written

␈↓ α∧␈↓4)␈↓ αD ␈↓↓K(Joe,Know(Pat,Telephone Mike))␈↓,

␈↓ α∧␈↓where ␈↓↓K(P,Q)␈↓ is the proposition that ␈↓↓P␈↓ knows that ␈↓↓Q.␈↓

␈↓ α∧␈↓␈↓ αTConsider the function ␈↓↓denot␈↓ and the statement

␈↓ α∧␈↓5)␈↓ αD ␈↓↓∀P1 P2.(denot P1 = denot P2 ⊃ denot Telephone P1 = denot Telephone P2)␈↓.

␈↓ α∧␈↓Here␈α␈↓↓denot␈αX␈↓␈αis␈αthe␈α␈↓↓denotation␈↓␈αof␈αthe␈αconcept␈α␈↓↓X,␈↓␈αand␈α5)␈αasserts␈αthat␈αthe␈αdenotation␈αof␈αthe␈αconcept
␈↓ α∧␈↓of␈α␈↓↓X␈↓'s␈αtelephone␈α
number␈αdepends␈αonly␈α
on␈αthe␈αdenotation␈α
of␈αconcept␈α␈↓↓X␈↓.␈α
 The␈αvariables␈αin␈α5)␈α
range


␈↓ α∧␈↓␈↓ ε|3␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓over␈αconcepts␈αof␈αpersons,␈αand␈αwe␈αregard␈α5)␈αas␈αasserting␈αthat␈α␈↓↓Telephone␈↓␈αis␈α␈↓↓extensional␈↓␈αwith␈αrespect
␈↓ α∧␈↓to␈α∞␈↓↓denot.␈↓␈α∂Note␈α∞that␈α∞our␈α∂␈↓↓denot␈↓␈α∞operates␈α∂on␈α∞concepts␈α∞rather␈α∂than␈α∞on␈α∞expressions,␈α∂but␈α∞a␈α∂theory␈α∞of
␈↓ α∧␈↓expressions will require a denotation function for them also.

␈↓ α∧␈↓␈↓ αT␈↓↓Know␈↓ is extensional with respect to ␈↓↓denot␈↓ in its first argument, and this expressed by

␈↓ α∧␈↓6)␈↓ αD ␈↓↓∀P1 P2 X.(denot P1 = denot P2 ⊃ denot Know(P1,X) = denot Know(P2,X))␈↓,

␈↓ α∧␈↓but␈αit␈αis␈αnot␈αextensional␈αin␈αits␈αsecond␈αargument.␈α (Note␈αthat␈αall␈αthese␈αpredicates␈αand␈αfunctions␈αare
␈↓ α∧␈↓entirely␈α∂extensional␈α⊂in␈α∂the␈α∂underlying␈α⊂logic,␈α∂and␈α⊂the␈α∂notion␈α∂of␈α⊂extensionality␈α∂presented␈α⊂here␈α∂is
␈↓ α∧␈↓relative to the function ␈↓↓denot.) ␈↓ The predicate ␈↓↓true␈↓ and the function ␈↓↓denot␈↓ are related by

␈↓ α∧␈↓7)␈↓ αD ␈↓↓∀Q.(true Q ≡ (denot Q = t))␈↓

␈↓ α∧␈↓provided␈αtruth␈αvalues␈αare␈αin␈αthe␈αrange␈αof␈α␈↓↓denot,␈↓␈αand␈α␈↓↓denot␈↓␈αmay␈αalso␈αbe␈αprovided␈αwith␈αa␈α␈↓↓(partial)
␈↓ α∧␈↓↓possible world␈↓ argument.

␈↓ α∧␈↓␈↓ αTIf␈α⊂we␈α⊂want␈α⊂a␈α⊂system␈α⊂in␈α⊂which␈α⊂not␈α∂all␈α⊂concepts␈α⊂have␈α⊂denotations,␈α⊂then␈α⊂we␈α⊂should␈α⊂use␈α∂a
␈↓ α∧␈↓predicate␈α⊃␈↓↓denotes(X,x)␈↓␈α⊃instead␈α⊃of␈α⊂a␈α⊃function.␈α⊃ The␈α⊃extensionality␈α⊂of␈α⊃␈↓↓Telephone␈↓␈α⊃would␈α⊃then␈α⊂be
␈↓ α∧␈↓written

␈↓ α∧␈↓8)␈↓ αD␈α⊗␈↓↓∀P1␈α⊗P2␈α⊗x␈α⊗u.(denotes(P1,x)∧denotes(P2,x)∧denotes(Telephone␈α⊗P1,u)␈α⊗⊃␈α∃denotes(Telephone
␈↓ α∧␈↓↓P2,u))␈↓.

␈↓ α∧␈↓␈↓ αTIn␈α
order␈αto␈α
combine␈αconcepts␈α
propositionally,␈α
we␈αneed␈α
analogs␈αof␈α
the␈αpropositional␈α
operators
␈↓ α∧␈↓such as ␈↓↓And,␈↓ which we shall use as an infix, and axiomatize by

␈↓ α∧␈↓9)␈↓ αD ␈↓↓∀Q1 Q2.(true (Q1 And Q2) ≡ true Q1 ∧ true Q2)␈↓, etc.

␈↓ α∧␈↓Assume that the corresponding formulas for ␈↓↓Or,␈↓ ␈↓↓Not,␈↓ ␈↓↓Implies,␈↓ and ␈↓↓Equiv␈↓ have been written.

␈↓ α∧␈↓␈↓ αTThe␈α
equality␈α∞symbol␈α
"="␈α
is␈α∞used␈α
with␈α
its␈α∞usual␈α
logical␈α
meaning␈α∞of␈α
identity,␈α
so␈α∞that␈α
␈↓↓X␈α∞=␈α
Y␈↓
␈↓ α∧␈↓asserts␈α⊃that␈α⊃␈↓↓X␈↓␈α⊃and␈α⊃␈↓↓Y␈↓␈α⊃are␈α⊃the␈α∩same␈α⊃concept.␈α⊃ To␈α⊃discuss␈α⊃concepts␈α⊃of␈α⊃particular␈α∩equalities,␈α⊃we
␈↓ α∧␈↓introduce␈α␈↓↓Equal,␈↓␈αand␈α␈↓↓Equal(X,Y)␈↓␈α
is␈αthe␈αproposition␈αthat␈α␈↓↓X␈↓␈α
and␈α␈↓↓Y␈↓␈αhave␈αequal␈α
denotations.␈α Thus
␈↓ α∧␈↓we have

␈↓ α∧␈↓10)␈↓ αD␈↓↓∀X Y.(true Equal(X,Y) ≡ denot X = denot Y)␈↓.

␈↓ α∧␈↓␈↓ αTPropositions␈αformed␈αby␈αquantification␈αpresent␈αmore␈αof␈αa␈αproblem.␈α We␈αwill␈αwant␈αa␈αfunction
␈↓ α∧␈↓␈↓↓All(var,exp),␈↓␈αwhere␈α␈↓↓var␈↓␈αis␈αa␈α"variable"␈αand␈α␈↓↓exp␈↓␈αis␈αsome␈αkind␈αof␈α"concept-valued␈αexpression".␈α We
␈↓ α∧␈↓will␈α
need␈α
objects␈α
called␈α
␈↓↓vars␈↓␈α
and␈α
variables␈α∞ranging␈α
over␈α
them␈α
as␈α
well␈α
as␈α
variables␈α∞ranging␈α
over
␈↓ α∧␈↓"concept-valued expressions".  In any case, the basic fact about quantifiers is something like

␈↓ α∧␈↓11)␈↓ αD ␈↓↓true All(x,E) ≡ ∀x'.(true Subst(x',x,E))␈↓,

␈↓ α∧␈↓where ␈↓↓subst(x,y,z)␈↓ is a suitable analog of the LISP ␈↓↓subst.␈↓



␈↓ α∧␈↓␈↓ ε|4␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓␈↓ αTThis␈α∩will␈α∩be␈α∩elaborated␈α∩subsequently␈α∩using␈α⊃the␈α∩notion␈α∩of␈α∩extensional␈α∩form.␈α∩ Note␈α⊃that
␈↓ α∧␈↓variables␈α∃ranging␈α∀over␈α∃concepts␈α∃and␈α∀quantifying␈α∃over␈α∃concepts␈α∀require␈α∃no␈α∃new␈α∀formalism;
␈↓ α∧␈↓problems arise only when the concept itself includes quantification.

␈↓ α∧␈↓␈↓ αTThe conceptual functions can be related to ordinary extensional functions.  Thus

␈↓ α∧␈↓12)␈↓ αD ␈↓↓∀P.(denot Telephone P = telephone denot P)␈↓,

␈↓ α∧␈↓and␈α␈↓↓telephone␈↓␈αcan␈α
be␈αused␈αin␈α
any␈αpurely␈αextensional␈α
context,␈αe.g.␈αin␈α
the␈αfollowing␈α"law"␈α
expressing
␈↓ α∧␈↓the effects of dialing someone's number in the notation of (McCarthy and Hayes 1970):

␈↓ α∧␈↓13)␈↓ αD ␈↓↓∀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␈↓␈α⊂and␈α⊂␈↓↓p2␈↓␈α⊂are␈α⊂speaking␈α⊂results␈α⊂from␈α⊂␈↓↓p1␈↓␈α⊃dialing␈α⊂␈↓↓p2␈↓'s
␈↓ α∧␈↓telephone number.

␈↓ α∧␈↓␈↓ αTIf␈α∂we␈α∂were␈α∞to␈α∂adopt␈α∂the␈α∞convention␈α∂that␈α∂a␈α∞proposition␈α∂appearing␈α∂at␈α∞the␈α∂outer␈α∂level␈α∂of␈α∞a
␈↓ α∧␈↓sentence␈αis␈αasserted␈αand␈αwere␈αto␈αregard␈αthe␈αdenotation-valued␈αfunction␈αas␈αstanding␈αfor␈αthe␈αsense-
␈↓ α∧␈↓valued␈αfunction␈αwhen␈αit␈αappears␈αas␈αthe␈αsecond␈α
argument␈αof␈α␈↓↓Know,␈↓␈αwe␈αwould␈αhave␈αa␈αnotation␈α
that
␈↓ α∧␈↓looks␈α
like␈α
ordinary␈αlanguage␈α
in␈α
handling␈αobliquity␈α
entirely␈α
by␈α
context.␈α There␈α
is␈α
no␈αguarantee␈α
that
␈↓ α∧␈↓general␈α⊂statements␈α⊂could␈α⊂be␈α⊂expressed␈α⊂unambiguously␈α⊂without␈α⊂circumlocution,␈α⊂but␈α⊂we␈α⊂take␈α∂the
␈↓ α∧␈↓possibility␈α∀as␈α∪an␈α∀additional␈α∀sign␈α∪that␈α∀we␈α∪are␈α∀moving␈α∀toward␈α∪the␈α∀expressiveness␈α∀of␈α∪natural
␈↓ α∧␈↓language.

␈↓ α∧␈↓␈↓ αTAs mentioned before, ␈↓↓"Pat knows Mike's telephone number"␈↓ is written

␈↓ α∧␈↓14)␈↓ αD ␈↓↓true Know(Pat,Telephone Mike)␈↓.

␈↓ α∧␈↓We can write ␈↓↓"Pat knows Mike's telephone number is 333-3333"␈↓

␈↓ α∧␈↓15)␈↓ αD ␈↓↓true K(Pat,Equal(Telephone Mike,Concept1 "333-3333")␈↓

␈↓ α∧␈↓where␈α⊃␈↓↓K(P,Q)␈↓␈α⊃is␈α⊃the␈α⊃proposition␈α∩that␈α⊃␈↓↓denot(P)␈↓␈α⊃knows␈α⊃the␈α⊃proposition␈α⊃␈↓↓Q␈↓␈α∩and␈α⊃␈↓↓Concept1("333-
␈↓ α∧␈↓↓3333")␈↓ is some standard concept of that telephone number.

␈↓ α∧␈↓␈↓ αTThe two ways of expressing knowledge are somewhat interdefinable, since we can write

␈↓ α∧␈↓16)␈↓ αD ␈↓↓K(P,Q) = And(Q,Know(P,Q))␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓17)␈↓ αD ␈↓↓true Know(P,X) ≡ ∃A.(constant A ∧ true K(P,Equal(X,A)))␈↓.

␈↓ α∧␈↓Here␈α␈↓↓constant␈αA␈↓␈αasserts␈αthat␈α␈↓↓A␈↓␈α
is␈αa␈αconstant,␈αi.e.␈αa␈αconcept␈αsuch␈α
that␈αwe␈αare␈αwilling␈αto␈αsay␈α
that␈α␈↓↓P␈↓
␈↓ α∧␈↓knows␈α⊂␈↓↓X␈↓␈α⊂if␈α∂he␈α⊂knows␈α⊂it␈α∂is␈α⊂equal␈α⊂to␈α⊂some␈α∂particular␈α⊂constant.␈α⊂ This␈α∂is␈α⊂clear␈α⊂enough␈α⊂for␈α∂some
␈↓ α∧␈↓domains␈αlike␈αintegers,␈αbut␈αit␈αis␈αnot␈αobvious␈αhow␈αto␈αtreat␈αknowing␈αa␈αperson.␈α Another␈αpossibility␈αis
␈↓ α∧␈↓to␈αintroduce␈αfor␈αthe␈αelements␈αof␈αcertain␈αdomains␈αa␈αfunction␈α␈↓↓Concept␈↓␈αthat␈αgives␈αa␈αsort␈αof␈α␈↓↓standard␈↓
␈↓ α∧␈↓␈↓↓concept␈↓ of an element of the domain.  Then we can rewrite 17) as

␈↓ α∧␈↓␈↓ ε|5␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓18)␈↓ αD ␈↓↓true Know(P,X) ≡ ∃a.true K(P,Equal(X,Concept a))␈↓.

␈↓ α∧␈↓␈↓ αT17)␈α∩and␈α∩18)␈α∩expresses␈α∩a␈α∩␈↓↓denotational␈↓␈α∪definition␈α∩of␈α∩␈↓↓Know␈↓␈α∩in␈α∩terms␈α∩of␈α∩␈↓↓K.␈↓␈α∪A␈α∩␈↓↓conceptual␈↓
␈↓ α∧␈↓definition seems to require something like

␈↓ α∧␈↓19)␈↓ αD ␈↓↓Know(P,X) = Exist(aA,K(P,Equal(X,Concept aA))␈↓,

␈↓ α∧␈↓where ␈↓↓aA␈↓ is a "variable", but we will postpone a discussion of the interpretation of 19).


␈↓ α∧␈↓αMODAL LOGIC (part 1)

␈↓ α∧␈↓␈↓ αTWe␈α∞will␈α∞divide␈α∞our␈α∞treatment␈α∞of␈α∞necessity␈α∞and␈α∞possibility␈α∞into␈α∞two␈α∞parts.␈α∂ In␈α∞␈↓↓unquantified
␈↓ α∧␈↓↓modal␈α∪logic␈↓,␈α∪the␈α∪arguments␈α∪of␈α∪the␈α∪modal␈α∪functions␈α∪will␈α∪not␈α∪involve␈α∪quantification␈α∪although
␈↓ α∧␈↓quantification occurs in the logic.

␈↓ α∧␈↓␈↓ αT␈↓↓Nec␈αQ␈↓␈αis␈αthe␈αproposition␈αthat␈αthe␈αproposition␈α␈↓↓Q␈↓␈αis␈αnecessary,␈αand␈α␈↓↓Poss␈αQ␈↓␈αis␈α
the␈αproposition
␈↓ α∧␈↓that␈α∞it␈α∂is␈α∞possible.␈α∂ To␈α∞assert␈α∂necessity␈α∞or␈α∂possibility␈α∞we␈α∞must␈α∂write␈α∞␈↓↓true␈α∂Nec␈α∞Q␈↓␈α∂or␈α∞␈↓↓true␈α∂Poss␈α∞Q␈↓.
␈↓ α∧␈↓This␈α
can␈α
be␈α
abbreviated␈α
by␈α
defining␈α
␈↓↓nec␈α
Q␈α
≡␈α
true␈α
Nec␈α
Q␈↓␈α
and␈α
␈↓↓poss␈α
Q␈↓␈α
correspondingly,␈α
but␈α
these␈α
are
␈↓ α∧␈↓predicates in the logic with ␈↓↓t␈↓ and ␈↓↓f␈↓ as values so that ␈↓↓nec␈↓ ␈↓↓Q␈↓ cannot be an argument of ␈↓↓nec␈↓ or ␈↓↓Nec.␈↓

␈↓ α∧␈↓␈↓ αTBefore␈αwe␈αeven␈αget␈αto␈αmodal␈αlogic␈αproper␈αwe␈α
have␈αa␈αdecision␈αto␈αmake␈α-␈αshall␈α␈↓↓Not␈αNot␈αQ␈↓␈α
be
␈↓ α∧␈↓considered the same proposition as ␈↓↓Q,␈↓ or is it merely extensionally equivalent?  The first is written

␈↓ α∧␈↓20)␈↓ αD␈↓↓∀Q. Not Not Q = Q␈↓,

␈↓ α∧␈↓and the second

␈↓ α∧␈↓21)␈↓ αD ␈↓↓ ∀Q.true Not Not Q ≡ true Q␈↓.

␈↓ α∧␈↓The second follows from the first by substitution of equals for equals.

␈↓ α∧␈↓␈↓ αTIf␈αwe␈αchoose␈α
the␈αfirst␈αalternative,␈α
then␈αwe␈αmay␈α
go␈αon␈αto␈α
identify␈αany␈αtwo␈α
propositions␈αthat
␈↓ α∧␈↓can␈α∩be␈α⊃transformed␈α∩into␈α∩each␈α⊃other␈α∩by␈α⊃Boolean␈α∩identities.␈α∩ This␈α⊃can␈α∩be␈α⊃assured␈α∩by␈α∩a␈α⊃small
␈↓ α∧␈↓collection␈α∃of␈α∃propositional␈α∃identities␈α∃like␈α⊗8)␈α∃including␈α∃associative␈α∃and␈α∃distributive␈α⊗laws␈α∃for
␈↓ α∧␈↓conjunction␈αand␈αdisjunction,␈αDe␈αMorgan's␈αlaw,␈αand␈αthe␈αlaws␈αgoverning␈αthe␈αpropositions␈α␈↓↓T␈↓␈αand␈α␈↓↓F.␈↓
␈↓ α∧␈↓In␈α
the␈α
second␈α
alternative␈αwe␈α
will␈α
want␈α
the␈αextensional␈α
forms␈α
of␈α
the␈αsame␈α
laws.␈α
 When␈α
we␈α
get␈αto
␈↓ α∧␈↓quantification␈α∀a␈α∀similar␈α∪choice␈α∀will␈α∀arise,␈α∪but␈α∀if␈α∀we␈α∪choose␈α∀the␈α∀first␈α∪alternative,␈α∀it␈α∀will␈α∪be
␈↓ α∧␈↓undecideable␈α∂whether␈α∞two␈α∂expressions␈α∞denote␈α∂the␈α∞same␈α∂concept.␈α∞ I␈α∂doubt␈α∞that␈α∂considerations␈α∞of
␈↓ α∧␈↓linguistic␈α∞usage␈α∞or␈α∞usefulness␈α∂in␈α∞AI␈α∞will␈α∞unequivocally␈α∂recommend␈α∞one␈α∞alternative,␈α∞so␈α∂both␈α∞will
␈↓ α∧␈↓have to be studied.

␈↓ α∧␈↓␈↓ αTThe␈α
same␈α
question␈α
arises␈α∞in␈α
modal␈α
logic.␈α
 Namely␈α∞we␈α
must␈α
choose␈α
between␈α∞the␈α
␈↓↓conceptual␈↓
␈↓ α∧␈↓␈↓↓identity␈↓

␈↓ α∧␈↓22)␈↓ αD ␈↓↓∀Q.(Poss Q = Not Nec Not Q)␈↓,

␈↓ α∧␈↓and the weaker extensional axiom

␈↓ α∧␈↓␈↓ ε|6␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓23)␈↓ αD ␈↓↓∀Q.(true Poss Q ≡ true Not Nec Not Q)␈↓.

␈↓ α∧␈↓We␈α∂will␈α∂write␈α∂the␈α∂rest␈α∂of␈α∂our␈α⊂modal␈α∂axioms␈α∂as␈α∂conceptual␈α∂identities,␈α∂but␈α∂their␈α⊂translation␈α∂into
␈↓ α∧␈↓extensional form is easy.

␈↓ α∧␈↓␈↓ αTWe have

␈↓ α∧␈↓24)␈↓ αD ␈↓↓∀Q.(Nec(Q) Implies Q) = T)␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓25)␈↓ αD ␈↓↓∀Q1 Q2.(Nec(Q1) And Nec(Q1 Implies Q2) Implies Nec(Q2)) = T)␈↓.

␈↓ α∧␈↓yielding a system equivalent to T.

␈↓ α∧␈↓␈↓ αTS4 is given by

␈↓ α∧␈↓26)␈↓ αD ∀Q.(␈↓↓Nec Q = Nec Nec Q)␈↓,

␈↓ α∧␈↓and S5 by

␈↓ α∧␈↓27)␈↓ αD ␈↓↓∀Q.(Poss Q = Nec Poss Q)␈↓.

␈↓ α∧␈↓␈↓ αTActually,␈αthere␈αmay␈αbe␈α
no␈αneed␈αto␈αcommit␈αourselves␈α
to␈αa␈αparticular␈αmodal␈αsystem.␈α
 We␈αcan
␈↓ α∧␈↓simultaneously have the functions ␈↓↓NecT,␈↓ ␈↓↓Nec4␈↓ and ␈↓↓Nec5,␈↓ related by axioms such as

␈↓ α∧␈↓28)␈↓ αD ␈↓↓∀Q.((Nec4 Q Implies Nec5 Q) = T)␈↓

␈↓ α∧␈↓which␈αwould␈αseem␈α
plausible␈αif␈αwe␈αregard␈α
S4␈αas␈αcorresponding␈αto␈α
provability␈αin␈αsome␈α
system␈αand
␈↓ α∧␈↓S5 as truth in the intended model of the system.

␈↓ α∧␈↓␈↓ αTPresumably we shall want to relate necessity and equality by the axiom

␈↓ α∧␈↓29)␈↓ αD ␈↓↓∀X.nec Equal(X,X)␈↓,

␈↓ α∧␈↓and we may want the stronger relation

␈↓ α∧␈↓30)␈↓ αD␈↓↓∀X Y.(X=Y ≡ nec Equal(X,Y))␈↓

␈↓ α∧␈↓which␈α
asserts␈α
that␈α
two␈α
concepts␈α
are␈α
the␈α
same␈α
if␈α
and␈α
only␈α
if␈α
the␈α
equality␈α
of␈α
what␈α
they␈α
may␈α
denote␈α
is
␈↓ α∧␈↓necessary.


␈↓ α∧␈↓αPHILOSOPHICAL EXAMPLES - MOSTLY WELL KNOWN

␈↓ α∧␈↓␈↓ αTSome␈α
sentences␈α
that␈α∞recur␈α
as␈α
examples␈α∞in␈α
the␈α
philosophical␈α∞literature␈α
will␈α
be␈α∞expressed␈α
in
␈↓ α∧␈↓our notation so the treatments can be compared.


␈↓ α∧␈↓␈↓ ε|7␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓␈↓ αTFirst␈α
we␈α
have␈α
␈↓↓"The␈α
number␈α
of␈α
planets␈α
=␈α
9"␈↓␈α
and␈α
␈↓↓"Necessarily␈α
9␈α
=␈α
9"␈↓␈α
from␈α
one␈α
doesn't␈α
want␈α
to
␈↓ α∧␈↓deduce␈α∂␈↓↓"Necessarily␈α∂the␈α∂number␈α⊂of␈α∂planets␈α∂=␈α∂9"␈↓.␈α∂ This␈α⊂example␈α∂is␈α∂discussed␈α∂in␈α⊂(Kaplan␈α∂1969).
␈↓ α∧␈↓Consider the sentences

␈↓ α∧␈↓31)␈↓ αD ␈↓↓¬nec Equal(Number Planets, Concept 9)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓32)␈↓ αD ␈↓↓nec Equal(Concept number planets,Concept 9)␈↓.

␈↓ α∧␈↓Both␈αare␈αtrue.␈α 31)␈αasserts␈αthat␈αit␈αis␈αnot␈αnecessary␈αthat␈αthe␈αnumber␈αof␈αplanets␈αbe␈α9,␈αand␈α32␈αasserts
␈↓ α∧␈↓that␈α
the␈α
number␈α∞of␈α
planets,␈α
once␈α
determined,␈α∞is␈α
a␈α
number␈α∞that␈α
is␈α
necessarily␈α
equal␈α∞to␈α
9.␈α
 It␈α∞is␈α
a
␈↓ α∧␈↓major␈α
virtue␈α
of␈α
our␈αformalism␈α
that␈α
both␈α
meanings␈α
can␈αbe␈α
expressed␈α
and␈α
are␈αreadily␈α
distinguished.
␈↓ α∧␈↓Leibniz's␈α
law␈α∞of␈α
the␈α
replacement␈α∞of␈α
equals␈α
by␈α∞equals␈α
causes␈α
no␈α∞trouble,␈α
because␈α
␈↓↓"The␈α∞number␈α
of
␈↓ α∧␈↓↓planets = 9"␈↓ may be written

␈↓ α∧␈↓33)␈↓ αD ␈↓↓number(planets) = 9␈↓

␈↓ α∧␈↓or, using concepts, as

␈↓ α∧␈↓34)␈↓ αD ␈↓↓true Equal(Number Planets, Concept 9)␈↓,

␈↓ α∧␈↓and ␈↓↓"Necessarily 9=9"␈↓ is written

␈↓ α∧␈↓35)␈↓ αD ␈↓↓nec Equal(Concept 9,Concept 9)␈↓,

␈↓ α∧␈↓and these don't yield the unwanted conclusion.

␈↓ α∧␈↓␈↓ αTThe␈αfollowing␈αsentence␈αattributed␈αto␈αRussell␈αis␈αis␈αdiscussed␈αby␈αKaplan:␈α␈↓↓"I␈αthought␈αthat␈αyour
␈↓ α∧␈↓↓yacht was longer than it is"␈↓.  We can write it

␈↓ α∧␈↓36)␈↓ αD ␈↓↓true Believed(I,Greater(Length YourYacht,Concept denot Length YourYacht))␈↓

␈↓ α∧␈↓where␈α
we␈α
are␈α
not␈αanalyzing␈α
the␈α
pronouns␈α
or␈α
the␈αtense,␈α
but␈α
are␈α
using␈α
␈↓↓denot␈↓␈αto␈α
get␈α
the␈α
real␈αlength␈α
of
␈↓ α∧␈↓the␈αyacht␈αand␈α
␈↓↓Concept␈↓␈αto␈αget␈α
back␈αa␈αconcept␈αof␈α
this␈αtrue␈αlength␈α
so␈αas␈αto␈α
end␈αup␈αwith␈αa␈α
proposition
␈↓ α∧␈↓that␈α∞the␈α∞length␈α
of␈α∞the␈α∞yacht␈α
is␈α∞greater␈α∞than␈α∞that␈α
number.␈α∞ This␈α∞looks␈α
problematical,␈α∞but␈α∞if␈α∞it␈α
is
␈↓ α∧␈↓consistent, it is probably useful, and I think it is consistent.

␈↓ α∧␈↓␈↓ αTThe␈αfunction␈α
␈↓↓Concept␈↓␈αused␈α
in␈αthe␈α
above␈αexamples␈αmerits␈α
further␈αstudy.␈α
 It␈αseems␈α
useful␈αto
␈↓ α∧␈↓provide␈α⊃such␈α⊂a␈α⊃function␈α⊂mapping␈α⊃integers␈α⊂into␈α⊃standard␈α⊂concepts␈α⊃of␈α⊂integers,␈α⊃and␈α⊂we␈α⊃used␈α⊂a
␈↓ α∧␈↓similar␈α∂function␈α∂for␈α∂mapping␈α∂telephone␈α∂numbers␈α⊂regarded␈α∂as␈α∂strings␈α∂of␈α∂digits␈α∂into␈α⊂concepts␈α∂of
␈↓ α∧␈↓them.␈α⊃ This␈α∩can␈α⊃be␈α∩extended␈α⊃to␈α∩other␈α⊃domains,␈α∩and␈α⊃there␈α∩is␈α⊃no␈α∩need␈α⊃to␈α∩look␈α⊃for␈α∩a␈α⊃unique
␈↓ α∧␈↓preferred map from objects to their concepts.  Any maps that are found useful can be used.

␈↓ α∧␈↓␈↓ αTRyle␈α∞used␈α∞the␈α∂sentences␈α∞␈↓↓"Baldwin␈α∞is␈α∂a␈α∞statesman"␈↓␈α∞and␈α∞␈↓↓"Pickwick␈α∂is␈α∞a␈α∞fiction"␈↓␈α∂to␈α∞illustrate
␈↓ α∧␈↓that␈αparallel␈αsentence␈αconstruction␈αdoes␈αnot␈αalways␈αgive␈αparallel␈αsense.␈α We␈αwould␈αrender␈αthe␈αfirst
␈↓ α∧␈↓as␈α␈↓↓true␈α
Statesman␈αBaldwin␈↓␈α
or␈α␈↓↓statesman␈α
denot␈αBaldwin␈↓␈α
or␈α␈↓↓statesman␈α
baldwin␈↓,␈αwhile␈α
the␈αsecond␈α
can
␈↓ α∧␈↓only be rendered as ␈↓↓true Fiction Pickwick␈↓ or ␈↓↓fiction Pickwick␈↓.

␈↓ α∧␈↓␈↓ ε|8␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓αEXAMPLES IN ARTIFICIAL INTELLIGENCE

␈↓ α∧␈↓␈↓ αTA␈α
computer␈α∞program␈α
with␈α∞general␈α
intelligence␈α∞must␈α
be␈α∞able␈α
to␈α∞represent␈α
facts␈α∞about␈α
what
␈↓ α∧␈↓information␈αit␈αlacks␈αand␈αwhere␈αand␈αhow␈αit␈αis␈αto␈αbe␈αobtained.␈α The␈αexample␈αproblem␈αI␈αhave␈αbeen
␈↓ α∧␈↓considering␈α∂is␈α∂that␈α∂of␈α∂representing␈α∂what␈α∂a␈α∂traveler␈α∂knows␈α∂about␈α∂the␈α∂information␈α∂airline␈α∞clerks,
␈↓ α∧␈↓travel␈α⊂agents,␈α∂and␈α⊂reservation␈α∂computers,␈α⊂and␈α∂airline␈α⊂guides␈α∂have␈α⊂relevant␈α∂to␈α⊂a␈α⊂proposed␈α∂trip.
␈↓ α∧␈↓This is still rather difficult, but the following considerations have emerged:

␈↓ α∧␈↓␈↓ αT1.␈α⊃Unless␈α∩we␈α⊃formalize␈α⊃␈↓↓knowing␈↓␈α∩␈↓↓what,␈↓␈α⊃we␈α∩add␈α⊃to␈α⊃our␈α∩heuristic␈α⊃difficulties,␈α∩because␈α⊃the
␈↓ α∧␈↓theorem prover or other reasoner has an extra existential quantifier to deal with.

␈↓ α∧␈↓␈↓ αT2.␈αSimilarly␈αin␈αtreating␈αbelief␈αwe␈αneed␈αsomething␈αlike␈α␈↓↓denot(Telephone␈αMike,Pat,s)␈↓␈αstanding
␈↓ α∧␈↓for␈αwhat␈αPat␈α
believes␈αMike's␈αtelephone␈α
number␈αto␈αbe␈α
in␈αthe␈αsituation␈α
␈↓↓s.␈↓␈αNeither␈αis␈α
formalized␈αin
␈↓ α∧␈↓the philosophical literature.

␈↓ α∧␈↓␈↓ αT3.␈α∞Modal␈α
logic␈α∞offers␈α
difficulties␈α∞especially␈α
as␈α∞we␈α
need␈α∞often␈α
need␈α∞multiple␈α∞modalitieπ␈α
like
␈↓ α∧␈↓␈↓↓"believes␈α⊃he␈α⊃wants␈α⊃to␈α∩know"␈↓␈α⊃in␈α⊃a␈α⊃single␈α⊃sentence,␈α∩and␈α⊃this␈α⊃makes␈α⊃the␈α⊃Kripke␈α∩possible␈α⊃worlds
␈↓ α∧␈↓semantics␈αfor␈αmodal␈αlogic␈αalmost␈αimpossibly␈αcumbersome.␈α Modal␈αlogic␈αis␈αespecially␈αtroublesome␈αif
␈↓ α∧␈↓oblique contexts are only a small part of the problem.

␈↓ α∧␈↓␈↓ αT4.␈α
For␈α
this␈α
reason,␈α
the␈α
most␈α
useful␈α
of␈α
the␈α
earlier␈α
treatments␈α
seemed␈α
to␈α
involve␈α
making␈α
the
␈↓ α∧␈↓argument␈α∞of␈α∞knowledge␈α∞or␈α
belief␈α∞a␈α∞sentence␈α∞or␈α∞term␈α
and␈α∞weakening␈α∞the␈α∞Montague␈α∞and␈α
Kaplan
␈↓ α∧␈↓(1963)␈αknowledge␈αaxioms␈αsuitably␈αto␈αavoid␈αtheir␈αparadox.␈α However,␈αit␈αis␈αnot␈αeasy␈αto␈αimplement␈αa
␈↓ α∧␈↓reasoning program that goes into quoted phrases.

␈↓ α∧␈↓␈↓ αTConsider the following easier example:

␈↓ α∧␈↓␈↓ αTJoe␈α∞wants␈α∞to␈α∞know␈α∂Mike's␈α∞telephone␈α∞number.␈α∞ He␈α∞knows␈α∂that␈α∞Pat␈α∞knows␈α∞it␈α∞and␈α∂that␈α∞Pat
␈↓ α∧␈↓likes␈α⊃Joe.␈α⊃ We␈α⊃want␈α⊃the␈α⊃program␈α⊃to␈α∩decide␈α⊃on␈α⊃Joe's␈α⊃behalf␈α⊃to␈α⊃ask␈α⊃Pat␈α⊃for␈α∩Mike's␈α⊃telephone
␈↓ α∧␈↓number.



␈↓ α∧␈↓αPHILOSOPHICAL ATTITUDES

␈↓ α∧␈↓␈↓ αTMy␈α⊗motivation␈α∃for␈α⊗introducing␈α⊗concepts␈α∃as␈α⊗objects␈α∃comes␈α⊗from␈α⊗artificial␈α∃intelligence.
␈↓ α∧␈↓Namely,␈α∞I␈α∞want␈α
computer␈α∞programs␈α∞that␈α∞can␈α
reason␈α∞intelligently␈α∞about␈α
who␈α∞wants␈α∞what␈α∞or␈α
who
␈↓ α∧␈↓knows␈α
what.␈α∞ This␈α
leads␈α∞to␈α
considering␈α∞examples␈α
like␈α∞that␈α
of␈α∞the␈α
previous␈α∞section␈α
and␈α∞seems␈α
to
␈↓ α∧␈↓have the following philosophical consequences:

␈↓ α∧␈↓␈↓ αT1.␈αSince␈αwe␈αcannot␈αhope␈αto␈αmake␈αa␈αprogram␈αcapable␈αof␈αunderstanding␈αthe␈αwhole␈αworld,␈αwe
␈↓ α∧␈↓try␈αto␈αformalize␈αknowledge␈αetc.␈αin␈αa␈αway␈αthat␈αenables␈αthe␈αprogram␈αto␈αact␈αintelligently␈αin␈αa␈αlimited
␈↓ α∧␈↓domain.

␈↓ α∧␈↓␈↓ αT2.␈αWe␈αare␈αnot␈αespecially␈αattached␈αto␈αthe␈αusages␈αof␈αnatural␈αlanguage␈αexcept␈αin␈αso␈αfar␈αas␈αthey
␈↓ α∧␈↓suggest useful formalizations.



␈↓ α∧␈↓␈↓ ε|9␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓␈↓ αT3.␈αThere␈αis␈αno␈αharm␈αin␈αintroducing␈αlots␈αof␈αabstract␈αentities␈αlike␈αconcepts␈αand␈αno␈αinclination
␈↓ α∧␈↓to␈α∪restrict␈α∪ourselves␈α∪to␈α∪entities␈α∪that␈α∪can␈α∪be␈α∪defined␈α∪finitistically.␈α∪ This␈α∪is␈α∪because␈α∪we␈α∩aren't
␈↓ α∧␈↓interested␈αin␈αmaking␈αour␈αown␈αknowledge␈αmore␈αsecure␈α(as␈αphilosophers␈αoften␈αare)␈αbut␈αrather␈αwant
␈↓ α∧␈↓to␈α⊂make␈α⊂a␈α⊂computer␈α∂program␈α⊂act␈α⊂effectively␈α⊂even␈α∂at␈α⊂the␈α⊂cost␈α⊂of␈α∂having␈α⊂it␈α⊂reason␈α⊂naively.␈α∂ In
␈↓ α∧␈↓designing such programs, we take for granted our own common sense views of the world.

␈↓ α∧␈↓␈↓ αTI␈α∞must␈α∞confess,␈α
however,␈α∞to␈α∞finding␈α
this␈α∞attitude␈α∞philosophically␈α
attractive,␈α∞i.e.␈α∞first␈α∞find␈α
a
␈↓ α∧␈↓formal␈α∂system␈α∂that␈α∂reproduces␈α∂common␈α∂sense␈α∂reasoning␈α∂-␈α∂naively␈α∂if␈α∂necessary,␈α∂and␈α∂then␈α⊂try␈α∂to
␈↓ α∧␈↓make it secure.


␈↓ α∧␈↓REFERENCES

␈↓ α∧␈↓Church,␈αAlonzo␈α
(1951),␈αThe␈αNeed␈α
for␈αAbstract␈α
Entities␈αin␈αSemantic␈α
Analysis,␈αin␈α
␈↓↓Contributions␈αto
␈↓ α∧␈↓↓the␈α⊂Analysis␈α⊂and␈α∂Synthesis␈α⊂of␈α⊂Knowledge␈↓,␈α∂Proceedings␈α⊂of␈α⊂the␈α∂American␈α⊂Academy␈α⊂of␈α⊂Arts␈α∂and
␈↓ α∧␈↓Sciences,␈α␈↓α80␈↓,␈αNo.␈α
1␈α(July␈α1951),␈α100-112.␈α
 Reprinted␈αin␈α␈↓↓The␈αStructure␈α
of␈αLanguage␈↓,␈αedited␈αby␈α
Jerry
␈↓ α∧␈↓A. Fodor and Jerrold Katz, Prentice-Hall 1964

␈↓ α∧␈↓Frege,␈α
Gottlob␈α
(1892),␈α
Uber␈α
Sinn␈α
und␈α
Bedeutung.␈α
␈↓↓Zeitschrift␈α
fur␈α
Philosophie␈α
und␈α
Philosophische
␈↓ α∧␈↓↓Kritik␈↓␈α100:25-50.␈α Translated␈αby␈αH.␈αFeigl␈αunder␈αthe␈αtitle␈α"On␈αSense␈αand␈αNominatum"␈αin␈αH.␈αFeigl
␈↓ α∧␈↓and␈α⊂W.␈α⊂Sellars␈α⊂(eds.)␈α⊃␈↓↓Readings␈α⊂in␈α⊂Philosophical␈α⊂Analysis␈↓,␈α⊃New␈α⊂York␈α⊂1949.␈α⊂ Translated␈α⊃by␈α⊂M.
␈↓ α∧␈↓Black␈αunder␈αthe␈αtitle␈α"On␈αSense␈αand␈αReference"␈αin␈αP.␈αGeach␈αand␈αM.␈αBlack,␈α␈↓↓Translations␈αfrom␈αthe
␈↓ α∧␈↓↓Philosophical Writings of Gottlob Frege␈↓, Oxford, 1952.

␈↓ α∧␈↓Kaplan,␈α∞David␈α∞(1969),␈α
Quantifying␈α∞In,␈α∞from␈α∞␈↓↓Words␈α
and␈α∞Objections:␈α∞Essays␈α
on␈α∞the␈α∞Work␈α∞of␈α
W.V.
␈↓ α∧␈↓↓Quine␈↓,␈α
edited␈α∞by␈α
D.␈α∞Davidson␈α
and␈α
J.␈α∞ Hintikka,␈α
(Dordrecht-Holland:␈α∞D.␈α
Reidel␈α∞Publishing␈α
Co.),
␈↓ α∧␈↓pp. 178-214.  Reprinted in (Linsky 1971).

␈↓ α∧␈↓Linsky,␈α∩Leonard,␈α⊃ed.(1971)␈α∩␈↓↓Reference␈α⊃and␈α∩Modality␈↓,␈α⊃Oxford␈α∩Readings␈α⊃in␈α∩Philosophy,␈α⊃Oxford
␈↓ α∧␈↓University Press.

␈↓ α∧␈↓McCarthy,␈α∪J.␈α∪and␈α∪Hayes,␈α∪P.J.␈α∀(1969)␈α∪Some␈α∪Philosophical␈α∪Problems␈α∪from␈α∪the␈α∀Standpoint␈α∪of
␈↓ α∧␈↓Artificial␈α∩Intelligence.␈α∪␈↓↓Machine␈α∩Intelligence␈α∩4␈↓,␈α∪pp.␈α∩463-502␈α∩(eds␈α∪Meltzer,␈α∩B.␈α∩and␈α∪Michie,␈α∩D.).
␈↓ α∧␈↓Edinburgh: Edinburgh University Press.

␈↓ α∧␈↓Montague, Richard (1974), ␈↓↓Formal Philosophy␈↓, Yale University Press


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

␈↓ α∧␈↓␈↓εThis draft of CONCEP[S76,JMC] PUBbed at 21:53 on June 10, 1976.␈↓





␈↓ α∧␈↓␈↓ εu10␈↓ ∧