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␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓␈↓ ε(draft





















␈↓ ↓H␈↓α␈↓ α
FIRST ORDER THEORIES OF INDIVIDUAL CONCEPTS AND PROPOSITIONS

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓modal operators.  Applications are given in philosophy and in artificial intelligence.

␈↓ ↓H␈↓␈↓εThis draft of CONCEP[E76,JMC] PUBbed at 11:12 on July 27, 1977.␈↓
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓αINTRODUCTION

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

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

␈↓ ↓H␈↓␈↓ α_According␈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
␈↓ ↓H␈↓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,␈α
whereas
␈↓ ↓H␈↓its␈α
meaning␈α
in␈α
the␈α
sentence␈α
␈↓↓"Pat␈α
dialed␈αMike's␈α
telephone␈α
number"␈↓␈α
is␈α
the␈α
number␈α
itself.␈α
 Thus␈αif␈α
we
␈↓ ↓H␈↓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
␈↓ ↓H␈↓↓telephone number"␈↓ follows, but ␈↓↓"Pat knows Mary's telephone number"␈↓ does not.

␈↓ ↓H␈↓␈↓ α_Frege␈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
␈↓ ↓H␈↓␈↓↓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␈↓
␈↓ ↓H␈↓␈↓↓contexts.␈↓␈α ␈↓↓Denotations␈↓␈αare␈αthe␈αbasis␈αof␈αthe␈αsemantics␈αof␈αfirst␈αorder␈αlogic␈αand␈αmodel␈αtheory␈αand␈αare
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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)
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓powerful, because it includes mappings from objects to concepts.

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

␈↓ ↓H␈↓␈↓ α_Our␈αidea␈αis␈αto␈αleave␈αthe␈αlogic␈αunchanged␈αand␈αto␈αtreat␈αconcepts␈αas␈αone␈αkind␈αof␈αobject␈αin␈αan
␈↓ ↓H␈↓ordinary␈αfirst␈αorder␈αtheory.␈α We␈αshall␈αhave␈αone␈αterm␈αthat␈αdenotes␈αMike's␈αtelephone␈αnumber␈αand␈α
a
␈↓ ↓H␈↓different␈α
term␈αdenoting␈α
the␈αconcept␈α
of␈α
Mike's␈αtelephone␈α
number␈αinstead␈α
of␈αhaving␈α
a␈α
single␈αterm
␈↓ ↓H␈↓whose␈αdenotation␈α
is␈αthe␈α
number␈αand␈α
whose␈αsense␈αis␈α
a␈αconcept␈α
of␈αit.␈α
 The␈αrelations␈αamong␈α
concepts
␈↓ ↓H␈↓and␈αbetween␈αconcepts␈αand␈αother␈αentities␈αare␈αexpressed␈αby␈αformulas␈αof␈αfirst␈αorder␈αlogic.␈α Ordinary
␈↓ ↓H␈↓model theory can then be used to study what spaces of concepts satisfy various sets of axioms.

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓␈↓↓Pegasus␈↓␈αand␈αnot␈αgeneral␈αconcepts␈αlike␈α␈↓↓telephone␈↓␈αor␈α␈↓↓unicorn.␈↓␈α Extension␈αto␈αgeneral␈αconcepts␈αseems
␈↓ ↓H␈↓feasible, but individual concepts provide enough food for thought for the present.

␈↓ ↓H␈↓␈↓ α_It seems surprising that such a straightforward and easy approach should be new.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓αKNOWING WHAT AND KNOWING THAT

␈↓ ↓H␈↓␈↓ α_To assert that Pat knows Mike's telephone number we write

␈↓ ↓H␈↓1)␈↓ α8  ␈↓↓true Know(Pat,Telephone Mike)␈↓

␈↓ ↓H␈↓with the following conventions:

␈↓ ↓H␈↓␈↓ α_1.␈α
Parentheses␈α
are␈α
often␈α
omitted␈α∞for␈α
one␈α
argument␈α
functions␈α
and␈α
predicates.␈α∞ This␈α
 purely
␈↓ ↓H␈↓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
␈↓ ↓H␈↓constant,␈αvariable␈αor␈αfunction␈αname␈αwhen␈αits␈αvalue␈αis␈αa␈αconcept.␈α (We␈αconsidered␈αalso␈αcapitalizing
␈↓ ↓H␈↓the last letter when the arguments are concepts, but it made the formulas ugly).

␈↓ ↓H␈↓␈↓ α_2.␈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
␈↓ ↓H␈↓himself.

␈↓ ↓H␈↓␈↓ α_3.␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓telephone number is not settled.  For the present, please suppose not.

␈↓ ↓H␈↓␈↓ α_4.␈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
␈↓ ↓H␈↓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,
␈↓ ↓H␈↓␈↓↓Know(Pat,Telephone␈αMike)␈↓␈αis␈αa␈αproposition␈αand␈αnot␈αa␈αtruth␈αvalue.␈α Note␈αthat␈αwe␈αare␈αformalizing
␈↓ ↓H␈↓␈↓↓knowing␈↓␈α∪␈↓↓what␈↓␈α∪rather␈α∪than␈α∪␈↓↓knowing␈↓␈α∪␈↓↓that␈↓␈α∀or␈α∪␈↓↓knowing␈↓␈α∪␈↓↓how.␈↓␈α∪ For␈α∪AI␈α∪and␈α∪for␈α∀other␈α∪practical
␈↓ ↓H␈↓purposes,␈α
␈↓↓knowing␈↓␈α
␈↓↓what␈↓␈α
seems␈α
to␈α
be␈α
the␈αmost␈α
useful␈α
notion␈α
of␈α
the␈α
three.␈α
 In␈α
English,␈α␈↓↓knowing␈α
what␈↓
␈↓ ↓H␈↓is written ␈↓↓knowing whether␈↓ when the "knowand" is a proposition.

␈↓ ↓H␈↓␈↓ α_5.␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓␈↓↓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
␈↓ ↓H␈↓introduce).

␈↓ ↓H␈↓␈↓ α_6.␈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,
␈↓ ↓H␈↓necessity,␈αetc.␈αwill␈αbe␈αdiscussed␈α
without␈αextending␈αthe␈αlogic␈αin␈α
any␈αway␈α-␈αsolely␈αby␈αthe␈α
introduction
␈↓ ↓H␈↓of␈αpredicate␈αand␈αfunction␈αsymbols␈αsubject␈αto␈αsuitable␈αaxioms.␈α In␈αthe␈αpresent␈αinformal␈αtreatement,
␈↓ ↓H␈↓we will not be explicit about sorts, but we will try to be typographically consistent.

␈↓ ↓H␈↓␈↓ α_The␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓order␈α
to␈α
solve␈α
problems.␈α From␈α
either␈α
point␈α
of␈αview,␈α
however,␈α
if␈α
(1)␈α
is␈αto␈α
be␈α
reasonable,␈α
it␈αmust␈α
not
␈↓ ↓H␈↓follow␈αfrom␈α(1)␈αand␈αthe␈αfact␈αthat␈αMary's␈αtelephone␈αnumber␈αis␈αthe␈αsame␈αas␈αMike's,␈αthat␈αPat␈αknows
␈↓ ↓H␈↓Mary's telephone number.

␈↓ ↓H␈↓␈↓ α_The proposition that Joe knows ␈↓↓whether␈↓ Pat knows Mike's telephone number, is written
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓2)␈↓ α8 ␈↓↓Know(Joe,Know(Pat,Telephone Mike))␈↓,

␈↓ ↓H␈↓and asserting it requires writing

␈↓ ↓H␈↓3)␈↓ α8 ␈↓↓true Know(Joe,Know(Pat,Telephone Mike))␈↓,

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

␈↓ ↓H␈↓4)␈↓ α8 ␈↓↓K(Joe,Know(Pat,Telephone Mike))␈↓,

␈↓ ↓H␈↓where␈α␈↓↓K(P,Q)␈↓␈αis␈αthe␈αproposition␈αthat␈α␈↓↓P␈↓␈αknows␈α␈↓↓that␈↓␈α␈↓↓Q.␈↓␈α While␈αEnglish␈αis␈αlogically␈αnot␈αuniform␈αin
␈↓ ↓H␈↓that␈αknowing␈αan␈αindividual␈αconcept␈αmeans␈αknowing␈αits␈αvalue␈αwhile␈αknowing␈αa␈αproposition␈αmeans
␈↓ ↓H␈↓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
␈↓ ↓H␈↓infirmity.

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓which ␈↓↓X␈↓ is a concept.  Then there is a function ␈↓↓denot␈↓ such that

␈↓ ↓H␈↓5)␈↓ α8  ␈↓↓x = denot X␈↓.

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

␈↓ ↓H␈↓6)␈↓ α8  ␈↓↓∀P1 P2.(denot P1 = denot P2 ⊃ denot Telephone P1 = denot Telephone P2)␈↓.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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)
␈↓ ↓H␈↓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
␈↓ ↓H␈↓respect␈αto␈α␈↓↓denot.␈↓␈αNote␈αthat␈αour␈α␈↓↓denot␈↓␈αoperates␈αon␈αconcepts␈αrather␈αthan␈αon␈αexpressions;␈αa␈αtheory␈αof
␈↓ ↓H␈↓expressions␈αwill␈αalso␈αneed␈αa␈αdenotation␈αfunction.␈α From␈α(6)␈αand␈αsuitable␈αlogical␈αaxioms␈αfollows␈α
the
␈↓ ↓H␈↓existence of a function ␈↓↓telephone␈↓ satisfying

␈↓ ↓H␈↓7)␈↓ α8 ␈↓↓∀P.(denot Telephone P = telephone denot P)␈↓.

␈↓ ↓H␈↓␈↓ α_␈↓↓Know␈↓ is extensional with respect to ␈↓↓denot␈↓ in its first argument, and this expressed by

␈↓ ↓H␈↓8)␈↓ α8  ␈↓↓∀P1 P2 X.(denot P1 = denot P2 ⊃ denot Know(P1,X) = denot Know(P2,X))␈↓,

␈↓ ↓H␈↓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)␈↓
␈↓ ↓H␈↓satisfying

␈↓ ↓H␈↓9)␈↓ α8 ␈↓↓∀P X.(true Know(P,X) ≡ know(denot P,X))␈↓.

␈↓ ↓H␈↓(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
␈↓ ↓H␈↓the notion of extensionality presented here is relative to ␈↓↓denot.)␈↓

␈↓ ↓H␈↓␈↓ α_The predicate ␈↓↓true␈↓ and the function ␈↓↓denot␈↓ are related by

␈↓ ↓H␈↓10)␈↓ α8  ␈↓↓∀Q.(true Q ≡ (denot Q = t))␈↓
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓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)
␈↓ ↓H␈↓↓possible world␈↓ argument.

␈↓ ↓H␈↓␈↓ α_When␈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)␈↓
␈↓ ↓H␈↓instead of a function.  The extensionality of ␈↓↓Telephone␈↓ would then be written

␈↓ ↓H␈↓11)␈↓ α8␈α␈↓↓∀P1␈αP2␈αx␈αu.(denotes(P1,x)∧denotes(P2,x)∧denotes(Telephone␈αP1,u)␈α⊃␈αdenotes(Telephone
␈↓ ↓H␈↓↓P2,u))␈↓.

␈↓ ↓H␈↓We now introduce the function ␈↓↓Exists␈↓ satisfying

␈↓ ↓H␈↓12)␈↓ α8 ␈↓↓∀X.(true Exists X ≡ ∃x.denotes(X,x))␈↓.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓do this by introducing the predicate ␈↓↓Ishorse␈↓ and writing

␈↓ ↓H␈↓13)␈↓ α8 ␈↓↓true Ishorse Pegasus␈↓

␈↓ ↓H␈↓which is related to the predicate ␈↓↓ishorse␈↓ by

␈↓ ↓H␈↓14)␈↓ α8 ␈↓↓∀X x.(denotes(X,x) ⊃ (ishorse x ≡ true Ishorse X))␈↓.

␈↓ ↓H␈↓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␈↓
␈↓ ↓H␈↓is␈αextensional␈αin␈αthis␈αsense,␈αbut␈αthe␈αcorresponding␈αpredicate␈α␈↓↓exists␈↓␈αis␈αidentically␈αtrue␈αand␈α
therefore
␈↓ ↓H␈↓dispensable.

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

␈↓ ↓H␈↓15)␈↓ α8  ␈↓↓∀Q1 Q2.(true(Q1 And Q2) ≡ true Q1 ∧ true Q2)␈↓.

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

␈↓ ↓H␈↓16)␈↓ α8 ␈↓↓∀Q1 Q2.(true(Q1 Or Q2) ≡ true Q1 ∨ true Q2)␈↓,

␈↓ ↓H␈↓17)␈↓ α8 ␈↓↓∀Q.(true(Not Q) ≡ ¬ true Q)␈↓,

␈↓ ↓H␈↓18)␈↓ α8 ␈↓↓∀Q1 Q2.(true(Q1 Implies Q2) ≡ true Q1 ⊃ true Q2)␈↓,

␈↓ ↓H␈↓and

␈↓ ↓H␈↓19)␈↓ α8 ␈↓↓∀Q1 Q2.(true(Q1 Equiv Q2) ≡ (true Q1 ≡ true Q2))␈↓.

␈↓ ↓H␈↓␈↓ α_The␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓proposition that ␈↓↓X␈↓ and ␈↓↓Y␈↓ denote the same thing if anything.  We shall want axioms

␈↓ ↓H␈↓20)␈↓ α8  ␈↓↓∀X.true Equal(X,X)␈↓,
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓21)␈↓ α8  ␈↓↓∀X Y.(true Equal(X,Y) ≡ true Equal(Y,X))␈↓,

␈↓ ↓H␈↓and

␈↓ ↓H␈↓22)␈↓ α8  ␈↓↓∀X Y Z.(true Equal(X,Y) ∧ true Equal(Y,Z) ⊃ true Equal(X,Z)␈↓

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

␈↓ ↓H␈↓23)␈↓ α8  ␈↓↓∀X Y x.(true Equal(X,Y) ∧ denotes(X,x) ⊃ denotes(Y,x))␈↓

␈↓ ↓H␈↓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␈↓
␈↓ ↓H␈↓symmetric by replacing (21) by

␈↓ ↓H␈↓24)␈↓ α8 ␈↓↓∀X Y.Equal(X,Y) = Equal(Y,X)␈↓,

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

␈↓ ↓H␈↓␈↓ α_The statement that Mary has the same telephone as Mike is asserted by

␈↓ ↓H␈↓25)␈↓ α8  ␈↓↓true Equal(Telephone Mary,Telephone Mike)␈↓,

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

␈↓ ↓H␈↓26)␈↓ α8  ␈↓↓true Know(Pat,Telephone Mary)␈↓.

␈↓ ↓H␈↓To draw this conclusion we need something like

␈↓ ↓H␈↓27)␈↓ α8  ␈↓↓true K(Pat,Equal(Telephone Mary,Telephone Mike))␈↓

␈↓ ↓H␈↓and suitable axioms about knowledge.

␈↓ ↓H␈↓␈↓ α_If␈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
␈↓ ↓H␈↓sentence␈αis␈αasserted␈αand␈αwere␈αto␈αregard␈αthe␈αdenotation-valued␈αfunction␈αas␈αstanding␈αfor␈αthe␈αsense-
␈↓ ↓H␈↓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
␈↓ ↓H␈↓resembles␈α
ordinary␈α
language␈α
in␈αhandling␈α
obliquity␈α
entirely␈α
by␈αcontext.␈α
 There␈α
is␈α
no␈αguarantee␈α
that
␈↓ ↓H␈↓general␈α∂statements␈α∂could␈α∂be␈α∂expressed␈α⊂unambiguously␈α∂without␈α∂circumlocution;␈α∂the␈α∂fact␈α⊂that␈α∂the
␈↓ ↓H␈↓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
␈↓ ↓H␈↓ordinary language for stating them.




␈↓ ↓H␈↓αFUNCTIONS FROM THINGS TO CONCEPTS OF THEM

␈↓ ↓H␈↓␈↓ α_While␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓number ␈↓↓n,␈↓ so that
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓28)␈↓ α8  ␈↓↓∀n.(denot Concept1 n = n)␈↓.

␈↓ ↓H␈↓We can then have simultaneously

␈↓ ↓H␈↓29)␈↓ α8 ␈↓↓true Not Knew(Kepler,Number Planets)␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓30)␈↓ α8 ␈↓↓true Knew(Kepler,Composite Concept1 denot Number Planets)␈↓.

␈↓ ↓H␈↓(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
␈↓ ↓H␈↓tense.)

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

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

␈↓ ↓H␈↓31)␈↓ α8 ␈↓↓Composite1 n = Composite Concept1 n␈↓

␈↓ ↓H␈↓getting

␈↓ ↓H␈↓32)␈↓ α8 ␈↓↓true Knew(Kepler,Composite1 denot Number Planets)␈↓.

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

␈↓ ↓H␈↓33)␈↓ α8 ␈↓↓Composite2 N = Composite Concept1 denot N␈↓,

␈↓ ↓H␈↓letting us write

␈↓ ↓H␈↓34)␈↓ α8 ␈↓↓true Knew(Kepler,Composite2 Number Planets)␈↓,

␈↓ ↓H␈↓which is true even though

␈↓ ↓H␈↓35)␈↓ α8 ␈↓↓true Knew(Kepler,Composite Number Planets)␈↓

␈↓ ↓H␈↓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"␈↓,
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓↓unbeknownst to Kepler) is the number of planets"␈↓.

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓like, ␈↓↓"Lassie knows the location of all her puppies"␈↓.  We write this

␈↓ ↓H␈↓36)␈↓ α8 ␈↓↓∀x.(ispuppy(x,lassie) ⊃ true Knowd(Lassie,Locationd Conceptd x))␈↓.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓␈↓↓Conceptd␈↓ can be tailored to our ideas of what dogs know.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓␈↓ α_A␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓sentence␈αwith␈αquantifiers␈αover␈αthings␈αrather␈αthan␈αover␈αconcepts.␈α However,␈αit␈αis␈αstill␈αpremature␈αto
␈↓ ↓H␈↓apply Occam's razor.


␈↓ ↓H␈↓αRELATIONS BETWEEN KNOWING WHAT AND KNOWING THAT

␈↓ ↓H␈↓␈↓ α_As mentioned before, ␈↓↓"Pat knows Mike's telephone number"␈↓ is written

␈↓ ↓H␈↓37)␈↓ α8  ␈↓↓true Know(Pat,Telephone Mike)␈↓.

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

␈↓ ↓H␈↓38)␈↓ α8 ␈↓↓true K(Pat,Equal(Telephone Mike,Concept1 "333-3333")␈↓

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

␈↓ ↓H␈↓␈↓ α_The two ways of expressing knowledge are somewhat interdefinable, since we can write

␈↓ ↓H␈↓39)␈↓ α8 ␈↓↓K(P,Q) = (Q And Know(P,Q))␈↓,

␈↓ ↓H␈↓and

␈↓ ↓H␈↓40)␈↓ α8 ␈↓↓true Know(P,X) ≡ ∃A.(constant A ∧ true K(P,Equal(X,A)))␈↓.

␈↓ ↓H␈↓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␈↓
␈↓ ↓H␈↓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␈αis
␈↓ ↓H␈↓not obvious how to treat knowing a person.

␈↓ ↓H␈↓␈↓ α_Using the ␈↓↓standard␈↓ ␈↓↓concept␈↓ function ␈↓↓Concept1, we␈↓ might replace (40) by

␈↓ ↓H␈↓41)␈↓ α8 ␈↓↓true Know(P,X) ≡ ∃a.true K(P,Equal(X,Concept1 a))␈↓

␈↓ ↓H␈↓with similar meaning.

␈↓ ↓H␈↓␈↓ α_(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␈↓
␈↓ ↓H␈↓definition seems to require something like

␈↓ ↓H␈↓42)␈↓ α8 ␈↓↓∀P X.(Know(P,X) = Exists X And K(P,Equal(X,Concept2 denot X)))␈↓,

␈↓ ↓H␈↓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
␈↓ ↓H␈↓sorts of objects.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓αUNQUANTIFIED MODAL LOGIC

␈↓ ↓H␈↓In␈α␈↓↓unquantified␈αmodal␈αlogic␈↓,␈αthe␈αarguments␈αof␈αthe␈αmodal␈αfunctions␈αwill␈αnot␈αinvolve␈αquantification
␈↓ ↓H␈↓although quantification occurs in the outer logic.

␈↓ ↓H␈↓␈↓ α_␈↓↓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
␈↓ ↓H␈↓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␈↓.
␈↓ ↓H␈↓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,
␈↓ ↓H␈↓since␈α
␈↓↓nec␈↓␈αis␈α
a␈α
predicates␈αin␈α
the␈α
logic␈αwith␈α
␈↓↓t␈↓␈αand␈α
␈↓↓f␈↓␈α
as␈αvalues,␈α
␈↓↓nec␈↓␈α
␈↓↓Q␈↓␈αcannot␈α
be␈α
an␈αargument␈α
of␈α␈↓↓nec␈↓␈α
or
␈↓ ↓H␈↓␈↓↓Nec.␈↓

␈↓ ↓H␈↓␈↓ α_Before␈αwe␈αeven␈αget␈αto␈αmodal␈αlogic␈αproper␈αwe␈α
have␈αa␈αdecision␈αto␈αmake␈α-␈αshall␈α␈↓↓Not␈αNot␈αQ␈↓␈α
be
␈↓ ↓H␈↓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
␈↓ ↓H␈↓written

␈↓ ↓H␈↓43)␈↓ α8␈↓↓∀Q. Not Not Q = Q␈↓,

␈↓ ↓H␈↓and the second

␈↓ ↓H␈↓44)␈↓ α8 ␈↓↓ ∀Q.true Not Not Q ≡ true Q␈↓.

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

␈↓ ↓H␈↓␈↓ α_In␈α∞␈↓↓Meaning␈α∞and␈α∞Necessity␈↓,␈α∞Carnap␈α∞takes␈α∞what␈α∞amounts␈α∞to␈α∞the␈α∞first␈α∞alternative,␈α
regarding
␈↓ ↓H␈↓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
␈↓ ↓H␈↓when␈αhe␈αwants␈αto␈αdiscuss␈αknowledge␈αwithout␈αassuming␈αthat␈αeveryone␈αknows␈αFermat's␈αlast␈αtheorem
␈↓ ↓H␈↓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
␈↓ ↓H␈↓equivalence classes of this relation.

␈↓ ↓H␈↓␈↓ α_If␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓collection␈α∪of␈α∪propositional␈α∪identities␈α∪like␈α∪(43)␈α∪including␈α∪associative␈α∪and␈α∪distributive␈α∪laws␈α∪for
␈↓ ↓H␈↓conjunction␈αand␈α
disjunction,␈αDe␈α
Morgan's␈αlaw,␈αand␈α
the␈αlaws␈α
governing␈αthe␈α
propositions␈α␈↓↓T␈↓␈αand␈α
␈↓↓F.␈↓
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓have to be studied.

␈↓ ↓H␈↓␈↓ α_Actually␈αthere␈αare␈αmore␈αthan␈αtwo␈αalternatives.␈α Let␈α
␈↓↓M␈↓␈αbe␈αthe␈αfree␈αalgebra␈αbuilt␈αup␈αfrom␈α
the
␈↓ ↓H␈↓"atomic"␈αconcepts␈αby␈αthe␈αconcept␈αforming␈αfunction␈α
symbols.␈α If␈α≡≡␈αis␈αan␈αequivalence␈αrelation␈αon␈α
␈↓↓M␈↓
␈↓ ↓H␈↓such that

␈↓ ↓H␈↓45)␈↓ α8␈↓↓∀X1 X2 ε M.((X1 ≡≡ X2) ⊃ (true X1 ≡ true X2))␈↓,

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

␈↓ ↓H␈↓␈↓ α_Similar possibilities arise in modal logic.  We can choose between the ␈↓↓conceptual␈↓ ␈↓↓identity␈↓
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓46)␈↓ α8 ␈↓↓∀Q.(Poss Q = Not Nec Not Q)␈↓,

␈↓ ↓H␈↓and the weaker extensional axiom

␈↓ ↓H␈↓47)␈↓ α8 ␈↓↓∀Q.(true Poss Q ≡ true Not Nec Not Q)␈↓.

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

␈↓ ↓H␈↓␈↓ α_We have

␈↓ ↓H␈↓48)␈↓ α8 ␈↓↓∀Q.(true Nec Q ⊃ true Q)␈↓,

␈↓ ↓H␈↓and

␈↓ ↓H␈↓49)␈↓ α8 ␈↓↓∀Q1 Q2.(true Nec Q1 ∧ true Nec(Q1 Implies Q2) ⊃ true Nec Q2)␈↓.

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

␈↓ ↓H␈↓␈↓ α_S4 is given by

␈↓ ↓H␈↓50)␈↓ α8 ∀Q.(␈↓↓true Nec Q ≡ true Nec Nec Q)␈↓,

␈↓ ↓H␈↓and S5 by

␈↓ ↓H␈↓51)␈↓ α8 ␈↓↓∀Q.(true Poss Q ≡ true Nec Poss Q)␈↓.

␈↓ ↓H␈↓␈↓ α_Actually,␈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
␈↓ ↓H␈↓simultaneously have the functions ␈↓↓NecT,␈↓ ␈↓↓Nec4␈↓ and ␈↓↓Nec5,␈↓ related by axioms such as

␈↓ ↓H␈↓52)␈↓ α8 ␈↓↓∀Q.(true Nec4 Q ⊃ true Nec5 Q)␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓S5 as truth in the intended model of the system.

␈↓ ↓H␈↓␈↓ α_Presumably we shall want to relate necessity and equality by the axiom

␈↓ ↓H␈↓53)␈↓ α8 ␈↓↓∀X.true Nec Equal(X,X)␈↓.

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

␈↓ ↓H␈↓54)␈↓ α8␈↓↓∀X Y.(X=Y ≡ true Nec Equal(X,Y))␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓necessary.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓αMORE PHILOSOPHICAL EXAMPLES - MOSTLY WELL KNOWN

␈↓ ↓H␈↓␈↓ α_Some␈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
␈↓ ↓H␈↓our notation so the treatments can be compared.

␈↓ ↓H␈↓␈↓ α_First␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓(1961) and (Kaplan 1969).  Consider the sentences

␈↓ ↓H␈↓55)␈↓ α8 ␈↓↓¬nec Equal(Number Planets, Concept1 9)␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓56)␈↓ α8 ␈↓↓nec Equal(Concept1 number planets,Concept1 9)␈↓.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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=
␈↓ ↓H␈↓↓9"␈↓ may be written

␈↓ ↓H␈↓57)␈↓ α8 ␈↓↓number(planets) = 9␈↓

␈↓ ↓H␈↓or, using concepts, as

␈↓ ↓H␈↓58)␈↓ α8 ␈↓↓true Equal(Number Planets, Concept1 9)␈↓,

␈↓ ↓H␈↓and ␈↓↓"Necessarily 9=9"␈↓ is written

␈↓ ↓H␈↓59)␈↓ α8 ␈↓↓nec Equal(Concept1 9,Concept1 9)␈↓,

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

␈↓ ↓H␈↓␈↓ α_Ryle␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓␈↓↓statesman1␈αBaldwin␈↓␈αwhere␈α
the␈αlast␈αasserts␈α
that␈αthe␈αconcept␈α
of␈αBaldwin␈αis␈α
one␈αof␈αa␈αstatesman.␈α
 The
␈↓ ↓H␈↓second can be rendered only as as ␈↓↓true Fiction Pickwick␈↓ or ␈↓↓fiction1 Pickwick␈↓.

␈↓ ↓H␈↓␈↓ α_Quine (1961) considers illegitimate the sentence

␈↓ ↓H␈↓60)␈↓ α8 ␈↓↓(∃x)(Philip is unaware that x denounced Catiline)␈↓

␈↓ ↓H␈↓obtained␈αfrom␈α␈↓↓"Philip␈αis␈αunaware␈αthat␈αTully␈αdenounced␈αCatiline"␈↓␈αby␈αexistential␈αgeneralization.␈α In
␈↓ ↓H␈↓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"␈↓.
␈↓ ↓H␈↓These␈αsentences␈αare␈αrelated␈α
to␈α(perhaps␈αeven␈αexplicated␈αby)␈α
several␈αsentences␈αin␈αour␈αsystem.␈α
 ␈↓↓Tully␈↓
␈↓ ↓H␈↓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
␈↓ ↓H␈↓we have
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓61)␈↓ α8 ␈↓↓tully = cicero␈↓,

␈↓ ↓H␈↓62)␈↓ α8 ␈↓↓denot Tully = cicero␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓63)␈↓ α8 ␈↓↓denot Cicero = cicero␈↓.

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓␈↓↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓conveniently expressed by

␈↓ ↓H␈↓64)␈↓ α8 ␈↓↓Concept2(philip,tully) = Cicero␈↓,

␈↓ ↓H␈↓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
␈↓ ↓H␈↓example also include

␈↓ ↓H␈↓65)␈↓ α8 ␈↓↓true K(Philip,Denounced(Cicero,Catiline))␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓66)␈↓ α8 ␈↓↓¬true K(Philip,Denounced(Tully,Catiline))␈↓,

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

␈↓ ↓H␈↓67)␈↓ α8 ␈↓↓∃P.true Denounced(P,Catiline) And Not K(Philip,Denounced(P,Catiline))␈↓,

␈↓ ↓H␈↓from (66) and others, and

␈↓ ↓H␈↓68)␈↓ α8 ␈↓↓¬∃p.(denounced(p,catiline) ∧ ¬true K(Philip,Denounced(Concept2(philip,p), Catiline)))␈↓

␈↓ ↓H␈↓using the additional hypotheses

␈↓ ↓H␈↓69)␈↓ α8 ␈↓↓∀p.(denounced(p,catiline) ⊃ p = cicero)␈↓,

␈↓ ↓H␈↓70)␈↓ α8 ␈↓↓denot Catiline = catiline␈↓,

␈↓ ↓H␈↓and

␈↓ ↓H␈↓71)␈↓ α8 ␈↓↓∀P1 P2.(denot Denounced(P1,P2) ≡ denounced(denot P1,denot P2))␈↓.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓If␈αwe␈αrefrain␈αfrom␈αusing␈αa␈αsilly␈αmap␈α
that␈αgives␈αsomething␈αlike␈α␈↓↓Denouncer(Catiline)␈↓␈αas␈αits␈αvalue,␈α
we
␈↓ ↓H␈↓can get results that correspond to intuition.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


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

␈↓ ↓H␈↓72)␈↓ α8 ␈↓↓true Believed(I,Greater(Length Youryacht,Concept1 denot Length Youryacht))␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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,
␈↓ ↓H␈↓but if it is consistent, it is probably useful.

␈↓ ↓H␈↓␈↓ α_In␈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
␈↓ ↓H␈↓␈↓↓Denot(Peter,X)␈↓ giving a concept of what Peter thinks the value of ␈↓↓X␈↓ is.  We now write

␈↓ ↓H␈↓73)␈↓ α8␈↓↓longer(youryacht,denot Denot(Peter,Length Youryacht))␈↓,

␈↓ ↓H␈↓but I am not certain this is a correct translation.

␈↓ ↓H␈↓␈↓ α_Quine␈α(1956)␈αdiscusses␈αan␈αexample␈αin␈αwhich␈αRalph␈αsees␈αBernard␈αJ.␈αOrtcutt␈αskulking␈αabout
␈↓ ↓H␈↓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
␈↓ ↓H␈↓same person.  The facts can be expresed in our formalism by equations

␈↓ ↓H␈↓74)␈↓ α8␈↓↓true Believe(Ralph, Isspy P1)␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓75)␈↓ α8  ␈↓↓true Believe(Ralph,Not Isspy P2)␈↓

␈↓ ↓H␈↓where ␈↓↓P1␈↓ and ␈↓↓P2␈↓ are concepts satisfying

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

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓␈↓↓believespy(ralph,ortcutt)␈↓, where we define

␈↓ ↓H␈↓76)␈↓ α8  ␈↓↓∀p1 p2.(believespy(p1,p2) ≡ true Believe(Concept1 p1,Isspy Concept7 p2)␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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)␈↓
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓␈↓↓notbelievespy(ralph,ortcutt)␈↓.␈α∞ However,␈α∞the␈α∂simple-minded␈α∞predicate␈α∞␈↓↓believespy,␈↓␈α∂suitably␈α∞defined,
␈↓ ↓H␈↓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
␈↓ ↓H␈↓circumstances.

␈↓ ↓H␈↓␈↓ α_Regarded␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓actions.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓αQUANTIFICATION

␈↓ ↓H␈↓␈↓ α_As␈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
␈↓ ↓H␈↓introducing␈α⊂standard␈α∂concept␈α⊂functions␈α∂makes␈α⊂"quantifying␈α∂in"␈α⊂rather␈α∂easy.␈α⊂ However,␈α∂forming
␈↓ ↓H␈↓propositions␈α⊗and␈α⊗individual␈α⊗concepts␈α⊗by␈α⊗quantification␈α⊗requires␈α⊗new␈α⊗ideas␈α↔and␈α⊗additional
␈↓ ↓H␈↓formalism.

␈↓ ↓H␈↓␈↓ α_We␈αwant␈αto␈αcontinue␈αdescribing␈αconcepts␈αwithin␈αfirst␈αorder␈αlogic␈αwith␈αno␈αlogical␈αextensions.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓↓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
␈↓ ↓H␈↓concept of ␈↓↓the V such that P␈↓.

␈↓ ↓H␈↓␈↓ α_Since␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓symbols␈αfor␈αinner␈αvariables␈α(got␈αthat?),␈αand␈αwe␈αwill␈αuse␈αdoubled␈αletters,␈αsometimes␈αwith␈αsubscripts,
␈↓ ↓H␈↓for these.  ␈↓↓XX␈↓ will be used for individual concepts, ␈↓↓PP␈↓ for persons, and ␈↓↓QQ␈↓ for propositions.

␈↓ ↓H␈↓␈↓ α_The␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓functions like ␈↓↓Telephone␈↓ and ␈↓↓Know␈↓ take the generalized concepts as arguments.

␈↓ ↓H␈↓␈↓ α_Thus

␈↓ ↓H␈↓77)␈↓ α8  ␈↓↓Child(Mike,PP) Implies Equal(Telephone PP,Telephone Mike)␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓his telephone number is the same as Mike's, and

␈↓ ↓H␈↓78)␈↓ α8  ␈↓↓All(PP,Child(Mike,PP) Implies Equal(Telephone PP,Telephone Mike))␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓propositions␈α
are␈α
formed␈α
similarly␈αto␈α
universal␈α
one's,␈α
but␈αthe␈α
function␈α
␈↓↓Exist␈↓␈α
introduced␈αhere␈α
should
␈↓ ↓H␈↓not be confused with the function ␈↓↓Exists␈↓ applied to individual concepts introduced earlier.

␈↓ ↓H␈↓␈↓ α_In␈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
␈↓ ↓H␈↓the object described exists.  Thus

␈↓ ↓H␈↓79)␈↓ α8  ␈↓↓The(PP,Child(Mike,PP))␈↓

␈↓ ↓H␈↓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
␈↓ ↓H␈↓described child exists.  We have

␈↓ ↓H␈↓80)␈↓ α8␈α$␈↓↓true␈α$Exists␈α$The(PP,Child(Mike,PP))␈α$≡␈α$true(Exist(PP,Child(Mike,PP)␈α$And
␈↓ ↓H␈↓↓All(PP1,Child(Mike,PP1) Implies Equal(PP,PP1))))␈↓,
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


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

␈↓ ↓H␈↓81)␈↓ α8␈α?␈αα␈↓↓Exists␈α?␈ααThe(PP,Child(Mike,PP))␈α?␈αα=␈α?␈ααExist(PP,Child(Mike,PP)␈α?␈ααAnd
␈↓ ↓H␈↓↓All(PP1,Child(Mike,PP1) Implies Equal(PP,PP1)))␈↓.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓the same.

␈↓ ↓H␈↓␈↓ α_In␈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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓the␈α
modal␈α∞operators,␈α
knowledge,␈α∞belief␈α
and␈α∞tense.␈α
 However,␈α∞for␈α
axiomatizing␈α∞quantification,␈α
we
␈↓ ↓H␈↓also need a function α such that

␈↓ ↓H␈↓82)␈↓ α8  π' = α(␈↓↓V,x␈↓,π)

␈↓ ↓H␈↓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
␈↓ ↓H␈↓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␈↓
␈↓ ↓H␈↓␈↓↓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
␈↓ ↓H␈↓of␈αmodal␈αlogic.␈α Later␈αwe␈αwill␈αuse␈αthis␈αCartesian␈αproduct␈αstructure␈αon␈αthe␈αspace␈αof␈αpossible␈αworlds
␈↓ ↓H␈↓to discuss counterfactual conditionals.

␈↓ ↓H␈↓␈↓ α_Let␈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
␈↓ ↓H␈↓possible world π.  Then

␈↓ ↓H␈↓83)␈↓ α8  ␈↓↓∀P.(true P ≡ true(P,␈↓π0)).

␈↓ ↓H␈↓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
␈↓ ↓H␈↓when that is defined.

␈↓ ↓H␈↓␈↓ α_The truth condition for ␈↓↓All(V,P)␈↓ is then given by

␈↓ ↓H␈↓84)␈↓ α8  ␈↓↓∀π V P.(true(All(V,P),π) ≡ ∀x.true(P,α(V,x,π))␈↓.

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

␈↓ ↓H␈↓85)␈↓ α8  ␈↓↓∀π V P.(true(Exist(V,P),π) ≡ ∃x.true(P,α(V,x,π))␈↓.

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

␈↓ ↓H␈↓86)␈↓ α8  ␈↓↓∀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))␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓87)␈↓ α8  ␈↓↓∀π V P.(¬∃!x.true(P,α(V,x,π)) ⊃ ¬true Exists The(V,P))␈↓.

␈↓ ↓H␈↓␈↓ α_We␈α∨also␈α have␈α∨the␈α following␈α∨"syntactic"␈α rules␈α∨governing␈α propositions␈α∨involving
␈↓ ↓H␈↓quantification:
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓88)␈↓ α8␈↓↓∀π Q1 Q2 V.(absent(V,Q1) ∧ true(All(V,Q1 Implies Q2),π) ⊃ true(Q1 Implies All(V,Q2),π))␈↓

␈↓ ↓H␈↓and

␈↓ ↓H␈↓89)␈↓ α8 ␈↓↓∀π V Q X.(true(All(V,Q),π) ⊃ true(Subst(X,V,Q),π))␈↓.

␈↓ ↓H␈↓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)␈↓
␈↓ ↓H␈↓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.␈↓
␈↓ ↓H␈↓␈↓↓absent␈↓ and ␈↓↓Subst␈↓ are characterized by the following axioms:

␈↓ ↓H␈↓90)␈↓ α8 ␈↓↓∀V1 V2.(absent(V1,V2) ≡ V1 ≠ V2)␈↓,

␈↓ ↓H␈↓91)␈↓ α8␈↓↓∀V P X.(absent(V,Know(P,X)) ≡ absent(V,P) ∧ absent(V,X))␈↓,

␈↓ ↓H␈↓axioms similar to (91) for other conceptual functions,

␈↓ ↓H␈↓92)␈↓ α8 ␈↓↓∀V Q.absent(V,All(V,Q))␈↓,

␈↓ ↓H␈↓93)␈↓ α8 ␈↓↓∀V Q.absent(V,Exist(V,Q))␈↓,

␈↓ ↓H␈↓94)␈↓ α8␈↓↓∀V Q.absent(V,The(V,Q))␈↓,

␈↓ ↓H␈↓95)␈↓ α8 ␈↓↓∀V X.Subst(V,V,X) = X␈↓,

␈↓ ↓H␈↓96)␈↓ α8 ␈↓↓∀X V.Subst(X,V,V) = X␈↓,

␈↓ ↓H␈↓97)␈↓ α8 ␈↓↓∀X V P Y.(Subst(X,V,Know(P,Y)) = Know(Subst(X,V,P),Subst(X,V,Y)))␈↓,

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

␈↓ ↓H␈↓98)␈↓ α8 ␈↓↓∀X V Q.(absent(V,Y) ⊃ Subst(X,V,Y) = Y)␈↓,

␈↓ ↓H␈↓99)␈↓ α8␈α$␈↓↓∀X␈α%V1␈α$V2␈α%Q.(V1␈α$≠␈α%V2␈α$∧␈α%absent(V2,X)␈α$⊃␈α%Subst(X,V1,All(V2,Q))␈α$ =
␈↓ ↓H␈↓↓All(V2,Subst(X,V1,Q)))␈↓,

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

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

␈↓ ↓H␈↓100)␈↓ α8 ␈↓↓∀V1 V2 Q.(All(V1,Q) = All(V2,Subst(V2,V1,Q)))␈↓.

␈↓ ↓H␈↓␈↓ α_The␈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
␈↓ ↓H␈↓and␈αdon't␈αappear␈αin␈α
the␈αconcepts␈αthemselves.␈α It␈α
seems␈αlikely␈αthat␈αthis␈α
is␈αharmless␈αuntil␈αwe␈αwant␈α
to
␈↓ ↓H␈↓form concepts of the laws of reasoning.

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓may require such concepts or else a formulation that doesn't use possible worlds.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓␈↓ α_Martin␈α∀Davis␈α∪(in␈α∀conversation)␈α∀pointed␈α∪out␈α∀the␈α∀advantages␈α∪of␈α∀an␈α∀alternate␈α∪treatment
␈↓ ↓H␈↓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
␈↓ ↓H␈↓standard concept.  Then we can write

␈↓ ↓H␈↓101)␈↓ α8 ␈↓↓∀V Q.(true All(V,Q) ≡ ∀x.true Subst(Concept1 x,V,Q))␈↓.





␈↓ ↓H␈↓αPOSSIBLE APPLICATIONS TO ARTIFICIAL INTELLIGENCE

␈↓ ↓H␈↓␈↓ α_The␈αforegoing␈αdiscussion␈αof␈αconcepts␈αhas␈αbeen␈αmainly␈αconcerned␈αwith␈αhow␈αto␈αtranslate␈αinto
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓the␈αformalization␈αis␈α
to␈αdevelop␈αan␈α
idea␈αof␈αwhat␈α
concepts␈αreally␈αare,␈α
but␈αthe␈αpossible␈α
formalizations
␈↓ ↓H␈↓have not been explored enough to draw even tentative conclusions about that.

␈↓ ↓H␈↓␈↓ α_For␈αartificial␈αintelligence,␈αthe␈αstudy␈αof␈αconcepts␈αhas␈αyet␈αa␈αdifferent␈αmotivation.␈α Our␈αsuccess
␈↓ ↓H␈↓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
␈↓ ↓H␈↓source␈αof␈αthe␈αlimitation␈αis␈αour␈α
inability␈αto␈αformalize␈αwhat␈αthe␈αworld␈α
is␈αlike␈αin␈αgeneral.␈α We␈αcan␈α
try
␈↓ ↓H␈↓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
␈↓ ↓H␈↓such␈αa␈αdescription␈αand␈αthe␈αfacts␈αof␈αa␈αsituation␈α
to␈αdiscover␈αa␈αstrategy␈αfor␈αachieving␈αa␈αgoal.␈α This␈α
is
␈↓ ↓H␈↓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
␈↓ ↓H␈↓and is discussed in (McCarthy and Hayes 1969).

␈↓ ↓H␈↓␈↓ α_We see the following potential uses for facts about knowledge:

␈↓ ↓H␈↓␈↓ α_1.␈αA␈αcomputer␈αprogram␈αthat␈αwants␈αto␈αtelephone␈αsomeone␈αmust␈αreason␈αabout␈αwho␈αknows␈αthe
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓persons.

␈↓ ↓H␈↓␈↓ α_2.␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓to␈αbe␈αproved␈αunknown␈αis␈αfalse.␈α Presumably␈αproving␈αone's␈αown␈αignorance␈αis␈αa␈αstimulus␈αto␈αlooking
␈↓ ↓H␈↓outside␈αfor␈αthe␈αinformation.␈α In␈αcompetitive␈αsituations,␈αit␈αmay␈αbe␈αimportant␈αto␈αshow␈αthat␈αa␈αcertain
␈↓ ↓H␈↓course of action will leave competitors ignorant.

␈↓ ↓H␈↓␈↓ α_3.␈αPrediction␈α
of␈αthe␈αbehavior␈α
of␈αothers␈αdepends␈α
on␈αdetermining␈αwhat␈α
they␈αbelieve␈αand␈α
what
␈↓ ↓H␈↓they want.

␈↓ ↓H␈↓␈↓ α_It␈αseems␈αto␈αme␈αthat␈αAI␈αapplications␈α
will␈αespecially␈αbenefit␈αfrom␈αfirst␈αorder␈αformalisms␈αof␈α
the
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓presence doesn't affect the formalization of the other parts.

␈↓ ↓H␈↓␈↓ α_In␈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
␈↓ ↓H␈↓↓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
␈↓ ↓H␈↓↓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
␈↓ ↓H␈↓towards␈αAI␈αapplications.␈α The␈αaxiomatization␈αis␈α␈↓↓quasi-static␈↓,␈αi.e.␈αeach␈αaction␈αtakes␈αa␈αsituation␈αinto
␈↓ ↓H␈↓a definite resulting situation and there are no concurrent processes.

␈↓ ↓H␈↓␈↓ α_The␈α special␈αpremisses␈αare␈αwritten␈α␈↓↓true(world,Want(Pat,Know(Joe,␈αTelephone␈αMike)))␈↓␈αand
␈↓ ↓H␈↓␈↓↓true(world,Know(Pat,Telephone␈α0Mike))␈↓,␈α1and␈α0the␈α0conclusion␈α1is␈α0␈↓↓true(world,Future
␈↓ ↓H␈↓↓Know(Joe,Telephone Mike))␈↓.

␈↓ ↓H␈↓␈↓ α_The␈α
proof␈α
from␈α
these␈α
axioms␈α
that␈α
Joe␈αwill␈α
know␈α
Mike's␈α
telephone␈α
number␈α
has␈α
has␈αabout␈α
15
␈↓ ↓H␈↓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
␈↓ ↓H␈↓(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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓because its statement doesn't say what remains unchanged when Pat tells Joe Mike's number.

␈↓ ↓H␈↓␈↓ α_Even␈αthe␈αfifteen␈αstep␈αproof␈αdoesn't␈αmodel␈αhuman␈αreasoning,␈αor␈αthe␈αway␈αcomputer␈αprograms
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓group␈αhas␈αemphasized,␈αthis␈αgeneral␈αfact␈αshould␈αbe␈αstored␈αso␈αas␈αto␈αbe␈αtriggered␈αspecifically␈α
by␈αthe
␈↓ ↓H␈↓desire that another person shall know something.


␈↓ ↓H␈↓α␈↓ α_ABSTRACT LANGUAGES

␈↓ ↓H␈↓␈↓ α_The␈αway␈αwe␈αhave␈αtreated␈αconcepts␈αin␈αthis␈αpaper,␈αespecially␈αwhen␈αwe␈αput␈αvariables␈αin␈αthem,
␈↓ ↓H␈↓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
␈↓ ↓H␈↓provided we use a suitable notion of ␈↓↓abstract␈↓ ␈↓↓language.␈↓

␈↓ ↓H␈↓␈↓ α_Ordinarily␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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-
␈↓ ↓H␈↓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
␈↓ ↓H␈↓sum␈αand␈αfunctions␈αfor␈αforming␈αsums␈αfrom␈αsummands␈αand␈αfunctions␈αfor␈αextracting␈αthe␈αsummands
␈↓ ↓H␈↓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
␈↓ ↓H␈↓languages,␈α
and␈αproving␈α
the␈αproperties␈α
of␈αinterpreters␈α
and␈α
compilers.␈α From␈α
that␈αpoint␈α
of␈αview,␈α
one
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓forming,␈αdiscriminating␈αand␈αtaking␈αapart␈α
␈↓↓abstract␈αexpressions␈↓.␈α However,␈αthe␈αlanguages␈α
considered
␈↓ ↓H␈↓at that time always admitted representations as strings of symbols.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓␈↓ α_If␈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
␈↓ ↓H␈↓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-
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓regarding concepts as strings of symbols becomes awkward even if possible.

␈↓ ↓H␈↓␈↓ α_It␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓case it would seem that concepts can be identified with expressions in an abstract language.





␈↓ ↓H␈↓α␈↓ α_BIBLIOGRAPHY

␈↓ ↓H␈↓␈↓ α_The␈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
␈↓ ↓H␈↓1976).␈α
 Church␈αintroduces␈α
what␈αmight␈α
be␈α
called␈αa␈α
two-dimensional␈αtype␈α
structure.␈α One␈α
dimension
␈↓ ↓H␈↓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
␈↓ ↓H␈↓dimension␈αis␈α
permits␈αconcepts␈α
of␈αconcepts,␈α
etc.␈α No␈α
examples␈αor␈α
applications␈αare␈α
given.␈α It␈αseems␈α
to
␈↓ ↓H␈↓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
␈↓ ↓H␈↓order logic.

␈↓ ↓H␈↓␈↓ α_Morgan'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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓without␈α⊗quantifiers␈α⊗-␈α⊗uses␈α∃operations␈α⊗like␈α⊗our␈α⊗␈↓↓And␈↓␈α∃to␈α⊗form␈α⊗compound␈α⊗propositions␈α∃from
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓expressed.

␈↓ ↓H␈↓␈↓ α_Although␈αthe␈α
formalism␈αleads␈α
in␈αalmost␈α
the␈αopposite␈α
direction,␈αthe␈α
present␈αpaper␈α
is␈αmuch␈α
in
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓intelligence applications.

␈↓ ↓H␈↓␈↓ α_We␈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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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
␈↓ ↓H␈↓paradoxes of naive set theory.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


␈↓ ↓H␈↓α␈↓ α_References:

␈↓ ↓H␈↓␈↓ α_␈↓αCarnap, Rudolf␈↓ (1956), ␈↓↓Meaning and Necessity␈↓, University of Chicago Press.

␈↓ ↓H␈↓␈↓ α_␈↓αChurch,␈α↔Alonzo␈↓␈α_(1951a),␈α↔The␈α_Need␈α↔for␈α_Abstract␈α↔Entities␈α_in␈α↔Semantic␈α_Analysis,␈α↔in
␈↓ ↓H␈↓␈↓↓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
␈↓ ↓H␈↓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␈↓,
␈↓ ↓H␈↓edited by Jerry A. Fodor and Jerrold Katz, Prentice-Hall 1964

␈↓ ↓H␈↓␈↓ α_            ␈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.),
␈↓ ↓H␈↓␈↓↓Essays in honor of Henry Sheffer␈↓, pp. 3-24.  New York.

␈↓ ↓H␈↓␈↓ α_␈↓αFrege,␈α_Gottlob␈↓␈α→(1892),␈α_Uber␈α→Sinn␈α_und␈α_Bedeutung.␈α→␈↓↓Zeitschrift␈α_fur␈α→Philosophie␈α_und
␈↓ ↓H␈↓↓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
␈↓ ↓H␈↓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.
␈↓ ↓H␈↓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,
␈↓ ↓H␈↓␈↓↓Translations from the Philosophical Writings of Gottlob Frege␈↓, Oxford, 1952.

␈↓ ↓H␈↓␈↓ α_␈↓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
␈↓ ↓H␈↓↓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
␈↓ ↓H␈↓Co.), pp. 178-214.  Reprinted in (Linsky 1971).

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

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

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

␈↓ ↓H␈↓␈↓ α_␈↓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
␈↓ ↓H␈↓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.).
␈↓ ↓H␈↓Edinburgh: Edinburgh University Press.

␈↓ ↓H␈↓␈↓ α_␈↓αMontague,␈α↔Richard␈↓␈α_(1963),␈α↔Syntactical␈α↔Treatments␈α_of␈α↔Modality,␈α↔with␈α_Corollaries␈α↔on
␈↓ ↓H␈↓Reflexion␈α≠Principles␈α≤and␈α≠Finite␈α≠Axiomatizability,␈α≤␈↓↓Acta␈α≠Philosophica␈α≤Fennica␈↓␈α≠␈↓α16␈↓:153-167.
␈↓ ↓H␈↓Reprinted in (Montague 1974).

␈↓ ↓H␈↓␈↓ α_␈↓αMontague, Richard␈↓ (1974), ␈↓↓Formal Philosophy␈↓, Yale University Press

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

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

␈↓ ↓H␈↓␈↓ α_␈↓αQuine, W.V.O.␈↓ (1961), ␈↓↓From a Logical Point of View␈↓, Harper and Row.
␈↓ ↓H␈↓␈↓ ε(draft␈↓ H


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