Singular Propositions and Modal Logic - Philebus

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University of Arkansas Press
Singular Propositions and Modal Logic
Author(s): Christopher Menzel
Source: Philosophical Topics, Vol. 21, No. 2, Philosophy of Logic (FALL 1993), pp. 113-148
Published by: University of Arkansas Press
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TOPICS
PHILOSOPHICAL
21 NO.2,SPRING
VOL.
1993
Singular
and
Modal
Propositions
Logic
ChristopherMenzel
TexasA&M University
THE RUSSELLIAN PUZZLE
Accordingto theprevailingview in thephilosophyof languagenothing
mediatessemantically
betweena propername(in a speaker'smouth)and
:; thereis
ofa nameis determined
thereference
itsreference.
Rather,
directly
A
that
referred
to.
thesis
name
than
the
ofa
no moretothesemantics
object
of
a
sentence
view
is
that
the
shared
oftenaccompanies
thiswidely
meaning
theproposition
[9] itexpresses.If9 containsa name,
cpis an abstract
entity,
theproposition
expressedis said to be singular.Thisthesisin turnis often
thesisthatpropositions,
withthemetaphysical
singular
proposupplemented
arestructurally
sitionsin particular,
; thatis, roughly,
(i) thatthey
complex
to
rather
thatcorresponds
structure
have,in somesense,an internal
directly
the
that
and
the
sentences
that
structure
of
thesyntactic
expressthem, (ii)
arethesemantic
orconstituents
, ofthatstructure
metaphysical
components,
ofthose
values themeanings ofthecorresponding
components
syntactic
orill,
for
better
related
theses
refer
to
these
three
Let us
sentences.1
jointly,
as Russelliansemantics.
Russellian semanticshas an importantconsequence.Considera
a name- thesentence'Quine is a philosopher',say.
sentencecontaining
it followsthatthesingularproposition
If Russelliansemanticsis correct,
- contains
thissentence
[Quineis a philosopher]
expresses theproposition
113
corstructure
inthatpartofitsmetaphysical
as a constituent
Quinehimself
wholes
in
the
realm
the
name
to
'Quine'.Now,although
physical
responding
theirconstituent
seemtopresuppose
do notgenerally
parts(it'sstillthesame
car despitethenew set of wipers),thesame does notseemto be trueof
ofa Russellianproposition
constituents
theindividual
Rather,
propositions.
butessentialtoit.
tothenatureoftheproposition,
seemnotat all incidental
Itis hard,forexample,to see howtheproposition
[Quineis a philosopher],
ifRussellian,couldpossiblyhaveexisted,couldpossiblyhavebeenwhatit
it fromthe
is, sans Quine.(What,forinstance,wouldhave distinguished
had
neither
is
a
existed?)Thus,Russellian
[Russell philosopher]
proposition
are ontologically
semanticsappearsto entailthatpropositions
dependent
that
i.e.,moreexactly,
upontheirconstituents;
ofa proposition
OD IfXis a constituent
y
y,then,
necessarily,
existsonlyifx does.
ButOD raisesa puzzle.Intuitively,
nothaveexisted;
(1) Quinemight
thatis,theproposition
[Quinedoesnotexist]couldhavebeentrue.However,
couldnothavebeentrue.For,as Adamsnotes,2
givenOD, thisproposition
mustbe in orderto be true."But by OD therewouldhave
"a proposition
as [Quinedoes notexist]ifQuinehadn'texisted,
beenno suchproposition
therewouldhavebeen
henceitwouldn'thavebeentheretobe true;rather,
Sinceofcourse[Quinedoesnotexist]
noinformation
aboutQuinewhatever.
also failsto be truewhenQuine does exist,it followsthat,necessarily,
thathe doesn'tis nottrue.Thus,
whether
he existsor nottheproposition
to ouriniis notreallypossible,andhence,contrary
Quine'snonexistence
tialintuition,
(1) isn'ttrueafterall.
Call thistheRussellianpuzzle.My goal inthispaperis todevelopsevviewof
boththestructured
tothispuzzlethatpreserve
erallogicalsolutions
- includingitsapparentconsequenceOD - and theintuition
propositions
inthewritings
ofArthur
oftheseis suggested
that(1) is true.Thefirst
Prior,
of
A simplemodification
is theleastattractive.
and,duetoitsawkwardness,
thesemantical
basisofPrior'ssolutionsuggested
byRobertAdamsguides
of
the second,muchmoredesirable,solution.Finally,a generalization
mostattractive
leads to whatI findto be thesimplest,
Adams'suggestion
solutiontotheRussellianpuzzle.
FRAMEWORK FOR A SOLUTION
theRussellianpuzzleis to proThe first
orderof businessin approaching
one capable of expressingthe
vide a properrepresentational
framework,
114
and makingthedistinctions
thatappearin thepuzzle. The
propositions
of
is
modal
bothcounterfactual
con, essentially
puzzle, course,
involving
ditionalsandthenotionsofpossibility
andnecessity.
Assuming,
reasonably,
thatthecounterfactuals
in thepuzzlearenecessarytruths
iftrueat all,the
canbe castintoanequivalent,
somewhat
morestilted
form
argument
though
thatreplacesthecounterfactual
conditionals
withstrict
entailment.
Wethereforeneed on thesegroundsadoptno morethana standardpropositional
modalbase. Althoughquantifiers
arenotexplicitly
involvedin thepuzzle,
does involvea distinguished
existencepredicate'E!', and
theargument
to characterize
itslogiccorrectly,
we willwantto be able to clarify
surely,
itsconnection
withtheexistential
Hence,an accuraterepresenquantifier.
tationof thepuzzle requiresthefullexpressivecapacitiesof a first-order
modallanguage.
Of coursewe cannotstopthere.For theheartof thepuzzleconcerns
Russellianpropositions.
More specifically,
thepuzzlerequiresthatvfebe
able to talkaboutsuchpropositions,
and in particular,
at least,thatwe be
ablesensiblytoascribebothtruth
andexistence
tothem.Furthermore,
since
thepuzzlehas to do withtheontologicaldependency
of singularpropositionson theirconstituents,
we shallhave to have someway of indicating
thata givenentity
is a constituent
of a givenproposition.
The mostnatural
of
this
last
is
to
show
as in
way meeting
requirement simply
constituency
theinformal
of
the
the
formation
of
presentation
puzzlesimplybyallowing
s
out
of
the
formulas
of
the
forproposition
denoting
expressions
language,
mulascontaining
namesin particular.
One mightthenexpressontological
dependency
bymeansoftheaxiomschema[II(E!e =) E!t), wheret is a term
thatoccurs(free)in s.
in theinformal
Propositiondenotingexpressionswere constructed
above
and
sentences, something
exposition
bysimplybracketing
alongthose
lineswilldo here.However,bracketing
ofsentences
is notsufficiently
genandrelationsas
eral,foranalogousRussellianpuzzlesariseforproperties
well.Justas therearesingular
therearealso singular
propositions,
propertiesandrelations
thatinvolveindividual
forexample,thepropconstituents;
fromQuine].Thus,itappearsthatOD generalizes
to
erty[beingdistinct
OD* IfXis a constituent
ofa property,
orproposition
relation,
y,
then,
yexistsonlyifx does.
necessarily,
Butthenwe can arguemuchas above.Forinstance,
intuitively,
existed
I wouldnonetheless
havebeendis(1*) IfQuinehadn't
tinct
from
him;
thatis, I would nonethelesshave had theproperty
[beingdistinctfrom
As
with
a
must
in
order
tobe exemplified.
be
Quine].
propositions,property
ButbyOD*, therewouldhavebeenno suchproperty
as [beingdistinct
from
115
Quine]ifQuinehadn'texisted,andhenceitwouldn'thavebeentheretobe
Hence,ifQuinehadn'texisted,I wouldn'thavehadthepropexemplified.
of
fromhim,andhenceitappearsthat(1*) isn'ttrueafter
erty beingdistinct
a
conclusion
no
less
thantheone in ouroriginalpuzzle.
all,
discomfiting
As withpropositions
in theoriginalpuzzle,in thisversionwe ascribe
existence
toproperties.
And,analogoustoourtalkintheoriginalofa proposition'sbeingtrue,herewe talkabouta property's
Thus,
beingexemplified.
a completesolutionto theentireclass ofRussellianpuzzleswillrequireof
us themeansfortalkingdirectly
aboutproperties,
and proposirelations,
tions(PRPs) generally.
Forthispurposewe willformalize
ourinformal
use
ofthebracket
notation
abovetoconstruct
a classofPRP-denoting
or
terms,
wherecpis anyfirst-order
intensional
abstracts.Specifically,
formulaand
Vj,. . . , vnanyvariables,theexpression[Xvj... vn9] is a termthat,intudenotesthe«-placerelation
itively,
expressedbycp.Thus,whereq is Quine,
the
1
^
is
fromQuine].
[Xjcèc q]
(i.e., -placerelation)[beingdistinct
property
Whenn = 0, intensional
abstracts
denotepropositions.
Thus,whereE! is
~E
the
is
the
existence,
[X 'q] (or,suppressing X,[~E!#]),
[Quine
proposition
doesn'texist].Its ontologicaldependenceon Quinecan thenbe expressed
as D(E! [X~E!g] z> E !#),andthedependence
of[beingdistinct
fromQuine]
on Quineby D(E![Xjcjc ï q] 3 E!#).Usingthisapparatus,then,we can
as DE![Xvcp]z>E!t, wheret is any
expressOD* forPRPs [Xv<p]
generally
in [Xvcp].3
constant
orfreevariableoccurring
tobeingabletoasserttheirexistence,
As noted,inaddition
we also need
to be able to expressthatcertainpropositions
aretrue,and,in thegeneral
PRPsareexemplified.
As withexistence,
we couldjustadd
case,thatcertain
truth
andexemplification
to
our
But
since
we areonly
predicates
language.
interested
in exploringthelogic of theRussellianpuzzle,it wouldbe far
ifwe couldavoidembroiling
morepreferable
ourselvesin theformidable
of
these
logicalcomplexities
predicates.4
Towardthisend,notethatPRPsplaytwometaphysical
roles:an objectualroleanda predicative
a
role
in
of
role,
theyplay beingtrulypredicated
otherobjects.In theirobjectualrole,PRPs canbe referred
toandquantified
overno less thananyotherkindofobject.Thisroleis reflected
in ouruse
of that-clauses,
and
other
infinitives,
gerunds,
PRP-denoting
expressions,
andin thelegitimacy
oversuchexpresof,e.g.,existentially
generalizing
sions,as in theinference
from,say,'JohnbelievesthatBach nevercomfor
to
'John
believes
. In theirpredicative
rolethey
posed guitar'
something'
arepredicated
ofotherobjects(or,in thecase ofpropositions,
theyare,so
to say,predicated
oftheworld),a rolereflected
in ordinary
Englishin our
use ofverbphrasesanddeclarative
sentences.In ourlogicallanguage,the
former
roleis ofcoursereflected
in ourcomplexterms,
andthelatterin the
usualwayin thepredicates
ofthelanguage.Now,withpossibleexceptions
to lightby Russell'sparadox(whichwe shallstudiously
brought
ignore),
116
areexactlythoseofthecorresponding
thetruth
conditions
fora predication
if
V
is
a predicateexpressing
someproperty
statement
exemplification
term
'A'
a
a
p, ^[Xjccpf complex
denoting
/?,
predi2-placeexemplification
'"tit'and
cate,andaTtheobjectdenotedbya termt, thenboththepredication
are truejust in
thecorresponding
statement
Â(['jccp],T)^
exemplification
in
since
the
involvesonly
case a7is theextension
ofp. However,
predication
of
the
theproperty
orrelation
and
the
beingpredicated
object(s)
predication
without
themediationof theexemplification
we can avoida sperelation,
inexpressing
thegeneralRussellianpuzzleif
cial exemplification
predicate
ofcomplexPRPs in their
we
have
a
robust
only
sufficiently
representation
role.
predicative
, our
However,as thingsstand,becausethereare onlycomplexterms
in
no
direct
of
the
has
logicallanguage
general
way expressing predication
of a givencomplexPRP of a givenobject;in particular,
one cannotin any
naturalwaypredicate
theproperty
distinct
from
Quine]ofme.What
[being
we need,ofcourse,is a representation
oftheunnominalized
verbphrase'is
to
the
term
x
distinct
fromQuine'corresponding complex
['x * q] thatrepWe
couldofcourse
resentsthenominalization
distinct
from
Quine'.
'being
tomeetthisneed,butthatwould
introduce
a classofnewcomplexpredicates
be unnecessary.
Forifwe acceptthatitis thesamePRP indicated
bya verb
indicated
in
different
then
there
and
its
albeit
roles,
nominalization,5
phrase
serve
as
both
is no reasonnottoletourcomplexPRP-denoting
expressions
nomandthereby
the
roles
of
both
termsandpredicates,
play grammatical
Call thisdualinalizedverbphrasesandtheirunnominalized
counterparts.
role syntax.Extendingthedutiesof complextermsin thisfashion,we
it
distinctness
fromQuinebysimplypredicating
myexemplifying
represent
as indicated,
anexplicitexemofmethus:[kxx & q]c; andinthisfashion,
difficulties
areavoided.6
plification
predicateanditsattendant
Truth
of as a speWhataboutthetruth
is
naturally
thought
predicate?
An « + 1-placeexemplification
cial case ofexemplification.
predicate
' +l
is trueofitsarguments
t, tv . . . , Tnjustin case t denotesan «-placerelationR thatis trueoftheobjectsav . . . ,andenotedbytv . . . , Tn.In thecase
wheren = 0, the 1-placeexemplification
predicateAj takesonlya single
termt as itsargument,
andis trueoft justin case t denotesa 0-placerela- thatis truesimpliciter.
tionP- i.e., a proposition
By allowing«-place
we avoidedtheneedforspecial
terms
to
serve
also
as predicates,
complex
thesecondformof theRussellian
exemplification
predicatesto represent
- suffices:
The
alone
indicated
puzzleabove;predication
byconcatenation
iff
.
.
.
.
.
t
is
true
...
.
, anexemplify
n-placepredication
av
^[Xvļ vn(p]jļ
J
. . . vncp]lTruthis just a special
(standin) therelationR denotedby
case. In thelimiting
case wheren = 0, a 0-placetermstandingalonewill
P itdenotes
suffice:
The0-placepredication
^[Xcp]lis trueifftheproposition
thata certainproposition
P is true,as
is true.Thus,assertions
to theeffect
117
in theoriginalformof thepuzzle,in ourlogicallanguagesimplytakethe
form^[Xcp]!,
where^[Xcp]!
expressesP. We can thushaveall theexpressive
we
need
for
power
examiningthelogic of theRussellianpuzzle without
ourselves
intheformidable
difficulties
oftruth
andexemplificaembroiling
tion.Forreadability,
we
introduce
an
eliminable
however,
"pseudo"truth
T'
in
predicate suchthat7ļcp]=df[9], so thatthepuzzle thegarbof dualrolesyntaxcorresponds
morecloselyto thelogicalformof thepuzzle as
expressedin ordinary
language.
THELOGICAL
STRUCTURE
OFPRPS
As notedabove,a sufficiently
robustnotionof a singularproperty
or
that
PRPs
exhibit
some
sort
of
structural
corproposition
requires
complexity
moreor less to thegrammatical
structure
of thesentencesthat
responding
expressthem.We shall cash thisnotionhereby adoptingtheview that
ofas being"builtup"logicallyfromlogically
complexPRPsarebestthought
less complexconstituents
These
bymeansof a groupoflogicalfunctions.7
can
be
into
classconsistsoflogical
operators
grouped threeclasses.Thefirst
to theclass of familiarsyntactic
functions
corresponding
operatorsfrom
modalpredicate
the
modal
and
boolean
andthequantilogic;viz.,
operators
theproperty
fiers.So, forinstance,
nonexistence
is
the
[Xjc~Elx]
negationof
theproperty
ofexistence;i.e., [Xjc~E!jc]= neg([Xjc
we
E!jc])(note areusing
rather
thanmentioning
ourabstracts
the
here); property
[Xjc0~E!jc]possible
nonexistence
is thepossibilization
of [Xjc~E!jc]; therelation
beingobjects
Xandy suchthatxis richandy is unhappyis theconjunction
ofthepropertiesbeingrichandbeingunhappy;i.e., [Xjcy
Rx A~Hy] = conj([XjcRx],
[Xy~//y])= conj([XjcRx],neg([Xy/fy]));andtheproperty
beingmarried
of(thesecondargument
(to someone)is theexistential
quantification
place
= exist2([Xjcy
relation;
i.e.,[XjcByMxy]
of)themarriage
Mxy]).
in thesecondclass reflect
certainpossiblerelations
Logicalfunctions
betweentheX-bound
variablesin an abstract
andoccurrences
[Kvļ. . . vn<p]
lack
of
in
in a complete
those
variables
(or
thereof)
cp.Thoughimportant
these
functions
are
irrelevant
for
our
account,
presentpurposes,and so we
won'texaminethemany further.
Not so our finalclass, thepredication
functions.
These are thefunctions
thatgive risemostdirectly
to singular
These
fall
into
two
propositions.
operators
naturally
camps: "simple"
functions
thatarereflected
in abstracts
like'[X[Xjc0~E!jc]#]'and
predication
a freeoccurrence
ofany(toplevel)
'[XjcBx[KSa]]' inwhichnotermcontains
and
inabstracts
X-bound
functions
reflected
variable; "complex"predication
like'[XjczBx[' &]]' and '[Xjc[kyCyx'b' inwhichthereis sucha term.
As it
both
and
functions
are
instances
of
a
single,general
happens, simple complex
of the
type.For purposeshere,however,we can avoid thecomplications
and
think
in
their
terms
of
brethren.
only
complexoperators
simpler
118
In thesimplestcase, a predication
function
pred.takesa singleobject
a and"plugs"itintotheIthargument
orrelation,
placeofa property
yielding
a PRP R withone fewerargument
place,and witha as a newconstituent.
inthecase ofa property
theresultis a singular
Thus,inparticular,
proposition about a. So, for example,pred1 takes the propertyof possible
- [Xjc0~E!;t]- and plugsQuine,say,intoitsfirst(and only)
nonexistence
thatQuine is possibly
argument
place, therebyyieldingtheproposition
= [X[Xjt
more
0~E !*]#].In
nonexistent;
i.e.,
formally,
predj([X*O-Eîjt],#)
a
can
involve
an
general,however, simplepredication
«-placerelationand
forexample,suchmulti-place
anyof itsargument
places; consider,
predicationsas theproperty
[XjcSaxb] of beingan x suchthata saysx to b. To
this,letor= (iv . . . , im)be anyfinite,
capture
increasing
sequenceofnumbers
than
0.
The
is
for
such
idea,then, that, any
o' we definea predicagreater
tionfunction
suchthatforanyn
ifR is an «-placerelation,
and
predCT
.
.
.
m
then
.
.
.
is
the
of
,
,
)
predication
av
amany objects,
predCT(Ä,
av
am
R ofav . . . , amrelativeto the/^, . . . , iJ* argument
of
R
, respecplaces
tively;thatis, to expressthisusingourlanguage,pred^/?,av . . . , am) =
M.x. +. ,l . . . x. MX.
['x .i ... xn-m
L
~J. Thus,inpartiJRx.l . . . x.iļ-ll/ļ
lm~ m
cular,pred^ 3)(5,a , b) = [kxSaxb].
Henceforth
we willassumea variety
ofaxiomsthatguarantee
thefineof PRPs. It is rather
tediousto lay thegroundwork
forstating
grainedness
theseaxiomsprecisely.
clarbe expressed
withsufficient
Theycan,however,
forpurposeshererather
ityandcompleteness
easily.First,itis assumedthat
P ^ Q, whereP is an «-placerelation,
and Q is an m-placerelation,
andn
^ m.Next,itis assumedthattheclassesofmodal,boolean,quantified,
and
are
all
PRPs
where
a
modal
PRP
is
one
that
is
predicative
pairwisedisjoint,
in therangeofthepossibilization
ornecessitation
a
boolean
PRP
function,
is in therangeofone ofthebooleanfunctions,
andso on.Thisaxiomguarthattheproposition
antees,forinstance,
[X[kx 0~E!jc]#]Quine is possibly
fromtheproposinonexistent,
thoughlogicallyequivalentto it,is distinct
tion[X 0~E!g] Possibly,Quine does notexist.The latter
is a modalpropoto
sition,derivedby applyingthepossibilization
operator theproposition
thatQuinedoes notexist,whiletheformer
is predicative,
derivedas noted
above.8The finalgroupof axiomscapturesthemostintuitive
elementof
that
PRPs
built
from
different
constituents
must
viz.,
fine-grainedness;
up
themselves
differ.
Thus,fromtheseaxiomsitfollows,forexample,thatthe
resultofpredicating
theproperty
beinga philosopherofQuine,[X Pq], is
a different
thantheone, [X Pg], thatresultsfrompredicating
proposition
thatproperty
ofPeterGeach,say,orofpredicating
theproperty
existenceof
[X
Quine, E!#].
119
THE PUZZLE REVISITED
Now thatwe havean appropriate
medium,letus use it to
representational
the
the
We
will
do
so
out
ordinary
languagerenbylaying
express puzzle.
the
in
its
form
of
followingeach
explicitly,
dering
puzzle
propositional
formal
with
its
formalized
Our
languagehas no councounterpart.
premise
the subjunctive
terfactual
conditional=>, and hencewe can't represent
all
the
are
elementoftheargument
since
However,
subjunctives
precisely.
z>
to
and
since
=»
is
D(9
i|>),9
i|/) logicallyequivalent '3(<p
necessary,
thelogicof
entailment
insteadwithout
we can substitute
affecting
ordinary
we let r[cp]=df[9]. Call the
theargument.
Also, as noted,forreadability,
RP.
following
argument
nothaveexisted.
(1) Quinemight
(Assumption.)
(10
0~E'q.
(2)
theproposition
[Quinedoesnotexist]is true.
Possibly,
(From(1).)
(20
0r[~E!tf].
(3)
ifQuinehadn't
existed,
[Quinedoesnot
Necessarily,
haveexisted.
exist]wouldn't
(ByOD.)
(30
CK~mqz>~m[~B'qì).
if[Quinedoesnotexist]hadn't
it
existed,
Necessarily,
wouldn't
havebeentrue.(Assumption.)
(4)
(40
D(~mi~mq]z>~T[~E'q').
ifQuinehadn't
existed,
(5) Hence,necessarily,
[Quine
havebeentrue.(From(3) and
doesnotexist]wouldn't
(4).)
(50 □(~E!^rz>~r[~E!#]).
ifQuinedoesexist,[Quinedoesnotexist]
(6) Necessarily,
is nottrue.
(Bylogic.)
(60
n(E'qz>~T[~E'q]).
ornot,
whether
(7) Hence,necessarily,
Quinehadexisted
[Quinedoesnotexist]wouldnothavebeentrue.(From
(5a) and(6a).)
(70 HQElqv ~E 'q 3 ~r[~E!tf]).
either
(8) Necessarily,
Quineexistsorhedoesn't.
(By
logic.)
(80 D(ß'q v ~E!tf).
(9)
that[Quinedoesnotexist]be true,
Itis notpossible
contradiction.
(2).)
(From(7) and(8),contradicting
120
(90
~07[~E!tf].
itis falsethatQuinemight
nothaveexisted.
(10) Therefore,
(1).)
(From(2) and(9),discharge
(100
THEUNDERLYING
LOGIC
Beforeexamining
solutionsto thepuzzle,itwillbe useful
prospective
tomakeexplicitthecentral
theargument.
First,
logicalprinciples
underlying
themovefrom(1) to(2), though
is nottrivial.
Itis based,ofcourse,
intuitive,
directionof the intuitiveequivalence0~E!<? =
upon the left-to-right
. Thisis
071-E!#],whichis an instanceofthegeneralschemaO9 = ÖTfXcp]
a modalizedversionof theprinciple9 = T['q>] that,roughlya statement
holdsif and onlyiftheproposition
it expressesis true.This principlein
- bearingin mindourstipulation
turn
that7[cp]is nothing
otherthan[cp]
:
is justa specialcase ofk-conversion
X-con: cpi= [kx <p]T,
wherex is an «-tupleofpairwisedistinct
variablesandt an«-tupleofterms.
direction
ofX-conappearsto be one ofthegenSo at leasttheleft-to-right
erallogicalprinciples
theargument.
How thendo we getfrom
the
underlying
~E
3
to
its
modalized
instance
'q T[~E'q]
0~E!#3
left-to-right
counterpart
~E
3
The
easiest
route
would
seem
to
be
this.
From
Ö7T-E!#]?
'q T[~E'q],
assumingtheruleofNecessitation
□I: h (p=> h Dep,
we have D(~E!^ 3 r[~E!#]). Fromthebasic modalpropositional
schema
K: D(cp3 iļj)3 (Dtp3 Q|j),
some propositional
logic,and theusual principlethat□ and 0 are interdefinable,
i.e.,
□/0: dep= ~0~9,
we can derivetheschemaD(9 3 iļ/)3 (O9 3 Oiļi).Thus,pluggingin ~E 'q
and 7ļ~E!#] for9 and iļ/,respectively,
we have Q~E!#3 Ô71-E!#],as
desired.
Now alreadyat thisearlystageofthegamedifficulties
beginto loom.
It is, presumably,
a logicaltruth
thatEbe = 3;y(y= x). Hence,because
Vjc3y(y = x) is a theorem
of classicalquantification
theory(CQT) with
DI andD/0
identity,
E!#is as well.So inthecontextofCQT withidentity,
alone are enoughto yield(100- Exploringthisproblemnow will getus
ahead of ourselves.So forthemomentwe will simplyassumesomereasonablefix,e.g.,switching
fromCQT to a freequantification
(i.e.,a
theory
thatallowsnondenoting
termsandhencedoesnothave
quantification
theory
E!jcas a theorem).
121
ofontological
Next,itwouldbe welltomaketheprinciple
dependence
- explicit;in fact,let us statea slightlymore
of which(3) is an instance
of a PRP are
generalprinciplethatcapturestheidea thattheconstituents
forits existenceas well as individually
also jointlysufficient
necessary.
letE!t abbreviate
Wheret = Tj . . . Tmis a nonrepeating
sequenceofterms,
in questionis:
E!tj A ... A EÎTm;thentheprinciple
T contains
all andonlythenoncomEx: E!['x (p]= E!t, where
andprimitive
variables,
(i.e.,constants,
prediplexterms
cates)thatoccurfreein['x cp].
is clearly
direction
theright-to-left
bytheproof,
Thoughnotstrictly
required
andhencewillhavetoholdin a coma partoftheunderlying
metaphysics,
we neednotexplicitly
pletelogicforit.Notealso that,givenNecessitation,
is calledEx becauseitis a reasonably
Theprinciple
modalizetheprinciple.
ofourlanguage)ofthe
boundaries
(withintheexpressive
precisestatement
.10
orworse)as existentialism
doctrine
thathascometobe known(forbetter
as theview thattheonly
Existentialism
is sometimescharacterized
- i.e.,
arethosethatareeither
PRPs thatexistnecessarily
purelyqualitative
individuals
thosethatcountnoconcrete
amongtheirconstituents
roughly,
all existnecessarily.
themselves
constituents
orwhoseconcrete
Thoughnot
- i.e., in theformexpressedby
entailedby existentialism
strictly
proper
andexistencearegenerSinceidentity
Ex- theviewis a naturalcorrelate.
tocapturethisviewinourlogicwe must
allytakentobe purelyqualitative,
matterswe will
To simplify
expresstheirnecessaryexistenceexplicitly.
E !, withtheproperty
existence,
[Xjc3y(x = j)] ofbeingidentical
identify
withsomething.
Thenwe needonlyadd theaxiom
E!=: E!=.
relationnowfollowsfromE!= and
The necessary
existenceoftheidentity
K.11
from
and
and
that
of
existence
Ex, DI,
□I,
let
us
the
(4) a littlemoreexplicassumption
Finally,
unpack important
cannotbe truewiththat
a
of
the
is
an
instance
proposition
itly.(4)
principle
outexisting:
(11) 7Tcp]=)E![cp],
or more generallystill,the principleused in the generalformof the
without
Russellianpuzzlethata PRP cannotbe exemplified
existing(where
is justforittobe true):
tobe exemplified
fora proposition
(12) tttidE!TT,
whereit is any«-placepredicateandT a sequenceofn terms.Intuitively,
hascalledseriousactuofwhatPlantinga
(12) expressesa classofinstances
or standin a relation
alism, theviewthatan objectcannothavea property
without
existing:
122
SA: 7tt=>E!T.
fromPtx.. . tnitfollowsthatP standsintheexemplification
For,intuitively,
relationA withtv . . . , tn, APtv . . . ,řn,
(orwheren = 0, thattthasthepropand
hence
that
of
and
E!P, bySA. However,becausetruth
erty beingtrue),
in
our
PRPs
are
not
equivalanguage,nothing
expressible
exemplification
fromSA. GivenEx, howlentto either(11) or (12) followsin thismanner
followfromthegeneralizedseriousactualism
ever,all threestatements
principle
GSA: 9 3 E![9],where
9 is atomic;12
cannotholdunlessitexpressesan existing
an assertion
i.e.,roughly,
propooftheexpressedproposisition,andhencebyEx unlesseveryconstituent
of(12)
we haveas a theorem
a succinctexpression
tionexists.In particular,
andSA:
(13) ittz) (E!ttAE!T).
Notealso thatbyGSA, Ex, andX-conwe havein general
(14) 9 d E![<p]foranycp.
ForbyX-conwe have9 z> [cp],andso since[9] is a 0-aryatomicformula,
of Ex in bothdirecwe have [9] z> E![[9]] by GSA. Severalapplications
tionsandsomepropositional
logicyieldthedesiredresult.
I willtakeGSA tobe thegeneralmetaphysical
(4).
underlying
principle
AN ACTUALIST ACCOUNT OF POSSIBLE WORLDS
In oursearchfora solutionto theRussellianPuzzle,itwillbe illuminating
notionofa possibleworld.Possibleworldsare
to makeuse ofthefamiliar
forourdisanditwillbe important
sortofentity,
ofcoursea controversial
cussion that whateveraccountof worlds we adopt be actualistically
FamiliaraccountsfromthelikesofAdams,Plantinga,
Pollock,
respectable.
1find
orstatesofaffairs.13
andotherstakeworldstobe (setsof)propositions
andso willuse insteada moremodel-theoretic
theseaccountsproblematic,
constructions
accountofpossibleworldsthattakesthemtobe set-theoretic
have
been.
as
the
structure
of
things theymight
exhibiting
As a firstcutat thisaccount,considera setD = u{D p D0, Dp . . .},
whereD j is a setofindividuals
(i.e.,non-PRPs)and,forn > 0, Dnis a setof
existence
E! in Dr Let extbe an extension
relations
that
includes
w-place
of
themembers
ofD0 intotruth
andfalsity,
function
thatmapsthemembers
>
memfor
n
into
of
the
members
of
0
into
subsets
of
and
D,
^-tuples
Dw
Dj
inparticular
thatext(E') = D. Thensaythatthepair
bersofD. We stipulate
123
S = (D, ext)is a stateofaffairs
ofthePRPs
justincase theactualextensions
inD, restricted
toD, areexactlyas extdepictsthem,andsaythatS is a possiblestateofaffairs
i.e.,
justincase itis possiblethatS be a stateofaffairs;
in
case
it
is
ofD exist(together)
andthat
just
possiblethatall themembers
extcorrectly
of thePRPs in D. Notethatsinceit is
depictstheextensions
constructed
outofactuallyexisting
S
objects,everypossiblestateofaffairs
even
if
it
not
even
if
extenis
ext
does
not
the
actual
exists,
actual;i.e.,
depict
sionsofthePRPs in S. We assumein additionthatpossiblestatesofaffairs
exhibit
ourunderlying
Russellianmetaphysics
withrespecttothePRPsthey
containas embodiedin theprinciple
so
that
an n-placePRP P e Dniff
Ex,
all itsconstituents
arein D. Giventhis*notealso thatsinceextmapsPRPs
intoD, eachpossiblestateofaffairs
is onein whichthegeneralized
serious
actualismprincipleGSA holds.Finally,say thatS is a possibleworld, or
worldstructure,
itis possible
justincase,amongthepossiblestatesofaffairs,
thatS be thelargest
. Thatis, letA be theclass ofpossiblestatesof affairs;
thenS is a possibleworldjustincase,possibly,
andfor
S is a stateofaffairs
=
e
S'
if
is
a
state
of
then
D/
for
all
z'.14
S'
ç D¿
affairs,
(D', exť) A,
any
A limitation
of thisconstruction
arisesfromthefactthat,intuitively,
therecouldhavebeenthingsotherthanthethingsthatactuallyexist;John
Paul II, forexample,underquitedifferent
circumstances,
mighthavehada
What
state
of
affairs
this
grandson.
possible
Though
represents possibility?
we willnotexplorethisissueindetailinthispaper,thecrucialidea is toset
aside someclass of actualistically
acceptableobjects- puresets,say- to
serveas surrogates
forsuch"possibilia"as JohnPaul IPs grandson.
These,
in turn,can serveas constituents
of "surrogate
PRPs"; thatis, PRPs conthattherefore
serveas surrogates
taining
surrogates
amongtheirconstituents
forPRPs thatwouldexistiftheindividuals
bythosesurrogates
represented
existed.Thus,iftheemptyset0 wereto be a surrogate
fora grandsonof
thentheproposition
JohnPaul II in somepossiblestateofaffairs,
[0 is sitwill
the
that
would
exist
ifthat
ting]
represent corresponding
proposition
were
to
A
exist.15
state
of
is
a
S
affairs,
then, pair = (D,
grandson
possible
such
that
it
for
there
is
to
be
some
ext)
possible
mappingmfromthesurroin
D
into
the
set
of
such
ext
objects16 that,underthatmapping,
gates
existing
D
the
extensions
in
of
the
PRPs
note
thereof;
(orsurrogates
correctly
depicts
thatonourtype-free
thePRPsforma subsetofD). Thefollowing
picture
figureillustrates
theidea in a simplefashion.WhereS = (D, ext),thepicture
indicatesa setDj ofproperties
anda setD j consisting
of fiveindividuals,
threeofwhicharesurrogates,
A
of
0.
including
portion extis indicatedby
thelinesfromthreeproperties
in thesetofPRPs to subsetsofD r M indicatesthesetofindividuals
thatwouldhaveexistedwereJohnPaulII tohave
hada grandchild,
andmis themappingtherewouldhavebeenfromD { into
mmapsJohnPaul II tohimselfand0 to hisgrandchild.
M; in particular,
124
n-
(
'
P
'
Q
'
R
^
J
cit(Py' / '
V
'ral(Q)
n-
f(
*
■
)}
1
vvWfeTy
»<mJ V
m(0)1
,„(■>
A possibleworld,then,is a possiblestateofaffairs
thatis possiblythelargest
understateofaffairs
inthesensedefined
above(exceptwith'stateofaffairs'
ofcourse).The important
stoodas in thecurrent
pointtonoteis
paragraph,
to playthe
thatby adoptingan actuallyexistingmodel-theoretic
structure
roletypically
accordedto possibleobjectsandpossibleworlds,we can use
ourselvesto an
theoftenhelpfullanguageofpossibiliawithout
committing
ungainly
possibilistmetaphysics.
SOLUTION 1: PRIORE AN INTERN ALISM
The first
solutionto thepuzzleI wantto consideris suggested
bythework
While
ofArthur
Prior.17
Prior'ssolutionis, at firstsightanyway,extreme:
ifwe take
thelogicunderlying
theRussellianpuzzleis flawed,nonetheless,
itsconclutheimplications
ofthenatureofsingular
seriously,
propositions
sionis stillunavoidable.
As Priornotes:
125
4Itcouldbethatcp'thatitistrueifandonlyif
Ifwesointerpret
could
then
is something
that
[9] couldbetrue,
mynon-existence
that
couldbetrue.18
notbe,since[I donotexist]... isnota thing
withtheunderlying
therearetworelatedproblems
So, forPrior,
logicofthe
Russellianpuzzle.The firstis thatfromthetrueconclusionthatQuine's
nonexistence
is impossible,it followsby D/0 thathis existenceis necesofcourse,appliestoevery
sary, thatheis boundtoexist.A similarargument,
andthat,Priornotes,"makesgodsof
existing
thing,
ostensibly
contingently
ina modallogic
an inappropriate
us all."19D/0 is therefore
logicalprinciple
forcontingent
beings.
The problemsdon'tstopthere,however.Considernextthefollowing
Fromthepropositional
tautology
argument.
(15) E!#v ~E'q
itfollowsfrom'-con thattheproposition
expressedis true,
(16) T'E'q v ~E 'q]è
As an instanceof(11), we have
(17) 71E!?v~E!tf]z>E![E!tf
v~E!tf],
byEx we have
(18) E![E!#v ~E!g]z>E!g,
andhencebypropositional
logic
(19) T'E'q v ~E!#]z>E!#,
andK we have
andso byNecessitation
(20) 'I'T[E'q v ~E 'q] z>'jE'q.
onceagain,from(16), we also have
By Necessitation
(21) an^lq v ~E 'ql
fallsoutonceagain:
thenecessity
ofQuine'sexistence
andsobyModusPonens,
(22) DE 'q.
is here.Since(15) is a tautology,
whattheculprit
Itis nomystery
(16) should
in anylogic withtermsdenotingthepropositions
be a theorem
expressed
by its sentencesand some meansof expressingtruth;forif a sentence
of logicthatthe
how couldit notalso be a truth
expressesa logicaltruth,
so expressedis true?20
That,in the0-placecase, is just what
proposition
itis a truth
oflogic,Priorargues,(15)
'-con guarantees.
However,although
thatwouldhavebeentrueno matis nota necessary
truth
; i.e.,a proposition
it appearsthatthatcouldbe
terwhat.Forby ourunderlying
metaphysics,
so onlyifitwerea necessary
being,andhenceonlyifall ofitsconstituents,
inregardto a
As Priorhimself
remarks
werenecessary.
Quineinparticular,
126
"ifit is necessarythatifI am a logicianthenI am a
similarlogicaltruth,
in theguiseof DI, is
logician,it is necessarythatI am."21Necessitation,
therefore
no moreappropriate
thanD/0.
: In termsoftheapparaPrioris thuswhatwe mightterman internalist
of a proposition
tusabove,theonlybasis forevaluatingthetruth
Q with
=
intheworld;i.e.,
respecttoa possibleworldw (Dw,extw)is itstruth-value
thevalueofextw
on Q. Becauseextw
is defined
onlyonthePRPs inw,itfolcan be evaluatedas true(orfalse)onlywithrespect
lowsthata proposition
to thoseworldsin whichitexists.Let us makethismoreexplicitbydrawbetweena proposition's
ingouttheconnections
logicalformanditsextension.For Prior,in accordancewithGSA, thefollowing
principlegoverns
theatomiccase:
= T,
PI extw([Paļ
. . . tfj)= T justincase(i) n = 0 andextw(P)
or(ii) n > 0 and(av...,an) e extw(P).
Thatis,foran atomicproposition
[Pax. . . an' (= [P] whenn = 0) tobe consideredtruein w,av . . . , anmuststandin therelationP (or,in the0-place
of extw
, theargucase, P mustbe true)in w, and hence,by thedefinition
mentsto therelationPav . . . , an (hence,byEx, P itself)mustall existin
workas expected.
w. The cases ofconjunctive
andquantified
propositions
The negatedandmodalcases areworthconsidering
overtly:
= T justincaseextw([
P2 ejciw([~<p])
9]) = F.
Giventhedefinition
ofextw
, onceagainitfollowsthata negatedproposition
mustbothexistin a givenworldandbe suchthatitsunnegated
counterpart
is falsetherein orderforit to be truein theworld;thisis crucialto the
Prioreanpicture.
we needclausesforeachofthem:
Since□ and0 arenotinterdefinable
= T justincaseextw,([<p])
= T forsomepossible
P3 £xiw([0<p])
worldw''
= T justincase¿*/^([9])= T forallpossible
P4 ex^flOcp])
worlds
w'.
Giventhedefinition
ofextwitis clearthat,inaccordancewithPI, thepropositionthatQuinedoes notexist,[~E!#],is nottruein anyworld,andhence
that[~Q~E!g] is true.But,as expected,[DE!#]is nottrue,since[~E!#]does
notexistin anyworldlackingQuine,andthusD/0 fails.Andsince[E'q v
~E!g] does notexistinanysuchworldas well,DI failsas well.P1-P4 thus
forpropositions
in Prior'sview.
conditions
appearto capturethetruth
If,as just noted,theconclusionof RP is soundforPrior,whatis an
abouthowthe
forit?Thebestwaytoansweris tothink
acceptableargument
logicaboveneedstobe revisedin lightofthefailureofD/0 andDI. Letus
notefirst
thatitis onlyone direction
ofED/0thatbreaksdown,theimplicationfrom~0~9 toCkp;as we'vejustseen,fromthefactthattheproposition
127
[9] couldnotpossiblyhavebeenfalseitdoesn'tfollowthatitwouldthereforehavebeennecessary;
thatexist
i.e.,countedamongthetruepropositions
ineveryworld.However,equallysurely,
if[9] is necessary,
ifitwouldhave
beenamongthetruepropositions
no matter
what,thenitcouldnotpossibly
havebeenfalse.~0~cpthusexpressesa sortofweaknecessity,
impliedby,
butnotimplying,
itsfull-blooded
To simplify
notation,
then,let
counterpart.
us introduce
a weaknecessity
operator
explicitly:
DefH: Hep =df~0~cp,
andnaildownitsrelationtoitsstronger
counterpart:
Qp =>ftp.
Whatabouttheotherdirection?
As justnoted,whatprevents
a weaklynecv
like
from
essaryproposition [E 'q ~E!g]
beingstrongly
necessaryis the
factthatitis nota necessary
It
is
it
being.
(looselyspeaking)truewhenever
- that'sjustwhatitis to be weaklynecessary
- butitis notthecase
exists
thatit wouldhavebeentrueno matter
what.Thus,fora proposition
to be
it
true
whenever
must
be
both
it
and
furthermore
exists,
strongly
necessary
suchthatitneverfailsto exist;i.e., a littlemoreprecisely,
itmustbe both
and
existent:
weaklynecessary necessarily
U/m2: BcpADE![(p]3 D(p.
Bothconditions
can be combinedintothesingleequivalence
□/■: Qp = HepA DE![cp].
The reasoningbehind□/■ also determinesthe fate of Necessitation:
Because of contingently
we are warranted
existinglogicaltruths,
onlyin
that
the
weak
necessitation
of
an
truth
is
a
inferring
arbitrary
logical
logical
truth.
Hence,in ourPrioreansystem,DI is simplyreplacedby its weak
counterpart:
■I: l-<p=>l-H<p.
To infermore,we have to knowmoreabouttheontologicalstatusof the
[<p],as capturedpreciselyin thederivedrule(from□/■ and
proposition
■d
DRD: h(p=» hDE![cp]z) dep;
ofa necessarily
is itselfa logical
i.e.,thenecessitation
existing
logicaltruth
truth.
Thisraisesthequestionofhowone provesthata givenproposition
P
existsnecessarily.
this
is
or
Generally,
accomplished
byshowing assuming
thatall ofitsconstituents
existnecessarily.
Thus,one appealsto (therightto-left
direction
of) a modalizedversionofEx:
□Ex: DE![Xx <p]s DE!t,
128
whereT is as in Ex. QEx is clearlytruein ourunderlying
metaphysics.
itis no
However,withstrongnecessitation
replacedbyweaknecessitation,
longerpossibletoproveitfromEx. Hence,IHExtooneedstobe addedas a
in itsownright.
separatelogicalprinciple
Finally,thoughK remainstrueinthisPriorean
logic,itis oflimiteduse
due toitsapplicability
Hence,we
onlyto necessarily
existing
propositions.
needa corresponding
to governweaknecessity.
We cannot,howprinciple
"If
ever,simplyreplace■ with□. Consider,forexample,theconditional
is human,"Hq 3 3xHx.Thisproposition
Quineis human,thensomething
is weaklynecessaryforPrior:In termsoftheusualjargon,in everyworld
in whichit exists,theproposition
thatif Quine is human,something
is
humanis true.Butnowsupposewe add H(cp 3 iļ/)3 (Bcp3 Bv)/)as an
axiom.Thenit followsthatMHq 3 ■ 3xHx.It is indeedplausiblethat
thatQuineis essentially
human.However,
itis falsethat
MHq; i.e.,ineffect,
MBxHx;itsurelycouldhavebeenthattherewereno humansat all.
Wherethingsgo wrongis that,unlikethecase ofK, [i(/]canexistwithout [cp],and in suchcircumstances
itstruthis notguaranteed.
So whatis
neededis a simplequalification
thatrulessuchcircumstances
out:
3 (Hep3 ■(E!t 3 iļ/)),
where
t contains
allthe
■K: H((p3 iļ/)
terms
thatoccurfreein<pbutnotiļ/.
noncomplex
ofMK tobe ofthesameformas K, only
(WhereT is null,we takeinstances
with■ replacing□.) Returning
to ourexample,all thatfollowsnowfrom
M{Hq 3 3xHx)andMHq is M(E 'q 3 3xHx),andthatis unproblematic.
Forinanysituation
inwhichQuineexistshe is human(byassumption),
and
so in anysituation
in whichhe existssomething
is human,as required.It
shouldbe notedthattheoriginalprinciple
K nowfollowsfromMK andthe
otherprinciples
above.
theRussellianpuzzle,ifwe simplychangeevery□ to
Now,regarding
■, RP is in factvalidin ourPrioreanlogic.However,we neednotgo to
suchlengths.
Foran advantageofthelogic,withitsweakerbrandofnecesis thatwe arefreetoreturn
toCQT. Thus,we caninfer
sitation,
Quine'sexisandhencebyMl theimpossibility
ofhisnonexistence,
tence,E!<?,directly,
his necessaryexistence,DE!#, in accordancewith
■E!#,butnotthereby
Prior'sinitialconception.
ifnotparticular
Now,thisis perhapsa serviceable,
comely,quantified
modallogic,anditcertainly
behindRP
appearsto capturethemetaphysics
withgreatprecision.However,something
is stillamiss.Forsurelythereis
a senseinwhich(1) is true;surelyitis possibleinsomesensethatQuinefail
to exist.AndindeedPrioragrees:22
. . . there
is a senseofThis might
nothaveexisted'
inwhich
whatitsayscouldbethecase(andgenerally
is),i.e.,thesense:
'Itis notthecasethat(itis necessary
that(xexists))'~DE!x.
129
To accountfortheintuitive
truth
of (1), then,corresponding
to theweak
in
Prior
effect
introduces
a
weak
■,
necessity
operator
possibility
operator
♦:
Def#: ♦(p=df
whilenotstrongly
therefore,
Quine'snonexistence,
possible,is nevertheless
that(1) is true.Priorthus
weaklypossible,andthataccountsforourintuition
ofsingular
appearstobe abletohaveitbothways:The metaphysics
propositionsdictatesthat(1) mustbe false;semantical
intuitions
tellus itis true.
The twosensesofpossibility
appearto letus hangon to boththedemands
ofmetaphysics
andtheappealsofintuition.
But notso. Considera logicallyfalsesingularproposition;
that,say,
Quinebothis andis nota logician,[Lq A ~Lq ]. Since [Lq A ~Lq] contains
it is notitselfa necessarybeing.Hence,
Quine amongits constituents,
falsewhenever
itexists,theproposition
failstobe falseinsituations
though
in whichitdoes notexist,andhenceitis notnecessarily
false.It is thusa
theorem
ofourPrioreanlogicthat-□ -{Lq A ~Lq)' i.e.,itis a theorem
that
thecontradiction
in questionis weaklypossible,♦(£#A ~ Lq).23Thus,in
thislogic,Quine'snonexistence
is possiblein precisely
thesamesensethat
hisbeingbotha logicianand a nonlogician
is possible.But surelytheformeris trueinsomesensethatthelatter
is not;surely,
insomesense,Quine's
nonexistence
is a waythingscouldhave been and his simultaneous
existenceand nonexistence
is not.Prior'slogic,however,is unableto distinthat(1) is true.24
Can we
guishthem,andhencefailstoexplainourintuition
do better
andstillremainwithintheboundsofactualism?
SOLUTION 2: ADAMS' PERSPECTIVALISM
Perhapswe can. In his seminalarticle"ActualismandThisness,"Robert
Adamssuggestsan actualistunderstanding
ofmodalpropositions
thatdoes
notgenerally
cashtheirtruth
conditions
intermsofwhatpropositions
could
or musthavebeentruein Prior'ssense.In particular,
on Adams'view,the
of
his
nonexistence
is nota matter
ofthetruth
oftheproposition
possibility
[Adamsdoes notexist]withinsomepossibleworld.He writes:
... I deny. . . that
Îtis possible
that
thatthe
p^ alwaysimplies
couldhavebeentrue.
haveoften
that-/?
proposition
Philosophers
found
itnatural
tocharacterize
andnecessities
in
possibilities
terms
ofwhatpropositions
wouldhavebeentrueinsomeorall
situations Thisseemsharmless
solongas it
possible
enough
isassumed
that
allpropositions
arenecessary
Butitismisbeings.
if(asI hold)somepropositions
existonlycontingently.25
leading
130
The properapproachto understanding
thetruthof modal propositions,
Adamssuggests,involvesa metaphor
ofperspective.Interestingly,
Prior
himselfsuggeststheidea in thefollowing
passage:
Thereare,then,
nopossiblestatesofaffairs
inwhichitis the
casethat~E!jc, andyetnotallpossible
states
ofaffairs
areones
inwhichE'x. Forthere
arepossiblestatesofaffairs
inwhich
there
arenofactsaboutx atall;andI don'tmeanonesinwhich
itis thecasethatthere
arenotfactsaboutx,butonessuchthat
itisn'tthecaseinthem
thatthere
arefactsaboutx.
Adamsexpressestheidea thus:
A [possible
nosingular
about
world]thatincludes
proposition
meconstitutes
anddescribes
a possible
inwhich
I would
world
notexist.Itrepresents
notbyincludnon-existence,
mypossible
I donotexistbutsimply
that
me.
ingtheproposition
byomitting
ThatI wouldnotexistifallthepropositions
itincludes,
andno
other
actualpropositions,
weretrueis nota factinternal
tothe
worldthatitdescribes,
butanobservation
thatwemakefrom
ourvantage
ofthat
pointintheactualworldabouttherelation
toanindividual
intheactualworld.26
world-story
Let w be a worldlackingtheproposition
thatQuineexists.The idea,then,
is that,eventhoughtheproposition
[~E!#]is notpartofw (oranyworldfor
thatmatter),
fromourperspective
intheactualworldwe cansee
nonetheless,
thatitsayssomething
trueaboutw; i.e.,abouthowthingswouldhavebeen
hadw beenactual.As Adamsputsit,whileitis nottrueinw, itis nonethelesstrueat w.Itis invirtueofthis- not,perimpossibile
ofitspos, invirtue
and not simplyin virtueof its
siblybeingamongthetruepropositions,
- thatwe
complement
failingtobe necessarily
amongthetruepropositions
takethemodalproposition
can,anddo, intuitively
[0~E!#]to be truesimplicités In a nutshell,then,Adams retainsan underlying
ontologyof
Russellianpropositions
RussellianPRPs) alongwithEx,
(ormoregenerally,
and henceretainsPrior'simplicitconceptionof otherpossiblestatesof
affairs.UnlikePrior,however,we determine
thetruth-values
of modal
themat worldsrather
thaninworlds.
propositions
byevaluating
To getfullyclearaboutwhatis goingonhere,letus capture
therelevant
conditionsfortruth-at
implicitinAdams'approacha littlemoreformally
anda littlemoregenerally.
Let w = (Dw, extj be a possibleworld.Adams
followsPriorinaccepting
echoes
SA fortheatomiccase,andso hisprinciple
Prior's:
= T,
Al [Pax. . . an]istrue
atwjustincase(i) n = 0 andextw(P)
or(ii) n > 0 and(av . . . , an) e extw(P).
notethatAl is toholdevenifP is a complexproperty
or
Now,importantly,
relationliketheproperty
ofbeinga non-fish:
131
that
isnota fish;
I amsomething
means
that
... 'I ama non-fish'
IfI didnot
ofbeinga non-fish.
itascribes
tometheproperty
thatis
I be something
I havethatproperty?
exist,
Might
might
atall,andwouldhaveno
nota fish?No,I wouldbe nothing
false
counted
isappropriately
Hence'lama non-fish'
properties.
I donotexist.27
inwhich
inworlds
at worldsw in whichhe doesn't
However,to denyAdams'non-fishiness
of ['x -Fx] in w andhenceto
he
is
in
the
extension
that
to
exist,i.e., deny
thattheatomicproposition
affirm
[['x ~Fx]a] is falseat w,is notto deny
thatAdamsis nota fishatthoseworlds;thatis to say,while[[X* ~Fx]a] is
falseat w, thenegatedproposition[-Fa] is trueat w, and indeedat any
world:Itis truein,henceat,thoseworldsinwhichAdamsexists(assuming
andat thoseworldsinwhichhefailstoexistsimheis essentially
non-fishy)
he
is
not
Either
then,
amongthefishythings.Moregenerally,
way
pliciter.
is nottrueatw.
A2 [~i|i]is trueatwjustincase[iļ/]
at a worldgivesus a senseoftruth
UnlikePrior,then,theconceptoftruth
can turnoutto be truewithrespectto the
on whichnegatedpropositions
worldsinwhichtheydon'texist.Thisis thesenseinwhichsuchpropositions
be countedpossible.Thatis, moregenerally:
can nonetheless
inparticular)
atw(theactualworld,
istrue
A3 [Ov|i]
justincase
w' suchthat[iļj]is trueatw'
there
is a world
oftheform
itfollowsthata proposition
FromA2 andA3 inparticular
is trueatw iff[0~i|;]is nottrueatw iffitis notthecase thatthereis a world
is trueat w' iffitis notthecase thatthereis a worldw'
w' suchthat
suchthat[i|;]is nottrueat w'; i.e.,iff[i|>]is trueat all worldsw'. Since [i'f]
neednotin generalexistata worldtobe trueatit,itfollowsthata propositioncan be trueat all worldswithout
beingtruein all worlds.In particular,
of propositional
logic are trueat all worlds.Consider,for
singulartruths
[Lq v ~Lq] thatQuineis eithera logicianornota
example,theproposition
is trueat a givenworldwjustin case either[Lq]
logician.Thisproposition
is trueatw or[-Lq] is. SupposeQuineexistsinw.Thenboth[Lq] and[-Lq]
existthereas well,andobviouslyone or theotheris true;i.e.,thepropositionis weaklynecessary.
However,supposeQuinedoesn'texistin w. Then
at all
[-Lq] is stilltrueat w, since[Lq] is not.Takingnecessityto be truth
worlds,then,itfollowsthat[{Lq v -Lq)] is necessary;i.e.,theproposition
neceshavetheoriginalscopeofstrong
[UiLq v -Lq)] is true.Wetherefore
to ouroriginal
to us intact,and this,in turn,signalsa return
sityreturned
DI (and hencealso suitable
modalprinciplesD/0 and fullnecessitation
to CQT, forreasonsnotedabove).
modifications
We can now see clearlywhatlies behindAdams' denialthat"^ltis
^
couldhavebeen
that-/?
possiblethatp alwaysimpliesthattheproposition
thatas a specialcase ofAl , whereP is a proposition
true."Notefirst
[iļi],we
132
= T. This
havethattheproposition
[[i|>]]is trueat wjustin case extw([''t])
whenwe use ourpseudo-truth
readsa bitmoreintuitively
predicate:[7ļiļ/]]
= T ; i.e.,theproposition
is trueat wjustin case extw([''t])
that[ iļi]is trueis
itselftrueat w just in case theproposition
[iļj]is truein w. Eitherway,the
to [i|i]ingeneral)
important
pointtonoticeis that[[i(/]](incontradistinction
is theproposition
andhence,likeall
expressedby a (0-place)predication,
fortheseriousactualist,
thecomponents
of [[iļi]],hence[[vļj]]
predications
itself,mustexistin a worldw in orderforittobe trueatw.
thatQuinehas essentially;
So considernow anyproperty
existenceis
theobviousexample.ThenbecausethereareworldsinwhichQuinedoesn't
exist,theprinciplesabove entailthatit is possiblethatQuinefailto exist,
0~E!#;i.e.,theproposition
[0~E!g] thatthisformula
expressesis true(i.e.,
trueat theactualworld):[0~E!g] is trueiff[~E!#]is trueat someworldw
(byA3) iff[E!#]is nottrueat someworldw (byA2) iffthereis a worldw
suchthatq č extw(
theclaim
E!), as we've supposed.Considerbycontrast
0[~E!#]thattheproposition
[~E!#]couldhavebeentrue.Thisclaimis true
ifftheproposition
[0[~E!#]]itexpressesis trueiffthereis a worldw such
thattheatomicproposition
[[~E!g]] is trueatw (byA3) iffextw([~E'q])= T
(byAl); i.e.,ifftheproposition
[~E!#]is trueinw, andhenceonlyifq e Dw
= extw(E'),byourconstruction.
Thisofcoursecan'tbe,sincew is a possible
stateof affairs.
So whileit is possiblethatAdamsfailto exist,0~E!g, the
[~E!g] thathe does notcouldnotbe true;i.e., ~0[~E!#].Thus,
proposition
totighten
thetruth
above,onthisapproachwe distinguish
upAdams'remark
ofa proposition
totheeffect
thata givenproposition
P is possiblefromthe
ofP's beingtrue.
possibility
The argument
andrelations:
We can showin
generalizesto properties
thesamefashionthat,e.g.,eventhoughitis possiblethatQuinefailtoexist,
thathe
0~E!g, it does notfollowthatit is possiblethathe be nonexistent,
ofbeingnonexistent,
theproperty
0['x ~E!jc]q. Forthisis so iff
exemplify
theproposition
is
[0[Xjc~E!jt]g]thatitis possiblethatQuinebe nonexistent
trueiffthereis a worldw suchthattheatomicproposition
[[Xjc~E'x]q] that
is trueat w iffq e extw(['x~E!jc]) andhenceonlyif
Quineis nonexistent
=
g
q Dw extjfi^ onceagain.
The generallogicallessonhereis thatthelaw of'-conversionappears
tobreakdownin modalcontexts.
Thatis, in particular,
whereasboth
(23) [0~E'q] = 0~E'q
and
(24) [Xjc0-ELx]q = 0~E'q
instancesof
appear to be valid (as theyought,being straightforward
^-conversion
in whichthemodaloperators
no
essential
both
role),
play
(25) 0[~E'q' = 0~E'q
133
and
(26) 0[Xjc~E'x]q = 0~E'q
arefalse,thedifference
beingthatbothsidesof (23) and (24) requireonly
that[~E!#]be trueat someworld(given0~E!g ), whiletheleftsidesof(25)
and (26) requireitto be truein someworld.Currently,
however,thelatter
twoareprovablein ourrevisedsystemas it stands:In thecase of (25), for
andpropositional
instance,
byX-conversion
logicwe have
(27) ~[~E 'ql m E 'q%
andso byDI we have
(28) □(-[-£!*] = E 'q)
whichcreatesthevexatiousmodalcontext,
and(25) followsbyK, D/0, and
contraposition.
The latterthreeprinciplesare unimpeachable
in thecurrent
context,
so theproblemappearsto lie witheitherX-conversion
or necessitation.
thantheprinX-conversion,
however,seemsscarcelymorecontroversial
In
above:
in
the
case
used
in
what
else could
ciples
(27),
particular,
0-place
itbe forcptoholdthanfortheproposition
itexpressestobe true?In fact,we
shallaffirm
thevalidity
ofX-conversion,
rather
thanan
thoughas a theorem
withnecessitation.
axiom;theproblemlies moredirectly
To getatthenatureoftheproblemanditssolution,
letus return
to the
of
in
modal
The
is
that
E!x
recall,
CQT
problem
quantified
logic.
problem,
is a theorem
ofCQT (letting
E'x =df3y(y = jc)),and so bynecessitation
it followsthatŒ'x is a theorem
as well; i.e., it becomesa truth
of logic
thateverything
is a necessary
A
common
around
this
being.
way
difficulty,
as noted,is tomovetoa freequantification
inparticular,
toreplace
theory;28
theusualuniversal
instantiation
axiom
z) (pí,foranyterm
UI: Vxcp
t thatis freeforjcincp
withitsfreecounterpart
FUI: Vjtxp
t thatis freeforx in9.
d(E!td 9Í), foranyterm
FromFUI, together
withthereflexivity
axiomforidentity
Id: jc = jc,
itis possibleonlytoproveEbe =) 3y (y = jc),i.e.,E!xd E!jc,notE!jc simand so insteadof necessitarianism
we deriveonlytheinnocuous
pliciter,
3
□(Ebe
E!*); i.e., it is trueat everyworldw thatifx existsin w, then
x existsin w; i.e., it is trueat everyworldw thateitherx existsthereorx
doesn'texistthere;i.e.,that[E!jc]e w or [E !jc]g w. So in thisrespectFUI
appearstobejustwhatwe need.However,thissolutionis in one important
behindfreelogic is to have a logical
wayunlovely.The chiefmotivation
134
ofnondenoting
terms.
Thatis
systemcapableofdealingwiththepossibility
all
well
and
but
as
a
solution
to
the
of
(perhaps)
good;
problem CQT in
modallogicitfoistsundesirable
us.
baggageupon FollowingPrior,we are
an
actualist
modal
beings, and hencewe
constructing
logic of contingent
wouldliketo have at ourdisposalat leastthepossibility
of signalingthis
factbyhavingE'x falloutas a theorem.
thus
us
FUI
gives therightresult
in modalcontexts,
butis undulyrestrictive
in nonmodalcontexts.
in thiscase at least,we can haveourcake andeat it too.
Fortunately,
ConsiderId once again.Ifwe take'= ' tobe a predicate
thatdenotesa fullrelation
I
we
then
we
do
not
want
itto follow
(as, think, ought),
fledged
- that□(* = jc); i.e., thattheproposition
as it does givennecessitation
true,foranyx. Forotherwise,
[x = jc]is necessarily
by(ii) inAl (andthe
definition
of extw),it followsthatx existsin everyworld.Equallyclearly,
we don'twantto lose Id as a logicaltruth;
'x = jc'is trueunder
however,
So
this
a
natural
restriction
onnecessitation;
viz.,
anyinterpretation. suggests
□I': h cp=> h □ cp,so longas cpprovable
without
instance
of
Id.
any
This of coursepreventstheinference
fromId to itsnecessitation.
At the
withSA, Id yieldsE'x and hence,in consame time,however,together
junctionwithFUI, fullUI. Butsincea proofofE'x requiresId, itsnecessitationis notprovable.Thus,by a well-motivated
(foractualists,
anyway)
we avoid thenecessitarian
restriction
on necessitation
problemsof CQT
without
it.
abandoning
We can framea preciselyanalogoussolutionto theproblemof Xthattheusualaxiomcanbe brokendownintotwo
conversion.
Considerfirst
conditionals:
'R:
[Kx <p]y=><Py
and
cp?z>[Ä.X<p]y,
wherex andy arenonrepeating
sequencesx{ . . . xnand . . . yn,
respectively,
andforall i < n,y.is freeforx. in cp.Applying
necessitation
to instances
of'R corresponding
to (25) and(26), we have
(29) D([~E 'q] ZD~E 'q)
and
(30) D([Xjc ~E!jt'q => ~E!$),
whichareclearlyunproblematic:
(29) is trueifftheproposition
[[~E!#] =)
~E 'q ] is trueateveryworldw iffeither[[~E!#]]is falseatw or[~E!#]is true
at w iffextw([~E'q])= F or [E!#]is nottrueat w iffextw('E'q')= T orq ë
q g extJE') or q £ extw(El).(30) followsin
extw(El)iff(by construction)
135
instancesof
muchthesameway.Notso therelevant
viz.,
(31) D(~E 'q 3 [~E'q')
and
(32) D(~E!^ =) ['x~E'x]q).
doesnotfolFromthefactthat[~E!g] is trueata givenworldw, itcertainly
itcannotfollow)thateither
low (andindeed,when'E!' denotesexistence,
[[~E!#]]is trueatw; i.e.,thatextJ'~E'q') = T, orthat[[Xjc ~E!jc]#]is true
atw; i.e.,thatq e extw([kx-E!*]). Forbothwouldrequireq's existencein
The problemin general,ofcourse,is thata negated
w byourconstruction.
atomicproposition
cfs existing
[~Pq] can be trueat a worldeitherthrough
inw butfailingtoexemplify
P , inwhichcase both[[~P#]]and[[Kx ~Px'q]
aretrueatw as well,orthrough
q's failingtoexistthere,in whichcase neitherproposition
is trueat w.
of
analoWhattheseobservations
suggest,then,is a qualification
of UI thatyieldsFUI: An instanceof thecondigous to thequalification
referred
to
tionalX^Lholdsin a givenworldw so longas all theindividuals
in theantecedent
existinw. Thatis, we replace with
Tcontains
where
thevariables
cpjd(E!td [Xxcp]y),
y¿and
freein<py,
andwhere
allother
terms
occurring
noncomplex
foralli < n,y. is freeforx. incp.
□r cannowbe appliedunproblematically
toVRL,butbecauseoftherestricilkwillno
tionon itsgeneralapplicability,
(31), (32), andtheirtroublesome
- thoughnot
fullX-conversion
longerfollow.ButbecauseE'x is a theorem,
- is provablefrom and
itsnecessitation
XLR ''RL, as desired.
This thenis thekeyto thesolutionof theRussellianpuzzle on our
reconstruction
ofAdams'approach.29
As notedabove,themovefrom(1) to
ofnecessitation
to~E!g 3 [~E!g] toyield(31)
theapplication
(2) requires
of~E'q 3 [~E!#]inthe
andthence0~E!g Z) 0[~E!#].Butthededuction
above logic requiresVRL,whichrequiresE!#,whichin turnrequiresId.
is blocked.
Thus,CU'cannotbe invokedto yield(31), andso theargument
We getonlytheinnocuous,
indeedpropositionally
trivial,
[H(~E!g =) (E!g z>
[~E !<?])).
SOLUTION 3: FULL PERSPECTIVALISM
TRUTHGENERALIZING
AT,I: RELATIONS
itis still
Preferable
as thislogicis tothePriorean
logicabove,however,
To see this,fora givennotionofextension
toomuchinthelatter'
s clutches.
P ispositivewithrespecttoa given
ofa proposition
e, saythattheextension
worldjust in case e(P , w) = T. Whattheconceptof truth-at
givesus, in
136
on whicha proposicontrast
to thePrioreanview,is a notionof extension
tionhas a positiveextension
withrespecttoworldsinwhichitdoesn'texist.
as we've seen,givesus a muchmorepalatablequanUsingthisdistinction,
tifiedmodallogicthatnonetheless
remainsfaithful
to theintuitions
behind
arelimiting
cases of
theRussellianviewof propositions.
Butpropositions
forrelations.
andtruthvalueslimiting
casesofextensions
n-placerelations,
we
can
the
above
definition:
For
notion
ofextenThus,
anygiven
generalize
R is positiveat a worldwjustin
sione, theextensionofan n-placerelation
Thenon ourreconcase e(R, w) = T ifn = 0, and e(R, w) ž 0 otherwise.
can havea
struction
ofAdams' approachas it stands,a relationgenerally
in
which
it
doesn't
exist
if
it
is
a proposiextension
at
a
world
only
positive
restriction.
For thenotionof truth-at
tion.But thisis an unwarranted
by
whichpositiveextensions
are assignedto propositions
at worldsin which
to a notionof holding-at
forntheydon'texistgeneralizesstraightaway
a
in
which
Adams
for
all
n.
For
consider
world
w
instance,
place relations,
existsbutQuinedoesnot.Justas itmakesgoodsenseto saythatthepropofromQuineis trueatw,itis equally
sition[Xa ž q] thatAdamsis distinct
fromQuine
the
jc
sensibleto say that property
[Xjc ^ q] of beingdistinct
holdsofAdamsat w as well.
Thisobservation
requiresthatwe definea generalnotionofa relation's
theextension
functions
extension
atworldsthatcomports
with,butextends,
to haveextensions
usedto defineworldsso as to allowrelationsgenerally
=
atworldsinwhichtheydon'texist.So let*Rw
>0D.andlet<R= KJWG
W^RW
whereWis theclassofall possibleworlds;% thatis,is theclassof"allpossible" PRPs (and hence, given our definitionof worlds,will include
PRPs"involving
"surrogate
surrogate
possibiliaamongtheirconstituents).30
be a function
thatmapstheeleThenfora givenworldw,we letexit 3 extw
ifR = [Xjc~<p],
extensions
in Dw.In particular,
mentsof*Rintoappropriate
=
theproperty
thenextw(R
[Xjcx ï q] andlet
extwiVvc
)
cp]).So consider
wbe theworldjustnotedinwhichAdamsexistsandQuinedoesnot.Thenwe
have extw([kxx ž q]) = T)w-extt{['x x = q]) = Dw- { b e w I (b, q) e
x = y])} = D^- { b e w I (b, q) e extj[hxyx = y])} = Dw- 0 =
exíÜ;([ÁJ9>
x ï q]).
Adamsg extwiV^x
Dw.So inparticular
PRPs are neededexplicitlybecausetheabove
Note that"surrogate"
thatforcesus toconsider
theextenina manner
canbe generalized
argument
Consideragaina
sionsof PRPs thatwouldexistifthingsweredifferent.
worldw in whichJohnPaul II- though,God forbid,notas pope- has a
andhence
worldu in whichthatgrandson,
andconsideranother
grandson,
theproperty
fromhim,does notexist,butAdamsdoesofbeingdistinct
theactualworlditselfwilldo. Then"lookingat" bothw andu we can see
fromJohnPaulII's
ofw,Adamsis no lessdistinct
that,fromtheperspective
137
as fromQuine;i.e.,figuratively,
fromourperspective
onceagain
grandson
andassignan extension
to
we can "project"ourselvesintow's perspective
theproperty
ofbeingdistinct
fromJohnPaulIPs grandson
atu thatincludes
tocomeouttrueevenifwe
Adams.Moregenerally,
we wantthefollowing
believetherecouldhavebeenobjectsotherthanthosethathappenactually
to exist:
(33) DVx(a * Xz>□((~E!jcAE!û) d [Kyx * y]a)).
ourrepresentation
ofthemodal
Thus,inthepossibleworldsthatconstitute
oftheproperties
therewouldbe ifthings
were
facts,we needrepresentations
as indicatedby modallyand quantificationally
embeddedterms
different,
like6[Xyx* yYin (33).
affects
thewaywe evaluate
Thisgeneralization
oftruth-at
significantly
withrespectto a givenpossibleworld.The keydifferatomicpropositions
encebetweentruth-in
andtruth-at
as definedaboveshowedup in theevaluationofnegatedpropositions:
UnlikeP2,A2- thata negatedproposition
[~i|/]is trueat wjustin case [vļi]is nottrueat w- permitted
negatedsingularpropositions
like [~E!#]to be truewithrespectto worldsin whichthey
- viz.,exit- thatallowsrelations
do notexist.Givena notionofextension
withrespecttoworldsinwhichtheydon'texist,
tobe exemplified
generally
we mustnowmodify
Al, which,beingdefinedin termsof extw,
prevents
this.Accordingly
all we needto do is replaceextw
inAl withexit
AI* [Paļ . . . an]is trueatwjustincase(i) n = 0 andext*(P)= T,
or(ii)n> 0 and(av . . . , an)e ext*(P).
A2 andA3 thenremainas before.If we nowthinkin termsofext„ rather
thanextw
thegeneralized
seriousactualismprinciple
GS A no longerholds;
moreexactly,
we can no longerinfer[E'R] fromthetruth
of [Rax. . . an]at
a givenworld.However,sinceext„ stillmapsextensions
intoD^, serious
actualismstillholds.Thatis, whilewe lose (12), tttz> E!tt,we retainSA,
7tT3 E!t. Furthermore,
we havetorelaxtherestriction
on X-conversion
in
allow
for
true
at
worlds
in
which
the
relato
accordingly
predications
tionpredicated
does notexist;so now,itappears,we haveinstead:
=) (E!y=>[Xx<p]y),
where
forall i < n, y.is freeforx.
'£l: cp*
in(p.
in
thatthedenotations
of
However,thiswon'tquitedo. The condition
incp*existintheworldofevaluationguartermsoccurring
all noncomplex
relationin thatworld.Our
anteedthat'[Xx<p]'wouldindicatea legitimate
observation
thatthemetaphor
ofperspective
enablesus toassignextensions
inworldsinwhichtheydon'tthemselves
existcausedus torelax
torelations
thisrestriction
in
this
relaxation
thattheembedHowever,
presupposes
fromsomeperspective,
a "possible"
dedterm'[Xxcp]'willalwaysrepresent,
138
relation;i.e., a relationthatexistsin someworld.Thereis no longerany
itseemsthattherecouldbe "incomthatthiswillbe so. Intuitively,
guarantee
itseemstobe possiblethattherebe some
Moreexactly,
possible"objects.31
x
objectX andpossibletherebe someotherobjecty suchthat,necessarily,
existsonlyify doesn'tand(hence)viceversa;i.e.,
(34) Q3x03y[J(E'xz)~E'y).
Thenthereis no worldw inwhichbothobjectsexist,andhenceitis notposas thefolbothobjectsas constituents,
siblethattherebe a PRP containing
a
theorem
of
our
is
logic:
lowing
(35) DVjOtyQE!* 3 ~Ely) 3 D~E!['z 9]),
V and cy'32Ifwe nowcounta predicawhere9 is anyformula
containing
- as falseatworldsinwhich
7r°,inparticular
tionitt- a 0-placepredication
certaininstancesof
it has no denotation,
yieldinvalidconsequences.
Consider,forexample,
(36) (E'x 3 ~E!y)3 [E!jc3 ~E 'yl
□I andK yield
(37) D(E!jc3 ~E 'y)3 D[E'x 3 ~E 'y],
fromwhichwe derive
(38) DVjOVyQE!* 3 ~E 'y)3 U[E'x 3 ~E'y])
and DI. But iftherecouldbe incompossible
via universalgeneralization
if
is
then
true,
i.e.,
(38) is false,as therecouldnotbe a propo(34)
objects,
sitionof theform[E!jc3 ~E!j] involvingthemas constituents.
Thinking
ofpossibleworlds,if'ť and y
intermsofourrepresentation
semantically
andmodalopervalueswhenwe unpackthequantifiers
takeincompossible
for'[Ebe3 ~E!;y]' no worldconatorsin (38), thenthereis no denotation
ofthe
involvingthosevalues,and so theantecedent
taininga proposition
false.
conditional
wouldbe trueandtheconsequent
of
Now we couldtake(36)-(38) as a logicalproofoftheimpossibility
decide
itself
shouldn't
but
that
seems
untoward;
logic
objects,
incompossible
into
condition
suchissues.A moreseemlysolutionis tobuildan additional
GivenEx, thefollowofincompossibles.
to accountforthepossibility
ingwilldo:
whereforall i < n,
(p£3 ((E!yA 0E!['x 9]) 3 ['x cp]y),
y.is freeforx. in9;
then
i.e.,if9y holds,thenifall they.existand ['x 9] is a possiblerelation,
[Xx 9]y.33Since 0E!['x 9] is provable,(36) now followsas a theorem.
- in particular,
theproof
However,šincetheproofof0E!['x 9] in general
of(36) is notprovable
of0E![E!x 3 ~E!y|- requiresId,34thenecessitation
139
in general,and so theabove proofof (38) is prevented.
Of course,if one
desirestoruleoutincompossibles
so
so
to
say,anyn possible
explicitly that,
are
then
becomes
Id, and
objects compossible, 0E!['x <p]
provablewithout
theaddedcondition
in
becomes
simply
superfluous.
WhatthenofRP in thisrevisedlogic?Notethatas a limiting
case of
we have
(39) ~E 'q 3 (0E![~E'q' 3 [~E'q]'
whichbypropositional
logic,DI', andK yields
(40) D0E![~E!^] z>D(~E!^ Z>[~E!#]).
But0>E![~E!#] is provablefromId, SA, theaxiomE!=, Ex, T, andtheS5
axiom0E![~E!#]z> D0E![~E!^]. Hence,unlikeon ourearlierpicture,
(31) n(~E'qz>[~E'q])
andthus
(41) Q~E'qz>0[~E'q]
are provable,and so themovefrom(1) to (2) is now valid.This is as it
shouldbe onourgeneralized
view,sinceitis nolongerrequired
perspectivai
thata PRP existin a worldtohavea positiveextension
there.However,the
premise
(40 □(~E![~E!tf]3~7T~E!<?])
in thepuzzlenowfails,sinceit requiresthediscredited
(12), specifically,
theinstance7ļ~E!#] z> E![~E!#]; i.e., [~E!g] 3 E![~E!g]. So thelogic
remainssafefromRP.
II: EXTENSIONS
GENERALIZING
TRUTH-AT,
The movefromPI first
toAl thentoAl has provideda broadersense
oftruth
forpropositions
withrespecttoa givenpossibleworldthatpreserves
a muchmorestandard
modallogicwithout
thebasic
quantified
abandoning
Russellianmetaphysics.
The modifications
thatled to Al and Al* were
drivenbythemetaphor
ofperspective,
whichprovidedan intuitive
underof positiveextensionsto PRPs withrespectto
pinningto theassignment
worldsin whichtheydon'texistwithout
violatingactualistscruples.Given
thatmetaphor,
itseemswe cango further
still.Forifwe do indeed
however,
have something
like a perspective
on otherworlds,can we not,fromour
in theactualworld,considera property's
extensionat a given
perspective
worldto encompassnotjust objectsin thatworld,butin otherworldsas
well?Consider,
onceagain,a worldw inwhichQuinedoes notexist.Then,
fromourperspective,
itseemsquitepossibleto considerQuinetohavethe
nonexistenceat- thoughnotin,to be sure- w. As theexample
property
tonegative
relations.
There
shows,onceagainthepointappliesmostdirectly
twowaysin whichI canbe includedin theextension
ofa
are,inparticular,
140
[Kx ~<p]at a worldw: I can existin w andbe amongthe
negativeproperty
that
have
[Xjc<p],orI can simplyfailtoexistinw, andhencefromthe
things
the
perspective
of actualworldcanbe countedamongthethingsthatexemplify[Xjc~Pjc]at w.35
To incorporate
thisidea intoourapparatus
ofpossibleworldsrequires
onlythatwe allow extt to map n-placerelationsintothenthCartesian
of "possibleindividuals".
productof the entiredomainT> =
in
continue
to
be
evaluated
at
worlds
accordancewiththeprinPropositions
Al
in
the
are
First,sinceobjects
ciples ,A2, andA3. Changes
logic minimal.
canbe intheextensions
atworldsinwhichtheydon'texist,SA
ofrelations
nowfails.Andsecond,we candropthecondition
on 'y' inA.^ thatthedenoin
the
world
of
tationsofall theyf.
exist
evaluation;thus,we havesimply
where
foralli < n, y.is free
(p£z>(0E!['x 9] =>[kxcp]y)
forX.in(p.
RP remainsblockedas above,as (12) is stillinvalid.However,without
SA,
itis no longerpossibletoprovesuchdesirabletheorems
as (31). Thiscanbe
remediedbysimplyaddingtherelevantspecialcase ofSA:
term
t
SA=: T = T d E!t,foranynoncomplex
totheeffect
thatanything
intheidentity
relation
withitselfexists.36
standing
All proofsintheprevioussystem
on
the
Id/SA
combination
nowgo
relying
as
before.
through
just
NotewellthattoabandontheaxiomSA is nottoabandonseriousactualismas a metaphysical
anymorethanabandoningPrior'sII
principle,
'~
for
DI'
is
to
abandon
themetaphysics
ofRussellianproposi=>
(l-cp
Hep)
aretheresultant
oftwovectors:ourmetaphysics
tions.Ourlogicalprinciples
we use to evaluate(thepropositions
and thesemanticprinciples
expressed
sentences
of
our
Russellianmetaby)
logical language.Our underlying
in thedefinition
of a possibleworld(Dw, extw)insofar
physicsis reflected
as everypossiblestateofaffairs
mustsatisfy
theprinciple
Ex. Ml reflects
modalsemantics.
botha commitment
to themetaphysics
and an internalist
themovetodi' reflected
As is evidentinthediscussion
ofAdams'approach,
the
no changein thebasic metaphysics;
thedefinition
of worldsremained
itremained
thecase thatall andonlythoseproposame,and,in particular,
sitionsexistwhoseconstituents
alsoexist.Rather,
itreflected
onlytheswitch
to a perspectivai
a changein thewaypropositions
areevaluated
semantics,
withrespectto worlds,as indicatedby theswitchfromthesemanticprinciplesP1-P4 toA1-A3.
of SA (and the
Preciselythesamepointappliesto ourabandonment
a
in
our
semanticprinmovefrom
to X.^). Bothreflect
only change
we
are
abletoderive
not
in
our
Given
the
metaphysics,
ciples,
metaphysics.
on
a broadernotionofexemplification
intermsofthenotionofperspective,
Al*.37
thebasisofwhichwe arriveat ournewsemanticprinciple
141
Of course,another
wayto abandonSA is to abandonthemetaphysics
the
and relationsin a givenworldto takevalues
by allowing properties
outsidethatworld;i.e., in termsof our apparatus,to allow extwto take
andrelations
toextensions
thatincludeobjectsthatdon'texistin
properties
w. Butwhynot?We defineextļ in sucha manner,
whynotextji How do
we showourselvesmorecommitted
toactualismbyrestricting
therangeof
and
then
not
without
such
restriction?
exit
defining
Why justallow
any
extw
the
itself
to
take
values
at
other
worlds
from
outset?
extw
The reasonforthisis thateach worldstructure
w = (Dw, extj in our
framework
in itselfrepresents
means
of surrogates)
a way
(in general,by
All
the
would
have
been
if
that
world
had
been
actual:
things
objectsthere
wouldhave been- theroleof Dw- and theway thatthoseobjectswould
- therole of ex An actualistwill therefore
have been configured
not
tw.
includeanything
in Dwthatis notin theextension
ofexistenceE! (thisis a
condition
onworldstructures,
willnotinclude
recall),anda seriousactualist
but
such
the
extensions
of
and
anything
existingthingsamong
properties
relations.
since
to
be
in
the
extension
of
a
or
relation
is
to
(Indeed,
property
be in somesense,it is hardto see how an actualistcouldnotbe a serious
itprovidesa fixed,
actualist.)Oncewe aregivenan arrayofsuchstructures,
of
the
modal
facts
in
terms
of
whichto cash the
objectiverepresentation
notionofperspective,
andthereby
the
semantic
foundations
forour
develop
alternative
modallogics.Again,though,
as
byrepresenting
things theycould
havebeenwithactuallyexistingobjects,we capturethemodalfactswithoutanyontological
commitment
and
toindividuals,
propositions,
properties,
relations
thatdon'texistinfactbutwouldhaveexistedhadthingsbeenotherwise.In thiswayactualism,
seriousactualism,and theRussellianmetaof
andpropositions
areall preserved
relations,
physics singular
properties,
a
robust
modal
alongside
quantified
logic.
APPENDIX: THE LOGICS
In thisappendixwe assemblethefouractualistlogicsin one place. Let E!
= jc)], and let x and y be arbitrary
abbreviate[Xjc
nonrepeating
sequencesofvariablesofanylengthn > 0. WhereT = Tj . . . Tmis a nonwe writeE!t to abbreviate
repeating
sequenceofterms,
E!tj A . . . A EÎTm.
THEPRIOREAN
SYSTEMQ
The following
modallogic,foris Prior'sfullsystemQ of quantified
mulatedin termsofourricherlanguage.Let M<p=df~0~cp.
Taut: Propositional
tautologies.
PK: ■ (9 =>iļj)3 (ŪE!t =>(■ cpz>■ iļ/)),
T contains
all
where
theterms
freein<pbutnoti|>.
occurring
142
PT:
P5:
cpz>O9.
O93 MO(p.
□/■: D<ps lep A DE! [9].
if* is notfreein9.
Qu: Vx((p3 iļj)3 (9 3 Vjciļi),
t is freefor* in(p.
3 cpi,where
UI:
Vxxp
where
GSA: 9 3 E![cp],
9 is atomic.
=
all andonlythenonEx:
E!['x 9] E!t, whereT contains
thanthex.thatoccurfreein9.
terms
other
complex
T is as inEx.
□Ex: DE!['x 9] = DE!t, where
□E!=: DE!=.
x = x.
Id:
and9' isjustlike9
x = y3 (9 3 9'), where
9 is atomic,
ofx
oneormore(free)occurrences
exceptthatyreplaces
in9, andwhere
yis freeforx in9.
forall i < n,y. is freeforx. in9.
'-con: 9* = [Xx 9] y,where
LL:
axiomsforPRPs.38
PRP: Fine-grainedness
RulesofInference
■I:
^ <p=> Iq m<p.
PMP: Iq<P,ÍqIP3>|i=>Î|>.
PGen: ^ 9 => ^ '/xņ.
THESYSTEMAl
It is easiestto definethissystemandtheonesbelowbasedonAdams'
We start
approachbymeansoftwologics,one a freesublogicoftheother.
ofthe
called
which
consists
a sublogicofbothsystems,
first
SI,
bydefining
following.
Taut: Propositional
tautologies.
K:
T:
5:
FUI:
Qui:
3 (Ū9 3 Q (/).
D(9 3 vļi)
Ū9 3 9.
O93 [HO9.
t thatis freeforx in9.
Vx93 (E!t 3 9?),foranyterm
Vx(93 iļi)3 (V*93
x is notfreein9.
Qu2: 9 3 V*9,where
GSA: 9 3 E! [9],where
9 is atomic.
all andonlythenonEx:
E!['x 9] = E!t, whereT contains
that
terms
other
than
the
complex
xř occurfreein9.
143
E!=:
NI:
Ind:
A^:
E!=.
T = t'z)D(E!T=) T = T').
and9' isjustlike
t = t' d (cpd (pO,where
9 is atomic,
of
oneormore(free)occurrences
that
t' replaces
9 except
t in9, andt' is freefort in9.
forall i < n,y.is freeforx. in9.
['x 9] y z>9* where
Tcontains
thevariables
yřand
X'RL: 9Jz>(E!t z>[Xx9]y),where
freein9J,andwhere
terms
allother
occurring
noncomplex
foralli < n,y.is freeforx. in9.
axiomsforPRPs.
PRP: Fine-grainedness
Let Gl be thesystemthatresultsfromaddingto SI thefollowingaxioms
andrulesofinference:
GId: 'fx(x = x).
term
t.
0E!t,foranynoncomplex
GMP: ^ļ9» »GÍ9=>
^GI^-
OB!:
GGen: ^9 =>^*9.
GNec: ^9 Î^DcpBy Al we willmeanthesystemthatresultsfromaddingto SI thefollowingaxiomsandrulesofinference:
t.
term
t = t,foranynoncomplex
Id:
Gen:
=>
inGl is
Nec: Iqi9 => ^Ckp. (N.B.:Anything
provable
inAl.)
necessary
THESYSTEMA2
Let S2 be thesystemthatresultsfromSI whenwe replaceGSA with
SA:
ttis anypredicate,
tp . . . , inany
iTTļ. . . Tnz>E!t.,where
and1 < i < n,
terms,
is replacedwith
andX.'RL
< n,
'™L: 9jz>((E!y A0E![Xx9])z>[Xx9]y),whereforall/
y.is freeforxiin9.
thatresultsfromaddingtoS2 theaxiomsandrulesof
Let G2 be thesystem
inference
GId, 0E!, GMP, GGen, and GNec above,exceptwiththesub'Gl'
script
replacedby 'G2' .
thatresultsbyaddingtoS2 theaxioms
A2
By we willmeanthesystem
'Gl'
and
andrulesofinference
Nec,exceptwiththesubscript
Id, MP, Gen,
'A2'
'Al'
and
the
subscripts
replacedby
replacedby 'G2'
144
THESYSTEMA3
Let S3 be thesystemthatresultsfromS2 whenwe replaceSA with
term
t.
SA=: T = tdE!t, foranynoncomplex
and
with
X*L:
z>(OE![Xxcp]=) [Xx(p]y),
where
foralli < n,y.isfree
<p*
forX.in(p.
Let G3 be thesystemthatresultsfromaddingtoS3 theaxiomsandrulesof
inference
GId, OE!, GMP, GGen, and GNec, exceptwiththesubscript
'Gl' replacedby»G3'
By A3 we willmeanthesystemthatresultsbyaddingtoS3 theaxioms
'Gl '
andrulesofinference
Id, MP, Gen,andNec,exceptwiththesubscript
4
39,40
'Al' replacedby A3'.
replacedby 'G3' andthesubscript
NOTES
andH.Wettstein,
inJ.Almog,
J.Perry,
1.See,e.g.,D. Kaplan,
eds.,
"Demonstratives,"
Themes
Oxford
Press,
1989);N.Salmon,
(NewYork:
Frege's
University
from
Kaplan
"Direct
Mass.:MITPress/Bradford
Puzzle
Books,
1986);andS. Soames,
(Cambridge,
inN.Salmon
andS.Soames,
andSemantic
Attitudes,
Content,"
Reference,
Prepositional
Oxford
197-239.
andAttitudes
Press,
1988),
eds.,Propositions
(NewYork:
University
49(1981):18.
2. R.Adams,
"Actualism
andThisness,"
Synthese
ofterms
metavariables
toindicate
I willoften
useboldface
3. Asillustrated
here,
sequences
ofthecorresponding
type.
in
inR.L.Martin,
"Truth
andParadox,"
4. See,for
A.Gupta,
ed.,Recent
Essays
example,
H.
Oxford
Truth
andtheLiarParadox
175-235;
Press,
1984),
(NewYork:
University
andR.
inR.L. Martin,
"Notes
onNaiveSemantics,"
op.cit.,133-174;
Herzberger,
455-472.
"ATheory
ofProperties,"
Journal
Turner,
ofSymbolic
Logic52(1987):
I find
their
andR.Turner,
5. Noteveryone
G.Chierchia
arguments
though
agrees,
notably
andPhilosophy
andProperty
seetheir
"Semantics
Linguistics
Theory,"
paper
unpersuasive;
"TheProper
Treatment
of
seeC. Menzel,
11(1988):261-302,
andfora response,
inJ.Tomberlin,
Intensional
Predication
inFine-Grained
ed.,Philosophical
Logic,"
Calif.:
andLogic(Atascadero,
Vol.7:Language
Press,
1993).
Ridgeview
Perspectives,
atlength.
value
and
ofsuch
a language
where
I
for
the
6. Cf.Menzel,
propriety
argue
op.cit.,
E. Zalta,
Oxford
7. SeeG.Bealer,
andConcept
Press,
1982);
(Oxford,
University
Quality
and"AComplete
C.Menzel,
Abstract
Reidel,
1983);
(Dordrecht:
Typeop.cit.,
Objects
no.CSLICSLIReport
FreeSecond-Order'
Foundations,"
LogicandItsPhilosophical
Stanford
andInformation,
fortheStudy
ofLanguage
Calif.:
Center
85-40(Stanford,
1985).
University,
distinction
between
that
theintuitive
ofdual-role
8. Itisanimportant
syntax
advantage
inprediIfX-terms
cannot
occur
canbeexpressed
these
twopropositions
syntactically.
andthemodalization
both
thepredication
then
oneistempted
toexpress
cateposition,
inparticular,
tosort
out
oneunable,
ofthesameabstract
'[X0~E!g]'leaving
bymeans
some
ofthedetails
oftheRussellian
puzzle.
ButasChris
for
ofsemantics
9. Intheusual
Lewis/Stalnaker
counterfactuals,
anyway.
types
necwith
for
some
counterfactuals
isnotintuitive
reminded
me,theequivalence
Swoyer
Athat
false
consider
thenecessarily
false
antecedents.
Forexample,
proposition
essarily
axiomatization.
Ontheusual
arithmetic
hasa complete
recursive
semantics,
anycounterif
false
that
likeAistrue.
Butitisintuitively
antecedent
factual
with
a necessarily
false
145
still
would
havebeenableto
recursive
axiomatization
Godei
hada complete
arithmetic
A'snecessary
falsehood
hissecond
However,
theorem,
notwithstanding.
important
prove
inthe
ofalltheconditionals
astheantecedents
asitis,thisissueisnotrelevant
here,
areintuitively
Russellian
contingent.
puzzle
44(1983):1-20.1should
Studies
"OnExistentialism,"
10.SeeA.Plantinga,
Philosophical
alsoprovide
solutions
toRPinthepresent
that
thethree
non-Priorean
note
replies
paper
on9-10ofthis
article.
constructs
totheanti-existentialist
important
argument
Plantinga
= is
= y)].ByExwehaveE!=z>E![Ax3>>(*= y)](since
E! =df
11. Recall
that
[Xjc3;y(jt
in['jc3y(x= y)],andsobyDI wehaveD(E!=
term
theonly
occurring
noncomplex
= y)]),and
that
z>E![Xjc3j(jc = y)]),andbyK itfollows
DE!= => IZIE!['*3;y(jt
= y)]).
ofE!= wehaveDE![Xjc3;y(;c
sobythenecessitation
nr.By
inthecaseof(12)(hence
12.Toseethis
case),suppose
(11)aswellinthe0-place
Ifitisa primitive
then
GSAwehaveE![ttt].
byEx'#ehaveE!irimmedipredicate,
t'
. Then
from
itis['x cp]
Sosuppose
E![['x <p]T],
byExwehaveE!t',where
ately.
occur
free
in[[Xx<p]
terms
that
allandonly
thenoncomplex
contains
t],andsobyprepoterms
t" contains
thenoncomplex
that
allandonly
sitional
logicwehaveE!t",where
wehaveE!['x cp]
SAfollows
in[Xx9] alone.
SobyExagain
occur
free
,asrequired.
toterms.
with
respect
bysimilar
reasoning
sec.6; A.
Nous8 (1974):211-231,
"Theories
ofActuality,"
13.See,e.g.,R. Adams,
ch.4; and
Oxford
TheNature
Press,
1974),
(Oxford:
ofNecessity
University
Plantinga,
Princeton
TheFoundations
Semantics
J.Pollock,
(Princeton:
University
ofPhilosophical
ch.3.
Press,
1984),
matters
willofcourse
havetoqualify
careful
account
14.Amore
somewhat;
e.g.,D , can't
bea set.Wecould
take
ittobea class,
¿ill
elseitwouldn't
itself
tocontain
beexpected
sets,
berevised
ofaffairs
asanordered
thedefinition
ofa state
butthen
(asclasses
pairmust
worlds
Another
istodefine
ofsetsinstandard
settheories).
cannot
bemembers
option
true
ofalltheelements
ofD p
relative
tosome
states
ofaffairs
tobemaximal
property
"TheTrueModalLogic,"
Journal
cf.C. Menzel,
Logic20(1991):
ofPhilosophical
of
wewillnothavetheluxury
asitis,however,
n.27.Important
331-374,
esp.371-2,
this
issuehere.
pursuing
ofaffairs,
itmight
ofpossible
states
canthemselves
beelements
15.Sincesuchsurrogates
tomapevery
settosomerepresentational
ingeneral
wellbenecessary
systematically
thesetperse.That
a setinitsrepresentational
rolewith
soasnottoconfuse
counterpart
wecanthen
thepure
setsbeoursurrogate
is,forinstance,
represent
possibilia,
letting
anordered
butnecessarily
setsthemselves
counterpart,
say,
pair
existing
bysome
impure
relanumbers
aresets,
theidentity
first
element
isthenumber
0 (or,ifyoubelieve
whose
inturn
berepresented
itself
Sucha pair(0,s) would
tion).
bythepair(0, (0,s)),andso
mentioned
inthedefinition
of
themapping
from
toexisting
on.Thus,
surrogates
objects
forsetsalways
bemapped
thecondition
that
must
alsomeet
states
ofaffairs
surrogates
carehastobetaken
when
conAdditional
totheir
correct
non-surrogate
counterparts.
soastoensure
that
the
forsemantical
worlds
setsofsuchsurrogate
purposes
sidering
are
ofthese
considerations
Thedetails
thesame
roleineachworld.
same
surrogate
plays
inC. Menzel,
fashion
worked
outinrather
"Actualism,
Ontological
excruciating
85(1990):355-389.
World
andPossible
Semantics,"
Commitment,
Ray
Synthese
Greg
Modal
in"Ontology-Free
these
ideashandsomely
hastightened
upandformalized
Arelated,
more
formal
intheJournal
Semantics,"
ofPhilosophical
Logic.
forthcoming
isfound
in"TheTrue
Modal
Logic,"
esp.350-2.
approach
hasnarrow
'thesetofexisting
that
theterm
tonote
16.Itisimportant
scopehere;
objects'
weare
within
thescopeofthepossibility
that
then,
is,itoccurs
Vulgarly
put,
operator.
a setofobjects
that
sethere,
butabout
about
anactually
notnecessarily
existing
talking
hadthings
beendifferent.
would
haveexisted
modal
ownquantified
follows
isvery
closetoPrior's
17.Thelogicthat
contingent
logicfor
inwhich
itis
duetothemore
butismuch
more
expressive
language
explicit
beings,
andprovide
a more
Prior's
I reconstruct
andcriticize
couched.
appealing
logicindetail,
Modal
in"TheTrue
alternative,
Logic."
146
48.1havemodTime
andModality
Oxford
18.A.N.Prior,
Press,
1957),
(Oxford:
University
ified
thequote
tocomport
with
slightly
mynotation.
19.Ibid.
inaddition
toPRPinlogics
a genuine
truth
20.Admittedly,
however,
containing
predicate
with
ofother
intuthis
isproblematic,
a couple
terms,
since,
together
denoting
principle
should
atleasthold
itleadstotheliarparadox.
theprinciple
itive
However,
principles,
don't
themselves
involve
thetruth
insuchlogics
forsentences
like(14)that
predicate.
"Toward
Useful
I,"
Theories,
See,e.g.,R.Turner,
Type-Free
op.cit.;alsoS. Feferman,
inR.L. Martin,
237-287.
op.cit.,
21.Prior,
Time
andModality
,48-9.
150.
22.A.N.Prior,
Clarendon
Past,Present
, andFuture
Press,
1967),
(Oxford:
in
individuals
isweakly
23.Moregenerally,
involving
contingent
possible
anyproposition
*a' wehave
a name
that
ourPriorean
~DE!a
is,where
9 isanysentence
involving
logic,
I- ~ū~9. Toseethis,
let9 besucha sentence
andassume
~DE!û.By□/■wehave
□~<p= ~0<pADE![-9]andhence
logic~D~cp= O9v -□E![~cp].
bypropositional
wehave~DE![~(p]3 ~ū~9. ByCExwehaveDEÍf-íp]z> DE 'a,andthus,
Thus,
sincebyassumption
-DE!a, bypropositional
logicagainwehave-DE![-9]andso
-□-9;i.e.,^9; i.e.,[9]isweakly
possible.
in A. Plantinga,
"On
24.Thisobjection
wasinspired
found
argument
bya similar
seeesp.18-19.
Existentialism,"
tosugandThisness,"
continues
oninthis
25.Robert
"Actualism
19.Adams
Adams,
passage
ofmodal
should
bethought
ofinterms
that
thetruth
ofmodal
gest
singular
propositions
Butbymylights
he
ofthose
bythesubjects
propositions.
properties
actually
possessed
instead
totheperspectivai
never
cashes
thisidea,turning
developed
really
metaphor
below.
22.
26.Ibid.,
27.Ibid.,
24.
for
Modal
Part
IalsoK.Fine,
"Model
28.Cf.,e.g.,R.Adams,
24ff;
Theory
Logic,
op.cit.,
125-156.
TheDERE/DE
Journal
1
DICTODistinction,"
(1978):
ofPhilosophical
Logic
inthis
section
seems
tobetheone
tobenoted
that
while
thesystem
29.Itought
developed
itismarkedly
different
from
theoneheactubehind
Adams'
lurking
intuitively
approach,
likethenonstandard
Priorean
endsuplooking
much
more
at,which
logic
allyarrives
I believe,
lies
Thechief
reason
forthis,
aboveinwhich
□ and0 arenotinterdefinable.
hislogic.
In
intheinadequate
ofthelanguage
Adams
usedtoframe
expressive
power
oftheform
lackofcomplex
ledhimtotakestatements
09(a)to
particular,
predicates
thefine
distinctions
about
a. Thisleft
himunable
tomake
express
singular
propositions
outthe
inlogical
andwhich
arecrucial
tosorting
form
that
ourricher
language
permits,
ina more
28ff.
attractive
fashion.
SeeAdams,
op.cit.,
paradox
- the
- andstudiously
fornow
butI acknowledge
30.Classtalkisconvenient
here,
ignore,
are
factthatsuchtalkis problematic,
sinceworlds
domains,
themselves,
i.e.,their
I assume
with
classtalkthat
involves
classes
themselves.
wecoulddispense
arguably
talkofclasses
ofclasses
ifneed
be.
in"Existence,"
31.See,e.g.,N.Salmon's
discussion
ofNothan
,
Philosophical
Perspectives
Vol.1:Metaphysics
Calif.:
Co.,1987),
49-108,
(Atascadero,
Publishing
esp.
Ridgeview
95ff.
- inthecontext
- easily
but
32.(35)hasinparticular
several
unusual
understood,
looking,
z>~E!y)z>EHE!*z>-Ely]).
instances;
e.g.,[ZIV;y(IIII(E!;t
into
hasbite
inmodal
contexts
inwhich
wequantify
33.Note
that
theadded
clause
here
only
intheactual
and
allthesimple
terms
in'['x 9]'take
values
otherwise
world,
'['x 9]',for
hence
thepredicate
anactual,
hence
relation.
denotes
possible,
wehaveEUAElyAE!=,
E!=that
34.Specifically,
exists,
identity
byId,SA,andtheaxiom
= *)])wehaveE![E!;t
=>~E!/|,andsobythe
andsobyEx(recall
that
E! = J[kxByCy
T axiom
=)-Ely].
schema
wehave0E![Ebc
147
inJ.Tomberlin
andP.vanInwagen,
35.Cf.J.Pollock,
onPossible
Worlds,"
eds.,
"Plantinga
121-144.
Alvin
Reidel,
(Dordrecht:
1985),
Plantinga
besoinourfully
semantics.
While
notobvious
that
this
should
36.Itisperhaps
perspectivai
I don't
relation
with
Pete
itseems
clear
that
stand
in,e.g.,theacquaintance-of
myfriend
winwhich
I don't
andthat
I dostand
initscomplement
with
Lundstrom
ata world
exist,
I don't
intheidentity
with
atw?Isitclear
him
atw,isitclear
that
stand
relation
myself
inwhich
I
that
I ambetter
characterized
asbeing
identical
with
ornotatworlds
myself
isconsistent.
Foritsimplifailtoexist?
sidewith
thelatter,
buttheformer
Myintuitions
cations
onthelogic,
seenote
39.
I think,
hasledsomeactualists
todeny
serious
actual37.Failure
toappreciate
thispoint,
in"Plantinga
onPossible
andN.
Worlds,"
126ff,
ism;see,e.g.,Pollock's
arguments
that
allthey
arereally
isdenying
SA
9Iff.I'dliketothink
"Existence,"
Salmon,
doing
asa logical
principle.
inparticular,
that
PRPswith
distinct
constituents
ordistinct
38.These
axioms
ensure,
logithedefinitions
calforms
arethemselves
distinct.
Sinceitisrather
tedious
tointroduce
needed
toexpress
these
axioms
andsincenothing
ontheir
hinges
precisely,
precise
forthepurposes
I willavoid
them
here.
ofthis
paper,
listing
expression
into
A3theprinciple
that
stand
intheidentity
relation
39.Toincorporate
contingent
objects
with
themselves
evenatwprlds
inwhich
don't
exist
wereplace
theaxiom
SA=in
they
theaxiom
schema
inA3)anddrop
NI(as
S3with
Id (soDt = t willnowbeprovable
from
the
thestronger
t = t' =>Dt = t' becomes
Id,Nec,andInd),replace
provable
inG3,andreplace
theaxiom
schema
axiom
VjcEbc
Id inA3with
E!t,
V*(jt= X) with
forallnoncomplex
terms
t.
toEdZaltafortypically
incisive
andtoJohn
Gibbon
andmy
40.Mythanks
comments,
comments
ona presentation
basedonanearly
draft
atTexas
A&Mfortheir
colleagues
Thanks
Hillfornumerous
a
ofthispaper.
alsotoChris
from
improvements
stemming
meticulous
ofthepenultimate
draft
ofthepaper,
andalsoforhisexemplary
reading
editorial
patience.
148

Singular Propositions and Modal Logic - Philebus

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