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␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓␈↓ εddraft





















␈↓ α∧␈↓α␈↓ αIFIRST ORDER THEORIES OF INDIVIDUAL CONCEPTS AND PROPOSITIONS

␈↓ 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∂that␈α⊂␈↓↓reify␈↓␈α⊂them.␈α⊂ This␈α⊂allows␈α⊂very␈α∂straightforward
␈↓ α∧␈↓formalizations␈α∞of␈α∞knowledge,␈α∂belief,␈α∞wanting,␈α∞and␈α∞necessity␈α∂in␈α∞ordinary␈α∞first␈α∞order␈α∂logic␈α∞without
␈↓ α∧␈↓modal operators.  Applications are given in philosophy and in artificial intelligence.
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓αINTRODUCTION

␈↓ 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∧␈↓␈↓ αTAdmitting␈α⊗individual␈α⊗concepts␈α⊗as␈α↔objects␈α⊗-␈α⊗with␈α⊗concept-valued␈α↔constants,␈α⊗variables,
␈↓ α∧␈↓functions␈αand␈αexpressions␈α-␈αallows␈αordinary␈α
first␈αorder␈αtheories␈αof␈αnecessity,␈αknowledge,␈αbelief␈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 for equals that either device makes necessary.

␈↓ α∧␈↓␈↓ αTAccording␈α∩to␈α∩the␈α∩ideas␈α∩of␈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␈α
Mike's␈αtelephone␈αnumber"␈↓␈αis␈αthe␈αconcept␈αof␈α
Mike's␈αtelephone
␈↓ α∧␈↓number,␈αwhereas␈αits␈αmeaning␈αin␈αthe␈αsentence␈α␈↓↓"Pat␈αdialed␈αMike's␈αtelephone␈αnumber"␈↓␈αis␈αthe␈α
number
␈↓ α∧␈↓itself.␈α∞ Thus␈α∂if␈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 Mary's 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␈↓␈αare␈αthe␈αbasis␈αof␈αthe␈αsemantics␈αof␈αfirst␈αorder␈αlogic␈αand␈αmodel␈αtheory␈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␈α
the␈α∞idea.␈α
 On␈α
the␈α∞other␈α
hand,␈α
logicians␈α∞such␈α
as␈α
Carnap␈α∞(1947␈α
and␈α
1956),␈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∪ All␈α∩these
␈↓ α∧␈↓formalizations␈α∪involve␈α∩modifying␈α∪the␈α∪logic␈α∩used;␈α∪ours␈α∩doesn't␈α∪modify␈α∪the␈α∩logic␈α∪and␈α∪is␈α∩more
␈↓ α∧␈↓powerful, because it includes mappings from objects to concepts.

␈↓ 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∧␈↓␈↓ αTOur␈α
approach␈α
involves␈α∞leaving␈α
the␈α
logic␈α∞unchanged␈α
and␈α
treating␈α∞concepts␈α
as␈α
one␈α∞kind␈α
of
␈↓ α∧␈↓object␈α
in␈α
an␈α
ordinary␈α
first␈α
order␈α
theory.␈α We␈α
shall␈α
have␈α
one␈α
term␈α
that␈α
denotes␈α
Mike's␈αtelephone
␈↓ α∧␈↓number␈αand␈αa␈αdifferent␈αterm␈αdenoting␈αthe␈αconcept␈αof␈αMike's␈αtelephone␈αnumber␈αinstead␈αof␈αhaving
␈↓ α∧␈↓a␈α
single␈αterm␈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␈αand␈αbetween␈αconcepts␈αand␈αother␈αentities␈αare␈αexpressed␈αby␈αformulas␈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␈αvarious
␈↓ α∧␈↓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␈α∞seems␈α∞surprising␈α∞that␈α∞such␈α∞a␈α
straightforward␈α∞and␈α∞easy␈α∞approach␈α∞should␈α∞not␈α∞have␈α
been
␈↓ α∧␈↓more fully explored than it apparently has.
␈↓ α∧␈↓␈↓ u3


␈↓ 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
is␈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
 ␈↓↓mike␈↓␈α
is␈αMike
␈↓ α∧␈↓himself.

␈↓ α∧␈↓␈↓ αT3.␈α␈↓↓Telephone␈↓␈α
is␈αa␈αfunction␈α
that␈αtakes␈αa␈α
concept␈αof␈αa␈α
person␈αinto␈αa␈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
␈↓ α∧␈↓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
Thus␈α
in␈α
(1)␈α
␈↓↓Know(Pat,Telephone
␈↓ α∧␈↓↓Mike)␈↓␈αis␈αa␈αproposition␈αand␈αnot␈αa␈αtruth␈αvalue.␈α Note␈αthat␈αwe␈αare␈αformalizing␈α␈↓↓knowing␈↓␈α␈↓↓what␈↓␈αrather
␈↓ α∧␈↓than␈α␈↓↓knowing␈↓␈α␈↓↓that␈↓␈αor␈α␈↓↓knowing␈↓␈α␈↓↓how.␈↓␈αFor␈αAI␈αand␈αfor␈αother␈αpractical␈αpurposes,␈α␈↓↓knowing␈↓␈α␈↓↓what␈↓␈αseems
␈↓ α∧␈↓to␈α
be␈α
the␈α
most␈α
useful␈α
notion␈α
of␈α
the␈αthree.␈α
 In␈α
English,␈α
␈↓↓knowing␈α
what␈↓␈α
is␈α
written␈α
␈↓↓knowing␈αwhether␈↓
␈↓ α∧␈↓when the "knowand" is a proposition.

␈↓ α∧␈↓␈↓ αT5.␈α␈↓↓true␈α
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␈αQ␈↓␈αin␈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∪another␈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∂suspect␈α∞will
␈↓ α∧␈↓eventually be found necessary).

␈↓ 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␈α
without␈αextending␈αthe␈αlogic␈αin␈α
any␈αway␈α-␈αsolely␈αby␈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 use different letters for variables of different sorts.

␈↓ 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∧␈↓many␈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∧␈↓Actually,␈α∞this␈α∞paper␈α∞isn't␈α
much␈α∞motivated␈α∞by␈α∞the␈α
philosophical␈α∞question␈α∞of␈α∞what␈α∞concepts␈α
really
␈↓ α∧␈↓are.␈α⊂ The␈α⊂goal␈α⊂is␈α⊂more␈α⊂to␈α⊂make␈α⊂a␈α⊃formal␈α⊂structure␈α⊂that␈α⊂can␈α⊂be␈α⊂used␈α⊂to␈α⊂represent␈α⊃facts␈α⊂about
␈↓ α∧␈↓knowledge␈αand␈αbelief␈αso␈αthat␈αa␈αcomputer␈α
program␈αcan␈αreason␈αabout␈αwho␈αhas␈αwhat␈α
knowledge␈αin
␈↓ α∧␈↓order␈αto␈αsolve␈αproblems.␈α From␈αeither␈αthe␈αphilosophical␈αor␈αthe␈αAI␈αpoint␈αof␈αview,␈αhowever,␈αif␈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␈α∞is␈α
the
␈↓ α∧␈↓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∧␈↓␈↓ u4


␈↓ 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␈α∪␈↓↓K(P,Q)␈↓␈α∪is␈α∪the␈α∪proposition␈α∪that␈α∪␈↓↓P␈↓␈α∩knows␈α∪␈↓↓that␈↓␈α∪␈↓↓Q.␈↓␈α∪English␈α∪does␈α∪not␈α∪treat␈α∪knowing␈α∩a
␈↓ α∧␈↓proposition␈αand␈αknowing␈αan␈αindividual␈αconcept␈αuniformly;␈αknowing␈αan␈αindividual␈αconcept␈αmeans
␈↓ α∧␈↓knowing␈α∂its␈α∂value␈α∂while␈α∂knowing␈α∂a␈α∂proposition␈α∂means␈α∂knowing␈α∂that␈α∂it␈α∂has␈α∂a␈α⊂particular␈α∂value,
␈↓ α∧␈↓namely ␈↓↓t.␈↓ There is no reason to impose this infirmity on robots.

␈↓ 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␈αwill␈αalso␈αneed␈αa␈αdenotation␈αfunction.␈α From␈α(6)␈αand␈αsuitable␈αlogical␈αaxioms␈αfollows␈α
the
␈↓ α∧␈↓existence of a function ␈↓↓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 is 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∧␈↓␈↓ u5


␈↓ α∧␈↓provided␈α
truth␈α
values␈α
are␈α
in␈α
the␈α
range␈α
of␈α
␈↓↓denot,␈↓␈α
and␈α
␈↓↓denot␈↓␈α
could␈α
also␈α
be␈α
provided␈α
with␈α
a␈α
␈↓↓(partial)
␈↓ α∧␈↓↓possible world␈↓ argument.

␈↓ α∧␈↓␈↓ αTWhen␈αwe␈αdon'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 function ␈↓↓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 predicate ␈↓↓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∧␈↓In␈αthis␈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␈αextensional␈αin␈αthis␈αsense,␈αbut␈αthe␈αcorresponding␈αpredicate␈α␈↓↓exists␈↓␈αis␈αidentically␈αtrue␈αand␈α
therefore
␈↓ α∧␈↓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 write as an infix and axiomatize by

␈↓ α∧␈↓15)␈↓ αt ␈↓↓∀Q1 Q2.(true(Q1 And Q2) ≡ true Q1 ∧ true Q2)␈↓.

␈↓ α∧␈↓The corresponding formulas for ␈↓↓Or,␈↓ ␈↓↓Not,␈↓ ␈↓↓Implies,␈↓ and ␈↓↓Equiv␈↓ are

␈↓ α∧␈↓16)␈↓ αt ␈↓↓∀Q1 Q2.(true(Q1 Or Q2) ≡ true Q1 ∨ true Q2)␈↓,

␈↓ α∧␈↓17)␈↓ αt ␈↓↓∀Q.(true(Not Q) ≡ ¬ true Q)␈↓,

␈↓ α∧␈↓18)␈↓ αt ␈↓↓∀Q1 Q2.(true(Q1 Implies Q2) ≡ true Q1 ⊃ true Q2)␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓19)␈↓ αt ␈↓↓∀Q1 Q2.(true(Q1 Equiv Q2) ≡ (true Q1 ≡ true Q2))␈↓.

␈↓ 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∧␈↓20)␈↓ αt ␈↓↓∀X.true Equal(X,X)␈↓,

␈↓ α∧␈↓21)␈↓ αt ␈↓↓∀X Y.(true Equal(X,Y) ≡ true Equal(Y,X))␈↓,
␈↓ α∧␈↓␈↓ u6


␈↓ α∧␈↓and

␈↓ α∧␈↓22)␈↓ 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∧␈↓23)␈↓ αt ␈↓↓∀X Y x.(true Equal(X,Y) ∧ denotes(X,x) ⊃ denotes(Y,x))␈↓

␈↓ α∧␈↓which␈α∪relates␈α∪it␈α∪to␈α∪equality␈α∪in␈α∪the␈α∀logic.␈α∪ We␈α∪can␈α∪make␈α∪the␈α∪concept␈α∪of␈α∀equality␈α∪␈↓↓essentially␈↓
␈↓ α∧␈↓symmetric by replacing (21) by

␈↓ α∧␈↓24)␈↓ αt ␈↓↓∀X Y.Equal(X,Y) = Equal(Y,X)␈↓,

␈↓ α∧␈↓i.e. making the two expressions denote the ␈↓↓same concept␈↓.

␈↓ α∧␈↓␈↓ αTThe statement that Mary has the same telephone as Mike is asserted by

␈↓ α∧␈↓25)␈↓ αt ␈↓↓true Equal(Telephone Mary,Telephone Mike)␈↓,

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

␈↓ α∧␈↓26)␈↓ αt ␈↓↓true Know(Pat,Telephone Mary)␈↓.

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

␈↓ α∧␈↓27)␈↓ 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 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⊃certain␈α∩concepts␈α⊃of␈α⊃them␈α∩seem␈α⊃useful.␈α∩ Some␈α⊃things␈α⊃such␈α∩as␈α⊃numbers␈α∩can␈α⊃be
␈↓ α∧␈↓regarded␈αas␈αhaving␈α␈↓↓standard␈↓␈α
concepts.␈α Suppose␈αthat␈α␈↓↓Concept1␈↓␈α␈↓↓n␈↓␈α
gives␈αa␈αstandard␈αconcept␈α
of␈αthe
␈↓ α∧␈↓number ␈↓↓n,␈↓ so that

␈↓ α∧␈↓28)␈↓ αt ␈↓↓∀n.(denot Concept1 n = n)␈↓.

␈↓ α∧␈↓We can then have simultaneously
␈↓ α∧␈↓␈↓ u7


␈↓ α∧␈↓29)␈↓ αt ␈↓↓true Not Knew(Kepler,Number Planets)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓30)␈↓ αt ␈↓↓true Knew(Kepler,Composite Concept1 denot Number Planets)␈↓.

␈↓ α∧␈↓(We␈α⊂have␈α⊂used␈α⊂␈↓↓Knew␈↓␈α⊃instead␈α⊂of␈α⊂␈↓↓Know,␈↓␈α⊂because␈α⊂we␈α⊃are␈α⊂not␈α⊂now␈α⊂concerned␈α⊃with␈α⊂formalizing
␈↓ α∧␈↓tense.)

␈↓ α∧␈↓(30) can be condensed using ␈↓↓Composite1␈↓ which takes

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

␈↓ α∧␈↓31)␈↓ αt ␈↓↓Composite1 n = Composite Concept1 n␈↓

␈↓ α∧␈↓getting

␈↓ α∧␈↓32)␈↓ αt ␈↓↓true Knew(Kepler,Composite1 denot Number Planets)␈↓.

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

␈↓ α∧␈↓33)␈↓ αt ␈↓↓Composite2 N = Composite Concept1 denot N␈↓,

␈↓ α∧␈↓letting us write

␈↓ α∧␈↓34)␈↓ αt ␈↓↓true Knew(Kepler,Composite2 Number Planets)␈↓,

␈↓ α∧␈↓which is true even though

␈↓ α∧␈↓35)␈↓ αt ␈↓↓true Knew(Kepler,Composite Number Planets)␈↓

␈↓ α∧␈↓is␈α
false.␈α
 (35)␈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⊃(30),␈α⊃(32),␈α⊂and␈α⊃(34)␈α⊃each␈α⊃expresses␈α⊂a␈α⊃proposition␈α⊃that␈α⊃can␈α⊂only␈α⊃be␈α⊃stated␈α⊃awkwardly␈α⊂in
␈↓ α∧␈↓English,␈α⊂e.g.␈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∧␈↓36)␈↓ αt ␈↓↓∀x.(ispuppy(x,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∧␈↓␈↓ αTA␈α⊂suitable␈α⊂collection␈α∂of␈α⊂functions␈α⊂from␈α∂things␈α⊂to␈α⊂concepts␈α∂might␈α⊂permit␈α⊂a␈α⊂language␈α∂that
␈↓ α∧␈↓omitted␈α∂some␈α∂individual␈α∂concepts␈α∂like␈α∂␈↓↓Mike␈↓␈α∂(replacing␈α∂it␈α∂with␈α∂␈↓↓Conceptx␈α∂mike␈↓)␈α∂and␈α∂wrote␈α∞many
␈↓ α∧␈↓sentences␈αwith␈αquantifiers␈αover␈αthings␈αrather␈αthan␈αover␈αconcepts.␈α However,␈αit␈αis␈αstill␈αpremature␈α
to
␈↓ α∧␈↓apply␈α
Occam's␈α
razor.␈α It␈α
may␈α
be␈α
possible␈αto␈α
avoid␈α
concepts␈α
as␈αobjects␈α
in␈α
expressing␈αparticular␈α
facts
␈↓ α∧␈↓but impossible to avoid them in stating general principles.
␈↓ α∧␈↓␈↓ u8


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

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

␈↓ α∧␈↓37)␈↓ αt ␈↓↓true Know(Pat,Telephone Mike)␈↓.

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

␈↓ α∧␈↓38)␈↓ 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∧␈↓39)␈↓ αt ␈↓↓K(P,Q) = (Q And Know(P,Q))␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓40)␈↓ 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
equals␈α␈↓↓A.␈↓␈α
This␈αis␈α
clear␈αenough␈α
for␈αsome␈α
domains␈αlike␈α
integers,␈αbut␈α
it␈αis
␈↓ α∧␈↓not obvious how to treat knowing a person.

␈↓ α∧␈↓␈↓ αTUsing the ␈↓↓standard␈↓ ␈↓↓concept␈↓ function ␈↓↓Concept1, we␈↓ might replace (40) by

␈↓ α∧␈↓41)␈↓ αt ␈↓↓true Know(P,X) ≡ ∃a.true K(P,Equal(X,Concept1 a))␈↓

␈↓ α∧␈↓with similar meaning.

␈↓ α∧␈↓␈↓ αT(40)␈α⊂and␈α⊃(41)␈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∧␈↓42)␈↓ αt ␈↓↓∀P X.(Know(P,X) = Exists X And K(P,Equal(X,Concept2 denot X)))␈↓,

␈↓ α∧␈↓where␈α∞␈↓↓Concept2␈↓␈α
is␈α∞a␈α∞suitable␈α
function␈α∞from␈α∞things␈α
to␈α∞concepts␈α
and␈α∞may␈α∞not␈α
be␈α∞available␈α∞for␈α
all
␈↓ α∧␈↓sorts of objects.


␈↓ α∧␈↓αUNQUANTIFIED MODAL LOGIC

␈↓ α∧␈↓␈↓ αTIn␈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 although quantification occurs in the outer 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 However,
␈↓ α∧␈↓since␈α␈↓↓nec␈↓␈αis␈αa␈αpredicate␈αin␈αthe␈α
logic␈αwith␈α␈↓↓t␈↓␈αand␈α␈↓↓f␈↓␈αas␈αvalues,␈α
␈↓↓nec␈↓␈α␈↓↓Q␈↓␈αcannot␈αbe␈αan␈αargument␈αof␈α␈↓↓nec␈↓␈α
or
␈↓ α∧␈↓␈↓↓Nec.␈↓
␈↓ α∧␈↓␈↓ u9


␈↓ 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∧␈↓43)␈↓ αt␈↓↓∀Q. Not Not Q = Q␈↓,

␈↓ α∧␈↓and the second

␈↓ α∧␈↓44)␈↓ 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∞what␈α∞amounts␈α∞to␈α∞the␈α∞first␈α∞alternative,␈α
regarding
␈↓ α∧␈↓concepts␈α
as␈α∞L-equivalence␈α
classes␈α
of␈α∞expressions.␈α
 This␈α
works␈α∞nicely␈α
for␈α
discussing␈α∞necessity,␈α
but
␈↓ α∧␈↓when␈αhe␈αwants␈αto␈αdiscuss␈αknowledge␈αwithout␈αassuming␈αthat␈αeveryone␈αknows␈αFermat's␈αlast␈αtheorem
␈↓ α∧␈↓if␈α
it␈α
is␈αtrue,␈α
he␈α
introduces␈αthe␈α
notion␈α
of␈α␈↓↓intensional␈↓␈α
␈↓↓isomorphism␈↓␈α
and␈αhas␈α
knowledge␈α
operate␈αon␈α
the
␈↓ α∧␈↓equivalence 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∧␈↓collection␈α∪of␈α∪propositional␈α∪identities␈α∪like␈α∪(43)␈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∧␈↓␈↓ αTActually␈αthere␈αare␈αmore␈αthan␈αtwo␈αalternatives.␈α Let␈α
␈↓↓M␈↓␈αbe␈αthe␈αfree␈αalgebra␈αbuilt␈αup␈αfrom␈α
the
␈↓ α∧␈↓"atomic"␈αconcepts␈αby␈αthe␈αconcept␈αforming␈αfunction␈α
symbols.␈α If␈α≡≡␈αis␈αan␈αequivalence␈αrelation␈αon␈α
␈↓↓M␈↓
␈↓ α∧␈↓such that

␈↓ α∧␈↓45)␈↓ 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∧␈↓46)␈↓ αt ␈↓↓∀Q.(Poss Q = Not Nec Not Q)␈↓,

␈↓ α∧␈↓and the weaker extensional axiom

␈↓ α∧␈↓47)␈↓ 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∧␈↓48)␈↓ αt ␈↓↓∀Q.(true Nec Q ⊃ true Q)␈↓,
␈↓ α∧␈↓␈↓ f10


␈↓ α∧␈↓and

␈↓ α∧␈↓49)␈↓ 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∧␈↓50)␈↓ αt ∀Q.(␈↓↓true Nec Q ≡ true Nec Nec Q)␈↓,

␈↓ α∧␈↓and S5 by

␈↓ α∧␈↓51)␈↓ 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∧␈↓52)␈↓ αt ␈↓↓∀Q.(true Nec4 Q ⊃ true Nec5 Q)␈↓

␈↓ 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∧␈↓53)␈↓ αt ␈↓↓∀X.true Nec Equal(X,X)␈↓.

␈↓ α∧␈↓Certain of Carnap's proposals translate to the stronger relation

␈↓ α∧␈↓54)␈↓ 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∧␈↓αMORE PHILOSOPHICAL 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␈αwe␈αhave␈α␈↓↓"The␈αnumber␈αof␈αplanets␈α=␈α9"␈↓␈αand␈α␈↓↓"Necessarily␈α9␈α=␈α9"␈↓␈αfrom␈αwhich␈αone␈αdoesn'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⊂by␈α∂Quine
␈↓ α∧␈↓(1961) and (Kaplan 1969).  Consider the sentences

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

␈↓ α∧␈↓and

␈↓ α∧␈↓56)␈↓ αt ␈↓↓nec Equal(Concept1 number planets,Concept1 9)␈↓.
␈↓ α∧␈↓␈↓ f11


␈↓ α∧␈↓Both␈α
are␈α
true.␈α
 (55)␈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(56)␈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∧␈↓57)␈↓ αt ␈↓↓number(planets) = 9␈↓

␈↓ α∧␈↓or, using concepts, as

␈↓ α∧␈↓58)␈↓ αt ␈↓↓true Equal(Number Planets, Concept1 9)␈↓,

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

␈↓ α∧␈↓59)␈↓ αt ␈↓↓nec Equal(Concept1 9,Concept1 9)␈↓,

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

␈↓ 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∞ The␈α
first␈α∞can␈α∞be␈α∞rendered␈α
in
␈↓ α∧␈↓four␈α∞ways,␈α
namely␈α∞␈↓↓true␈α∞Statesman␈α
Baldwin␈↓␈α∞or␈α
␈↓↓statesman␈α∞denot␈α∞Baldwin␈↓␈α
or␈α∞␈↓↓statesman␈α∞baldwin␈↓␈α
or
␈↓ α∧␈↓␈↓↓statesman1␈αBaldwin␈↓␈αwhere␈α
the␈αlast␈αasserts␈α
that␈αthe␈αconcept␈α
of␈αBaldwin␈αis␈α
one␈αof␈αa␈αstatesman.␈α
 The
␈↓ α∧␈↓second can be rendered only as as ␈↓↓true Fiction Pickwick␈↓ or ␈↓↓fiction1 Pickwick␈↓.

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

␈↓ α∧␈↓60)␈↓ 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∧␈↓61)␈↓ αt ␈↓↓tully = cicero␈↓,

␈↓ α∧␈↓62)␈↓ αt ␈↓↓denot Tully = cicero␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓63)␈↓ αt ␈↓↓denot Cicero = cicero␈↓.

␈↓ α∧␈↓␈↓ αTWe␈α~can␈α~discuss␈α→Philip's␈α~concept␈α~of␈α→the␈α~person␈α~Tully␈α→by␈α~introducing␈α~a␈α→function
␈↓ α∧␈↓␈↓↓Concept2(p1,p2)␈↓␈α
giving␈α
for␈αsome␈α
persons␈α
␈↓↓p1␈↓␈α
and␈α␈↓↓p2,␈↓␈α
␈↓↓p1␈↓'s␈α
concept␈α
of␈α␈↓↓p2.␈↓␈α
Such␈α
a␈α
function␈αneed
␈↓ α∧␈↓not␈α⊃be␈α∩unique␈α⊃or␈α⊃always␈α∩defined,␈α⊃but␈α∩in␈α⊃the␈α⊃present␈α∩case,␈α⊃some␈α⊃of␈α∩our␈α⊃information␈α∩may␈α⊃be
␈↓ α∧␈↓conveniently expressed by

␈↓ α∧␈↓64)␈↓ αt ␈↓↓Concept2(philip,tully) = Cicero␈↓,
␈↓ α∧␈↓␈↓ f12


␈↓ 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∧␈↓65)␈↓ αt ␈↓↓true K(Philip,Denounced(Cicero,Catiline))␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓66)␈↓ αt ␈↓↓¬true K(Philip,Denounced(Tully,Catiline))␈↓,

␈↓ α∧␈↓From (62), ... ,(66) we can deduce

␈↓ α∧␈↓67)␈↓ αt ␈↓↓∃P.true Denounced(P,Catiline) And Not K(Philip,Denounced(P,Catiline))␈↓,

␈↓ α∧␈↓from (66) and others, and

␈↓ α∧␈↓68)␈↓ αt ␈↓↓¬∃p.(denounced(p,catiline) ∧ ¬true K(Philip,Denounced(Concept2(philip,p), Catiline)))␈↓

␈↓ α∧␈↓using the additional hypotheses

␈↓ α∧␈↓69)␈↓ αt ␈↓↓∀p.(denounced(p,catiline) ⊃ p = cicero)␈↓,

␈↓ α∧␈↓70)␈↓ αt ␈↓↓denot Catiline = catiline␈↓,

␈↓ α∧␈↓and

␈↓ α∧␈↓71)␈↓ αt ␈↓↓∀P1 P2.(denot Denounced(P1,P2) ≡ denounced(denot P1,denot P2))␈↓.

␈↓ α∧␈↓Presumably␈α∞(67)␈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⊂(68)␈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∧␈↓␈↓ 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∧␈↓72)␈↓ αt ␈↓↓true Believed(I,Greater(Length Youryacht,Concept1 denot Length Youryacht))␈↓

␈↓ α∧␈↓where␈αwe␈αare␈α
not␈αanalyzing␈αthe␈α
pronouns␈αor␈αthe␈α
tense,␈αbut␈αare␈α
using␈α␈↓↓denot␈↓␈αto␈α
get␈αthe␈αactual␈α
length
␈↓ 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.

␈↓ α∧␈↓␈↓ αTIn␈α
order␈α
to␈α
express␈α
␈↓↓"Your␈αyacht␈α
is␈α
longer␈α
than␈α
Peter␈αthinks␈α
it␈α
is."␈↓,␈α
we␈α
need␈α
the␈αexpression
␈↓ α∧␈↓␈↓↓Denot(Peter,X)␈↓ giving a concept of what Peter thinks the value of ␈↓↓X␈↓ is.  We now write

␈↓ α∧␈↓73)␈↓ αt␈↓↓longer(youryacht,denot Denot(Peter,Length Youryacht))␈↓,

␈↓ α∧␈↓but I am not certain this is a correct translation.
␈↓ α∧␈↓␈↓ f13


␈↓ α∧␈↓␈↓ αTQuine␈α(1956)␈αdiscusses␈αan␈αexample␈αin␈αwhich␈αRalph␈αsees␈αBernard␈αJ.␈αOrtcutt␈αskulking␈αabout
␈↓ α∧␈↓and␈αconcludes␈αthat␈αhe␈αis␈αa␈αspy,␈αand␈αalso␈αsees␈αhim␈αon␈αthe␈αbeach,␈αbut␈αdoesn't␈αrecognize␈αhim␈αas␈αthe
␈↓ α∧␈↓same person.  The facts can be expresed in our formalism by equations

␈↓ α∧␈↓74)␈↓ αt␈↓↓true Believe(Ralph, Isspy P1)␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓75)␈↓ αt ␈↓↓true Believe(Ralph,Not Isspy P2)␈↓

␈↓ α∧␈↓where ␈↓↓P1␈↓ and ␈↓↓P2␈↓ are concepts satisfying

␈↓ α∧␈↓␈↓↓denot␈αP1␈α=␈αortcutt␈↓␈αand␈α␈↓↓denot␈αP2␈α=␈αortcutt␈↓.␈α ␈↓↓P1␈↓␈αand␈α␈↓↓P2␈↓␈αare␈αfurther␈αdescribed␈αby␈αsentences␈αrelating
␈↓ α∧␈↓them to the circumstances under which Ralph formed them.

␈↓ α∧␈↓␈↓ αTWe␈α∃can␈α∃still␈α⊗consider␈α∃a␈α∃simple␈α⊗sentence␈α∃involving␈α∃the␈α∃persons␈α⊗as␈α∃things␈α∃-␈α⊗write␈α∃it
␈↓ α∧␈↓␈↓↓believespy(ralph,ortcutt)␈↓, where we define

␈↓ α∧␈↓76)␈↓ αt ␈↓↓∀p1 p2.(believespy(p1,p2) ≡ true Believe(Concept1 p1,Isspy Concept7 p2)␈↓

␈↓ α∧␈↓using␈α
suitable␈α
mappings␈α␈↓↓Concept1␈↓␈α
and␈α
␈↓↓Concept7␈↓␈αfrom␈α
persons␈α
to␈αconcepts␈α
of␈α
persons.␈α
 We␈αmight
␈↓ α∧␈↓also␈αchoose␈αto␈αdefine␈α␈↓↓believespy␈↓␈αin␈αsuch␈αa␈αway␈αthat␈αit␈αrequires␈α␈↓↓true␈αBelieve(Concept1␈αp1,␈αIsspy␈αP)␈↓
␈↓ α∧␈↓for␈α∞several␈α∞concepts␈α∞␈↓↓P␈↓␈α∞of␈α∂␈↓↓p2,␈↓␈α∞e.g.␈α∞the␈α∞concepts␈α∞arising␈α∞from␈α∂all␈α∞␈↓↓p1␈↓'s␈α∞encounters␈α∞with␈α∞␈↓↓p2␈↓␈α∂or␈α∞his
␈↓ α∧␈↓name.␈α↔ In␈α↔this␈α↔case␈α↔␈↓↓believespy(ralph,ortcutt)␈↓␈α↔will␈α↔be␈α↔false␈α↔and␈α↔so␈α↔would␈α↔a␈α⊗corresponding
␈↓ α∧␈↓␈↓↓notbelievespy(ralph,ortcutt)␈↓.␈α∞ However,␈α∞the␈α∂simple-minded␈α∞predicate␈α∞␈↓↓believespy,␈↓␈α∂suitably␈α∞defined,
␈↓ α∧␈↓may␈α∞be␈α∞quite␈α
useful␈α∞for␈α∞expressing␈α∞the␈α
facts␈α∞necessary␈α∞to␈α
predict␈α∞someone's␈α∞behavior␈α∞in␈α
simpler
␈↓ α∧␈↓circumstances.

␈↓ α∧␈↓␈↓ αTRegarded␈α
as␈α
an␈α
attempt␈αto␈α
explicate␈α
the␈α
sentence␈α
␈↓↓"Ralph␈αbelieves␈α
Ortcutt␈α
is␈α
a␈α
spy"␈↓,␈αthe␈α
above
␈↓ α∧␈↓may␈α∂be␈α∞considered␈α∂rather␈α∂tenuous.␈α∞ However,␈α∂we␈α∂are␈α∞proposing␈α∂it␈α∂as␈α∞a␈α∂notation␈α∂for␈α∞expressing
␈↓ α∧␈↓Ralph's␈α⊂beliefs␈α⊃about␈α⊂Ortcutt␈α⊃so␈α⊂that␈α⊃correct␈α⊂conclusions␈α⊃may␈α⊂be␈α⊃drawn␈α⊂about␈α⊃Ralph's␈α⊂future
␈↓ α∧␈↓actions.  For this it seems to be adequate.


␈↓ α∧␈↓αPROPOSITIONS EXPRESSING QUANTIFICATION

␈↓ α∧␈↓␈↓ αTAs␈α∂the␈α∞examples␈α∂of␈α∞the␈α∂previous␈α∞sections␈α∂have␈α∞shown,␈α∂admitting␈α∞concepts␈α∂as␈α∂objects␈α∞and
␈↓ α∧␈↓introducing␈α⊂standard␈α∂concept␈α⊂functions␈α∂makes␈α⊂"quantifying␈α∂in"␈α⊂rather␈α∂easy.␈α⊂ However,␈α∂forming
␈↓ α∧␈↓propositions␈α⊗and␈α⊗individual␈α⊗concepts␈α⊗by␈α⊗quantification␈α⊗requires␈α⊗new␈α⊗ideas␈α↔and␈α⊗additional
␈↓ α∧␈↓formalism.

␈↓ α∧␈↓␈↓ αTWe␈αwant␈αto␈αcontinue␈αdescribing␈αconcepts␈αwithin␈αfirst␈αorder␈αlogic␈αwith␈αno␈αlogical␈αextensions.
␈↓ α∧␈↓Therefore,␈α∀in␈α∀order␈α∃to␈α∀form␈α∀new␈α∀concepts␈α∃by␈α∀quantification␈α∀and␈α∀description,␈α∃we␈α∀introduce
␈↓ α∧␈↓functions␈α␈↓↓All,␈↓␈α
␈↓↓Exist,␈↓␈αand␈α
␈↓↓The␈↓␈αsuch␈α
that␈α␈↓↓All(V,P)␈↓␈αis␈α
(approximately)␈αthe␈α
proposition␈αthat␈α
␈↓↓for␈αall
␈↓ α∧␈↓↓values␈αof␈αV␈α
P␈αis␈αtrue␈↓,␈α
␈↓↓Exist(V,P)␈↓␈αis␈αthe␈α
corresponding␈αexistential␈αproposition,␈α
and␈α␈↓↓The(V,P)␈↓␈αis␈α
the
␈↓ α∧␈↓concept of ␈↓↓the V such that P␈↓.

␈↓ α∧␈↓␈↓ αTSince␈α∩␈↓↓All␈↓␈α∩is␈α∩to␈α∩be␈α∩a␈α∩function,␈α∩␈↓↓V␈↓␈α∩and␈α∩␈↓↓P␈↓␈α∩must␈α∩be␈α∩objects␈α∩in␈α∩the␈α∩logic.␈α∩ However,␈α∩␈↓↓V␈↓␈α⊃is
␈↓ α∧␈↓semantically␈α∞a␈α∞variable␈α∞in␈α∞the␈α
formation␈α∞of␈α∞␈↓↓All(V,P),␈α∞etc.,␈↓␈α∞and␈α
we␈α∞will␈α∞call␈α∞such␈α∞objects␈α
␈↓↓inner
␈↓ α∧␈↓␈↓ f14


␈↓ α∧␈↓↓variables␈↓␈α∞so␈α
as␈α∞to␈α
distinguish␈α∞them␈α
from␈α∞variables␈α
in␈α∞the␈α
logic.␈α∞ We␈α
will␈α∞use␈α
␈↓↓V,␈↓␈α∞sometimes␈α
with
␈↓ α∧␈↓subscripts,␈α∩for␈α⊃a␈α∩logical␈α⊃variable␈α∩ranging␈α⊃over␈α∩inner␈α⊃variables.␈α∩ We␈α⊃also␈α∩need␈α∩some␈α⊃constant
␈↓ α∧␈↓symbols␈αfor␈αinner␈αvariables␈α(got␈αthat?),␈αand␈αwe␈αwill␈αuse␈αdoubled␈αletters,␈αsometimes␈αwith␈αsubscripts,
␈↓ α∧␈↓for these.  ␈↓↓XX␈↓ will be used for individual concepts, ␈↓↓PP␈↓ for persons, and ␈↓↓QQ␈↓ for propositions.

␈↓ α∧␈↓␈↓ αTThe␈α∪second␈α∪argument␈α∪of␈α∪␈↓↓All␈↓␈α∪and␈α∪friends␈α∪is␈α∪a␈α∪"proposition␈α∪with␈α∪variables␈α∪in␈α∪it",␈α∪but
␈↓ α∧␈↓remember␈α
that␈αthese␈α
variables␈αare␈α
inner␈α
variables␈αwhich␈α
are␈αconstants␈α
in␈α
the␈αlogic.␈α
 Got␈αthat?␈α
 We
␈↓ α∧␈↓won't␈α∪introduce␈α∪a␈α∪special␈α∩term␈α∪for␈α∪them,␈α∪but␈α∪will␈α∩generally␈α∪allow␈α∪concepts␈α∪to␈α∪include␈α∩inner
␈↓ α∧␈↓variables.␈α∂ Thus␈α∂concepts␈α∂now␈α∞include␈α∂inner␈α∂variables␈α∂like␈α∞␈↓↓XX␈↓␈α∂and␈α∂␈↓↓PP,␈↓␈α∂and␈α∂concept␈α∞forming
␈↓ α∧␈↓functions like ␈↓↓Telephone␈↓ and ␈↓↓Know␈↓ take the generalized concepts as arguments.

␈↓ α∧␈↓␈↓ αTThus

␈↓ α∧␈↓77)␈↓ αt ␈↓↓Child(Mike,PP) Implies Equal(Telephone PP,Telephone Mike)␈↓

␈↓ α∧␈↓is␈αa␈αproposition␈αwith␈αthe␈αinner␈αvariable␈α␈↓↓PP␈↓␈αin␈αit␈αto␈αthe␈αeffect␈αthat␈αif␈α␈↓↓PP␈↓␈αis␈αa␈αchild␈αof␈αMike,␈αthen
␈↓ α∧␈↓his telephone number is the same as Mike's, and

␈↓ α∧␈↓78)␈↓ αt ␈↓↓All(PP,Child(Mike,PP) Implies Equal(Telephone PP,Telephone Mike))␈↓

␈↓ α∧␈↓is␈αthe␈αproposition␈αthat␈αall␈αMike's␈αchildren␈αhave␈αthe␈αsame␈αtelephone␈αnumber␈αas␈αMike.␈α Existential
␈↓ α∧␈↓propositions␈αare␈αformed␈α
similarly␈αto␈αuniversal␈α
ones,␈αbut␈αthe␈α
function␈α␈↓↓Exist␈↓␈αintroduced␈αhere␈α
should
␈↓ α∧␈↓not be confused with the function ␈↓↓Exists␈↓ applied to individual concepts introduced earlier.

␈↓ α∧␈↓␈↓ αTIn␈α
forming␈αindividual␈α
concepts␈αby␈α
the␈α
description␈αfunction␈α
␈↓↓The,␈↓␈αit␈α
doesn't␈α
matter␈αwhether
␈↓ α∧␈↓the object described exists.  Thus

␈↓ α∧␈↓79)␈↓ αt ␈↓↓The(PP,Child(Mike,PP))␈↓

␈↓ α∧␈↓is␈αthe␈α
concept␈αof␈α
Mike's␈αonly␈α
child.␈α ␈↓↓Exists␈α
The(PP,Child(Mike,PP))␈↓␈αis␈α
the␈αproposition␈α
that␈αthe
␈↓ α∧␈↓described child exists.  We have

␈↓ α∧␈↓80)␈↓ αt␈α$␈↓↓true␈α$Exists␈α$The(PP,Child(Mike,PP))␈α$≡␈α$true(Exist(PP,Child(Mike,PP)␈α$And
␈↓ α∧␈↓↓All(PP1,Child(Mike,PP1) Implies Equal(PP,PP1))))␈↓,

␈↓ α∧␈↓but we may want the equality of the two propositions, i.e.

␈↓ α∧␈↓81)␈↓ αt␈α?␈αα␈↓↓Exists␈α?␈ααThe(PP,Child(Mike,PP))␈α?␈αα=␈α?␈ααExist(PP,Child(Mike,PP)␈α?␈ααAnd
␈↓ α∧␈↓↓All(PP1,Child(Mike,PP1) Implies Equal(PP,PP1)))␈↓.

␈↓ α∧␈↓This␈α
is␈αpart␈α
of␈α
general␈αproblem␈α
of␈α
when␈αtwo␈α
logically␈αequivalent␈α
concepts␈α
are␈αto␈α
be␈α
regarded␈αas
␈↓ α∧␈↓the same.

␈↓ α∧␈↓␈↓ αTIn␈α∀order␈α∀to␈α∀discuss␈α∀the␈α∀truth␈α∀of␈α∀propositions␈α∀and␈α∀the␈α∀denotation␈α∀of␈α∀descriptions,␈α∀we
␈↓ α∧␈↓introduce␈α
␈↓↓possible␈↓␈α
␈↓↓worlds␈↓␈α
reluctantly␈α
and␈α∞with␈α
an␈α
important␈α
difference␈α
from␈α
the␈α∞usual␈α
treatment.
␈↓ α∧␈↓We␈αneed␈α
them␈αto␈α
give␈αvalues␈α
to␈αthe␈α
inner␈αvariables,␈α
and␈αwe␈α
can␈αalso␈α
use␈αthem␈α
for␈αaxiomatizing
␈↓ α∧␈↓the␈α
modal␈α∞operators,␈α
knowledge,␈α∞belief␈α
and␈α∞tense.␈α
 However,␈α∞for␈α
axiomatizing␈α∞quantification,␈α
we
␈↓ α∧␈↓also need a function α such that

␈↓ α∧␈↓82)␈↓ αt π' = α(␈↓↓V,x␈↓,π)
␈↓ α∧␈↓␈↓ f15


␈↓ α∧␈↓is␈α∞the␈α∞possible␈α∞world␈α∞that␈α∞is␈α∂the␈α∞same␈α∞as␈α∞the␈α∞world␈α∞π␈α∂except␈α∞that␈α∞the␈α∞inner␈α∞variable␈α∞␈↓↓V␈↓␈α∂has␈α∞the
␈↓ α∧␈↓value␈α∞␈↓↓x␈↓␈α∞instead␈α
of␈α∞the␈α∞value␈α∞it␈α
has␈α∞in␈α∞π.␈α
 In␈α∞this␈α∞respect␈α∞our␈α
possible␈α∞worlds␈α∞resemble␈α∞the␈α
␈↓↓state␈↓
␈↓ α∧␈↓␈↓↓vectors␈↓␈α
or␈α␈↓↓environments␈↓␈α
of␈α
computer␈αscience␈α
more␈α
than␈αthe␈α
possible␈α
worlds␈αof␈α
the␈αKripke␈α
treatment
␈↓ α∧␈↓of␈αmodal␈αlogic.␈α This␈αCartesian␈α
product␈αstructure␈αon␈αthe␈αspace␈α
of␈αpossible␈αworlds␈αcan␈αalso␈αbe␈α
used
␈↓ α∧␈↓to treat counterfactual conditional sentences.

␈↓ α∧␈↓␈↓ αTLet␈α⊂π0␈α⊂be␈α⊃the␈α⊂actual␈α⊂world.␈α⊃ Let␈α⊂␈↓↓true(P,π)␈↓␈α⊂mean␈α⊃that␈α⊂the␈α⊂proposition␈α⊃␈↓↓P␈↓␈α⊂is␈α⊂true␈α⊃in␈α⊂the
␈↓ α∧␈↓possible world π.  Then

␈↓ α∧␈↓83)␈↓ αt ␈↓↓∀P.(true P ≡ true(P,␈↓π0)).

␈↓ α∧␈↓Let␈α
␈↓↓denotes(X,x,␈↓π)␈α
mean␈α
that␈α␈↓↓X␈↓␈α
denotes␈α
␈↓↓x␈↓␈α
in␈απ,␈α
and␈α
let␈α
␈↓↓denot(X,π)␈↓␈α
mean␈αthe␈α
denotation␈α
of␈α
␈↓↓X␈↓␈αin␈α
π
␈↓ α∧␈↓when that is defined.

␈↓ α∧␈↓␈↓ αTThe truth condition for ␈↓↓All(V,P)␈↓ is then given by

␈↓ α∧␈↓84)␈↓ αt ␈↓↓∀π V P.(true(All(V,P),π) ≡ ∀x.true(P,α(V,x,π))␈↓.

␈↓ α∧␈↓Here␈α␈↓↓V␈↓␈αranges␈αover␈αinner␈αvariables,␈α␈↓↓P␈↓␈αranges␈αover␈αpropositions,␈αand␈α␈↓↓x␈↓␈αranges␈αover␈αthings.␈α There
␈↓ α∧␈↓seems to be no harm in making the domain of ␈↓↓x␈↓ depend on π.  Similarly

␈↓ α∧␈↓85)␈↓ αt ␈↓↓∀π V P.(true(Exist(V,P),π) ≡ ∃x.true(P,α(V,x,π))␈↓.

␈↓ α∧␈↓The meaning of ␈↓↓The(V,P)␈↓ is given by

␈↓ α∧␈↓86)␈↓ αt ␈↓↓∀π V P x.(true(P,α(V,x,π)) ∧ ∀y.(true(P,α(V,y,π)) ⊃ y = x) ⊃ denotes(The(V,P),x,π))␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓87)␈↓ αt ␈↓↓∀π V P.(¬∃!x.true(P,α(V,x,π)) ⊃ ¬true Exists The(V,P))␈↓.

␈↓ α∧␈↓␈↓ αTWe␈α∨also␈α have␈α∨the␈α following␈α∨"syntactic"␈α rules␈α∨governing␈α propositions␈α∨involving
␈↓ α∧␈↓quantification:

␈↓ α∧␈↓88)␈↓ αt␈↓↓∀π Q1 Q2 V.(absent(V,Q1) ∧ true(All(V,Q1 Implies Q2),π) ⊃ true(Q1 Implies All(V,Q2),π))␈↓

␈↓ α∧␈↓and

␈↓ α∧␈↓89)␈↓ αt ␈↓↓∀π V Q X.(true(All(V,Q),π) ⊃ true(Subst(X,V,Q),π))␈↓.

␈↓ α∧␈↓where␈α␈↓↓absent(V,X)␈↓␈αmeans␈αthat␈αthe␈αvariable␈α␈↓↓V␈↓␈αis␈αnot␈αpresent␈αin␈αthe␈αconcept␈α␈↓↓X,␈↓␈αand␈α␈↓↓Subst(X,V,Y)␈↓
␈↓ α∧␈↓is␈α
the␈α∞concept␈α
that␈α∞results␈α
from␈α
substituting␈α∞the␈α
concept␈α∞␈↓↓X␈↓␈α
for␈α
the␈α∞variable␈α
␈↓↓V␈↓␈α∞in␈α
the␈α∞concept␈α
␈↓↓Y.␈↓
␈↓ α∧␈↓␈↓↓absent␈↓ and ␈↓↓Subst␈↓ are characterized by the following axioms:

␈↓ α∧␈↓90)␈↓ αt ␈↓↓∀V1 V2.(absent(V1,V2) ≡ V1 ≠ V2)␈↓,

␈↓ α∧␈↓91)␈↓ αt␈↓↓∀V P X.(absent(V,Know(P,X)) ≡ absent(V,P) ∧ absent(V,X))␈↓,

␈↓ α∧␈↓axioms similar to (91) for other conceptual functions,
␈↓ α∧␈↓␈↓ f16


␈↓ α∧␈↓92)␈↓ αt ␈↓↓∀V Q.absent(V,All(V,Q))␈↓,

␈↓ α∧␈↓93)␈↓ αt ␈↓↓∀V Q.absent(V,Exist(V,Q))␈↓,

␈↓ α∧␈↓94)␈↓ αt␈↓↓∀V Q.absent(V,The(V,Q))␈↓,

␈↓ α∧␈↓95)␈↓ αt ␈↓↓∀V X.Subst(V,V,X) = X␈↓,

␈↓ α∧␈↓96)␈↓ αt ␈↓↓∀X V.Subst(X,V,V) = X␈↓,

␈↓ α∧␈↓97)␈↓ αt ␈↓↓∀X V P Y.(Subst(X,V,Know(P,Y)) = Know(Subst(X,V,P),Subst(X,V,Y)))␈↓,

␈↓ α∧␈↓axioms similar to (97) for other functions,

␈↓ α∧␈↓98)␈↓ αt ␈↓↓∀X V Q.(absent(V,Y) ⊃ Subst(X,V,Y) = Y)␈↓,

␈↓ α∧␈↓99)␈↓ αt␈α%␈↓↓∀X␈α&V1␈α%V2␈α%Q.(V1␈α&≠␈α%V2␈α%∧␈α&absent(V2,X)␈α%⊃␈α&Subst(X,V1,All(V2,Q))␈α%=
␈↓ α∧␈↓↓All(V2,Subst(X,V1,Q)))␈↓,

␈↓ α∧␈↓and corresponding axioms to (99) for ␈↓↓Exist␈↓ and ␈↓↓The.␈↓

␈↓ α∧␈↓Along with these comes the axiom that binding kills variables, i.e.

␈↓ α∧␈↓100)␈↓ αt ␈↓↓∀V1 V2 Q.(All(V1,Q) = All(V2,Subst(V2,V1,Q)))␈↓.

␈↓ α∧␈↓␈↓ αTThe␈α
functions␈α␈↓↓absent␈↓␈α
and␈α␈↓↓Subst␈↓␈α
play␈α
a␈α"syntactic"␈α
role␈αin␈α
describing␈αthe␈α
rules␈α
of␈αreasoning
␈↓ α∧␈↓and␈αdon't␈αappear␈αin␈α
the␈αconcepts␈αthemselves.␈α It␈α
seems␈αlikely␈αthat␈αthis␈α
is␈αharmless␈αuntil␈αwe␈αwant␈α
to
␈↓ α∧␈↓form concepts of the laws of reasoning.

␈↓ α∧␈↓␈↓ αTWe␈α⊂used␈α∂the␈α⊂Greek␈α∂letter␈α⊂π␈α∂for␈α⊂possible␈α∂worlds,␈α⊂because␈α∂we␈α⊂did␈α∂not␈α⊂want␈α∂to␈α⊂consider␈α∂a
␈↓ α∧␈↓possible␈α
world␈αas␈α
a␈αthing␈α
and␈αintroduce␈α
concepts␈αof␈α
possible␈αworlds.␈α
 Reasoning␈α
about␈αreasoning
␈↓ α∧␈↓may require such concepts or else a formulation that doesn't use possible worlds.

␈↓ α∧␈↓␈↓ αTMartin␈α∀Davis␈α∪(in␈α∀conversation)␈α∀pointed␈α∪out␈α∀the␈α∀advantages␈α∪of␈α∀an␈α∀alternate␈α∪treatment
␈↓ α∧␈↓avoiding␈α∂possible␈α∂worlds␈α∂in␈α∂case␈α∂there␈α∂is␈α∂a␈α∂single␈α∂domain␈α∂of␈α∂individuals␈α∂each␈α∂of␈α∂which␈α⊂has␈α∂a
␈↓ α∧␈↓standard concept.  Then we can write

␈↓ α∧␈↓101)␈↓ αt ␈↓↓∀V Q.(true All(V,Q) ≡ ∀x.true Subst(Concept1 x,V,Q))␈↓.





␈↓ α∧␈↓αPOSSIBLE APPLICATIONS TO ARTIFICIAL INTELLIGENCE

␈↓ α∧␈↓␈↓ αTThe␈αforegoing␈αdiscussion␈αof␈αconcepts␈αhas␈αbeen␈αmainly␈αconcerned␈αwith␈αhow␈αto␈αtranslate␈αinto
␈↓ α∧␈↓a␈α_suitable␈α_formal␈α_language␈α_certain␈α_sentences␈α_of␈α_ordinary␈α_language.␈α_ The␈α_success␈α_of␈α_the
␈↓ α∧␈↓formalization␈α
is␈α
measured␈α
by␈αthe␈α
extent␈α
to␈α
which␈α
the␈αlogical␈α
consequences␈α
of␈α
these␈α
sentences␈αin␈α
the
␈↓ α∧␈↓formal␈α
system␈α
agree␈α
with␈α
our␈α
intuitions␈α
of␈α
what␈α
these␈α
consequences␈α
should␈α
be.␈α
 Another␈α
goal␈α
of
␈↓ α∧␈↓␈↓ f17


␈↓ α∧␈↓the␈αformalization␈αis␈α
to␈αdevelop␈αan␈α
idea␈αof␈αwhat␈α
concepts␈αreally␈αare,␈α
but␈αthe␈αpossible␈α
formalizations
␈↓ α∧␈↓have not been explored enough to draw even tentative conclusions about that.

␈↓ α∧␈↓␈↓ αTFor␈αartificial␈αintelligence,␈αthe␈αstudy␈αof␈αconcepts␈αhas␈αyet␈αa␈αdifferent␈αmotivation.␈α Our␈αsuccess
␈↓ α∧␈↓in␈α⊃making␈α∩computer␈α⊃programs␈α∩with␈α⊃␈↓↓general␈α⊃intelligence␈↓␈α∩has␈α⊃been␈α∩extremely␈α⊃limited,␈α∩and␈α⊃one
␈↓ α∧␈↓source␈αof␈αthe␈αlimitation␈αis␈αour␈α
inability␈αto␈αformalize␈αwhat␈αthe␈αworld␈α
is␈αlike␈αin␈αgeneral.␈α We␈αcan␈α
try
␈↓ α∧␈↓to␈αseparate␈αthe␈αproblem␈αof␈αdescribing␈αthe␈αgeneral␈αaspects␈αof␈αthe␈αworld␈αfrom␈αthe␈αproblem␈αof␈αusing
␈↓ α∧␈↓such␈αa␈αdescription␈αand␈αthe␈αfacts␈αof␈αa␈αsituation␈α
to␈αdiscover␈αa␈αstrategy␈αfor␈αachieving␈αa␈αgoal.␈α This␈α
is
␈↓ α∧␈↓called␈α
separating␈α∞the␈α
␈↓↓epistemological␈↓␈α∞and␈α
the␈α
␈↓↓heuristic␈↓␈α∞parts␈α
of␈α∞the␈α
artificial␈α∞intelligence␈α
problem
␈↓ α∧␈↓and is discussed in (McCarthy and Hayes 1969).

␈↓ α∧␈↓␈↓ αTWe see the following potential uses for facts about knowledge:

␈↓ α∧␈↓␈↓ αT1.␈αA␈αcomputer␈αprogram␈αthat␈αwants␈αto␈αtelephone␈αsomeone␈αmust␈αreason␈αabout␈αwho␈αknows␈αthe
␈↓ α∧␈↓number.␈α∪ More␈α∪generally,␈α∪it␈α∪must␈α∪reason␈α∪about␈α∪what␈α∪actions␈α∪will␈α∪obtain␈α∀needed␈α∪knowledge.
␈↓ α∧␈↓Knowledge␈α
in␈α
books␈α
and␈αcomputer␈α
files␈α
must␈α
be␈αtreated␈α
in␈α
a␈α
parallel␈αway␈α
to␈α
knowledge␈α
held␈αby
␈↓ α∧␈↓persons.

␈↓ α∧␈↓␈↓ αT2.␈α
A␈α
program␈α
must␈α
often␈α
determine␈α
that␈α
it␈α
does␈α
not␈α
know␈α
something␈α
or␈α
that␈α
someone␈α
else
␈↓ α∧␈↓doesn't.␈α∩ This␈α⊃has␈α∩been␈α⊃neglected␈α∩in␈α⊃the␈α∩usual␈α⊃formalizations␈α∩of␈α⊃knowledge,␈α∩and␈α∩methods␈α⊃of
␈↓ α∧␈↓proving␈αpossibility␈αhave␈αbeen␈α
neglected␈αin␈αmodal␈αlogic.␈α
 Christopher␈αGoad␈α(to␈αbe␈α
published)␈αhas
␈↓ α∧␈↓shown␈αhow␈αto␈αprove␈αignorance␈αby␈αproving␈αthe␈αexistence␈αof␈αpossible␈αworlds␈αin␈αwhich␈αthe␈αsentence
␈↓ α∧␈↓to␈αbe␈αproved␈αunknown␈αis␈αfalse.␈α Presumably␈αproving␈αone's␈αown␈αignorance␈αis␈αa␈αstimulus␈αto␈αlooking
␈↓ α∧␈↓outside␈αfor␈αthe␈αinformation.␈α In␈αcompetitive␈αsituations,␈αit␈αmay␈αbe␈αimportant␈αto␈αshow␈αthat␈αa␈αcertain
␈↓ α∧␈↓course of action will leave competitors ignorant.

␈↓ α∧␈↓␈↓ αT3.␈αPrediction␈α
of␈αthe␈αbehavior␈α
of␈αothers␈αdepends␈α
on␈αdetermining␈αwhat␈α
they␈αbelieve␈αand␈α
what
␈↓ α∧␈↓they want.

␈↓ α∧␈↓␈↓ αTIt␈αseems␈αto␈αme␈αthat␈αAI␈αapplications␈α
will␈αespecially␈αbenefit␈αfrom␈αfirst␈αorder␈αformalisms␈αof␈α
the
␈↓ α∧␈↓kind␈α
described␈α
above.␈α First,␈α
many␈α
of␈αthe␈α
present␈α
problem␈αsolvers␈α
are␈α
based␈αon␈α
first␈α
order␈αlogic.
␈↓ α∧␈↓Morgan␈α(1976)␈αin␈αdiscussing␈αtheorem␈αproving␈αin␈αmodal␈αlogic␈αalso␈αtranslates␈αmodal␈αlogic␈αinto␈αfirst
␈↓ α∧␈↓order␈α
logic.␈α Second,␈α
our␈αformalisms␈α
leaves␈αthe␈α
syntax␈αand␈α
semantics␈αof␈α
statements␈α
not␈αinvolving
␈↓ α∧␈↓concepts␈αentirely␈α
unchanged,␈αso␈α
that␈αif␈αknowledge␈α
or␈αwanting␈α
is␈αonly␈αa␈α
small␈αpart␈α
of␈αa␈αproblem,␈α
its
␈↓ α∧␈↓presence doesn't affect the formalization of the other parts.

␈↓ α∧␈↓␈↓ αTIn␈α∞Appendix␈α∞I,␈α
we␈α∞give␈α∞a␈α∞set␈α
of␈α∞axioms␈α∞for␈α∞knowledge␈α
that␈α∞permits␈α∞deduction␈α∞from␈α
␈↓↓"Pat
␈↓ α∧␈↓↓knows␈α
Mike's␈α
telephone␈α
number"␈↓␈αand␈α
␈↓↓Pat␈α
wants␈α
Joe␈α
to␈αknow␈α
Mike's␈α
telephone␈α
number"␈↓␈α
that␈α␈↓↓Joe
␈↓ α∧␈↓↓will␈αknow␈αMike's␈αtelephone␈αnumber"␈↓.␈α Treatments␈αof␈αthe␈α"dynamics"␈αof␈αknowledge␈αare␈αa␈αfirst␈αstep
␈↓ α∧␈↓towards␈αAI␈αapplications.␈α The␈αaxiomatization␈αis␈α␈↓↓quasi-static␈↓,␈αi.e.␈αeach␈αaction␈αtakes␈αa␈αsituation␈αinto
␈↓ α∧␈↓a definite resulting situation, and there are no concurrent processes.

␈↓ α∧␈↓␈↓ αTThe␈α
special␈α
premisses␈α
are␈α
written␈α
␈↓↓true(world,Want(Pat,Know(Joe,␈α
Telephone␈α
Mike)))␈↓␈αand
␈↓ α∧␈↓␈↓↓true(world,Know(Pat,Telephone␈α0Mike))␈↓,␈α1and␈α0the␈α0conclusion␈α1is␈α0␈↓↓true(world,Future
␈↓ α∧␈↓↓Know(Joe,Telephone Mike))␈↓.

␈↓ α∧␈↓␈↓ αTThe␈α∞proof␈α∞from␈α∞these␈α∞axioms␈α∞that␈α∞Joe␈α∞will␈α∞know␈α∞Mike's␈α∞telephone␈α∞number␈α∞has␈α∞about␈α∞15
␈↓ α∧␈↓steps.␈α
 Since␈α
there␈α
is␈α
only␈α
one␈α
action␈α
-␈α
Pat␈α
telling␈α
Joe␈α
Mike's␈α
telephone␈α
number,␈α
the␈α
frame␈α
problem
␈↓ α∧␈↓(McCarthy␈α
and␈αHayes␈α
1969)␈α
doesn't␈αarise.␈α
 A␈α
more␈αelaborate␈α
example␈α
in␈αwhich␈α
Joe␈α
wants␈αto␈α
know
␈↓ α∧␈↓␈↓ f18


␈↓ α∧␈↓Mike's␈αtelephone␈αnumber,␈αtells␈αPat␈αthat␈αfact,␈αand␈αleading␈αto␈αPat␈αtelling␈αJoe␈αthe␈αnumber␈α
has␈αbeen
␈↓ α∧␈↓partially␈α∪worked␈α∩out.␈α∪ but␈α∩the␈α∪treatment␈α∩is␈α∪not␈α∩very␈α∪satisfactory.␈α∩ Several␈α∪frame␈α∪axioms␈α∩are
␈↓ α∧␈↓required,␈α⊂the␈α⊂proof␈α⊂would␈α∂be␈α⊂quite␈α⊂long,␈α⊂and␈α∂the␈α⊂previous␈α⊂result␈α⊂cannot␈α∂be␈α⊂used␈α⊂as␈α⊂a␈α∂lemma
␈↓ α∧␈↓because its statement doesn't say what remains unchanged when Pat tells Joe Mike's number.

␈↓ α∧␈↓␈↓ αTEven␈αthe␈αfifteen␈αstep␈αproof␈αdoesn't␈αmodel␈αhuman␈αreasoning,␈αor␈αthe␈αway␈αcomputer␈αprograms
␈↓ α∧␈↓should␈α∞be␈α∞designed␈α∞to␈α∂reason.␈α∞ Namely,␈α∞the␈α∞particular␈α∞result␈α∂is␈α∞obtained␈α∞by␈α∞substitution␈α∂from␈α∞a
␈↓ α∧␈↓general␈α∂statement␈α∂about␈α∂what␈α∂to␈α∂do␈α∂when␈α∂a␈α∞person␈α∂or␈α∂machine␈α∂wants␈α∂another␈α∂to␈α∂know␈α∂a␈α∞fact.
␈↓ α∧␈↓Therefore,␈α∞there␈α∞is␈α∞no␈α∂reason␈α∞to␈α∞deduce␈α∞it␈α∂each␈α∞time␈α∞it␈α∞is␈α∂needed.␈α∞ Moreover,␈α∞as␈α∞the␈α∂M.I.T.␈α∞AI
␈↓ α∧␈↓school␈αhas␈αemphasized,␈αthis␈αgeneral␈αfact␈αshould␈αbe␈αstored␈αso␈αas␈αto␈αbe␈αtriggered␈αspecifically␈αby␈αthe
␈↓ α∧␈↓desire that another person shall know something.


␈↓ α∧␈↓α␈↓ αTABSTRACT LANGUAGES

␈↓ α∧␈↓␈↓ αTThe␈αway␈αwe␈αhave␈αtreated␈αconcepts␈αin␈αthis␈αpaper,␈αespecially␈αwhen␈αwe␈αput␈αvariables␈αin␈αthem,
␈↓ α∧␈↓suggests␈α
trying␈α
to␈α
identify␈α
them␈α
with␈α
terms␈α
in␈αsome␈α
language.␈α
 It␈α
seems␈α
to␈α
me␈α
that␈α
this␈α
can␈αbe␈α
done
␈↓ α∧␈↓provided we use a suitable notion of ␈↓↓abstract␈↓ ␈↓↓language.␈↓

␈↓ α∧␈↓␈↓ αTOrdinarily␈α∩a␈α∩language␈α∩is␈α∩identified␈α∩with␈α∩a␈α∩set␈α∩of␈α∩strings␈α∩of␈α∩symbols␈α∩taken␈α∩from␈α⊃some
␈↓ α∧␈↓alphabet.␈α
 McCarthy␈α
(1963)␈α
introduces␈α
the␈α
idea␈α∞of␈α
␈↓↓abstract␈α
syntax␈↓,␈α
the␈α
idea␈α
being␈α
that␈α∞it␈α
doesn't
␈↓ α∧␈↓matter␈αwhether␈αsums␈αare␈αrepresented␈α␈↓↓a+b␈↓␈αor␈α␈↓↓+ab␈↓␈αor␈α␈↓↓ab+␈↓␈αor␈αby␈αthe␈αinteger␈α␈↓↓2␈↓∧a␈↓↓3␈↓∧b␈↓␈αor␈αby␈αthe␈αLISP␈αS-
␈↓ α∧␈↓expression␈α
(PLUS␈αA␈α
B),␈αso␈α
long␈αas␈α
there␈α
are␈αpredicates␈α
for␈αdeciding␈α
whether␈αan␈α
expression␈α
is␈αa
␈↓ α∧␈↓sum␈αand␈αfunctions␈αfor␈αforming␈αsums␈αfrom␈αsummands␈αand␈αfunctions␈αfor␈αextracting␈αthe␈αsummands
␈↓ α∧␈↓from␈α∂the␈α∂sum.␈α∞ In␈α∂particular,␈α∂abstract␈α∞syntax␈α∂facilitates␈α∂defining␈α∞the␈α∂semantics␈α∂of␈α∞programming
␈↓ α∧␈↓languages,␈α
and␈αproving␈α
the␈αproperties␈α
of␈αinterpreters␈α
and␈α
compilers.␈α From␈α
that␈αpoint␈α
of␈αview,␈α
one
␈↓ α∧␈↓can␈αrefrain␈α
from␈αspecifying␈α
any␈αconcrete␈α
representation␈αof␈α
the␈α"expressions"␈α
of␈αthe␈α
language␈αand
␈↓ α∧␈↓consider␈α⊃it␈α⊃merely␈α⊃a␈α⊃collection␈α⊃of␈α⊃abstract␈α⊃synthetic␈α⊃and␈α⊃analytic␈α⊃functions␈α⊃and␈α⊃predicates␈α⊂for
␈↓ α∧␈↓forming,␈αdiscriminating␈αand␈αtaking␈αapart␈α
␈↓↓abstract␈αexpressions␈↓.␈α However,␈αthe␈αlanguages␈α
considered
␈↓ α∧␈↓at that time always admitted representations as strings of symbols.

␈↓ α∧␈↓␈↓ αTIf␈α∞we␈α∂consider␈α∞concepts␈α∞as␈α∂a␈α∞free␈α∂algebra␈α∞on␈α∞basic␈α∂concepts,␈α∞then␈α∞we␈α∂can␈α∞regard␈α∂them␈α∞as
␈↓ α∧␈↓strings␈α⊂of␈α⊂symbols␈α⊂on␈α⊂some␈α⊂alphabet␈α⊂if␈α⊂we␈α∂want␈α⊂to,␈α⊂assuming␈α⊂that␈α⊂we␈α⊂don't␈α⊂object␈α⊂to␈α⊂a␈α∂non-
␈↓ α∧␈↓denumerable␈αalphabet␈αor␈αinfinitely␈αlong␈αexpressions␈αif␈αwe␈αwant␈αstandard␈αconcepts␈αfor␈αall␈αthe␈αreal
␈↓ α∧␈↓numbers.␈α
 However,␈αif␈α
we␈αwant␈α
to␈αregard␈α
␈↓↓Equal(X,Y)␈↓␈αand␈α
␈↓↓Equal(Y,X)␈↓␈αas␈α
the␈αsame␈α
concept,␈αand
␈↓ α∧␈↓hence␈α∞as␈α∞the␈α∞same␈α∂"expression"␈α∞in␈α∞our␈α∞language,␈α∞and␈α∂we␈α∞want␈α∞to␈α∞regard␈α∞expressions␈α∂related␈α∞by
␈↓ α∧␈↓renaming␈α
bound␈αvariables␈α
as␈αdenoting␈α
the␈αsame␈α
concept,␈αthen␈α
the␈αalgebra␈α
is␈αno␈α
longer␈α
free,␈αand
␈↓ α∧␈↓regarding concepts as strings of symbols becomes awkward even if possible.

␈↓ α∧␈↓␈↓ αTIt␈α∪seems␈α∩better␈α∪to␈α∩accept␈α∪the␈α∩notion␈α∪of␈α∩␈↓↓abstract␈α∪language␈↓␈α∩defined␈α∪by␈α∩the␈α∪collection␈α∩of
␈↓ α∧␈↓functions␈α
and␈αpredicates␈α
that␈α
form,␈αdiscriminate,␈α
and␈α
extract␈αthe␈α
parts␈α
of␈αits␈α
"expressions".␈α In␈α
that
␈↓ α∧␈↓case it would seem that concepts can be identified with expressions in an abstract language.
␈↓ α∧␈↓␈↓ f19


␈↓ α∧␈↓α␈↓ αTBIBLIOGRAPHY

␈↓ α∧␈↓␈↓ αTThe␈α
treatment␈α
given␈α
here␈α
should␈α
be␈αcompared␈α
with␈α
that␈α
in␈α
(Church␈α
1951b)␈α
and␈αin␈α
(Morgan
␈↓ α∧␈↓1976).␈α
 Church␈αintroduces␈α
what␈αmight␈α
be␈α
called␈αa␈α
two-dimensional␈αtype␈α
structure.␈α One␈α
dimension
␈↓ α∧␈↓permits␈α∞higher␈α∞order␈α∞functions␈α∞and␈α∞predicates␈α∞as␈α∞in␈α∞the␈α∞usual␈α∞higher␈α∞order␈α∞logics.␈α∂ The␈α∞second
␈↓ α∧␈↓dimension␈α
permits␈α
concepts␈αof␈α
concepts,␈α
etc.␈α
 No␈αexamples␈α
or␈α
applications␈αare␈α
given.␈α
 It␈α
seems␈αto
␈↓ α∧␈↓me␈α∞that␈α∞concepts␈α∞of␈α∞concepts␈α∞will␈α∞be␈α∞eventually␈α∞required,␈α∞but␈α∞this␈α∞can␈α∞still␈α∞be␈α∞done␈α∂within␈α∞first
␈↓ α∧␈↓order logic.

␈↓ α∧␈↓␈↓ αTMorgan's␈α∞motivation␈α∂is␈α∞to␈α∂use␈α∞first␈α∂order␈α∞logic␈α∂theorem␈α∞proving␈α∂programs␈α∞to␈α∂treat␈α∞modal
␈↓ α∧␈↓logic.␈α⊂ He␈α⊃gives␈α⊂two␈α⊂approaches.␈α⊃ The␈α⊂syntactic␈α⊂approach␈α⊃-␈α⊂which␈α⊂he␈α⊃applies␈α⊂only␈α⊃to␈α⊂systems
␈↓ α∧␈↓without␈α⊗quantifiers␈α⊗-␈α⊗uses␈α∃operations␈α⊗like␈α⊗our␈α⊗␈↓↓And␈↓␈α∃to␈α⊗form␈α⊗compound␈α⊗propositions␈α∃from
␈↓ α∧␈↓elementary␈αones.␈α Provability␈αis␈αthen␈αaxiomatized␈αin␈αthe␈αouter␈αlogic.␈α His␈αsemantic␈α
approach␈αuses
␈↓ α∧␈↓axiomatizations␈α
of␈α
the␈αKripke␈α
accessibility␈α
relation␈αbetween␈α
possible␈α
worlds.␈α It␈α
seems␈α
to␈α
me␈αthat
␈↓ α∧␈↓our␈α
treatment␈α
can␈αbe␈α
used␈α
to␈α
combine␈αboth␈α
of␈α
Morgan's␈α
methods,␈αand␈α
has␈α
two␈αfurther␈α
advantages.
␈↓ α∧␈↓First,␈α∂concepts␈α∂and␈α⊂individuals␈α∂can␈α∂be␈α∂separately␈α⊂quantified.␈α∂ Second,␈α∂functions␈α∂from␈α⊂things␈α∂to
␈↓ α∧␈↓concepts␈α∀of␈α∃them␈α∀permit␈α∀relations␈α∃between␈α∀concepts␈α∃of␈α∀things␈α∀that␈α∃could␈α∀not␈α∃otherwise␈α∀be
␈↓ α∧␈↓expressed.

␈↓ α∧␈↓␈↓ αTAlthough␈αthe␈α
formalism␈αleads␈α
in␈αalmost␈α
the␈αopposite␈α
direction,␈αthe␈α
present␈αpaper␈α
is␈αmuch␈α
in
␈↓ α∧␈↓the␈α∂spirit␈α∞of␈α∂(Carnap␈α∂1956).␈α∞ We␈α∂appeal␈α∂to␈α∞his␈α∂ontological␈α∂tolerance␈α∞in␈α∂introducing␈α∂concepts␈α∞as
␈↓ α∧␈↓objects,␈α
and␈αhis␈α
section␈αon␈α
intensions␈αfor␈α
robots␈αexpresses␈α
just␈αthe␈α
attitude␈αrequired␈α
for␈αartificial
␈↓ α∧␈↓intelligence applications.

␈↓ α∧␈↓␈↓ αTWe␈α∞have␈α
not␈α∞yet␈α
investigated␈α∞the␈α
matter,␈α∞but␈α
plausible␈α∞axioms␈α
for␈α∞necessity␈α∞or␈α
knowledge
␈↓ α∧␈↓expressed␈α∞in␈α∞terms␈α∞of␈α∞concepts␈α∞may␈α∂lead␈α∞to␈α∞the␈α∞paradoxes␈α∞discussed␈α∞in␈α∞(Montague␈α∂and␈α∞Kaplan
␈↓ α∧␈↓1961)␈α
and␈α
(Montague␈α
1963).␈α Our␈α
intuition␈α
is␈α
that␈α
the␈αparadoxes␈α
can␈α
be␈α
avoided␈α
by␈αrestricting␈α
the
␈↓ α∧␈↓axioms␈α
concerning␈α
knowledge␈α
of␈α
facts␈α
about␈α
knowledge␈α
and␈α
necessity␈α
of␈α
statements␈α
about␈α
necessity.
␈↓ α∧␈↓The␈α∪restrictions␈α∩will␈α∪be␈α∪somewhat␈α∩unintuitive␈α∪as␈α∩are␈α∪the␈α∪restrictions␈α∩necessary␈α∪to␈α∪avoid␈α∩the
␈↓ α∧␈↓paradoxes of naive set theory.

␈↓ α∧␈↓␈↓ αTChee␈αK.␈αYap␈α(1977)␈αproposes␈α␈↓↓Virtual␈αSemantics␈↓␈αfor␈αintensional␈αlogics␈αas␈αa␈αgeneralization␈αof
␈↓ α∧␈↓Carnap's␈α∞individual␈α∞concepts.␈α∞ Apart␈α∞from␈α∞the␈α∂fact␈α∞that␈α∞Yap␈α∞does␈α∞not␈α∞stay␈α∂within␈α∞conventional
␈↓ α∧␈↓first order logic, we don't yet know the relation between his work and that described here.





␈↓ α∧␈↓α␈↓ αTREFERENCES:

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

␈↓ α∧␈↓␈↓ αT␈↓αChurch,␈α↔Alonzo␈↓␈α_(1951a),␈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␈↓,␈α∞No.␈α
1␈α∞(July␈α∞1951),␈α
100-112.␈α∞ Reprinted␈α∞in␈α
␈↓↓The␈α∞Structure␈α∞of␈α
Language␈↓,
␈↓ α∧␈↓edited by Jerry A. Fodor and Jerrold Katz, Prentice-Hall 1964

␈↓ α∧␈↓␈↓ αT            ␈α∞(1951b),␈α
A␈α∞formulation␈α
of␈α∞the␈α∞logic␈α
of␈α∞sense␈α
and␈α∞denotation.␈α
 In:␈α∞P.␈α∞Henle␈α
(ed.),
␈↓ α∧␈↓␈↓↓Essays in honor of Henry Sheffer␈↓, pp. 3-24.  New York.
␈↓ α∧␈↓␈↓ f20


␈↓ α∧␈↓␈↓ αT␈↓αFrege,␈α_Gottlob␈↓␈α→(1892),␈α_Uber␈α→Sinn␈α_und␈α_Bedeutung.␈α→␈↓↓Zeitschrift␈α_fur␈α→Philosophie␈α_und
␈↓ α∧␈↓↓Philosophische␈α↔Kritik␈↓␈α⊗100:25-50.␈α↔ Translated␈α⊗by␈α↔H.␈α⊗Feigl␈α↔under␈α⊗the␈α↔title␈α⊗"On␈α↔Sense␈α⊗and
␈↓ α∧␈↓Nominatum"␈α
in␈α
H.␈α
Feigl␈α
and␈α
W.␈αSellars␈α
(eds.)␈α
␈↓↓Readings␈α
in␈α
Philosophical␈α
Analysis␈↓,␈α
New␈αYork␈α
1949.
␈↓ α∧␈↓Translated␈α∞by␈α∞M.␈α∂Black␈α∞under␈α∞the␈α∂title␈α∞"On␈α∞Sense␈α∂and␈α∞Reference"␈α∞in␈α∂P.␈α∞Geach␈α∞and␈α∂M.␈α∞Black,
␈↓ α∧␈↓␈↓↓Translations from the Philosophical Writings of Gottlob Frege␈↓, Oxford, 1952.

␈↓ α∧␈↓␈↓ αT␈↓αKaplan,␈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.), pp. 178-214.  Reprinted in (Linsky 1971).

␈↓ α∧␈↓␈↓ αT␈↓αKaplan,␈αDavid␈↓␈αand␈α␈↓αMontague,␈αRichard␈↓␈α(1960),␈αA␈αParadox␈αRegained,␈α␈↓↓Notre␈αDame␈αJournal
␈↓ α∧␈↓↓of Formal Logic␈↓ 1:79-90.  Reprinted in (Montague 1974).

␈↓ α∧␈↓␈↓ αT␈↓αLinsky,␈α∃Leonard␈↓,␈α∃ed.(1971)␈α∃␈↓↓Reference␈α∃and␈α∃Modality␈↓,␈α∃Oxford␈α∃Readings␈α∃in␈α∀Philosophy,
␈↓ α∧␈↓Oxford University Press.

␈↓ α∧␈↓␈↓ αT␈↓αMcCarthy,␈α∞J.␈↓␈α∞(1963),␈α∞Towards␈α∞a␈α∞Mathematical␈α∞Science␈α∞of␈α∞Computation,␈α∞in␈α∂␈↓↓Proceedings␈α∞of
␈↓ α∧␈↓↓IFIP Congress 1962␈↓, North-Holland Publishing Co., Amsterdam.

␈↓ α∧␈↓␈↓ αT␈↓αMcCarthy,␈αJ.␈αand␈αHayes,␈αP.J.␈↓␈α(1969)␈αSome␈αPhilosophical␈αProblems␈αfrom␈αthe␈αStandpoint␈αof
␈↓ 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∧␈↓␈↓ αT␈↓αMontague,␈α↔Richard␈↓␈α_(1963),␈α↔Syntactical␈α↔Treatments␈α_of␈α↔Modality,␈α↔with␈α_Corollaries␈α↔on
␈↓ α∧␈↓Reflexion␈α≠Principles␈α≤and␈α≠Finite␈α≠Axiomatizability,␈α≤␈↓↓Acta␈α≠Philosophica␈α≤Fennica␈↓␈α≠␈↓α16␈↓:153-167.
␈↓ α∧␈↓Reprinted in (Montague 1974).

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

␈↓ α∧␈↓␈↓ αT␈↓αMorgan,␈α∩Charles␈α∩G.␈↓␈α∩(1976),␈α∩Methods␈α∩for␈α∩Automated␈α∩Theorem␈α∩Proving␈α∩in␈α⊃Nonclassical
␈↓ α∧␈↓Logics, ␈↓↓IEEE Transactions on Computers␈↓, vol. C-25, No. 8, August 1976

␈↓ α∧␈↓␈↓ αT␈↓αQuine,␈α
W.V.O.␈↓␈α
(1956),␈α
Quantifiers␈α
and␈α
Propositional␈α
Attitudes,␈α
␈↓↓Journal␈α
of␈α
Philosophy␈↓,␈α
53.
␈↓ α∧␈↓Reprinted in (Linsky 1971).

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

␈↓ α∧␈↓␈↓ αT␈↓αYap,␈α∞Chee␈α∞K.␈↓␈α∂(1977),␈α∞␈↓↓A␈α∞Semantical␈α∞Analysis␈α∂of␈α∞Intensional␈α∞Logics␈↓,␈α∞Research␈α∂Report,␈α∞IBM
␈↓ α∧␈↓Thomas␈α
J.␈α
Watson␈α∞Research␈α
Center,␈α
Yorktown␈α
Heights,␈α∞New␈α
York.␈α
RC␈α
6893␈α∞(#29538)␈α
12/13/77,
␈↓ α∧␈↓47 pp.