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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→␈↓↓individual␈↓␈α_␈↓↓concepts␈↓␈α→are␈α→admitted␈α_as
␈↓ α∧␈↓mathematical␈αobjects␈α
along␈αwith␈α
the␈α␈↓↓things␈↓␈α
they␈αdenote.␈α
 This␈αleads␈α
to␈αvery␈α
straightforward␈αfirst
␈↓ α∧␈↓order␈αformalizations␈αof␈αknowledge,␈αbelief,␈αwanting,␈αand␈αnecessity␈αwithout␈αputting␈αmodal␈αoperators
␈↓ α∧␈↓into the logic.  Applications are given in philosophy and in artificial intelligence.

␈↓ α∧␈↓␈↓εThis draft of CONCEP[S76,JMC] PUBbed at 16:35 on July 11, 1976.␈↓



















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


␈↓ α∧␈↓␈↓↓"...it␈αseems␈αthat␈αhardly␈αanybody␈αproposes␈αto␈αuse␈αdifferent␈αvariables␈αfor␈αpropositions␈αand␈αfor␈αtruth-
␈↓ α∧␈↓↓values, or different variables for individuals and individual concepts."␈↓ - (Carnap 1956, p. 113).


␈↓ α∧␈↓αINTRODUCTION

␈↓ α∧␈↓␈↓ αTAdmitting␈α⊗individual␈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 equals 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's␈α∀telephone␈α∀number␈α∃=␈α∀Mike's␈α∀telephone␈α∀number"␈↓,␈α∀then␈α∀␈↓↓"Pat␈α∃dialed␈α∀Mary's
␈↓ α∧␈↓↓telephone 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␈↓␈αare␈αthe␈αbasis␈αof␈αTarski's␈αsemantics␈αof␈αfirst␈αorder␈αlogic␈αand␈αmodel␈αtheory␈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␈αidea.␈α On␈αthe␈αother␈αhand,␈αlogicians␈αsuch␈αas␈αCarnap␈α(1947␈αand␈α1956),␈αChurch
␈↓ 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 been very satisfactory.

␈↓ α∧␈↓␈↓ αTThe␈αproblem␈αidentified␈αby␈α
Frege␈α-␈αof␈αsuitably␈α
limiting␈αthe␈αapplication␈αof␈αthe␈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␈αfor
␈↓ α∧␈↓any␈αsystem␈αthat␈αmust␈αrepresent␈αinformation␈αabout␈αbeliefs,␈αknowledge,␈αdesires,␈αor␈αlogical␈αnecessity␈α-
␈↓ α∧␈↓regardless␈αof␈αwhether␈αthe␈αrepresentation␈αis␈αdeclarative␈αor␈αprocedural␈α(as␈αin␈αPLANNER␈αand␈αother
␈↓ α∧␈↓AI formalisms).

␈↓ α∧␈↓␈↓ αTThe␈αpresent␈α
idea␈αis␈α
to␈αleave␈αthe␈α
logic␈αunchanged␈α
and␈αto␈αtreat␈α
concepts␈αas␈α
one␈αkind␈αof␈α
object
␈↓ α∧␈↓in␈α⊃an␈α⊂ordinary␈α⊃first␈α⊂order␈α⊃theory.␈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
expression␈α
whose␈αvalue␈α
is␈α
the␈α
concept␈α
of␈α
Mike's␈α
telephone␈αnumber␈α
instead
␈↓ α∧␈↓of␈αhaving␈αa␈αsingle␈αexpression␈αwhose␈αdenotation␈αis␈αthe␈αnumber␈αand␈αwhose␈αsense␈αis␈αa␈αconcept␈αof␈αit.
␈↓ α∧␈↓The␈αrelations␈αamong␈αconcepts␈αand␈αbetween␈αconcepts␈αand␈αother␈αentities␈αare␈αexpressed␈αby␈αformulas
␈↓ α∧␈↓of␈α∞first␈α∞order␈α∞logic.␈α∞ Ordinary␈α∞model␈α∞theory␈α∞can␈α∞then␈α∞be␈α∞used␈α∞to␈α∞study␈α∞what␈α∞spaces␈α∞of␈α
concepts
␈↓ α∧␈↓satisfy various sets of axioms.

␈↓ α∧␈↓␈↓ αTWe␈α
treat␈α
primarily␈α
what␈α
Carnap␈α∞calls␈α
␈↓↓individual␈↓␈α
␈↓↓concepts␈α
like␈↓␈α
␈↓↓Mike's␈α
telephone␈α∞number␈↓␈α
or
␈↓ α∧␈↓␈↓↓Pegasus␈↓␈α
and␈αnot␈α
general␈αconcepts␈α
like␈α␈↓↓telephone␈↓␈α
or␈α␈↓↓unicorn.␈↓␈α
Extension␈αto␈α
general␈α
concepts␈αseems
␈↓ α∧␈↓feasible, but individual concepts provide enough food for thought for the present.

␈↓ α∧␈↓␈↓ αTIt seems surprising that such a straightforward and easy approach should be new.





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


␈↓ 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∧␈↓␈↓ 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␈α∞itself.␈α
 Whether␈α∞the␈α
function␈α∞␈↓↓Telephone␈↓␈α
can␈α∞be␈α
identified␈α∞with␈α
the␈α∞general␈α
concept␈α∞of␈α
a
␈↓ α∧␈↓person's telephone number is not settled.  For the present, please suppose not.

␈↓ 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␈↓␈αrather␈α
than␈α␈↓↓knowing␈↓␈α
␈↓↓that␈↓␈αor␈α
␈↓↓knowing␈αhow.␈α
For␈αAI␈α
and␈αfor␈α
other␈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.␈α
The␈α
formulas␈α
are␈α
in␈α
a␈α
sorted␈α
first␈α
order␈α
logic␈α
with␈α
functions␈α
and␈α
equality.␈α
 Knowledge,
␈↓ α∧␈↓necessity,␈αetc.␈αwill␈αbe␈αdiscussed␈αwithout␈αextending␈αthe␈α
logic␈αin␈αany␈αway␈α-␈αsolely␈αbe␈αthe␈α
introduction
␈↓ α∧␈↓of␈αpredicate␈αand␈αfunction␈αsymbols␈αsubject␈αto␈αsuitable␈αaxioms.␈α In␈αthe␈αpresent␈αinformal␈αtreatement,
␈↓ α∧␈↓we will not be explicit about sorts, but we will try to be typographically 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∧␈↓␈↓ ε|3␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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␈α∂␈↓↓K(P,Q)␈↓␈α∂is␈α∂the␈α∂proposition␈α∂that␈α∂␈↓↓P␈↓␈α∂knows␈α∂␈↓↓that␈↓␈α∂␈↓↓Q.␈↓␈α∂English␈α∂is␈α∂logically␈α∂not␈α∂uniform,␈α∂since
␈↓ α∧␈↓knowing␈α∞an␈α∞individual␈α∞concept␈α
means␈α∞knowing␈α∞its␈α∞value␈α
whereas␈α∞knowing␈α∞a␈α∞proposition␈α
means
␈↓ α∧␈↓knowing␈α∞that␈α∞it␈α∞has␈α∞a␈α∞particular␈α∞value,␈α∞namely␈α∞␈↓↓t.␈↓␈α∞There␈α∞is␈α∞no␈α∞reason␈α∞to␈α∞make␈α∞robots␈α∞with␈α
this
␈↓ α∧␈↓infirmity.

␈↓ 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␈α∂will␈α∂also␈α∂need␈α∂a␈α∂denotation␈α⊂function.␈α∂ From␈α∂(6)␈α∂follows␈α∂the␈α∂existence␈α∂of␈α⊂a␈α∂function
␈↓ α∧␈↓␈↓↓telephone␈↓ satisfying

␈↓ α∧␈↓7)␈↓ αt ␈↓↓∀P.(denot Telephone P = telephone denot p)␈↓.

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

␈↓ α∧␈↓8)␈↓ αt ␈↓↓∀P1 P2 X.(denot P1 = denot P2 ⊃ denot Know(P1,X) = denot Know(P2,X))␈↓,

␈↓ α∧␈↓but␈αit␈αis␈αnot␈αextensional␈αin␈αits␈αsecond␈αargument.␈α We␈αcan␈αtherefore␈αdefine␈αa␈αpredicate␈α␈↓↓know(p,X)␈↓
␈↓ α∧␈↓satisfying

␈↓ α∧␈↓9)␈↓ αt ␈↓↓∀P X.(true Know(P,X) ≡ know(denot P,X))␈↓.

␈↓ α∧␈↓(Note␈αthat␈αall␈αthese␈α
predicates␈αand␈αfunctions␈αare␈αentirely␈α
extensional␈αin␈αthe␈αunderlying␈α
logic,␈αand
␈↓ α∧␈↓the notion of extensionality presented here is relative to ␈↓↓denot.)␈↓

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

␈↓ α∧␈↓10)␈↓ 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∧␈↓␈↓ ε|4␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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∧␈↓11)␈↓ 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∧␈↓12)␈↓ 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∧␈↓13)␈↓ αt ␈↓↓true Ishorse Pegasus␈↓

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

␈↓ α∧␈↓14)␈↓ αt ␈↓↓∀X x.(denotes(X,x) ⊃ (ishorse x ≡ true Ishorse X))␈↓.

␈↓ α∧␈↓which␈α∂is␈α∞the␈α∂way␈α∂we␈α∞assert␈α∂extensionality␈α∞without␈α∂assuming␈α∂that␈α∞all␈α∂concepts␈α∂have␈α∞denotations.
␈↓ α∧␈↓␈↓↓Exists␈↓␈α∞is␈α
extensional␈α∞in␈α
this␈α∞sense,␈α∞but␈α
the␈α∞corresponding␈α
predicate␈α∞␈↓↓exists␈↓␈α
is␈α∞identically␈α∞true␈α
and
␈↓ α∧␈↓therefore dispensable.

␈↓ 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∧␈↓15)␈↓ 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 that ␈↓↓X␈↓ and ␈↓↓Y␈↓ denote the same thing if anything.  We shall want axioms

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

␈↓ α∧␈↓17)␈↓ αt ␈↓↓∀X Y.(true Equal(X,Y) ≡ true Equal(Y,X))␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓18)␈↓ 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∧␈↓19)␈↓ 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∧␈↓␈↓ ε|5␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


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

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

␈↓ α∧␈↓21)␈↓ αt ␈↓↓true Know(Pat,Telephone Mary)␈↓.

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

␈↓ α∧␈↓22)␈↓ αt ␈↓↓true K(Pat,Equal(Telephone Mary,Telephone Mike)␈↓

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

␈↓ 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∂the␈α∂fact␈α⊂that␈α∂the
␈↓ α∧␈↓principles␈α∞of␈α∞intensional␈α∞reasoning␈α∂haven't␈α∞yet␈α∞been␈α∞stated␈α∂is␈α∞evidence␈α∞against␈α∞the␈α∂suitability␈α∞of
␈↓ α∧␈↓ordinary language for stating them.


␈↓ 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∧␈↓23)␈↓ αt ␈↓↓∀n.(denot Concept1 n = n)␈↓.

␈↓ α∧␈↓We can then have simultaneously

␈↓ α∧␈↓24)␈↓ αt ␈↓↓true Not Knew(Kepler,Number Planets)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓25)␈↓ αt ␈↓↓true Knew(Kepler,Composite Concept1 denot Number Planets)␈↓.

␈↓ α∧␈↓(25) can be condensed using ␈↓↓Composite1␈↓ which takes

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

␈↓ α∧␈↓26)␈↓ αt ␈↓↓Composite1 n = Composite Concept1 n␈↓

␈↓ α∧␈↓getting

␈↓ α∧␈↓27)␈↓ αt ␈↓↓true Knew(Kepler,Composite1 denot Number Planets)␈↓.


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


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

␈↓ α∧␈↓28)␈↓ αt ␈↓↓Composite2 N = Composite Concept1 denot N␈↓,

␈↓ α∧␈↓letting us write

␈↓ α∧␈↓29)␈↓ αt ␈↓↓true Knew(Kepler,Composite2 Number Planets)␈↓,

␈↓ α∧␈↓which is true even though

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

␈↓ α∧␈↓is␈α
false.␈α
 (30)␈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∂(25),␈α∂(27),␈α⊂and␈α∂(29)␈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∀is␈α∀composite,␈α∃where␈α∀this␈α∃number␈α∀(perhaps
␈↓ α∧␈↓↓unbeknownst to Kepler) is the number of planets"␈↓.

␈↓ α∧␈↓␈↓ αTWe␈αmay␈αalso␈αwant␈αa␈αmap␈αfrom␈αthings␈αto␈αconcepts␈αof␈αthem␈αin␈αorder␈αto␈αformalize␈αa␈αsentence
␈↓ α∧␈↓like, ␈↓↓"Lassie knows the location of all her puppies"␈↓.  We write this

␈↓ α∧␈↓31)␈↓ αt ␈↓↓∀x.(x ε puppies(lassie) ⊃ true Knowd(Lassie,Locationd Conceptd x))␈↓.

␈↓ α∧␈↓Here␈α␈↓↓Conceptd␈↓␈αtakes␈αa␈αpuppy␈αinto␈αa␈αdog's␈αconcept␈αof␈αit,␈αand␈α␈↓↓Locationd␈↓␈αtakes␈αa␈αdog's␈αconcept␈α
of␈αa
␈↓ α∧␈↓puppy␈α∩into␈α⊃a␈α∩dog's␈α⊃concept␈α∩of␈α⊃its␈α∩location.␈α⊃ The␈α∩axioms␈α⊃satisfied␈α∩by␈α⊃␈↓↓Knowd,␈↓␈α∩␈↓↓Locationd␈↓␈α⊃and
␈↓ α∧␈↓␈↓↓Conceptd␈↓ can be tailored to our ideas of what dogs know.


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


␈↓ 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 like integers, but it is not obvious how to treat knowing a person.

␈↓ α∧␈↓␈↓ αTIf there is a ␈↓↓standard␈↓ ␈↓↓concept␈↓ function ␈↓↓Concept1,␈↓ then we can rewrite (35) as

␈↓ α∧␈↓36)␈↓ αt ␈↓↓true Know(P,X) ≡ ∃a.true K(P,Equal(X,Concept1 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,Concept1 AA))␈↓,

␈↓ α∧␈↓where ␈↓↓AA␈↓ is an "inner 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∧␈↓␈↓ αTIn␈α⊂␈↓↓Meaning␈α⊃and␈α⊂Necessity␈↓,␈α⊂Carnap␈α⊃takes␈α⊂the␈α⊂first␈α⊃alternative,␈α⊂regarding␈α⊂concepts␈α⊃as␈α⊂L-
␈↓ α∧␈↓equivalence␈αclasses␈αof␈αexpressions.␈α This␈αworks␈αnicely␈αfor␈αdiscussing␈αnecessity,␈αbut␈αwhen␈αhe␈αwants
␈↓ α∧␈↓to␈αdiscuss␈αknowledge␈αwithout␈αassuming␈αthat␈αeveryone␈αknows␈αFermat's␈αlast␈αtheorem␈αif␈αit␈αis␈αtrue,␈αhe
␈↓ α∧␈↓introduces␈α
the␈α∞notion␈α
of␈α
␈↓↓intensional␈↓␈α∞␈↓↓isomorphism␈↓␈α
and␈α
has␈α∞knowledge␈α
operate␈α
on␈α∞the␈α
equivalence
␈↓ α∧␈↓classes of this relation.

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


␈↓ α∧␈↓collection␈α∪of␈α∪propositional␈α∪identities␈α∪like␈α∪(11)␈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∧␈↓␈↓ αTOf␈αcourse,␈αthere␈αare␈αmore␈αthan␈αtwo␈αalternatives.␈α If␈αdifferently␈αLet␈α␈↓↓M␈↓␈αbe␈αthe␈αa␈αfree␈αalgebra
␈↓ α∧␈↓built␈α∪up␈α∩from␈α∪the␈α∩"atomic"␈α∪concepts␈α∩by␈α∪the␈α∩concept␈α∪forming␈α∩function␈α∪symbols.␈α∩ If␈α∪≡≡␈α∪is␈α∩an
␈↓ α∧␈↓equivalence relation on ␈↓↓M␈↓ such that

␈↓ α∧␈↓40)␈↓ αt␈↓↓∀X1 X2 ε M.((X1 ≡≡ X2) ⊃ (true X1 ≡ true X2))␈↓,

␈↓ α∧␈↓then the set of equivalence classes under ≡≡ may be taken as the set of concepts.

␈↓ α∧␈↓␈↓ αTSimilar possibilities arise in modal logic.  We can choose between the ␈↓↓conceptual␈↓ ␈↓↓identity␈↓

␈↓ α∧␈↓41)␈↓ αt ␈↓↓∀Q.(Poss Q = Not Nec Not Q)␈↓,

␈↓ α∧␈↓and the weaker extensional axiom

␈↓ α∧␈↓42)␈↓ αt ␈↓↓∀Q.(true Poss Q ≡ true Not Nec Not Q)␈↓.

␈↓ α∧␈↓We will write the rest of our modal axioms in extensional form.

␈↓ α∧␈↓␈↓ αTWe have

␈↓ α∧␈↓43)␈↓ αt ␈↓↓∀Q.(true Nec Q ⊃ true Q)␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓44)␈↓ αt ␈↓↓∀Q1 Q2.(true Nec Q1 ∧ true Nec(Q1 Implies Q2) ⊃ true Nec Q2)␈↓.

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

␈↓ α∧␈↓␈↓ αTS4 is given by

␈↓ α∧␈↓45)␈↓ αt ∀Q.(␈↓↓true Nec Q ≡ true Nec Nec Q)␈↓,

␈↓ α∧␈↓and S5 by

␈↓ α∧␈↓46)␈↓ αt ␈↓↓∀Q.(true Poss Q ≡ true 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∧␈↓47)␈↓ αt ␈↓↓∀Q.(true Nec4 Q ⊃ true Nec5 Q)␈↓

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


␈↓ 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∧␈↓48)␈↓ αt ␈↓↓∀X.true Nec Equal(X,X)␈↓,

␈↓ α∧␈↓and Carnap's system translates to the stronger relation

␈↓ α∧␈↓49)␈↓ αt␈↓↓∀X Y.(X=Y ≡ true 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∧␈↓αQUANTIFIERS AND ABSTRACT FORMS

␈↓ α∧␈↓␈↓ αTIn␈αthis␈αsection␈αwe␈αintroduce␈αpropositions␈αformed␈αby␈αquantification␈αand␈α
individual␈αconcepts
␈↓ α∧␈↓formed␈α∨by␈α∨description.␈α≡ The␈α∨formalism␈α∨does␈α≡this␈α∨within␈α∨first␈α≡order␈α∨logic,␈α∨so␈α≡that
␈↓ α∧␈↓␈↓↓All(XX1,CChild(MMike,XX1)␈α≡IImplies␈α≡KKnow(MMike,TTelephone␈α≡XX1))␈↓␈α≡which␈α∨is␈α≡the
␈↓ α∧␈↓proposition␈αthat␈αMike␈α
knows␈αthe␈αtelephone␈α
numbers␈αof␈αall␈α
his␈αchildren,␈αis␈α
formed␈αby␈αapplying␈α
the
␈↓ α∧␈↓function␈α␈↓↓All␈↓␈αto␈αtwo␈αarguments␈αwhich␈αin␈αturn␈αare␈αbuilt␈αup␈αfrom␈αtheir␈αelementary␈αconstituents.␈α We
␈↓ α∧␈↓start␈α⊃with␈α⊃a␈α⊃class␈α⊃of␈α⊃␈↓↓objects␈↓␈α⊃which␈α⊃we␈α⊃will␈α⊃call␈α⊃␈↓↓inner␈α⊃variables␈↓␈α⊃and␈α⊃build␈α⊃up␈α⊃from␈α⊃them␈α⊂and
␈↓ α∧␈↓constants␈α∀a␈α∃class␈α∀of␈α∀␈↓↓abstract␈α∃forms␈↓␈α∀using␈α∃certain␈α∀functions␈α∀with␈α∃names␈α∀like␈α∃␈↓↓ttelephone␈↓␈α∀and
␈↓ α∧␈↓␈↓↓TTelepone.␈α⊃ ␈↓␈α⊃As␈α∩far␈α⊃as␈α⊃the␈α⊃outer␈α∩logic␈α⊃is␈α⊃concerned␈α⊃the␈α∩␈↓↓inner␈α⊃variables␈↓␈α⊃are␈α⊃objects␈α∩and␈α⊃not
␈↓ α∧␈↓variables␈α⊂at␈α∂all,␈α⊂and␈α⊂therefore␈α∂we␈α⊂will␈α∂need␈α⊂variables␈α⊂that␈α∂can␈α⊂take␈α∂inner␈α⊂variables␈α⊂as␈α∂values.
␈↓ α∧␈↓Moreover,␈α
␈↓↓inner␈αvariables␈↓␈α
can␈α
either␈αrange␈α
over␈αconcepts␈α
or␈α
things.␈α To␈α
keep␈αall␈α
this␈α
straight␈αwe
␈↓ α∧␈↓make the following typographical conventions:

␈↓ α∧␈↓1.␈αAs␈αbefore,␈αan␈αexpression␈αwhose␈α
value␈αis␈αa␈αconcept␈αbegins␈αin␈α
upper␈αcase␈αand␈αone␈αwhose␈αvalue␈α
is
␈↓ α∧␈↓a thing is in lower case.

␈↓ α∧␈↓2.␈αAn␈α␈↓↓inner␈αvariable␈↓␈αis␈αrepresented␈αby␈αa␈αdoubled␈αletter␈αfollowed␈αby␈αdigits,␈αand␈αa␈αvariable␈αranging
␈↓ α∧␈↓over␈α∩␈↓↓abstract␈α⊃forms␈↓␈α∩is␈α⊃represented␈α∩by␈α⊃doubled␈α∩letters␈α⊃without␈α∩digits.␈α⊃ Thus␈α∩␈↓↓XX0␈↓␈α⊃is␈α∩an␈α⊃inner
␈↓ α∧␈↓variable␈αranging␈αover␈αconcepts,␈αand␈α␈↓↓XX␈↓␈αis␈αa␈αvariable␈αranging␈αover␈αconcept-valued␈αabstract␈αforms
␈↓ α∧␈↓which␈α⊃will␈α⊃include␈α⊂inner␈α⊃variables␈α⊃like␈α⊂␈↓↓XX0.␈↓␈α⊃Likewise␈α⊃␈↓↓xx0␈↓␈α⊂is␈α⊃an␈α⊃inner␈α⊃variable␈α⊂representing
␈↓ α∧␈↓things, and ␈↓↓xx␈↓ is a variable ranging thing-valued abstract forms.

␈↓ α∧␈↓␈↓ αTInner␈α
variables␈α
are␈α
one␈α
kind␈α
of␈α
abstract␈α
form,␈α
and␈α
another␈α
kind␈α
are␈α∞constants.␈α
 Constants
␈↓ α∧␈↓are␈αtreated␈αsimilarly␈αto␈αvariables.␈α Thus␈αwe␈αhave␈α␈↓↓MMike␈↓␈α-␈αa␈αconstant␈αwhose␈αvalue␈αis␈αthe␈αconcept
␈↓ α∧␈↓represented by ␈↓↓Mike␈↓ and ␈↓↓mmike␈↓ a constant whose value is Mike himself.

␈↓ α∧␈↓␈↓ αTAbstract␈αforms␈α
are␈αbuilt␈α
from␈αconstants␈α
and␈αvariables␈α
by␈αfunctions.␈α
 Thus␈α␈↓↓TTelephone␈↓␈α
takes
␈↓ α∧␈↓an␈αabstract␈αform␈αrepresenting␈α
a␈αperson␈αinto␈αan␈α
abstract␈αform␈αrepresenting␈αhis␈α
telephone␈αnumber.
␈↓ α∧␈↓Thus␈α
we␈α
have␈α
␈↓↓TTelephone␈α
PP0␈↓␈α
is␈α
an␈α
abstract␈α
form␈α
and␈α
so␈α
is␈α
␈↓↓TTelephone␈α∞MMike␈↓.␈α
 ␈↓↓TTelephone
␈↓ α∧␈↓↓PP␈↓␈α
is␈α
an␈α
expression␈α
with␈α
a␈α
variable␈α
␈↓↓PP␈↓␈α
ranging␈α
over␈α
abstract␈α
forms.␈α
 Corresponding␈α
to␈α
all␈αthe
␈↓ α∧␈↓functions␈α∞of␈α∞the␈α∞previous␈α∞sections␈α∞we␈α∞have␈α∞form-making␈α∞functions␈α∞represented␈α∞by␈α∞doubling␈α∞the
␈↓ α∧␈↓initial letters.

␈↓ α∧␈↓␈↓ εu10␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓␈↓ αTWe␈α∩have␈α∩two␈α∩additional␈α∪ways␈α∩of␈α∩making␈α∩abstract␈α∩forms.␈α∪ Namely,␈α∩if␈α∩␈↓↓XX␈↓␈α∩is␈α∪an␈α∩inner
␈↓ α∧␈↓variable␈α⊗and␈α∃␈↓↓QQ␈↓␈α⊗is␈α⊗an␈α∃abstract␈α⊗form,␈α∃then␈α⊗␈↓↓AAll(XX,QQ)␈↓␈α⊗is␈α∃an␈α⊗abstract␈α∃form␈α⊗and␈α⊗so␈α∃is
␈↓ α∧␈↓␈↓↓TThe(XX,QQ).␈αThus␈↓␈α␈↓↓TThe(PP0,KKnow(PP0,TTelephone␈αMMike))␈↓␈αis␈αan␈αabstract␈αform,␈αand␈αwe
␈↓ α∧␈↓intend␈α∞that␈α
it␈α∞shall␈α∞have␈α
as␈α∞value␈α∞a␈α
concept␈α∞of␈α
the␈α∞unique␈α∞person␈α
who␈α∞knows␈α∞Mike's␈α
telephone
␈↓ α∧␈↓number.␈α⊃ In␈α∩the␈α⊃logic␈α∩␈↓↓AAll␈↓␈α⊃and␈α⊃␈↓↓TThe␈↓␈α∩are␈α⊃functions.␈α∩ In␈α⊃order␈α⊃to␈α∩avoid␈α⊃trouble␈α∩with␈α⊃bound
␈↓ α∧␈↓variables, we shall want

␈↓ α∧␈↓50)␈↓ αt␈α⊂␈↓↓TThe(PP0,KKnow(PP0,TTelephone␈α⊂MMike))␈α⊂=␈α∂TThe(PP5,KKnow(PP5,TTelephone
␈↓ α∧␈↓↓Mike))␈↓␈α∞i.e.␈α∂the␈α∞abstract␈α∂form␈α∞produced␈α∂by␈α∞␈↓↓AAll␈↓␈α∞is␈α∂invariant␈α∞under␈α∂alphabetic␈α∞change␈α∂of␈α∞bound
␈↓ α∧␈↓variable.  This is part of the reason for calling them ␈↓↓abstract.␈↓

␈↓ α∧␈↓␈↓ αTIn␈α
order␈αto␈α
go␈α
from␈αconcept-valued␈α
abstract␈αforms␈α
to␈α
concepts,␈αwe␈α
introduce␈α
two␈αfunctions
␈↓ α∧␈↓␈↓↓Value␈αEE␈↓␈αand␈α␈↓↓Value(EE,)␈↓␈αwhere␈αis␈αcalled␈αa␈αstate␈αvector␈αand␈αassigns␈αvalues␈αto␈αall␈αinner␈αvariables.
␈↓ α∧␈↓If␈α∞the␈α∂abstract␈α∞form␈α∂␈↓↓EE␈↓␈α∞has␈α∞no␈α∂inner␈α∞variables,␈α∂then␈α∞␈↓↓Value␈α∞EE␈↓␈α∂can␈α∞be␈α∂used.␈α∞ The␈α∂inner␈α∞form
␈↓ α∧␈↓␈↓↓TThe(PP0,KKnow(PP0,TTelephone␈α
MMike))␈↓␈α
has␈α
no␈α
variables;␈α
the␈α
function␈α
␈↓↓TThe␈↓␈α
gets␈α
rid␈α
of␈α
the
␈↓ α∧␈↓variable␈α
␈↓↓PP0␈↓␈α
in␈α
its␈α
second␈αargument,␈α
so␈α
␈↓↓Value␈α
TThe(PP0,KKnow(PP0,TTelephone␈α
MMike))␈↓␈αis
␈↓ α∧␈↓the␈αconcept␈αof␈αthe␈αunique␈αperson␈αwho␈αknows␈αMike's␈αtelephone␈αnumber.␈α If␈αthere␈αis␈αsuch␈αa␈αperson,
␈↓ α∧␈↓then ␈↓↓denote Value TThe(PP0,KKnow(PP0,TTelephone MMike))␈↓ is that person.

␈↓ α∧␈↓␈↓ αTSimilarly,

␈↓ α∧␈↓51)␈↓ αt␈α∩␈↓↓Value␈α∩AAll(PP0,KKnow(PP0,TTelephone␈α∩MMike)␈α∩Implies␈α∩KKnow(PP0,TTelephone
␈↓ α∧␈↓↓MMary))␈↓

␈↓ α∧␈↓is the proposition that everyone who knows Mike's telephone number also knows Mary's.

␈↓ α∧␈↓(If␈α⊃the␈α⊃reader␈α⊂is␈α⊃getting␈α⊃impatient␈α⊃with␈α⊂all␈α⊃this␈α⊃formalism,␈α⊃let␈α⊂him␈α⊃remember␈α⊃that␈α⊃the␈α⊂above
␈↓ α∧␈↓formulas␈α∞are␈α
constants␈α∞in␈α∞a␈α
first␈α∞order␈α
logic␈α∞formalism,␈α∞i.e.␈α
are␈α∞formed␈α
by␈α∞applying␈α∞functions␈α
to
␈↓ α∧␈↓arguments.  This will pay off later, I promise).

␈↓ α∧␈↓␈↓ αTActually,␈αwe␈αmay␈αhave␈α
made␈αmore␈αdistinctions␈αin␈αthe␈α
notation␈αthan␈αare␈αnecessary,␈αbut␈α
it␈αis
␈↓ α∧␈↓easier to remove them than add them later.





␈↓ 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
by␈αQuine␈α
(1961)␈αand
␈↓ α∧␈↓(Kaplan 1969).  Consider the sentences

␈↓ α∧␈↓52)␈↓ αt ␈↓↓¬nec Equal(Number Planets, Concept1 9)␈↓


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


␈↓ α∧␈↓and

␈↓ α∧␈↓53)␈↓ αt ␈↓↓nec Equal(Concept1 number planets,Concept1 9)␈↓.

␈↓ α∧␈↓Both␈α
are␈α
true.␈α
 (52)␈αasserts␈α
that␈α
it␈α
is␈α
not␈αnecessary␈α
that␈α
the␈α
number␈α
of␈αplanets␈α
be␈α
9,␈α
and␈α(53)␈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∧␈↓Sustitutivity␈αof␈αequals␈αholds␈αin␈αthe␈αlogic␈αbut␈αcauses␈αno␈αtrouble,␈αbecause␈α␈↓↓"The␈αnumber␈αof␈αplanets␈α=
␈↓ α∧␈↓↓9"␈↓ may be written

␈↓ α∧␈↓54)␈↓ αt ␈↓↓number(planets) = 9␈↓

␈↓ α∧␈↓or, using concepts, as

␈↓ α∧␈↓55)␈↓ αt ␈↓↓true Equal(Number Planets, Concept1 9)␈↓,

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

␈↓ α∧␈↓56)␈↓ αt ␈↓↓nec Equal(Concept1 9,Concept1 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∧␈↓57)␈↓ αt ␈↓↓true Believed(I,Greater(Length YourYacht,Concept1 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␈α∩yacht␈α∩and␈α∩␈↓↓Concept1␈↓␈α∩to␈α∩get␈α∪back␈α∩a␈α∩concept␈α∩of␈α∩this␈α∩true␈α∪length␈α∩so␈α∩as␈α∩to␈α∩end␈α∩up␈α∪with␈α∩a
␈↓ 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 if it is consistent, it is probably useful, and I think it is consistent.

␈↓ 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∧␈↓␈↓ αTQuine (1961) considers illegitimate the sentence

␈↓ α∧␈↓58)␈↓ αt ␈↓↓(∃x)(Philip is unaware that x denounced Catiline)␈↓

␈↓ α∧␈↓obtained␈αfrom␈α␈↓↓"Philip␈αis␈αunaware␈αthat␈αTully␈αdenounced␈αCatiline"␈↓␈αby␈αexistential␈αgeneralization.␈α In
␈↓ α∧␈↓the␈α
example,␈α
we␈α
are␈α
also␈α
supposing␈α
the␈αtruth␈α
of␈α
␈↓↓Philip␈α
is␈α
aware␈α
that␈α
Cicero␈α
denounced␈αCatiline"␈↓.
␈↓ α∧␈↓These␈αsentences␈αare␈αrelated␈α
to␈α(perhaps␈αeven␈αexplicated␈αby)␈α
several␈αsentences␈αin␈αour␈αsystem.␈α
 ␈↓↓Tully␈↓
␈↓ α∧␈↓and␈α␈↓↓Cicero␈↓␈α
are␈αtaken␈αas␈α
distinct␈αconcepts.␈α The␈α
person␈αis␈αcalled␈α
␈↓↓tully␈↓␈αor␈α␈↓↓cicero␈↓␈α
in␈αour␈αlanguage,␈α
and
␈↓ α∧␈↓we have

␈↓ α∧␈↓59)␈↓ αt ␈↓↓tully = cicero␈↓,

␈↓ α∧␈↓␈↓ εu12␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓60)␈↓ αt ␈↓↓denot Tully = cicero␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓61)␈↓ αt ␈↓↓denot Cicero = cicero␈↓.

␈↓ α∧␈↓Going from persons to concepts, we may suppose

␈↓ α∧␈↓62)␈↓ αt ␈↓↓Concept2(philip,cicero) = Cicero␈↓,

␈↓ α∧␈↓asserting␈α∞that␈α
Philip's␈α∞concept␈α
of␈α∞the␈α
person␈α∞Cicero␈α
is␈α∞␈↓↓Cicero.␈α
The␈↓␈α∞basic␈α
assumptions␈α∞of␈α
Quine's
␈↓ α∧␈↓example also include

␈↓ α∧␈↓63)␈↓ αt ␈↓↓true K(Philip,Denounced(Cicero,Catiline))␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓64)␈↓ αt ␈↓↓¬true K(Philip,Denounced(Tully,Catiline))␈↓,

␈↓ α∧␈↓From (60), ... ,(64) we can deduce

␈↓ α∧␈↓65)␈↓ αt ␈↓↓∃P.true Denounced(P,Catiline) And Not K(Philip,Denounced(P,Catiline))␈↓,

␈↓ α∧␈↓from (64) and others, and

␈↓ α∧␈↓66)␈↓ αt ␈↓↓¬∃p.(denounced(p,catiline) ∧ ¬true K(Philip,Denounced(Concept2(philip,p), Catiline)))␈↓

␈↓ α∧␈↓using the additional hypotheses

␈↓ α∧␈↓67)␈↓ αt ␈↓↓∀p.(denounced(p,catiline) ⊃ p = cicero)␈↓,

␈↓ α∧␈↓68)␈↓ αt ␈↓↓denot Catiline = catiline␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓69)␈↓ αt ␈↓↓∀P1 P2.(denot Denounced(P1,P2) ≡ denounced(denot P1,denot P2))␈↓.

␈↓ α∧␈↓Presumably␈α∞(65)␈α∂is␈α∞always␈α∂true,␈α∞because␈α∂we␈α∞can␈α∞always␈α∂construct␈α∞a␈α∂concept␈α∞whose␈α∂denotation␈α∞is
␈↓ α∧␈↓Cicero␈α⊂unbeknownst␈α⊂to␈α⊂Philip.␈α⊂ The␈α⊂truth␈α⊂of␈α⊂(66)␈α⊂depends␈α⊂on␈α⊂Philip's␈α⊂knowing␈α⊃that␈α⊂someone
␈↓ α∧␈↓denounced␈αCatiline␈αand␈αon␈α
the␈αmap␈α␈↓↓Concept2(p1,p2)␈↓␈αthat␈α
gives␈αone␈αperson's␈αconcept␈α
of␈αanother.
␈↓ α∧␈↓If␈αwe␈αrefrain␈αfrom␈αusing␈αa␈αsilly␈αmap␈α
that␈αgives␈αsomething␈αlike␈α␈↓↓Denouncer(Catiline)␈↓␈αas␈αits␈αvalue,␈α
we
␈↓ α∧␈↓can get results that correspond to intuition.







␈↓ α∧␈↓␈↓ εu13␈↓ ∧
␈↓ 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␈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∧␈↓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∧␈↓␈↓ εu14␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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∧␈↓αREMARKS


␈↓ α∧␈↓REFERENCES

␈↓ α∧␈↓Carnap, Rudolf (1956), ␈↓↓Meaning and Necessity␈↓, University of Chicago Press.

␈↓ 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∧␈↓␈↓ εu15␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ 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∧␈↓Quine, W.V.O. (1961), ␈↓↓From a Logical Point of View␈↓, Harper and Row.


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

































␈↓ α∧␈↓␈↓ εu16␈↓ ∧