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␈↓ α∧␈↓α␈↓ ¬∪CONCEPT-VALUED FUNCTIONS

␈↓ α∧␈↓α␈↓ εεby John McCarthy


␈↓ α∧␈↓αIntroduction

␈↓ α∧␈↓␈↓ αTThe␈αintroduction␈α
of␈αconcepts␈α
as␈αvalues␈α
of␈αfunctions␈α
allows␈αus␈α
to␈αformalize␈αknowledge,␈α
belief,
␈↓ α∧␈↓wanting, necessity and possibily and other modalities in a way that has the following advantages:

␈↓ α∧␈↓␈↓ αT1. It uses only first order logic without modal operators or quotation.

␈↓ α∧␈↓␈↓ αT2. It is compatible with most of the approaches to the controversial issues.

␈↓ α∧␈↓␈↓ αT3. The formulas are shorter than in most approaches.

␈↓ α∧␈↓␈↓ αTSuppose we wish to assert that Pat knows Mike's telephone number.  We write

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

␈↓ α∧␈↓with the following conventions:

␈↓ α∧␈↓␈↓ αT1.␈α∂Parentheses␈α∂are␈α∂omitted␈α⊂for␈α∂one␈α∂argument␈α∂functions␈α⊂and␈α∂predicates.␈α∂ This␈α∂is␈α⊂a␈α∂purely
␈↓ α∧␈↓syntactic convention and is not important.

␈↓ α∧␈↓␈↓ αT2.␈α␈↓↓Mike␈↓␈αdenotes␈αthe␈αconcept␈αof␈αMike;␈αi.e.␈αit␈αis␈αthe␈α"sense"␈αof␈αthe␈αexpression␈α"Mike".␈α We␈αwill
␈↓ α∧␈↓use ␈↓↓mike␈↓ when we wish to denote Mike himself.

␈↓ α∧␈↓␈↓ αT3.␈α∩␈↓↓Telephone␈↓␈α∩is␈α⊃a␈α∩function␈α∩that␈α⊃takes␈α∩the␈α∩concept␈α⊃of␈α∩a␈α∩person␈α⊃into␈α∩the␈α∩concept␈α∩of␈α⊃his
␈↓ α∧␈↓telephone number.

␈↓ α∧␈↓␈↓ αT4.␈α␈↓↓Know␈↓␈αtakes␈αa␈αperson␈αconcept␈αand␈αanother␈αconcept␈αand␈αproduces␈αthe␈αan␈αassertion␈αconcept
␈↓ α∧␈↓to␈αthe␈αeffect␈αthat␈αthe␈αperson␈αknows␈αthe␈αvalue␈αof␈αthe␈αconcept.␈α Our␈αnotion␈αof␈αknowledge␈αtherefore
␈↓ α∧␈↓corresponds␈αto␈αknowing␈αwhat␈αrather␈α
than␈αknowing␈αthat␈αor␈αknowing␈α
how.␈α For␈αAI,␈αit␈αseems␈α
to␈αbe
␈↓ α∧␈↓the most useful notion of the three.

␈↓ α∧␈↓␈↓ αT␈↓↓true␈↓␈αasserts␈αthat␈αan␈αassertion␈αconcept␈αis␈αtrue.␈α Its␈αargument␈αis␈αa␈αconcept␈αand␈αits␈αvalue␈αis␈α
␈↓αtrue␈↓
␈↓ α∧␈↓or␈α⊂␈↓αfalse␈↓.␈α⊂In␈α⊂some␈α⊂of␈α⊂the␈α⊂formalisms␈α⊂we␈α∂shall␈α⊂consider,␈α⊂␈↓↓true␈↓␈α⊂will␈α⊂have␈α⊂a␈α⊂second␈α⊂argument␈α⊂-␈α∂a
␈↓ α∧␈↓␈↓↓possible␈↓ ␈↓↓world␈↓ or even a ␈↓↓partial possible world␈↓ (a notion we hope to introduce).

␈↓ α∧␈↓␈↓ αTThe␈α
reader␈α
is␈α
doubtless␈α
nervous␈α
about␈α
what␈α
we␈α
mean␈α
by␈α
␈↓↓concept.␈↓␈α
He␈α
will␈α
have␈α∞to␈α
remain
␈↓ α∧␈↓nervous;␈α
no␈αdefinite␈α
commitment␈α
will␈αbe␈α
made␈α
in␈αthis␈α
paper,␈α
and␈αthe␈α
formalism␈α
will␈αbe␈α
compatible
␈↓ α∧␈↓with␈αa␈αvariety␈αof␈αpossibilities.␈α However,␈αif␈α1)␈αis␈αto␈αbe␈αreasonable,␈αit␈αmust␈αnot␈αfollow␈αfrom␈α1)␈αand
␈↓ α∧␈↓the␈α∞fact␈α∞that␈α∞Mary's␈α∞telephone␈α∞number␈α∞is␈α
the␈α∞same␈α∞as␈α∞Mike's,␈α∞that␈α∞Pat␈α∞knows␈α∞Mary's␈α
telephone
␈↓ α∧␈↓number.

␈↓ α∧␈↓␈↓ αTThe␈α
approach␈αtaken␈α
here␈αis␈α
perhaps␈α
in␈αthe␈α
spirit␈αof␈α
Frege␈α
and␈αof␈α
Church␈α(1951).␈α
 However,


␈↓ α∧␈↓␈↓ ε|1␈↓ ∧



␈↓ α∧␈↓instead␈α∪of␈α∪treating␈α∩the␈α∪sense␈α∪and␈α∩denotation␈α∪of␈α∪an␈α∩expression,␈α∪we␈α∪introduce␈α∩concept-valued
␈↓ α∧␈↓expressions␈α
and␈α
concept-valued␈α
variables.␈α
 These␈α
allow␈α
us␈α
to␈α
stay␈α
within␈α
first␈α
order␈α
logic␈α∞at␈α
the
␈↓ α∧␈↓price␈αof␈αreplacing␈αmost␈αnames␈αby␈αpairs␈αof␈αnames,␈αand␈αeven␈αthis␈αprice␈αcan␈αbe␈αavoided␈αat␈αthe␈αprice
␈↓ α∧␈↓(which we don't recommend paying) of a context dependent notation.

␈↓ α∧␈↓␈↓ αTWe now introduce the function ␈↓↓denot␈↓ and the assertion

␈↓ α∧␈↓2) ␈↓↓∀P1 P2.(denot P1 = denot P2 ⊃ denot Telephone P1 = denot Telephone P2)␈↓.

␈↓ α∧␈↓Here␈α␈↓↓denot␈αX␈↓␈αis␈αthe␈α␈↓↓denotation␈↓␈αof␈αthe␈αconcept␈α␈↓↓X,␈↓␈αand␈α2)␈αasserts␈αthat␈αthe␈αdenotation␈αof␈αthe␈αconcept
␈↓ α∧␈↓of␈α␈↓↓X␈↓'s␈αtelephone␈αnumber␈αdepends␈αonly␈αon␈αthe␈α
denotation␈αof␈αconcept␈α␈↓↓X␈↓.␈α We␈αregard␈α2)␈αas␈α
asserting
␈↓ α∧␈↓that␈α∞␈↓↓Telephone␈↓␈α∂is␈α∞␈↓↓extensional␈↓␈α∂with␈α∞respect␈α∂to␈α∞␈↓↓denot.␈↓␈α∂The␈α∞variables␈α∂in␈α∞2)␈α∂range␈α∞over␈α∂concepts␈α∞of
␈↓ α∧␈↓persons.␈α∞ We␈α∞will␈α∞use␈α∞capitals␈α∂for␈α∞concept␈α∞valued␈α∞variables␈α∞and␈α∞begin␈α∂concept-valued␈α∞functions
␈↓ α∧␈↓with capital letters.

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

␈↓ α∧␈↓3) ␈↓↓∀p1 p2 e.(denot p1 = denot p2 ⊃ denot Know(p1,e) = denot Know(p2,e))␈↓,

␈↓ α∧␈↓but␈α
it␈αis␈α
not␈αextensional␈α
with␈αrespect␈α
to␈α
its␈αsecond␈α
argument.␈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␈α∀notion␈α∀of␈α∀extensionality
␈↓ α∧␈↓presented␈α∀here␈α∀is␈α∪relative␈α∀to␈α∀the␈α∪function␈α∀␈↓↓denot.␈↓␈α∀We␈α∪envisage␈α∀introducing␈α∀other␈α∪denotation
␈↓ α∧␈↓functions later and considering extensionality relative to them).

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

␈↓ α∧␈↓4) ␈↓↓∀E.(true E ≡ (denot E = ␈↓αtrue␈↓))

␈↓ α∧␈↓provided␈αtruth␈αvalues␈αare␈αin␈αthe␈αrange␈αof␈α␈↓↓denot,␈↓␈αand␈α␈↓↓denot␈↓␈αmay␈αalso␈αbe␈αprovided␈αwith␈αa␈α␈↓↓(partial)
␈↓ α∧␈↓↓possible world␈↓ argument.

␈↓ 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␈↓ axiomatized by

␈↓ α∧␈↓5) ␈↓↓∀Q1 Q2.(true And(Q1,Q2) ≡ true Q1 ∧ true Q2)␈↓, etc.

␈↓ α∧␈↓Assume that the rest of these formulas have been written.

␈↓ α∧␈↓␈↓ αTQuantifiers␈α⊃present␈α⊃more␈α⊃of␈α⊃a␈α⊃problem.␈α⊂ We␈α⊃want␈α⊃to␈α⊃introduce␈α⊃a␈α⊃function␈α⊂␈↓↓All(var,exp),␈↓
␈↓ α∧␈↓where␈α
␈↓↓var␈↓␈α
is␈α
a␈α
variable␈α
and␈α
␈↓↓exp␈↓␈α
is␈α
a␈α
concept.␈α
 Therefore,␈α
we␈α
need␈α
objects␈α
called␈α
␈↓↓vars␈↓␈αand␈α
variables
␈↓ α∧␈↓in␈αthe␈αouter␈αlogic␈αranging␈αover␈αthem.␈α We␈αalso␈αuse␈αvariables␈αranging␈αover␈αconcepts.␈α In␈αany␈αcase,
␈↓ α∧␈↓the basic fact about quantifiers is something like

␈↓ α∧␈↓6) ␈↓↓true All(x,E) ≡ ∀x'.(true Subst(x',x,E))␈↓,

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

␈↓ α∧␈↓␈↓αThis requires some clarification in later drafts.␈↓

␈↓ α∧␈↓␈↓ ε|2␈↓ ∧



␈↓ α∧␈↓␈↓ αTThe␈αabove␈αconceptual␈α
functions␈αcan␈αbe␈αrelated␈α
to␈αordinary␈αextensional␈α
functions.␈α Suppose
␈↓ α∧␈↓␈↓↓Telephone␈↓␈α∩is␈α∩as␈α∩above,␈α∩and␈α∩␈↓↓telephone␈↓␈α∩is␈α∩the␈α∩ordinary␈α∩telephone-number␈α∩function.␈α∩ They␈α∩are
␈↓ α∧␈↓related by

␈↓ α∧␈↓7) ␈↓↓denot Telephone x = telephone denot x␈↓,

␈↓ α∧␈↓and ␈↓↓telephone␈↓ can be used in any purely extensional context.

␈↓ α∧␈↓␈↓ αTIf␈αwe␈αwere␈αto␈α
introduce␈αthe␈αconvention␈αthat␈αassertion␈α
concepts␈αappearing␈αat␈αthe␈α
outer␈αlevel
␈↓ α∧␈↓of␈αa␈αsentence␈αare␈αasserted␈αand␈αwere␈αto␈αregard␈αthe␈αdenotation-valued␈αfunction␈αas␈αstanding␈αfor␈αthe
␈↓ α∧␈↓sense-valued␈αfunction␈αwhen␈αit␈αappears␈αas␈αthe␈αargument␈αof␈α␈↓↓Know,␈↓␈αwe␈αwould␈αhave␈αa␈αnotation␈αthat
␈↓ α∧␈↓looks␈α
like␈αordinary␈α
language␈α
in␈αhandling␈α
opaqueness␈αentirely␈α
by␈α
context.␈α There␈α
is␈α
no␈αguarantee
␈↓ α∧␈↓that␈α
general␈α
statements␈α
could␈α
be␈αexpressed␈α
unambiguously␈α
in␈α
it,␈α
but␈αit␈α
as␈α
an␈α
indication␈α
that␈αwe␈α
are
␈↓ α∧␈↓on␈α
the␈α
right␈α
track.␈α
 In␈α
fact,␈α
Robert␈α
C.␈α
Moore␈α
points␈α
out␈α
to␈α
me␈α
that␈α
distinguishing␈αsense-values␈α
and
␈↓ α∧␈↓denotation-values by context was recommended by Frege.  REFERENCE5

␈↓ α∧␈↓Church,␈αAlonzo␈α
(1951),␈αThe␈αNeed␈α
for␈αAbstract␈α
Entities␈αin␈αSemantic␈α
Analysis,␈αin␈α
␈↓↓Contributions␈αto
␈↓ α∧␈↓↓the␈α⊂Analysis␈α⊂and␈α∂Synthesis␈α⊂of␈α⊂Knowledge␈↓,␈α∂Proceedings␈α⊂of␈α⊂the␈α∂American␈α⊂Academy␈α⊂of␈α⊂Arts␈α∂and
␈↓ α∧␈↓Sciences,␈α␈↓α80␈↓,␈αNo.␈α
1␈α(July␈α1951),␈α100-112.␈α
 Reprinted␈αin␈α␈↓↓The␈αStructure␈α
of␈αLanguage␈↓,␈αedited␈αby␈α
Jerry
␈↓ α∧␈↓A. Fodor and Jerrold Katz, Prentice-Hall 1964

␈↓ α∧␈↓This version of SENSE[S76,JMC]@SU-AI PUBbed on May 14, 1976.



























␈↓ α∧␈↓␈↓ ε|3␈↓ ∧