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␈↓ 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∧␈↓␈↓εThis draft of CONCEP[S76,JMC] PUBbed at 12:39 on July 29, 1976.␈↓



















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


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

␈↓ α∧␈↓␈↓ αTFrege␈αfurther␈αproposed␈αthat␈α
a␈αphrase␈αhas␈αa␈α
␈↓↓sense␈↓␈αwhich␈αis␈αa␈α
␈↓↓concept␈↓␈αand␈αis␈αits␈α
␈↓↓meaning␈↓␈αin
␈↓ α∧␈↓␈↓↓oblique␈↓␈α∩␈↓↓contexts␈↓␈α∪like␈α∩knowing␈α∪and␈α∩wanting,␈α∪and␈α∩a␈α∩␈↓↓denotation␈↓␈α∪which␈α∩is␈α∪its␈α∩␈↓↓meaning␈↓␈α∪in␈α∩␈↓↓direct␈↓
␈↓ α∧␈↓␈↓↓contexts.␈↓␈α␈↓↓Denotations␈↓␈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
but␈α∞none␈α
have
␈↓ α∧␈↓been very satisfactory.

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

␈↓ α∧␈↓␈↓ αTThe␈αpresent␈α
idea␈αis␈α
to␈αleave␈αthe␈α
logic␈αunchanged␈α
and␈αto␈αtreat␈α
concepts␈αas␈α
one␈αkind␈αof␈α
object
␈↓ α∧␈↓in␈αan␈αordinary␈αfirst␈αorder␈αtheory.␈α We␈αshall␈αhave␈αone␈αterm␈αthat␈αdenotes␈αMike's␈αtelephone␈αnumber
␈↓ α∧␈↓and␈αa␈αdifferent␈αterm␈αdenoting␈αthe␈αconcept␈αof␈αMike's␈αtelephone␈αnumber␈αinstead␈αof␈αhaving␈αa␈αsingle
␈↓ 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␈αof
␈↓ α∧␈↓axioms.

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

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






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


␈↓ α∧␈↓αKNOWING WHAT AND KNOWING THAT

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

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

␈↓ α∧␈↓with the following conventions:

␈↓ α∧␈↓␈↓ αT1.␈α∞Parentheses␈α∞are␈α∞often␈α∞omitted␈α∞for␈α∞one␈α∞argument␈α∞functions␈α∞and␈α∞predicates.␈α∞ This␈α∞purely
␈↓ α∧␈↓syntactic␈α⊂convention␈α∂is␈α⊂not␈α∂important.␈α⊂ Another␈α∂convention␈α⊂is␈α∂to␈α⊂capitalize␈α∂the␈α⊂first␈α∂letter␈α⊂of␈α∂a
␈↓ α∧␈↓constant,␈αvariable␈αor␈αfunction␈αname␈αwhen␈αits␈αvalue␈αis␈αa␈αconcept.␈α (We␈αconsidered␈αalso␈αcapitalizing
␈↓ α∧␈↓the last letter when the arguments are concepts, but it made the formulas ugly).

␈↓ α∧␈↓␈↓ αT2.␈α␈↓↓Mike␈↓␈αis␈αthe␈αconcept␈αof␈αMike;␈αi.e.␈αit␈αis␈αthe␈α␈↓↓sense␈↓␈αof␈αthe␈αexpression␈α␈↓↓"Mike"␈↓.␈α ␈↓↓mike␈↓␈αstands␈αfor
␈↓ α∧␈↓Mike 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≡In␈α∨(1),␈α≡therefore,
␈↓ α∧␈↓␈↓↓Know(Pat,Telephone␈αMike)␈↓␈αis␈αa␈αproposition␈αand␈αnot␈αa␈αtruth␈αvalue.␈α Note␈αthat␈αwe␈αare␈αformalizing
␈↓ α∧␈↓␈↓↓knowing␈↓␈α␈↓↓what␈↓␈α
rather␈αthan␈α␈↓↓knowing␈↓␈α
␈↓↓that␈↓␈αor␈α
␈↓↓knowing␈↓␈α␈↓↓how.␈↓␈αFor␈α
AI␈αand␈α
for␈αother␈αpractical␈α
purposes,
␈↓ α∧␈↓␈↓↓knowing␈↓␈α
␈↓↓what␈↓␈α
seems␈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␈↓␈αis␈α
written
␈↓ α∧␈↓␈↓↓knowing whether␈↓ when the "knowand" is a proposition.

␈↓ α∧␈↓␈↓ αT5.␈α∞␈↓↓true(Q)␈↓␈α∞is␈α∞the␈α∞truth␈α∞value,␈α∞␈↓↓t␈↓␈α∞or␈α∂␈↓↓f,␈↓␈α∞of␈α∞the␈α∞proposition␈α∞␈↓↓Q,␈↓␈α∞and␈α∞we␈α∞must␈α∞write␈α∂␈↓↓true(Q)␈↓␈α∞in
␈↓ α∧␈↓order␈α∞to␈α∞assert␈α∞␈↓↓Q.␈↓␈α∞Later␈α∞we␈α∞will␈α∞consider␈α∞formalisms␈α∞in␈α∞which␈α∞␈↓↓true␈↓␈α∞has␈α∞a␈α∞second␈α∞argument␈α∂-␈α∞a
␈↓ α∧␈↓␈↓↓situation,␈↓␈α∩a␈α⊃␈↓↓story,␈↓␈α∩a␈α∩␈↓↓possible␈↓␈α⊃␈↓↓world,␈↓␈α∩or␈α∩even␈α⊃a␈α∩␈↓↓partial␈α⊃possible␈α∩world␈↓␈α∩(a␈α⊃notion␈α∩we␈α∩hope␈α⊃to
␈↓ α∧␈↓introduce).

␈↓ α∧␈↓␈↓ αT6.␈α
The␈α
formulas␈α
are␈α
in␈α
a␈α
sorted␈α
first␈α
order␈α
logic␈α
with␈α
functions␈α
and␈α
equality.␈α
 Knowledge,
␈↓ α∧␈↓necessity,␈αetc.␈αwill␈αbe␈αdiscussed␈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 try to be typographically consistent.

␈↓ α∧␈↓␈↓ αTThe␈α⊃reader␈α⊂may␈α⊃be␈α⊃nervous␈α⊂about␈α⊃what␈α⊂is␈α⊃meant␈α⊃by␈α⊂␈↓↓concept.␈↓␈α⊃He␈α⊂will␈α⊃have␈α⊃to␈α⊂remain
␈↓ α∧␈↓nervous;␈α
no␈α∞final␈α
commitment␈α∞will␈α
be␈α∞made␈α
in␈α∞this␈α
paper.␈α∞ The␈α
formalism␈α∞is␈α
compatible␈α∞with␈α
a
␈↓ α∧␈↓variety␈α
of␈αpossibilities,␈α
and␈αthese␈α
can␈αbe␈α
compared␈αusing␈α
the␈αmodels␈α
of␈αtheir␈α
first␈α
order␈αtheories.
␈↓ α∧␈↓However,␈αif␈α(1)␈αis␈αto␈αbe␈αreasonable,␈αit␈αmust␈αnot␈αfollow␈αfrom␈α(1)␈αand␈αthe␈αfact␈αthat␈αMary's␈αtelephone
␈↓ α∧␈↓number is the same as Mike's, that Pat knows Mary's telephone number.

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

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


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


␈↓ 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␈αproposition␈αthat␈α␈↓↓P␈↓␈αknows␈α␈↓↓that␈↓␈α␈↓↓Q.␈↓␈αWhile␈αEnglish␈αis␈αlogically␈αnot␈αuniform␈αin
␈↓ α∧␈↓that␈αknowing␈αan␈αindividual␈αconcept␈αmeans␈αknowing␈αits␈αvalue␈αwhile␈αknowing␈αa␈αproposition␈αmeans
␈↓ α∧␈↓knowing␈α∂that␈α∞it␈α∂has␈α∞a␈α∂particular␈α∞value,␈α∂namely␈α∞␈↓↓t,␈↓␈α∂there␈α∞is␈α∂no␈α∞reason␈α∂to␈α∞make␈α∂robots␈α∂with␈α∞this
␈↓ α∧␈↓infirmity.

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

␈↓ α∧␈↓5)␈↓ αt ␈↓↓x = denot X␈↓.

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

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

␈↓ α∧␈↓We␈α
call␈α
␈↓↓denot␈α
X␈↓␈α
the␈α
␈↓↓denotation␈↓␈α
of␈α
the␈α
concept␈α
␈↓↓X,␈↓␈α
and␈α
(6)␈α
asserts␈α
that␈α
the␈α
denotation␈α
of␈α
the␈α
concept
␈↓ α∧␈↓of␈α␈↓↓P␈↓'s␈αtelephone␈α
number␈αdepends␈αonly␈αon␈α
the␈αdenotation␈αof␈αthe␈α
concept␈α␈↓↓P␈↓.␈α The␈αvariables␈α
in␈α(6)
␈↓ α∧␈↓range␈αover␈αconcepts␈αof␈αpersons,␈αand␈αwe␈α
regard␈α(6)␈αas␈αasserting␈αthat␈α␈↓↓Telephone␈↓␈αis␈α
␈↓↓extensional␈↓␈αwith
␈↓ α∧␈↓respect␈αto␈α␈↓↓denot.␈↓␈αNote␈αthat␈αour␈α␈↓↓denot␈↓␈αoperates␈αon␈αconcepts␈αrather␈αthan␈αon␈αexpressions;␈αa␈αtheory␈αof
␈↓ α∧␈↓expressions␈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 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∧␈↓␈↓ ε|4␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


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


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

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


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

␈↓ α∧␈↓We can then have simultaneously

␈↓ 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∂express␈α∂a␈α⊂proposition␈α∂that␈α∂can␈α∂only␈α⊂be␈α∂stated␈α∂awkwardly␈α⊂in␈α∂English,
␈↓ α∧␈↓perhaps␈α∀as␈α∃␈↓↓"Kepler␈α∀knew␈α∀that␈α∃a␈α∀certain␈α∃number␈α∀is␈α∀composite,␈α∃where␈α∀this␈α∃number␈α∀(perhaps
␈↓ α∧␈↓↓unbeknownst to Kepler) is the number of planets"␈↓.

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

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


␈↓ 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∧␈↓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
␈↓ α∧␈↓apply Occam's razor.


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


␈↓ α∧␈↓αMODAL LOGIC (part 1)

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

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

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

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


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


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


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


␈↓ 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∧␈↓αQUANTIFICATION

␈↓ 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∧␈↓↓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∧␈↓74)␈↓ αt ␈↓↓Child(Mike,PP) Implies Equal(Telephone PP,Telephone Mike)␈↓


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


␈↓ 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∧␈↓75)␈↓ 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␈α
are␈α
formed␈α
similarly␈αto␈α
universal␈α
one's,␈α
but␈αthe␈α
function␈α
␈↓↓Exist␈↓␈α
introduced␈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∧␈↓76)␈↓ 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∧␈↓77)␈↓ 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∧␈↓78)␈↓ 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∧␈↓79)␈↓ αt π' = α(␈↓↓V,x␈↓,π)

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

␈↓ 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∧␈↓80)␈↓ αt ␈↓↓∀P.(true P ≡ true(P,␈↓π0)).



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


␈↓ 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∧␈↓81)␈↓ 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∧␈↓82)␈↓ 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∧␈↓83)␈↓ 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∧␈↓84)␈↓ 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∧␈↓85)␈↓ 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∧␈↓86)␈↓ 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∧␈↓↓87)␈↓ αt ∀V1 V2.(absent(V1,V2) ≡ V1 ≠ V2)␈↓,

␈↓ α∧␈↓88)␈↓ αt␈↓↓∀V P X.(absent(V,Know(P,X)) ≡ absent(V,P) ∧ absent(V,X))␈↓,

␈↓ α∧␈↓axioms similar to (88) for other conceptual functions,

␈↓ α∧␈↓89)␈↓ αt ␈↓↓∀V Q.absent(V,All(V,Q))␈↓,

␈↓ α∧␈↓90)␈↓ αt ␈↓↓∀V Q.absent(V,Exist(V,Q))␈↓,

␈↓ α∧␈↓91)␈↓ αt␈↓↓∀V Q.absent(V,The(V,Q))␈↓,

␈↓ α∧␈↓92)␈↓ αt ␈↓↓∀V X.Subst(V,V,X) = X␈↓,

␈↓ α∧␈↓93)␈↓ αt ␈↓↓∀X V.Subst(X,V,V) = X␈↓,

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


␈↓ α∧␈↓94)␈↓ αt ␈↓↓∀X V P Y.(Subst(X,V,Know(P,Y)) = Know(Subst(X,V,P),Subst(X,V,Y)))␈↓,

␈↓ α∧␈↓axioms similar to (94) for other functions,

␈↓ α∧␈↓95)␈↓ αt ␈↓↓∀X V Q.(absent(V,Y) ⊃ Subst(X,V,Y) = Y)␈↓,

␈↓ α∧␈↓96)␈↓ 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 (96) for ␈↓↓Exist␈↓ and ␈↓↓The.␈↓

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

␈↓ α∧␈↓97)␈↓ 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∧␈↓αEXAMPLES IN 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∧␈↓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∧␈↓␈↓ αTThe␈α
epistemological␈α
part␈α
of␈α∞the␈α
problem␈α
is␈α
very␈α
difficult␈α∞in␈α
general␈α
so␈α
it␈α
is␈α∞worthwhile␈α
to
␈↓ α∧␈↓invent␈α∩formalisms␈α∪that␈α∩are␈α∪valid␈α∩for␈α∪limited␈α∩classes␈α∪of␈α∩problems.␈α∪ Therefore␈α∩let␈α∪us␈α∩consider
␈↓ α∧␈↓formalizing␈α∞the␈α∞following␈α∞fact␈α∞about␈α∞purposeful␈α
beings:␈α∞␈↓αPurposeful␈α∞beings␈α∞do␈α∞what␈α∞they␈α
think
␈↓ α∧␈↓αwill achieve their goals.␈↓

␈↓ α∧␈↓␈↓ αTWe␈αattempt␈αthis␈αformalization␈αusing␈αour␈α
formalizations␈αof␈αconcepts␈αand␈αwithin␈αwhat␈αwe␈α
call
␈↓ α∧␈↓the␈α∞␈↓↓quasi-static␈↓␈α∂model␈α∞of␈α∂action.␈α∞ The␈α∂quasi-static␈α∞model␈α∂assumes␈α∞that␈α∂actions␈α∞are␈α∂discrete␈α∞and


␈↓ α∧␈↓␈↓ εu16␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓happen␈αone␈αat␈αa␈αtime,␈αEach␈αaction␈αchanges␈αthe␈αsituation␈αinto␈αa␈αnew␈αsituation,␈αand␈αthe␈αnext␈αaction
␈↓ α∧␈↓by␈αthe␈αsame␈αor␈α
a␈αdifferent␈αactor␈αtakes␈α
place␈αin␈αthis␈αnew␈αsituation.␈α
 Most␈αof␈αthe␈αwork␈α
in␈αartificial
␈↓ α∧␈↓intelligence␈α∪so␈α∪far␈α∩has␈α∪been␈α∪within␈α∩the␈α∪quasi-static␈α∪model␈α∩-␈α∪usually␈α∪without␈α∪recognizing␈α∩its
␈↓ α∧␈↓limitations.

␈↓ α∧␈↓␈↓ αTThe␈αfirst␈αproblem␈α
we␈αshall␈αdiscuss␈αin␈α
this␈αframework␈αis␈αextremely␈α
simple.␈α We␈αwould␈αlike␈α
to
␈↓ α∧␈↓deduce␈α
from␈α
the␈α
facts␈α
that␈α
Pat␈α
knows␈α
Mike's␈α
telephone␈α
number␈α
and␈α
that␈α
he␈α
wants␈α
Joe␈α∞to␈α
know
␈↓ α∧␈↓Mike's␈α∂telephone␈α∞number␈α∂the␈α∂conclusion␈α∞that␈α∂Joe␈α∞will␈α∂come␈α∂to␈α∞know␈α∂Mike's␈α∂telephone␈α∞number,
␈↓ α∧␈↓(because Pat will tell it to him).  All sorts of simplifications will be made.  ****

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

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

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

␈↓ α∧␈↓␈↓ αT3.␈α∞Modal␈α∞logic␈α∞offers␈α∞difficulties␈α∞especially␈α∞as␈α∞we␈α∞need␈α∞often␈α∞need␈α∞multiple␈α∞modalities␈α
like
␈↓ α∧␈↓␈↓↓"believes␈α⊃he␈α∩wants␈α⊃to␈α⊃know"␈↓␈α∩in␈α⊃a␈α⊃single␈α∩sentence,␈α⊃and␈α⊃this␈α∩makes␈α⊃the␈α⊃Kripke␈α∩possible␈α⊃worlds
␈↓ α∧␈↓semantics␈α∂for␈α∂modal␈α∂logic␈α∞rather␈α∂cumbersome.␈α∂ Modal␈α∂logic␈α∞is␈α∂especially␈α∂troublesome␈α∂if␈α∞oblique
␈↓ α∧␈↓contexts␈αare␈αonly␈αa␈α
small␈αpart␈αof␈αthe␈α
problem.␈α Moreover,␈αit␈αis␈α
preferable␈αto␈αbe␈αable␈α
to␈αintroduce
␈↓ α∧␈↓new modalities by introducing new predicates instead of having to change the logic.

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

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

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

␈↓ α∧␈↓*****

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

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

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


␈↓ α∧␈↓␈↓ εu17␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓and

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

␈↓ α∧␈↓we will be able to deduce

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

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





␈↓ α∧␈↓αABSTRACT 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
(1962)␈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∧␈↓␈↓ εu18␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓αPHILOSOPHICAL REMARKS

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

␈↓ α∧␈↓␈↓ αT1.␈αSince␈α
we␈αcan't␈α
immediately␈αmake␈α
programs␈αcapable␈α
of␈αunderstanding␈α
the␈αwhole␈αworld,␈α
we
␈↓ α∧␈↓are interested in formalizations that allow programs to act intelligently in a limited domains.

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

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

␈↓ α∧␈↓␈↓ αTI␈α∞must␈α∞confess,␈α
however,␈α∞to␈α∞finding␈α
this␈α∞attitude␈α∞philosophically␈α
attractive,␈α∞i.e.␈α∞first␈α∞find␈α
a
␈↓ α∧␈↓formal␈αsystem␈αthat␈αallows␈αexpressing␈αcommon␈αsense␈αreasoning␈α-␈αnaively␈αif␈αnecessary,␈αand␈αthen␈αtry
␈↓ α∧␈↓to make it secure.



␈↓ α∧␈↓αNOTES






















␈↓ α∧␈↓␈↓ εu19␈↓ ∧
␈↓ α∧␈↓␈↓ εddraft␈↓ ∧


␈↓ α∧␈↓αREFERENCES

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

␈↓ α∧␈↓␈↓αChurch,␈αAlonzo␈↓␈α(1951a),␈αThe␈αNeed␈αfor␈αAbstract␈αEntities␈αin␈αSemantic␈αAnalysis,␈αin␈α␈↓↓Contributions␈αto
␈↓ α∧␈↓↓the␈α⊂Analysis␈α⊂and␈α∂Synthesis␈α⊂of␈α⊂Knowledge␈↓,␈α∂Proceedings␈α⊂of␈α⊂the␈α∂American␈α⊂Academy␈α⊂of␈α⊂Arts␈α∂and
␈↓ α∧␈↓Sciences,␈α␈↓α80␈↓,␈αNo.␈α
1␈α(July␈α1951),␈α100-112.␈α
 Reprinted␈αin␈α␈↓↓The␈αStructure␈α
of␈αLanguage␈↓,␈αedited␈αby␈α
Jerry
␈↓ α∧␈↓A. Fodor and Jerrold Katz, Prentice-Hall 1964

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

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

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

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

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

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


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











␈↓ α∧␈↓␈↓ εu20␈↓ ∧