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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∞concepts.␈α
 This␈α
allows␈α∞quite␈α
straightforward␈α∞first␈α
order
␈↓ α∧␈↓formalizations␈α
of␈α∞knowledge,␈α
belief,␈α
wanting,␈α∞and␈α
necessity␈α
without␈α∞putting␈α
modal␈α∞operators␈α
into
␈↓ α∧␈↓the logic.  Applications are given in philosophy and in artificial intelligence.

␈↓ α∧␈↓␈↓εThis draft of CONCEP[S76,JMC] PUBbed at 1:32 on July 1, 1976.␈↓



















␈↓ 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␈α∪and␈α∩wanting,␈α∪and␈α∩a␈α∩␈↓↓denotation␈↓␈α∪which␈α∩is␈α∪its␈α∩␈↓↓meaning␈↓␈α∪in␈α∩␈↓↓direct␈↓
␈↓ α∧␈↓␈↓↓contexts.␈↓␈α␈↓↓Denotations␈↓␈α
are␈αthe␈α
basis␈αof␈α
the␈αTarskian␈α
semantics␈αof␈α
first␈αorder␈α
logic␈αand␈αmodel␈α
theory
␈↓ α∧␈↓and␈α
are␈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␈α∂the␈α∂idea.␈α∂ On␈α∂the␈α∂other␈α∂hand,␈α∂logicians␈α∂such␈α∂as␈α∂Church␈α∂(1951)␈α∂and␈α∞Montague
␈↓ α∧␈↓(1974)␈α∩see␈α⊃a␈α∩need␈α∩for␈α⊃␈↓↓concepts␈↓␈α∩and␈α∩have␈α⊃proposed␈α∩formalizations,␈α∩but␈α⊃none␈α∩have␈α∩been␈α⊃very
␈↓ α∧␈↓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␈α⊂necessity␈α⊂-␈α⊂regardless␈α⊂of␈α⊂whether␈α⊃the␈α⊂representation␈α⊂is␈α⊂declarative␈α⊂or␈α⊂procedural␈α⊃(as␈α⊂in
␈↓ α∧␈↓PLANNER and other AI formalisms).

␈↓ 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⊂one␈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∧␈↓␈↓ αTThe␈α
concepts␈αtreated␈α
here␈α
are␈αthose␈α
of␈α
individual␈αthings␈α
like␈α
␈↓↓the␈αconcept␈α
of␈αMike's␈α
telephone
␈↓ α∧␈↓↓number␈↓ and not general concepts like ␈↓↓the concept of telephone␈↓.



␈↓ α∧␈↓αKNOWING WHAT AND KNOWING THAT

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

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

␈↓ α∧␈↓with the following conventions:


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


␈↓ 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,␈αvariable␈αor␈αfunction␈αname␈αwhen␈αits␈αvalue␈αis␈αa␈αconcept.␈α (We␈αconsidered␈αalso␈αcapitalizing
␈↓ α∧␈↓the last letter when the arguments are concepts, but it made the formulas ugly).

␈↓ 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␈α∨that␈α≡␈↓↓P␈α∨knows␈↓␈α≡the␈α∨value␈α≡of␈α∨␈↓↓X.␈↓␈α≡In␈α∨(1),␈α≡therefore,
␈↓ α∧␈↓␈↓↓Know(Pat,Telephone␈αMike)␈↓␈αis␈αa␈αproposition␈αand␈αnot␈αa␈αtruth␈αvalue.␈α Note␈αthat␈αwe␈αare␈αformalizing
␈↓ α∧␈↓␈↓↓knowing␈↓␈α␈↓↓what␈↓␈α
rather␈αthan␈α␈↓↓knowing␈↓␈α
␈↓↓that␈↓␈αor␈α
␈↓↓knowing␈↓␈α␈↓↓how.␈↓␈αFor␈α
AI␈αand␈α
for␈αother␈αpractical␈α
purposes,
␈↓ α∧␈↓␈↓↓knowing␈↓ ␈↓↓what␈↓ seems to be the most useful notion of the three.

␈↓ α∧␈↓␈↓ αT5.␈α∞␈↓↓true(Q)␈↓␈α∞is␈α∞the␈α∞truth␈α∞value,␈α∞␈↓↓t␈↓␈α∞or␈α∂␈↓↓f,␈↓␈α∞of␈α∞the␈α∞proposition␈α∞␈↓↓Q,␈↓␈α∞and␈α∞we␈α∞must␈α∞write␈α∂␈↓↓true(Q)␈↓␈α∞in
␈↓ α∧␈↓order␈α∞to␈α∞assert␈α∞␈↓↓Q.␈↓␈α∞Later␈α∞we␈α∞will␈α∞consider␈α∞formalisms␈α∞in␈α∞which␈α∞␈↓↓true␈↓␈α∞has␈α∞a␈α∞second␈α∞argument␈α∂-␈α∞a
␈↓ α∧␈↓␈↓↓situation,␈↓␈α∩a␈α⊃␈↓↓story,␈↓␈α∩a␈α∩␈↓↓possible␈↓␈α⊃␈↓↓world,␈↓␈α∩or␈α∩even␈α⊃a␈α∩␈↓↓partial␈α⊃possible␈α∩world␈↓␈α∩(a␈α⊃notion␈α∩we␈α∩hope␈α⊃to
␈↓ α∧␈↓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,␈αif␈α(1)␈αis␈αto␈αbe␈αreasonable,␈αit␈αmust␈αnot␈αfollow␈αfrom␈α(1)␈αand␈αthe␈αfact␈αthat␈αMary's␈αtelephone
␈↓ 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)␈↓ αt ␈↓↓Know(Joe,Know(Pat,Telephone Mike))␈↓,

␈↓ α∧␈↓and asserting it requires writing

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

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

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

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



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


␈↓ α∧␈↓␈↓ αTWe␈αfirst␈αconsider␈αsystems␈αin␈αwhich␈αcorresponding␈αto␈αeach␈αconcept␈α␈↓↓X,␈↓␈αthere␈αis␈αa␈αthing␈α
␈↓↓x␈↓␈αof
␈↓ α∧␈↓which ␈↓↓X␈↓ is a concept.  Then there is a function ␈↓↓denot␈↓ such that

␈↓ α∧␈↓5)␈↓ αt ␈↓↓x = denot X␈↓.

␈↓ α∧␈↓Functions like ␈↓↓Telephone␈↓ are then related to ␈↓↓denot␈↓ by equations like

␈↓ α∧␈↓6)␈↓ αt ␈↓↓∀P1 P2.(denot P1 = denot P2 ⊃ denot Telephone P1 = denot Telephone P2)␈↓.

␈↓ α∧␈↓We␈α
call␈α
␈↓↓denot␈α
X␈↓␈α
the␈α
␈↓↓denotation␈↓␈α
of␈α
the␈α
concept␈α
␈↓↓X,␈↓␈α
and␈α
(6)␈α
asserts␈α
that␈α
the␈α
denotation␈α
of␈α
the␈α
concept
␈↓ α∧␈↓of␈α␈↓↓P␈↓'s␈αtelephone␈α
number␈αdepends␈αonly␈αon␈α
the␈αdenotation␈αof␈αthe␈α
concept␈α␈↓↓P␈↓.␈α The␈αvariables␈α
in␈α(6)
␈↓ α∧␈↓range␈αover␈αconcepts␈αof␈αpersons,␈αand␈αwe␈α
regard␈α(6)␈αas␈αasserting␈αthat␈α␈↓↓Telephone␈↓␈αis␈α
␈↓↓extensional␈↓␈αwith
␈↓ α∧␈↓respect␈αto␈α␈↓↓denot.␈↓␈αNote␈αthat␈αour␈α␈↓↓denot␈↓␈αoperates␈αon␈αconcepts␈αrather␈αthan␈αon␈αexpressions;␈αa␈αtheory␈αof
␈↓ α∧␈↓expressions will also need a denotation function.

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

␈↓ α∧␈↓7)␈↓ αt ␈↓↓∀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 ␈↓↓denot.)␈↓

␈↓ α∧␈↓␈↓ αTThe predicate ␈↓↓true␈↓ and the function ␈↓↓denot␈↓ are related by

␈↓ α∧␈↓8)␈↓ αt ␈↓↓∀Q.(true Q ≡ (denot Q = t))␈↓

␈↓ α∧␈↓provided␈αtruth␈αvalues␈αare␈αin␈αthe␈αrange␈αof␈α␈↓↓denot,␈↓␈αand␈α␈↓↓denot␈↓␈αmay␈αalso␈αbe␈αprovided␈αwith␈αa␈α
␈↓↓(partial)
␈↓ α∧␈↓↓possible world␈↓ argument.

␈↓ α∧␈↓␈↓ αTWhen␈αwe␈α
can't␈αassume␈α
that␈αall␈α
concepts␈αhave␈α
denotations,␈αwe␈α
use␈αa␈α
predicate␈α␈↓↓denotes(X,x)␈↓
␈↓ α∧␈↓instead of a function.  The extensionality of ␈↓↓Telephone␈↓ would then be written

␈↓ α∧␈↓9)␈↓ αt␈α␈↓↓∀P1␈αP2␈αx␈αu.(denotes(P1,x)∧denotes(P2,x)∧denotes(Telephone␈αP1,u)␈α⊃␈αdenotes(Telephone
␈↓ α∧␈↓↓P2,u))␈↓.

␈↓ α∧␈↓We now introduce the ␈↓↓property␈↓ ␈↓↓Exists␈↓ satisfying

␈↓ α∧␈↓10)␈↓ αt ␈↓↓∀X.(true Exists X ≡ ∃x.denotes(X,x))␈↓.

␈↓ α∧␈↓Suppose␈αwe␈αwant␈αto␈αassert␈αthat␈αPegasus␈αis␈αa␈αhorse␈αwithout␈αasserting␈αthat␈αPegasus␈αexists.␈α We␈αcan
␈↓ α∧␈↓do this by introducing the ␈↓↓property␈↓ ␈↓↓Ishorse␈↓ and writing

␈↓ α∧␈↓11)␈↓ αt ␈↓↓true Ishorse Pegasus␈↓

␈↓ α∧␈↓which is related to the predicate ␈↓↓ishorse␈↓ by

␈↓ α∧␈↓12)␈↓ αt ␈↓↓∀X x.(denotes(X,x) ⊃ (ishorse x ≡ true Ishorse X))␈↓.

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


␈↓ α∧␈↓We␈α
call␈α
␈↓↓Ishorse␈↓␈α
a␈α
␈↓↓regular␈↓␈α␈↓↓property␈↓␈α
because␈α
it␈α
is␈α
related␈α
to␈αa␈α
predicate␈α
␈↓↓ishorse␈↓␈α
as␈α
(12).␈α The␈α
property
␈↓ α∧␈↓␈↓↓Exists␈↓ is not regular.

␈↓ 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∧␈↓13)␈↓ αt ␈↓↓∀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␈αsymbol␈α
"="␈α
is␈αpart␈α
of␈α
the␈αlogic␈α
so␈α
that␈α␈↓↓X␈α
=␈α
Y␈↓␈αasserts␈α
that␈α
␈↓↓X␈↓␈αand␈α
␈↓↓Y␈↓␈α
are␈αthe␈α
same
␈↓ α∧␈↓concept.␈α⊗ To␈α↔write␈α⊗propositions␈α↔expressing␈α⊗equality,␈α↔we␈α⊗introduce␈α↔␈↓↓Equal(X,Y)␈↓␈α⊗which␈α↔is␈α⊗a
␈↓ α∧␈↓proposition␈α⊂that␈α∂␈↓↓X␈↓␈α⊂and␈α∂␈↓↓Y␈↓␈α⊂have␈α∂the␈α⊂same␈α⊂denote␈α∂the␈α⊂same␈α∂thing␈α⊂gf␈α∂anything.␈α⊂ We␈α⊂shall␈α∂want
␈↓ α∧␈↓axioms

␈↓ α∧␈↓14)␈↓ αt ␈↓↓∀X.true Equal(X,X)␈↓,

␈↓ α∧␈↓15)␈↓ αt ␈↓↓∀X Y.(true Equal(X,Y) ≡ true Equal(Y,X))␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓16)␈↓ αt ␈↓↓∀X Y Z.(true Equal(X,Y) ∧ true Equal(Y,Z) ⊃ true Equal(X,Z)␈↓

␈↓ α∧␈↓making ␈↓↓true Equal(X,Y)␈↓ an equivalence relation, and

␈↓ α∧␈↓17)␈↓ αt ␈↓↓∀X Y x.(true Equal(X,Y) ∧ denotes(X,x) ⊃ denotes(Y,x))␈↓

␈↓ α∧␈↓which␈α
relates␈αit␈α
to␈αequality␈α
in␈αthe␈α
logic.␈α The␈α
statement␈αthat␈α
Mary␈αhas␈α
the␈αsame␈α
telephone␈αas␈α
Mike
␈↓ α∧␈↓is asserted by

␈↓ α∧␈↓18)␈↓ αt ␈↓↓true Equal(Telephone Mary,Telephone Mike)␈↓,

␈↓ α∧␈↓and it obviously doesn't follow from this and (1) that

␈↓ α∧␈↓19)␈↓ αt ␈↓↓true Know(Pat,Telephone Mary)␈↓.

␈↓ α∧␈↓To draw this conclusion we need something like

␈↓ α∧␈↓20)␈↓ αt ␈↓↓true K(Pat,Equal(Telephone Mary,Telephone Mike)␈↓

␈↓ α∧␈↓and suitable axioms about knowledge.

␈↓ α∧␈↓␈↓ αTPropositions␈α⊂formed␈α⊂by␈α∂quantification␈α⊂present␈α⊂more␈α⊂difficulty.␈α∂ We␈α⊂will␈α⊂want␈α⊂a␈α∂function
␈↓ α∧␈↓␈↓↓All(var,exp),␈↓␈αwhere␈α
␈↓↓var␈↓␈αis␈α
a␈α"variable"␈αand␈α
␈↓↓exp␈↓␈αis␈α
some␈αkind␈α
of␈α"concept-valued␈αexpression".␈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∧␈↓21)␈↓ αt ␈↓↓true All(x,E) ≡ ∀x'.(true Subst(x',x,E))␈↓,

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


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

␈↓ 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∧␈↓␈↓ αTThese functions of concepts can be related to corresponding functions of things.  Thus

␈↓ α∧␈↓22)␈↓ αt ␈↓↓∀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∧␈↓23)␈↓ αt ␈↓↓∀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␈α
function␈α
when␈α
it␈α
appears␈αas␈α
the␈α
second␈α
argument␈α
of␈α␈↓↓Know,␈↓␈α
we␈α
would␈α
have␈α
a␈αnotation␈α
that
␈↓ α∧␈↓resembles␈α
ordinary␈α
language␈α
in␈αhandling␈α
obliquity␈α
entirely␈α
by␈αcontext.␈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∧␈↓αFUNCTIONS FROM THINGS TO STANDARD CONCEPTS OF THEM

␈↓ α∧␈↓␈↓ αTWhile␈α
the␈αrelation␈α
␈↓↓denotes(X,x)␈↓␈αbetween␈α
concepts␈αand␈α
things␈αis␈α
many-one,␈α
functions␈αgoing
␈↓ α∧␈↓from␈α
things␈α
to␈α∞standard␈α
concepts␈α
of␈α
them␈α∞are␈α
often␈α
useful.␈α
 Presumably␈α∞not␈α
all␈α
classes␈α∞of␈α
things
␈↓ α∧␈↓have␈α∂standard␈α∂concepts,␈α⊂but␈α∂numbers␈α∂certainly␈α⊂do.␈α∂ Suppose␈α∂that␈α⊂␈↓↓Concept1␈↓␈α∂␈↓↓n␈↓␈α∂gives␈α⊂a␈α∂standard
␈↓ α∧␈↓concept of the number ␈↓↓n,␈↓ so that

␈↓ α∧␈↓24)␈↓ αt ␈↓↓∀n.(denot Concept1 n = n)␈↓.

␈↓ α∧␈↓We can then have simultaneously

␈↓ α∧␈↓25)␈↓ αt ␈↓↓true Not Knew(Kepler,Number Planets)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓26)␈↓ αt ␈↓↓true Knew(Kepler,Composite Concept1 denot Number Planets)␈↓.

␈↓ α∧␈↓(26) can be condensed using ␈↓↓Composite1␈↓ which takes

␈↓ α∧␈↓a number into the proposition that it is composite, i.e.


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


␈↓ α∧␈↓27)␈↓ αt ␈↓↓Composite1 n = Composite Concept1 n␈↓

␈↓ α∧␈↓getting

␈↓ α∧␈↓28)␈↓ αt ␈↓↓true Knew(Kepler,Composite1 denot Number Planets)␈↓.

␈↓ α∧␈↓A further condensation can be achieved using ␈↓↓Composite2␈↓ defined by

␈↓ α∧␈↓29)␈↓ αt ␈↓↓Composite2 N = Composite Concept1 denot N␈↓,

␈↓ α∧␈↓letting us write

␈↓ α∧␈↓30)␈↓ αt ␈↓↓true Knew(Kepler,Composite2 Number Planets)␈↓

␈↓ α∧␈↓which is true even though

␈↓ α∧␈↓31)␈↓ αt ␈↓↓true Knew(Kepler,Composite Number Planets)␈↓

␈↓ α∧␈↓is␈α
false.␈α
 (31)␈α
is␈α
our␈α∞formal␈α
expression␈α
of␈α
␈↓↓"Kepler␈α
knew␈α∞that␈α
the␈α
number␈α
of␈α
planets␈α∞is␈α
composite"␈↓,
␈↓ α∧␈↓while␈α∂(26),␈α∂(28),␈α⊂and␈α∂(30)␈α∂express␈α∂a␈α⊂proposition␈α∂that␈α∂can␈α∂only␈α⊂be␈α∂stated␈α∂awkwardly␈α⊂in␈α∂English,
␈↓ α∧␈↓perhaps␈α∩as␈α∩␈↓↓"Kepler␈α⊃knew␈α∩that␈α∩a␈α∩certain␈α⊃number␈α∩n␈α∩is␈α⊃composite,␈α∩where␈α∩this␈α∩number␈α⊃(perhaps
␈↓ α∧␈↓↓unbeknownst to Kepler) is the number of planets"␈↓.


␈↓ α∧␈↓αRELATIONS BETWEEN KNOWING WHAT AND KNOWING THAT

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

␈↓ α∧␈↓32)␈↓ αt ␈↓↓true Know(Pat,Telephone Mike)␈↓.

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

␈↓ α∧␈↓33)␈↓ αt ␈↓↓true K(Pat,Equal(Telephone Mike,Concept1 "333-3333")␈↓

␈↓ α∧␈↓where␈α␈↓↓K(P,Q)␈↓␈αis␈α
the␈αproposition␈αthat␈α
␈↓↓denot(P)␈↓␈αknows␈αthe␈α
proposition␈α␈↓↓Q␈↓␈αand␈α
␈↓↓Concept1("333-3333")␈↓
␈↓ α∧␈↓is some standard concept of that telephone number.

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

␈↓ α∧␈↓34)␈↓ αt ␈↓↓K(P,Q) = (Q And Know(P,Q))␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓35)␈↓ αt ␈↓↓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∧␈↓␈↓ ε|7␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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 (35) as

␈↓ α∧␈↓36)␈↓ αt ␈↓↓true Know(P,X) ≡ ∃a.true K(P,Equal(X,Concept a))␈↓.

␈↓ α∧␈↓␈↓ αT(35)␈α⊂and␈α⊃(36)␈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∧␈↓37)␈↓ αt ␈↓↓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 (37).


␈↓ 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␈α⊃the␈α⊃same␈α⊃proposition␈α⊃as␈α⊃␈↓↓Q,␈↓␈α⊃or␈α⊂is␈α⊃it␈α⊃merely␈α⊃extensionally␈α⊃equivalent?␈α⊃ The␈α⊃first␈α⊂is
␈↓ α∧␈↓written

␈↓ α∧␈↓38)␈↓ αt␈↓↓∀Q. Not Not Q = Q␈↓,

␈↓ α∧␈↓and the second

␈↓ α∧␈↓39)␈↓ αt ␈↓↓ ∀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∃(9)␈α∀including␈α∀associative␈α∃and␈α∀distributive␈α∃laws␈α∀for
␈↓ α∧␈↓conjunction␈αand␈α
disjunction,␈αDe␈α
Morgan's␈αlaw,␈αand␈α
the␈αlaws␈α
governing␈αthe␈α
propositions␈α␈↓↓T␈↓␈αand␈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∧␈↓␈↓ ε|8␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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∧␈↓40)␈↓ αt ␈↓↓∀Q.(Poss Q = Not Nec Not Q)␈↓,

␈↓ α∧␈↓and the weaker extensional axiom

␈↓ α∧␈↓41)␈↓ αt ␈↓↓∀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∧␈↓42)␈↓ αt ␈↓↓∀Q.(Nec(Q) Implies Q) = T)␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓43)␈↓ αt ␈↓↓∀Q1 Q2.(Nec(Q1) And Nec(Q1 Implies Q2) Implies Nec(Q2)) = T)␈↓.

␈↓ α∧␈↓yielding a system equivalent to von Wright's T.

␈↓ α∧␈↓␈↓ αTS4 is given by

␈↓ α∧␈↓44)␈↓ αt ∀Q.(␈↓↓Nec Q = Nec Nec Q)␈↓,

␈↓ α∧␈↓and S5 by

␈↓ α∧␈↓45)␈↓ αt ␈↓↓∀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∧␈↓46)␈↓ αt ␈↓↓∀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∧␈↓47)␈↓ αt ␈↓↓∀X.nec Equal(X,X)␈↓,

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

␈↓ α∧␈↓48)␈↓ αt␈↓↓∀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∧␈↓␈↓ ε|9␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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∧␈↓␈↓ αTFirst␈αwe␈α
have␈α␈↓↓"The␈αnumber␈α
of␈αplanets␈α
=␈α9"␈↓␈αand␈α
␈↓↓"Necessarily␈α9␈α
=␈α9"␈↓␈αfrom␈α
one␈αdoesn't␈αwant␈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∧␈↓49)␈↓ αt ␈↓↓¬nec Equal(Number Planets, Concept 9)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓50)␈↓ αt ␈↓↓nec Equal(Concept number planets,Concept 9)␈↓.

␈↓ α∧␈↓Both␈αare␈αtrue.␈α
 (49)␈αasserts␈αthat␈α
it␈αis␈αnot␈α
necessary␈αthat␈αthe␈α
number␈αof␈αplanets␈α
be␈α9,␈αand␈α(50␈α
asserts
␈↓ 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∧␈↓51)␈↓ αt ␈↓↓number(planets) = 9␈↓

␈↓ α∧␈↓or, using concepts, as

␈↓ α∧␈↓52)␈↓ αt ␈↓↓true Equal(Number Planets, Concept 9)␈↓,

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

␈↓ α∧␈↓53)␈↓ αt ␈↓↓nec Equal(Concept 9,Concept 9)␈↓,

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

␈↓ α∧␈↓␈↓ αTThe␈αfollowing␈αsentence␈αattributed␈αto␈αRussell␈αis␈αis␈αdiscussed␈αby␈αKaplan:␈α␈↓↓"I␈αthought␈αthat␈αyour
␈↓ α∧␈↓↓yacht was longer than it is"␈↓.  We can write it

␈↓ α∧␈↓54)␈↓ αt ␈↓↓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∧␈↓␈↓ εu10␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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␈αillustrate␈α
that
␈↓ α∧␈↓parallel␈α
sentence␈α∞construction␈α
does␈α∞not␈α
always␈α∞give␈α
parallel␈α
sense.␈α∞ We␈α
would␈α∞render␈α
the␈α∞first␈α
as
␈↓ α∧␈↓␈↓↓true␈α
Statesman␈α
Baldwin␈↓␈α∞or␈α
␈↓↓statesman␈α
denot␈α∞Baldwin␈↓␈α
or␈α
␈↓↓statesman␈α∞baldwin␈↓,␈α
while␈α
the␈α∞second␈α
can
␈↓ α∧␈↓only be rendered as ␈↓↓true Fiction Pickwick␈↓ or ␈↓↓fiction Pickwick␈↓.


␈↓ 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␈αMike,Pat,s)␈↓␈α
standing
␈↓ α∧␈↓for␈αwhat␈αPat␈αbelieves␈αMike's␈αtelephone␈αnumber␈αto␈αbe␈αin␈αthe␈αsituation␈α␈↓↓s.␈↓␈αNeither␈αis␈αformalized␈αin
␈↓ 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∧␈↓*****

␈↓ α∧␈↓␈↓ αTThis section will be completed with a set of axioms from which together with the premisses

␈↓ α∧␈↓␈↓ αT␈↓↓true Want(Joe,Know(Joe,Telephone Mike)),

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

␈↓ α∧␈↓and

␈↓ α∧␈↓␈↓ αT␈↓↓true K(Joe,Like(Pat,Joe))␈↓,

␈↓ α∧␈↓␈↓ εu11␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓we will be able to deduce

␈↓ α∧␈↓␈↓ αT␈↓↓true Future Know(Joe,Telephone Mike)␈↓

␈↓ α∧␈↓entirely within the FOL formalism for first order logic.



␈↓ α∧␈↓αPHILOSOPHICAL REMARKS

␈↓ 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.␈αSince␈α
we␈αcan't␈α
immediately␈αmake␈α
programs␈αcapable␈α
of␈αunderstanding␈α
the␈αwhole␈αworld,␈α
we
␈↓ α∧␈↓are interested in formalizations that allow programs to act intelligently in a limited domains.

␈↓ 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∧␈↓␈↓ 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␈α
our␈α
own␈α
knowledge␈αmore␈α
secure␈α
(as␈α
philosophers␈α
sometimes␈α
define␈αtheir␈α
task)
␈↓ α∧␈↓but␈αrather␈αwant␈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␈α
such␈α∞programs,␈α∞we␈α∞take␈α
for␈α∞granted␈α∞our␈α∞own␈α
common␈α∞sense␈α∞views␈α∞of␈α
the
␈↓ α∧␈↓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␈αthat␈αallows␈αexpressing␈αcommon␈αsense␈αreasoning␈α-␈αnaively␈αif␈αnecessary,␈αand␈αthen␈αtry
␈↓ α∧␈↓to 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∧␈↓␈↓ εu12␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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∧␈↓␈↓ εu13␈↓ ∧