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ALVIN PLANTINGA AND PATRICK GRIM
TRUTH, OMNISCIENCE, AND CANTORIAN
ARGUMENTS: AN EXCHANGE
(Received6 May, 1992)
INTRODUCTION (GRIM)
In "Logicand Limitsof KnowledgeandTruth"Nous 22 (1988), 341-
367) I offered a Cantorianargumentagainsta set of all truths,against
an approachto possible worlds as maximalsets of propositions,and
againstomniscience.'The basicargument gainsta set of all truths s as
follows:
Supposethere werea set T of all truths,and considerall subsetsof T
- all membersof the powerset 9 T. To each elementof thispowersetwill corresponda truth. To each set of the power set, for example,a
particularruthT1eitherwill or will not belongas a member.In either
case we will have a truth: hatT, is a memberof that set, or that it is
not.
There will then be at least as many truths as there are elements of
the power set S T. But by Cantor'spower set theorem we know that
thepower
set ofany
set willbe larger hanthe original.Therewill thenbe more truths hanthere aremembersof T, and for anyset of truthsT
therewillbe some truth eftout. Therecanbe no set of all truths.
One thing this gives us, I said, is "a short and sweet Cantorian
argumentagainstomniscience."Were there an omniscientbeing, what
that being would know would constitutea set of all truths.But there
can be no setof alltruths,and so canbe no omniscientbeing.
Such s thesetting orthefollowing xchange.2
1. PLANTINGA TO GRIM
My mainpuzzleis this:whydo you think the notion of omniscience,orof knowledgehavingan intrinsicmaximum,demands hat there be a setof all truths?As you point out, it's plausibleto think there is no such
PhilosophicalStudies71: 267-306, 1993.? 1993 KluwerAcademicPublishers.Printed n the Netherlands.
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268 ALVIN PLANTINGA AND PATRICK GRIM
set. Still, thereare truthsof the sort:everyproposition s true or false
(or if you don't think that'sa truth,everyproposition s either true or
not-true).Thisdoesn'trequire hattherebe a set of all truths:whybuy
the dogma that quantificationessentially involves sets? Perhaps it
requires hat there be a propertyhadby all andonlythosepropositionsthataretrue;but so far as I can see there'sno difficulty here.Similarly,
then,we may supposethat an omniscientbeing like God (one that has
the maximaldegree of knowledge)knows every true propositionand
believesno false ones. We mustthen concede thatthereis no set of all
the propositionsGod knows.I can'tsee that thereis a problemherefor
God's knowledge; n the same way, the fact that there is no set of all
truepropositions onstitutesno problem, o far asI cansee,for truth.So I'm inclined to agree that there is no set of all truths,and no
recursivelyenumerablesystem of all truths. But how does that show
thatthere s a problem orthe notionof abeingthatknowsalltruths?
2. GRIM TO PLANTINGA
Here are somefurtherhoughtson theissuesyouraise:
1. The immediatetarget of the Cantorianargument n the Nous
piece is of course a set of all truths,or a set of all that an omniscient
being would have to know. I think the argument will also apply,
however,againstany classor collectionof all truthsas well.In the Nous
piece the issue of classes was addressed by pointing out intuitive
problems and chronictechnical imitations hat seem to plagueformal
class theories.But I also thinkthe issue can be broachedmoredirectly- I think something ike the Cantorianargumentcan be constructedagainstany class, collection, or totalityof all truths,and that such an
argumentcan be constructedwithoutany explicit use of the notion of
membership...
2. I take your suggestion,however,to be more radicalthansimply
an appeal to some other type of collection 'beyond'sets. What youseem to want to do is to appealdirectly o propositionalquantification,
and of the options available n response to the Cantorianargumentthink hat s clearly hemostplausible.
In the finalsection of the Nous piece, however,I tried to hedge myclaimhereabit:
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 269
Is omnisciencempossible?Withinany logic wehave,I think, he answer s 'yes'.
The immediateproblemI see for any appealto quantification s a
way out - withinany logic we have - is that the only semantics wehave for quantifications in terms of sets.A set-theoretical emantics
for any genuine quantificationover all propositions,however,would
demanda set of all propositions,and any such supposedset will fall
victim to preciselythe same type of argumentevelledagainsta set of
all truths.Withinanylogic we have there seemsto be no placefor any
genuine quantificationover 'all propositions', hen, for precisely the
samereasons hat here s no placefor a set of alltruths.One might of course construct a class-theoreticalsemantics for
quantification.But if I'm rightthat the same Cantorianproblemsface
classes, that won't give us an acceptablesemanticsfor quantification
over 'allpropositions' ither.
Givenany availablesemantics or quantification,hen - and in that
sense 'within any logic we have' - it seems that even appeal to
propositionalquantification ails to give us an acceptablenotion of
omniscience. What is a defender of omniscience to do? I see two
optionshere:
(A) One mightseriously ry to introducea new and bettersemantics
for quantification. thinkthis is a genuinepossibility, houghwhat I've
been able to do in the area so far seems to indicatethat a semantics
with the requisitefeatureswould have to be radicallyunfamiliar n a
number of importantways. (I've talkedto ChristopherMenzel about
this in terms of my notion of 'plenums',but furtherworkremains o bedone.) I wouldalso want to emphasize hat I thinkthe onus here is on
the defenderof omniscienceor similarnotions to actuallyproduce such
a semantics anoffhandpromissory oteisn'tenough.
(B) One might, on the other hand, propose that we do without
formalsemanticsas we know it. I takesuch a moveto be characteristic
of, for example,Boolos' direct appeal to pluralnoun phrases of our
mothertongue n dealingwithsecond-orderquantifiers.Butwith an eyeto omniscience I'd say something ike this would be a proposal for anotionof omnisciencewithout' ny ogicwe have,rather han within'.
I'malsounsure hateven anappeal o quantificationwithoutstandard
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270 ALVIN PLANTINGA AND PATRICK GRIM
semanticswill work as a responseto the Cantoriandifficultiesat issue
regarding'all truths'. Boolos' proposal seems to me to face some
importantdifficulties,butthey maynot be relevanthere.Morerelevant,I think, is the prospectthat the Cantorianargumentagainst all truths'
can be constructedusingonly quantification nd some basic intuitionsregarding ruths - without,in particular,any explicitappeal to sets,
classes,orcollectionsof any kind.
3. Consider or examplean argumentalongthe following ines,with
regard o yoursuggestion hat theremightbe a property hadby all and
onlythosepropositionshatare true:
ConsideranypropertyT which is proposedas applying o all andonly truths.Withoutyet decidingwhetherT does in fact do what it is supposedto do, we'll call all thosethings o whichT doesapply 's.
Consider urther 1) a propertywhich in factappliesto nothing,and(2) all proper-ties that applyto one or more t's - to one or more of the thingsto which T in factapplies.[we couldtechnicallydo without 1) here,butno matter.]
We can now show thatthere are strictlymorepropertiesreferred o in (1) and (2)above than thereare t's to whichour originalpropertyT applies.The argumentmightrunasfollows:
Supposeany wayg of mapping 'sone-to-one to propertiesreferred o in (1) and(2)above. Can any such mappingassigna t to everysuch property?No. For considerin
particularhepropertyD:
D: the propertyof beinga t to whichg(t) - the property t is mapped ontoby g - does not apply.
Whatt couldg maponto propertyD? None. For suppose D is g(t*) for someparticulart*;does g(t*)applyto t*or not? If it does, since D appliesto only those t for whichg(t)does not apply,it does not applyto t*. If it doesn't,since D appliesto all those t forwhichg(t) does not apply,it does apply.Eitheralternative,hen, gives us a contradic-tion.Thereis no wayof mapping 's one-to-oneto propertiesreferred o in (1) and(2)thatdoesn't eave somepropertyout: herearemore suchproperties hanthereare t's.
Note thatfor eachof the propertiesreferred o in (1) and (2) above,however, herewill be a distinct truth:a truthof the form 'propertyp is a property', or example,or'propertyp is referredto in (1) or (2)'. There are as many truths as there are suchproperties, hen,butwe've also shownthattherearemore suchproperties han t's,andthus there must be more truths thanthereare t's - more truths thanour propertyT,supposed o apply o all truths, n factapplies o.
This form of the Cantorianargument, think,relies in no way onsets or any other explicitnotion of collections.It seems to be phrased
entirely n terms of quantificationndturnssimplyon notions of truths,of properties,and the fact thatthe hypothesisof a one-to-onemappingof a certain sort leads to contradiction. t is this type of argument hatleads me to believe that Cantoriandifficultiesregarding all truths'go
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 271
deeperthan is sometimessupposed; hat the argument ppliesnot only
to setsbut to all typesof collectionsand thatultimately venquantifica-
tion failsto offer awayout.
4. Letmeturn,however, o anotherpassage nyourresponse:
Still, hereare truthsof thesorteverypropositions true or false ...
What the type of argumentoffered above seems to suggest, of
course,is that there can be no realquantificationver 'allpropositions'.
One casualtyof such an argumentwould be anyquantificationalutline
of omniscience.It must be admitted that anothercasualtywould be
'logical aws'of theformyou indicate.
5. By the way,it's sometimesraisedas a difficulty hat an argumentsuchas the one I'vetriedto sketch above itselfinvolveswhatappear o
be quantificationsver all propositions. thinksuchan objectioncould
be avoided,however,by judiciouslyemploying carequotesin orderto
phrase the entire argument n terms of mere mentions of supposed
'quantificationsverallpropositions', orexample.
3. PLANTINGA TO GRIM
Let me just say this much.Your argument eems to me to show, not
that thereis a paradox n the ideathatthereis somepropertyhadby all
true propositions,but ratherthat the notion of quantifications not to
be understood n terms of sets. Your argumentproceeds in terms of
mappings,1-1 mappings,and the like ("Supposeany way g of mapping
t's one-to-one to properties referred to in (1) and (2) above . . ."); but
these notions are ordinarily hought of in termsof sets and functions.
Furthermore,you invoke the notion of cardinality;you propose to
argue that "thereare more propertiesreferredto in (1) and (2) than
there are t's to whichour originalpropertyT applies";but cardinality
too is ordinarily houghtof in terms of sets. (And of course we are
agreeingfrom the outset that there is no set of all truths).I don't see
anywayof statingyour argument on settheoretically.
If we thinkwe have to employ the notion of set in order to explainor understandquantification, hen some of the problems you mention
do indeedarise;butwhythinkthat? The semanticsordinarily iven for
quantificationalready presupposes the notions of quantification;we
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272 ALVIN PLANTINGA AND PATRICK GRIM
speak of the domainD for the quantifierand then say that '(z) Az' is
truejust in case every memberof D has (or is assignedto) A. So the
semanticsobviouslydoesn't ell us whatquantifications.
Further, t tellsus falsehood:what it reallytells us is that'Everything
is F' expresses he proposition hat each of the thingsthatactuallyexistsis F (and is hence equivalent o a vastconjunctionwhere for eachthing
in the domain,there is a conjunct o the effect that thatthingis F). But
that sn't n fact true.If I say'Alldogsaregood-natured'heproposition
expresscould be false even if thatconjunctionwere true. (Considera
state of affairs 3 in whicheverything hat existsin a (the actualworld)exists, plus a few moreobjectsthatareevil-tempereddogs;in thatstate
of affairs the propositionI express when I say 'All dogs are good-natured' s false,but the conjunctionn question s true.)The proposi-
tion to whichthe semanticsdirectsourattentions materially quivalent
to the propositionexpressedby 'All dogs are good-natured'but not
equivalent o it in thebroadlyogicalsense.
So I don't think we need a set theoretical emantics or quantifiers;
don't think the ones we have actuallyhelp us understandquantifiers
(theydon'tget thingsrightwithrespect to the quantifiers);ndif I haveto choose between set-theoreticalsemantics for quantifiersand thenotionthat it makesperfectlygood sense to say,for example, hatevery
proposition s either rueornot-true, 'llgiveuptheformer.
4. GRIM TO PLANTINGA
You pointout thatthe argument offeredin termsof properties s still
phrasedusingmappingsor functions,one-to-onecorrespondences, nda notionof cardinality, nd thatthese areordinarilyhoughtof in termsof sets."Idon'tsee anyway,"you say,"ofstatingyourargumentnon set
theoretically."
I do. InfactI don'tconsider heargument o be statedset-theoreticallyas it stands,strictlyspeaking; t's a philosophicalratherthan a formalargument. n orderto escape anylingering uggestionof sets, however,
we can also outline all of the notions you mention entirely in termsmerely of relations - properties applying to pairs of things - and
quantification. don'tsee anyreasonforyou to objectto that;you seem
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 273
quite happy with both propertiesand quantificationover properties
generally.
A relationR gives us a one-to-one mapping rom those thingsthathavea propertyPI into thosethings hathave a propertyp2 ust ncase:
VxVy[Plx&ply &3z(P2z&Rxz&Ryz) - x =y]
& Vx[Plx - 3yVz(p2z& Rxz z = y)].
A relation R gives us a mappingfrom those thingsthat are PI that is
one-to-one and onto those things that are p2 just in case (here we
merelyadd aconjunct):
VxVy[Plx& Ply & 3z(p2z & Rxz & Ryz) - x = y]
&Vx[Plx - 3yVz(p2z&Rxz z = y)]& Vy[P2y 31xPx& Rxy)].
We can outlinecardinality,inally,simplyin terms of whetherthere is
or is not a relation that satisfies the first condition but doesn'tsatisfythe second. I'mnot sure thatwe mightnot be able to do withouteven
that - I'mnot sure we couldn'tphrasethe argumentas a reductioon
the assumptionof a certainrelation,for example, withoutusing anynotionof cardinalitywithin heargument t all.
I don't agree, then, that the argumentdepends on importingsome
kind of major and philosophically oreign set-theoreticalmachinery.
Notions of functionsor mappingsand one-to-onecorrespondencesare
central to the argument,but in the sense that these are required heycanbe outlinedpurely n terms of relations or propertiesapplying o
pairsof things- andquantification.Cardinality,f we need it at all,can
be introducedna similarlynnocuousmanner.You also suggestseveralother reasons to be unhappywith a set-
theoreticalsemantics for quantification, nd end by sayingthat "if Ihave to choose betweenset-theoreticalemantics or quantifiers ndthenotionthat it makesperfectlygood sense to say, for example, hatevery
proposition s either rueornot-true, 'llgive uptheformer."Theremay or may not be independentreasonsto be unhappywith
set-theoretical emanticsfor quantifiers I thinkthe points you raiseare interestingones, and I'll want to think about them further.Myimmediatereactionis that the first point you make does raise a very
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274 ALVIN PLANTINGA AND PATRICK GRIM
importantquestionas to what formal semanticscan honestlyclaimor
be expectedto do, and I'mvery sympathetic o the notion that it has
sometimesbeen treatedas something hatit neither is nor can be. My
guess is that the second issue mightbe handledin a numberof ways
familiar romdifferentapproaches o possibleworlds,withoutany deepthreat to set-theoretical emantics.But I could be wrongaboutthat -
asI say,I'llwant o thinkabout hesequestions urther.
Even if there are independent reasons to be unhappywith set-
theoreticalsemantics,however,I think your final characterization f
availableoptions is off the mark.For reasonsindicatedabove,I think
sets aren'tessentialto the type of Cantorianargumentat issue - the
argumentcan for examplebe phrasedentirelyin termsof properties,relations,and quantification.f that'sright,however, the basic issue is
not one to be settledby some choice betweenset-theoretical emantics
and, say, 'all propositions'.The problemsare deeper than that:even
abandoning set-theoreticalsemanticsentirely, it seems, wouldn'tbe
enough o avoidbasic Cantorian ifficulties.
5. PLANTINGA TO GRIM
Right:we can definemappingsandcardinalities s you suggest, n terms
of properties rather than sets. We can then develop the property
analogueof Cantor'sargument or the conclusionthat for any set S,
P(S) (the powersetof S) > S asfollows.
Say thatA* is a subpropertyof a propertyA iff everything hat has
A*has A; and say thatthe powerpropertyP(A) of a propertyA is the
propertyhad by all andonlythesubproperties f A.
Now suppose thatfor some A andits powerpropertyP(A),there is a
mapping 1-1 function) from A onto P(A).Let B be the propertyof A
suchthata thingx has B if andonlyif it does not havef(x).There must
be aninverse magey of B underf;and y willhave B iffy does not have
B, which s too muchto putup with.
But if P(A)exceedsA in cardinality or anyA, thentherewon't be a
propertyA had by everything; or if therewere, it would have a powerpropertythat exceeds it in cardinality,which is impossible.So there
won'tbe a propertyhad by everyobject, andtherewon't be a property
hadby everyproposition.Hence if we thinkquantifiersmustrangeover
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 275
something,eithera set or a property,we won'tbe able to speakof all
propositionsor of allthings.
But of course the Cantorianpropertyargumenthas premises,andit
mightbe that some of the premisesare such that one is less sure of
them thanof the proposition,e.g., thatevery proposition s eithertrueor not true, or that everythinghas the propertyof self-identity.In
particular,nepremiseof the Cantorian rgument s stated s
(a) For any propertiesA and B and mapping from A onto B,
there exists the subpropertyC of A such thatfor any x, x has
C if and onlyifx has A andx does not havef(x)
This doesn't seem at all obvious. In particular,suppose there areuniversalproperties not being a marriedbachelor,for example,and
suppose the mappingis the identity mapping.Then there exists that
subpropertyC if and only if there is such a propertyas the property
non-selfexemplification which we alreadyknow is at best extremely
problematic.
So which is more likely: that we can speak of all propositions,
propertiesand the like (and if we can'tjust how are we understanding
(a)?), or that (a) is true?I thinkI can more easily get alongwithout
(a).
One finalnote. These problems don't seem to me to have anything
special to do with omniscience.One who wants to say what omnis-
cience is will have difficulties,of course, in so doing withouttalking
about all propositions.But the same goes for someone who wants to
hold thatthere aren'tany marriedbachelors,or thateverything s self-
identical. f we accept the Cantorianargument,we shallhave to engagein uncomfortablecircumlocutionsn all these cases, circumlocutions
suchthatit isn'tat all clear thatwe can use them to saywhat we take to
be the truth. But the problemwon't be any worse in theology than
anywhere lse.
The best coursethough Ithink) s to reject a).
6. GRIM TO PLANTINGA
I think your response to the Cantorianpropertyargument s an inter-
estingone. Here howeveraresome further houghtson theissue.
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276 ALVIN PLANTINGA AND PATRICK GRIM
Letmestartwitha reminder s to wherewe stand.
The Cantorian ropertyargument syou present t is asfollows:
Say thatA* is a subpropertyof a propertyA iff everything hathas A* has A; and saythat the power propertyP(A)of a propertyA is the propertyhad by all and only thesubproperties f A.
Now suppose that for some A andits power propertyP(A),there is a mapping 1-1function)f from A onto P(A).Let B be the propertyof A such thata thingx has B ifand only if it does not have f(x).There mustbe an inverse mage y of B underf; andywill haveB iffy does not have B, which s too muchto put upwith.
As this stands,of course, it is merely an argument hat the power
propertyP(A)of anypropertyA will have a wider extensionthandoes
A. But if we suppose A to be a propertyhad by all properties,or apropertyhad by all things,we willget a contradiction.There can be no
suchproperty.. . or so theargumenteemsto tellus.
The escape you propose here is essentiallya denial of the diagonal
property required in the argument.Given some favored universal
propertyA and a chosen function f, what the argumentdemands is a
propertyB 'that s a subproperty f A such thata thingx has B if and
only if it does not have f(x).' But there is no such property.The
argumentdemandsthat thereis, and so is unsound.Or so the strategy
goes.
Somewhatmore generally, he strategy s to deny any principlesuch
as(a) thattells us thattherewillbe a property uch as B:
(a) For anypropertiesA andB mapping from A onto B, there
exists the subpropertyC of A such that for any x, x has C if
and onlyifx has A andx does not have f(x).
(a) "doesn't eem at all obvious,"you say."Ithink we can easily get
along without (a)."
I don'tbelieve that thingsare by any means that simple.Here I have
two fairly nformalcommentsto make, followed by some more formal
considerations:
1. As phrasedabove, I agree, (a) is hardlyso obvious as to compel
immediateand unwaveringassent. The diagonal propertiesdemandedin formsof the argument imilarto yours above - propertiessuch as
'B, a subpropertyof A, the property being a property', hat applies to
all and only those thingswhichdo-not have the property (x) mapped
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 277
onto them by our chosen functionf' - may similarly ack immediate
intuitiveappeal.
I think this is largely an artifactof the particular orm in which
you've presented the Cantorianargument,however. Yours follows
standardset-theoreticalargumentsvery closely, completefor examplewith a notationof 'powerproperty'.The argumentbecomes formally
remoteand symbolicallypricklyas a result,and the diagonalproperty
called for is offered in termswhich by their mere technicalformality
maydullrelevantphilosophicalntuitions.
But the Cantorianargumentdoesn'thave to be presentedthatway.It can, for example, be phrasedwithoutany notion of power set or
power propertyat all-
on this see "OnSets and Worlds,a Reply toMenzel" .. . .4 When the argument is more smoothly presented, more-
over, the diagonalconstructed n the argumentbecomes significantlyharder to deny. Consider for examplean extractfrom a form of the
argumenthatappeared arliernourcorrespondence:
ConsideranypropertyT which is proposed as applying o all andonlytruths.Withoutyet deciding whetherT does in fact do whatit is supposedto do, we'll call all thosethings o whichT doesapply 's.
Considerfurther(1) a propertywhichin fact appliesto nothing,and (2) all prop-erties thatapply to one or more t's - to one or more of the thingsto which T in factapplies ..
We can now show thatthere are strictlymore propertiesreferred o in (1) and (2)above thanthereare t's to whichouroriginalpropertyT applies ..
Supposeanyway g of mapping 's one-to-oneto properties eferred o in (1) and (2)above. Can any such mappingassigna t to everysuch property?No. For consider inparticularhepropertyD:
D: the propertyof beinga t to whichg(t) - the property t is mapped ontoby g - does not apply.
What couldgmapontopropertyD? None ...
Consideralso thefollowingCantorian rgument:
Cantherebe apropositionwhich s genuinelyabout allpropositions?No. For supposeanypropositionP, andconsiderall propositions t is about.These
we willtermP-propositions.Were P genuinelyabout all propositions,of course, there would be a one-to-one
mapping fromP-propositionsonto propositions impliciter: mapping whichassigns
P-propositions to propositions one-to-one and leaves no proposition without anassignedP-proposition.
But there can be no such mapping.For suppose there were, and considerall P-propositionsp such that the propositionto which they are assignedby our chosenmapping- their (p)- is notabout hem.
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278 ALVIN PLANTINGA AND PATRICK GRIM
Certainlywe can form a propositionabout preciselythese - using propositionalquantificationnd 'A' o represent about', propositionof thefollowing orm:
Vp((Pp & - A(f(p))p - ... p ... )
ConsideranysuchpropositionPd.WhatP-proposition ouldf maponto it?
None ...
In the first argument, think,we have an eminently ntuitiveprop-
erty: he propertyof beinga t to which a corresponding ropertywe've
imagineddoes not apply.In the second,we have an eminently ntuitive
proposition.Thereare,it seems clear, P-propositionswhichwon'thave
a correspondingpropositionthat happensto be about them. Isn't that
itself a proposition hatis about hem?
The general point is this. In order for a strategyof denying thediagonal o proveeffectiveagainstall offending orms of the Cantorian
argument,one would have to deny an entire rangeof propertiesand
propositionsand conditions and truths liable to turnup in a diagonal
role. Some of these, I think, will have an intuitive plausibilityfar
strongerthan that of the formal constructionyou offer in your more
formalrenditionof theargument bove.
The diagonalsat issuewillalways nvolvea function or a relationRsupposed one-to-one from one batch of things onto another. Such
functionsor relationsalone, we've agreed, seem entirely nnocent.But
passagessuch as the following,fromotherimaginableCantorianargu-
ments,seemintuitivelynnocentas well:
f is proposed as a mappingbetween known truths or truthsknown by some individualG) andall truths.Some known truthswillhave a corresponding-truthon thatmapping
that is about them.Somewon't.Surely herewill be a truthabout all those that don'tthe truth hattheyall aretruths, orexample.
f is proposedas a mappingbetween a group G of propertiesand all properties.SomeG-propertieswill have correspondingpropertiesby f that in fact apply to them. Somewon't.Considerall those thatdon't,and consider he property hey thereby hare ..
f is proposed as a mappingbetween (i) the thingsa certain act F is a fact about and (ii)all satisfiableconditions.Some thingsF is about will thereby be mapped onto condi-tions they themsleves satisfy. Some won't. Consider the condition of being somethingthathasanf-correlatet doesn'tsatisfy ..
Each of these is the diagonalcore of a Cantorianargument: gainst
the possibility of all truths being known truths, against any compre-
hensive grouping of all properties, and against any fact about all
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 279
satisfiableconditions.Whenpassagessuch as these are offeredstep by
step and in full philosophicalform, I think, the truth,property,and
satisfiableconditionthey call for are very ntuitive.How, one wantsto
ask, could there not be such a truth,or such a property,or such a
condition?I don'tbelieve, therefore, hat the situation s one in whichI have a
formal argumenton my side and you have the intuitionson yours.
Althoughsomewhatcomplex,the Cantorian rgument anbe presented
as a fully philosophicalargumentwith significant ntuitive force. I'm
also willing o admit hatthere are at least initial ntuitions hatsomehow
truths should collect into some totality,or thatthere should be an 'all'
to the propositions.Whatwe seem to face, then,is a clash of intuitions.But it is a genuineclash of intuitions,I think,withgenuinelyforceful
intuitionsonboth sides.
2. There s also a furtherdifficulty.Consideragain he basicstructure
of our earlierargumentagainsta propertyhad by all and only truths.
Essentially:
1. We consider a propertyT, proposedas applying o all and only truths,and call
thethings t doesapply o t's.2. We can showthat thereare strictlymore propertieswhichapplyto one or more
t's than there are t's. For suppose any way g of mappingt's one-to-one onto suchproperties, ndconsider nparticularhepropertyD:
D: thepropertyof beinga t towhichg(t) does notapply.
What could g mapontopropertyD? None ...3. There are thenmorepropertieswhichapply to one or more t's than thereare t's.
But for each suchproperty here is a distinct truth.Thus there are moretruths han t's:contraryo hypothesis,T cannotapply o all truths.
Here the strategy you propose would have us deny the diagonal
propertyD.
If there is no suchproperty,however, he conditions aid downin (2)
above are conditionswithout a correspondingproperty. Being a t to
which...' is merely a stipulation, r a set of conditions,or a specifica-
tion that ailsof propertyhood.
Given any of these, however,we will be able to frame a Cantorianargumentwithmuchthe sameformand to preciselythe sameeffect asthe original.For at (2) we can showthat thereare strictlymorestipula-
tions, or sets of conditions, or specifications property-specifying r
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280 ALVIN PLANTINGA AND PATRICK GRIM
not - than thereare t's.But (in 3) therewillbe a distinct ruth or each
of these,and thusmore truths hant's. WhateverpropertyT appliesto,
we concludeasbefore, t cannotapply o all truths.
The problemseems to grow.If we deny a supposed diagonalprop-
erty D property-hood,or a proposed diagonal truth D truth, or aproposed diagonalpropositionD propositionality,we'llstillwant to say
whatD in each case is instead- a propertyless onditionor a pseudo-
truthor a mere logical form short of propositionality r the like. But
given any answer here, it appears,we'll be able to frame a further
Cantorianargumentof muchthe same form and to preciselythe same
effect astheoriginal.
I think of these as Strengthenedorms of the Cantorianargument,analogousnimportantwaysto Strengthenedormsof the Liar.
3. Let me also offer some thoughts rom a somewhatmore formal
angle. Here I'll start with a considerationthat is admittedlymerely
suggestive:
The argumentswe're dealing with, of course, parallel Cantorian
argumentsof major importance n set theory and numbertheory.A
strategy f 'denialof thediagonal'willalso have aparallel here.
Deny the diagonal n number theory, however,and you face some
devastating onsequences.The doorto Cantor'sparadise s immediately
closed.Standard onstructionof the reals fromthe rationals s blocked,
we renounce canonicalresults regarding he transfiniteand fixed point
theoremsand the like, and the locus classicusof Godel's and Lob's
theoremsvanish.Ourmathematical orldshrinks.
We haven'ttakenthat pathin mathematics, nd I think there would
be general agreementwe should not. But why then choose the analo-
gous pathhere?
4. There are also some significantly trongerarguments rom a more
formalperspective.
Your proposal,phrasedwith respect to a particularCantorianargu-
ment, is to deny the existence of a diagonalpropertystipulated n the
courseof thatargument.
If such a strategy is to apply to offending Cantorian argumentssystematicallyand in general, however, rather than being applied
merelyad hoc on the whimof the wielder,we need a principle hat will
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 281
tell us whichdiagonalproperties,propositions, ruths,or conditionsto
reject.Youdon't,quiteclearly,want o rejectall diagonals.
Here there s a nestedpairof problems.
The first is simplythatwe'vebeen offered no satisfactory rincipleof
such a sort, and I doubtvery muchthatanyonewill in fact be able toproduceone.
The second problemis deeper.There are of course deep affinities
between the Cantorianresultsat issue here and certainaspectsof the
classical paradoxes.If those affinitieshold, I think, we can bet that
any principlethat was proposed as an exhaustiveconditionof what
diagonals to accept and what to reject would face a crucial and
devastatingest case constructed n its own
terminology.f
so,it's
notmerelythata comprehensiveprincipleas to whichdiagonals o accept
and which to reject hasn'tin fact been offered.If importantparallels
hold,it'srather hatno suchprinciplecouldbe offered.
If we are in fact givenno principle o guide a strategyof denying he
diagonal,I think, such a strategy can only be applied in a manner
bound to be rejected as unprincipledand ad hoc. If I'm right that no
adequate principle can be given, of course, any such strategywill
moreoverbe essentiallyandinescapably d hoc.
5. As you note, a principlewhich tells us that there is the diagonal
propertyyourformof theargumet equires s (a):
(a) For any propertiesA and B and mapping from A onto B,
there existsthe subpropertyC of A such thatfor any x, x has
C if and onlyifx hasAandx does not havef(x).
Youwantto deny the diagonal,and so deny (a).
(a) itself, however- like similarprinciplesrelevantto otherforms
of the argument strictlyfollows from some very elementaryand
strongly ntuitiveassumptions.
In set- and class-theory, he straightanalogue or (a) follows essen-
tially from just the power set and separationaxioms alone. Lurkingust
beneath the surface in our correspondence has been an incipientpropertytheory. But whichof the following intuitiveprincipleswould
you denyto properties?:
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282 ALVIN PLANTINGA AND PATRICK GRIM
(1) PropertyComprehensionor subset,orseparation):
If there is a propertyPI, there is also the propertyp2 that
applies to just those P1 things that are 0. (for expressible
conditions 6)
(2) PowerProperty:If there is a propertyPI, there is also the propertyp2 that
applies to the subpropertiesof P1. (here we can use your
definitionof 'subproperty')
Each of these is just as intuitiveregardingproperties,I think, as
regarding ets. There seems no stronger ntuitivegroundto deny either
here than nthe caseof sets.The situation s even tighter hanthis,however.
Of the two principlesabove,the moreplausiblecandidate or denial
is surely (2). But as it turns out, 'powerproperty' sn'tin fact required
for a form of the argumentagainst,say,a notion of truth's otalityor a
property distinctivelycharacteristic f truths.Comprehensionalone is
sufficient(on this once again I call your attentionto "On Sets and
Worlds").
In order to escape a Cantorianargumentand save a propertyof all
and only truths, say, we would have to put majorlimitationson any
principleof comprehensionorproperties.
That is of course essentiallywhatwas done in avoidingRussell'sand
Cantor'sparadoxesby the creationof ZF set theory.Here our restric-
tions on Comprehensionwould have to be still tighter,however,tied to
properties,propositions, ruths,andthe like.
The resultof such a restriction n ZF, however, s an explicitsacrifice
of Cantor's set of all sets'.Trya similarrestrictionn property heory,I
think,andcomprehensionherewill fail to countenanceany propertyof
all properties, any proposition regardingall propositions, any truth
aboutalltruths,and the like.
The point can also be put another way. If one could specify a
restrictionon property comprehensionwhich would intuitivelyescape
our Cantorianarguments hroughoutand yet would allow for a prop-ertyof all properties,a propositionabout all propositions,and the like,
we could predictably ead off it a formof set theory that would escape
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 283
Cantor'sparadoxwhile includingCantor'sset. I'm sure that the set
theoristswould ove to hearabout t.
6. Finally,however, et me agreeon an importantpoint.The prob-
lems at issue here are not uniqueto theology,andI've never said they
were. These are problems quite generallyregardingmetaphysicalandepistemologicalnotions of the widest scope. Such concepts appearin
philosophical heologyas well as elsewhere.
At the end of your comments,you suggest hat Cantorian rguments
may force us to "uncomfortableircumlocutions"n a wide rangeof
cases.I want to emphasize hat the centralproblemsat issue here seem
to me fardeeperthan that.These are realconceptualproblems,as solid
as contradiction.They're problemsfor metaphysicsand epistemology
generally,ratherthan for philosophical heology alone,but they'renot
problems hatany circumlocution, oweveruncomfortable,s genuinely
goingto resolve.
7. PLANTINGA TO GRIM
I think we may have gone about as far as we can go here;from here on
we may find ourselvessimplyrepeatingourselves;perhapswe shalljust
have to agree to disagree.I'd like to summarizebriefly how I see the
situation as a result of our discussion, and introduceone additional
consideration.
First,a Cantorianargument or the conclusion that no propositions
are about all propositions or properties (that no propositions are
genuinelyuniversal)will typically nvolve a diagonalpropertyor propo-
sition;I propose that in every case it will be less unlovely, intuitivelyspeaking, o deny the relevantdiagonalpremisethan to accede in the
conclusion.(More on the unlovelinessof the conclusion below.) You
remark,quite correctly, that not all diagonal propositions and prop-
erties are to be rejected, and it seems at best extremelydifficult and
maybe impossible to give general directions as to which diagonal
propositionsand properties o accept and which to reject.You go on to
say, however, that the rejection of a given diagonalpremise will be"unprincipled",atthe whim of the wielder"and "adhoc".This doesn't
seem to me to follow. First, the course I suggest is not unprincipled.
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284 ALVIN PLANTINGA AND PATRICK GRIM
The principle nvolved s this:of a numberof propositions hattogether
lead to a contradictionby impeccableargumentforms, give up the
propositions hat have the least intuitivesupport.In the examplesI can
think of, it seems to me much more intuitiveto reject the relatively
complex and obscurediagonalpremisethan to reject the propositionthat some propositionsaregenuinelyuniversal.Obviously hisalso isn't
a matter of whim; and while it is ad hoc, it is not ad hoc in an
objectionablesense. It would indeed be nice to have such general
directions;but here as in most areas of philosophyand logic we don't
have anything ike a satisfactoryalgorithm or determiningwhat will
and what won't get us into trouble. That'sjust part of the human
condition.I do agreewithyou, though, hat thereis indeeda cost here. It seems
as if there should be the diagonalpropertiesor propositions nvolved.
So I agree with you when you say "But it is a genuine clash of
intuitions, I think, with genuinelyforceful intuitionson both sides."
Whatwe havehere, after all, is a paradox,andanywayout of a genuine
paradox exacts a price. But the intuitivesupportfor the existence of
genuinelyuniversalpropositions(as I see it) is strongerthan for the
relevantdiagonalpremises; o the price s right.
Finally, (and here I'mintroducing omethingnew, not just summa-
rizing)it seems to me that there is self-referential roublewith your
position; t is in a certainway self-defeating.First,it seemshard to see
how to stateyourargument.Consider, or example, he mainstatement
of yourCantorianargumenton p. 52 and also on p. 59. This argument
begins:
ConsideranypropertyT which s proposedasapplying o alland only truths.
Then it allegesthat any suchpropertyT will havesomefurtherproperty
Q. Butthen thenext step of theargumentmustbe somethingike:
So anypropertyTwhich s proposedas applyingo all andonly truthswillhaveQ.
And that is a quantification ver all properties which,according o
yourconclusion, s illicit.Sohow is theargument o be stated?Perhaps as follows. I believe that there are propositions that are
genuinelyuniversal; .e., that quantifyover all propositionsor prop-
erties. You propose various premises which I am inclined to some
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TRUTH, OMNISCIENCE,AND CANTORIAN ARGUMENTS 285
degreeto acceptand whichtogether (and by way of argumentormsI
accept) yield a conclusion. You yourselfdon't take any responsibility
for any of the premises,of course;your arguments strictlydialectical.
(It isn't even a reductio,becauseyour conclusionimpliesnot that the
supposition o be reducedto absurditys absurd,but that it doesn'tsomuch as exist.) But you are enablingme to apprehendan argument
whichshowsthatsomething believe s mistaken.
But whatis the conclusionto be drawn?Thatis, what conclusion s
it thatI am supposedto draw:what is the conclusionyou suggest s the
right one for me to draw, from the argument you suggest? (This
conclusion is also one you will have presumablydrawnupon offering
the same argument o yourself.)Now here we must be careful: t istempting, f course, o saythatthe conclusion s that
Thereare no genuinelyuniversalpropositions.
But of course that is itself a genuinelyuniversalproposition, tatingas it
doesthateveryproposition s non(genuinely niversal).
You suggestthatwe can avoid the problemhere by judicioususe of
scare quotes. ("It'ssometimes raised as a difficultythat an argument
such as the one I'vetried to sketch aboveitselfinvolves whatappear o
be quantificationsver all propositions. think such an objectioncould
be avoided,however,by judiciouslyemploying care quotesin order to
phrase the entire argument n terms of mere mentions of supposed
'quantifications . . .'.") But how is that supposed to work? The conclu-
sion will be expressedin a sentence, presumablyone involving scare
quotes. Either that sentenceexpressesa propositionor it does not. If it
does not, we won't makeany advanceby usingthe sentence; f it does,we should be able to remove the scare quotes. But how can we remove
the scare quotes? What is the conclusion of the argument,straight-
forwardly tated?
The use of quotes suggeststhat the conclusionhas something o do
with some phrase,or sentence, or perhaps some linguisticterm. But
whatwouldthatbe? Consider, or example, hesentence
(a) Everypropositions either rueor not-true
which is in some way unsatisfactory on this version of your view. What
is the problem? It isn't that this form of words expresses a proposition,
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286 ALVIN PLANTINGA AND PATRICK GRIM
whichproposition s necessarily alse:for your conclusion,putmy way,
is that there aren'tany genuinelyuniversalpropositions there isn't
any such thing as quantificationover all propositionsor properties.
Shall we say that (a) is ill-formed,does not conform to the rules of
Englishsentence formation?That seems clearlyfalse.Shall we say that(a) does not (contrary o appearances)ucceed in expressinga proposi-
tion? That can't be right;for that conclusion is again a genuinely
universalproposition, ayingof eachproposition hat t has theproperty
of not being expressedby (a).Shallwe saythat(a) is meaningless?That
isn't right either;we certainlyunderstand t, and can deduce from it
(from the proposition t expresses)with the otherpremisesyou suggest
the conclusionsneededto make the Cantorianargumentwork. So whatwouldbe theproblemwith(a)?
And in any event, the conclusion of your argument, take it, isn't
really about linguistic items, expressions of English or of any other
language.It is really an ontological conclusion about propositions,
sayingthat there is a certainkind of proposition the kindgenuinely
universal such that once we get really clear about that kind,we see
that it can'thave any examples.That'sreally the conclusion;but that
conclusion,sadly enough, s also self-referentiallyncoherent n that it is
an example of the kind it says has no examples;it quantifiesover
propositionsgenerally,sayingthat each of them lacks the propertyof
beinggenuinelyuniversal.
So the right course, as I see it, is to perseverein the original and
intuitiveview that there are indeed propositions hat quantifyover all
propositions: or example, everyproposition s either true or not true,
andfor any propositionsP and Q, if, if P then Q, and P, then Q. The
alternative eemsto be to say that there simplyare no suchpropositions
(no propositions hat quantifyover all propositionsor properties):but
thatpropositionseems to be self referentially bsurd,and also (given a
couple of other plausible premises) necessarily alse. We should then
try to avoid paradox by refusingto assert the premises that (together
with the above) yield paradox; we don't get into trouble, after all,
simply by makingthe above assertion.And one reason for resistingsome of the premisesof those paradox-concluding rguments s just
that, togetherwith other things that seem acceptable, hey lead to the
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 287
conclusion that no propositionsare about all propositions,which is
itselfdeeplyparadoxical.
It mustbe grantedthat some of those premisesseem initially nno-
cent, and even to have a certain degree of intuitive warrant.The
conclusion has to be, I think,that they don't have as much intuitivewarrant as does the proposition that some propositionsare indeed
aboutallpropositions.
Finally,I return o the point that the problemhere, insofaras it is a
problem, sn't really a problemfor traditional heology.It is a general
problemwitha life of its own;andyou don'tget a problem or theology
by takinga problemwith a life of its own and nailing t to theology.If
there aren'tany really general propositions(and notice that the ante-cedent looks like a really generalproposition) henthe thesisthatGod
is omniscientwill have to be stated in some other way, as will such
paradigms f goodsense as thatno propositions both trueandfalse.
8. GRIM TO PLANTINGA
At the core of the issue, we agree, is a clash of intuitions.On the oneside are the intuitionsthat fuel the Cantorianargument n its various
forms. On the other side is the lingering eelingthat there nonetheless
somehowought o be some totalityof all truthsor of all propositions.
Whatyou propose as a way out is that we pickand choose, argument
by argument,which to give up: the totalityof truthsor propositionsor
thingsknownthat the argument xplicitlyattacks,or the diagonal ruth
or propositionor thing known that it uses to attack that totality.Our
universalprinciple,you propose, is this:case by case we let intuitionbe
our guide. I'm afraid that seems to me to be neither universalnor a
genuine principle.
I also don't want the basic clash of intuitions o be misportrayed. f
we treat this as a choice, the choice is not between (1) the intuitive
appealof an idea of omniscience, ay, and (2) the intuitiveappealof an
awkwardlyphrased diagonal 'piece of knowledge'proposed in the
Cantorianargumentagainst omniscience.When properly understood,the choice is rather between (1) the intuitiveappeal of an idea of
omniscience and (2) some very basic principlesregardingtotalities,
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288 ALVIN PLANTINGA AND PATRICK GRIM
truth,and knowledge.Given those basic intuitiveprinciples, t follows
that there will be a diagonal 'piece of knowledge'of the sort the
argumentcalls for. One can't then just deny the diagonal;one would
haveto denyone or moreof theintuitiveprinciplesbehind taswell.
Considerfor examplethe argument hat there can be no totalityofthe things that an omniscientbeing would have to know. Here the
Cantonan argumenthas us envisage sub-totalitiesof that supposed
totality, and proceeds by showingthat for any proposed one-to-one
mappingf from individual hingsknown to subtotalitiesof the whole
there will be some subtotalityleft out. In particular,perhaps, we
envisagethat'diagonal'bunch of individual hingsknown which do not
appear n thesubtotalitieso which mapsthem.Are we to deny that there really is such a diagonalsubtotality?
simplydon't see how. We startedby supposinga certain otality a big
bunchof things.Once we havethose, the 'subtotality' t issue is simply
a bunch of thingswe alreadyhad. We didn't create them. The most
we've done is to specify them, without ambiguity,one way among
others, ntermsof themapping.
Denialof the diagonalat thisstage doesn'tthusseem verypromising.
Since there are these things,however, it seems there must be a truth
about these things.Otherwisetruth would be somethingfar cheaper
and more paltrythan we take it to be. Truthwouldn'tbe the whole
story: herewouldbe thingsout therewithoutanytruthsabout hem.5
If there is a truthabout these things,however,an omniscientbeing
would have to know that truth. Otherwise omniscience would be
somethingfar cheaper and more paltry than we take it to be: an
omniscientbeing would be said to know everything,perhaps, even iftherearesometruthshe doesn'tknow.
Denyingthe diagonal n a case like thisis thus not simplya matterof
denyingsome one awkwardlyphrasedpiece of proposed knowledge.
To deny the diagonalwe must deny that if we have the things of a
groupwe stillhave themwhenwe talkaboutsub-groups,perhaps,or to
deny thatanything here is is something here is a truthabout, or that
knowledgeis knowledgeof truths,or that universalknowledge wouldbeknowledgeof all truths.
Another basic difficultyin selectively denying diagonals,which it
seems to me you haven'taddressed, s the strengthenedCantorian rgu-
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 289
ment:the problemof the 'reappearingdiagonal'.If in some case we
choose to deny that there is a diagonaltruth or propositionof some
sort - a propositionaboutall those propositionswhich are not about
the propositionsmapped onto them by a particular unction f, for
example- we will have to do so by claiming hat the specificationatissuefails to giveus a proposition,or that the diagonalcondition ailsof
propositionhood,or the like. But then there will be a Cantorian
argumentparallel o the originalwhich reliesmerelyon the fact thatfor
every specification r conditionthere willbe a truthor proposition.We
will thus still have an argumentwhich shows that there can be no
totalityof truthsor propositionsor the like.Denyingthe diagonal n the
arguments t issuesimplydoesn'tseem to work.Let me turnbriefly o your new point,thoughI thinkthata complete
treatmentof this issue wouldtakeus well beyondour exchange and
perhaps our abilities - here.
The purestform of the argument, think, s one whichyou represent
withbeautiful larity:
Youproposevariouspremiseswhich I aminclinedto some degreeto acceptand which
together (and by way of argument orms I accept)yield a contradiction.You yourselfdon't takeanyresponsibilityor anyof thepremises,of course;yourarguments strictlydialectical ... But you are enablingme to apprehendan argumentwhichshows thatsomething believe s mistaken.pp.66-67)
But given such an argument,you ask, what positive conclusion
shouldbe drawn?These,as you rightlypointout, aredangerouswaters.
I can'tclaimto have navigated hem all, nor can I claimto be able to
anticipate llpossibledangers.Let menonetheless ketchsome ideas:
One possibility,of course, is that I shouldn't attempt a positiveconclusion.Perhaps it is enough for me to guide the opposition into
their own conceptual mazes of consternation and confusion. That
wouldteachthema lesson,even if not apropositional ne.
I'm not yet convinced that we can't have some kind of positive
conclusion,however.In the past I've proposed phrasingsuch a con-
clusion, in at least some cases, by using scare quotes or some other
means of indirect speech.At this point you say that if our conclusionexpresses a proposition "we should be able to remove the scare
quotes."But I'm not sure why you think that. Scare quotes serve a
varietyof importantandlittle-understoodunctions,andit may be that
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290 ALVIN PLANTINGA AND PATRICK GRIM
sometimeswhatwe want to say can only be expressedby such means.
Why thinkthat they are alwaysavoidable?At this point you also say
that the use of quotes"suggestshat the conclusionhas something o dowith some phrase,or sentence,or perhapssome linguisticterm."But
isn't that buyinginto a fairlyshallowlogician'snotion of quotationasnaming? (As I remember,Anscombe and Haack have some fairly
tellingpointsto makeagainst uchatreatment.)6For the moment, however, let me try to avoid the difficultiesof
quotesbyproposinganotherpossibilityorapositiveconclusion.
A simpleexamplehelps. We can convinceourselves,I think,that the
concept of round squares is an incoherent one. It is temptingto
conclude on that basis that round squares don't exist. But this lastposition brings with it the well-knownphilosophicaldifficulties of
negativeexistentials.Perhapsthe apparentdifficulties hereare merely
apparent.But at any rate we can avoidthemby stoppingwithour firstclaim: hattheconceptof roundsquaress anincoherentone.
Perhaps that is how we shouldphrase our positiveconclusionhereas well: the concept of omniscience s an incoherentconcept,as is thenotion of a totality of truthor of a propositionabout all propositions.
Havingconvincedourselvesthat the notion of a propositionabout allpropositions s an incoherentone, we are temptedto concludethatno
propositionsare genuinelyuniversal.The phrasingof this last position
brings with it all the philosophicaldifficultiesyou point out. But
perhapswe could avoid them, while still having a positiveconclusion,by stoppingwithourfirstclaim: hat theconceptsat issueareincoherentones.
My fallbackand first love remainsthe pure form of the argumentabove, offered withoutpositive conclusion.If a positive conclusionisdemanded,this suggestion s perhapsworth a try. It must be addedimmediately,however, hatwe'llbe ableto takethissuggestion eriouslyonly if we're willing to give up a few things:at least (1) a Russellian
treatmentof definitedescriptionsand (2) the idea thatsimplepredica-tionssomehow nvolvehiddenquantifications. utit is perhaps imewe
gaveup thoseanyway.
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 291
9. PLANTINGA TO GRIM
I thinkwe are makingprogress,but perhapswe are also approachingtheends of ourrespective opes;here is myfinalsalvo.
First, I reiterate hat the problemwe have on our hands,whateverexactly it is, isn't really a problem about omniscience.Omniscience(above, p. 50) should be thoughtof as a maximaldegreeof knowl-edge, or better, as maximal perfection with respect to knowledge.Historically, this perfection has often been understood in such a way
that a being x is omniscientonly if for every propositionp, x knowswhetherp is true. (I understand t that way myself.)This of courseinvolves
quantificationover all propositions.Now you suggest thatthereis a problemhere:we can'tquantifyoverallpropositions,becauseCantorianargumentsshow that there aren'tany propositionallyuni-versal propositions(propositionsabout all propositions- 'universalpropositions' or short),and also aren'tany propertieshad by all andonly propositions. Note, by the way that each of these conclusions sitselfa universalproposition.)But supposeyou areright:whatwe have,then, is a difficulty,not for omniscienceas such, but for one way ofexplicatingomniscience,one way of sayingwhat this maximalperfec-tion with respect to knowledge s. A person who agreeswithyou willthen be obligedto explainthis maximalperfection n some otherway;but she won't be obliged, at any ratejust by these considerations, ogiveupthenotionof omiscience tself.
Second, you and I agree that what we have here is a clash ofintuitions;but I am not quite satisfied with your outlining of the
attractions n eachside.Youputit likethis:
... the choice is not between(1) the intuitiveappealof an ideaof omniscience, ay,and(2) the intuitiveappeal of an awkwardlyphraseddiagonal'piece of knowledge'pro-posed in the Cantorianargumentagainstomniscience.Whenproperlyunderstood, hechoice is ratherbetween (1) the intuitiveappeal to an idea of omniscience,and (2)some verybasicprinciplesregardingotalities, ruthand knowledge.Giventhose basicintuitiveprinciples, t follows thatthere will be a diagonal pieceof knowledge'of thesorttheargumentallsfor(pp.69-70).
You also suggest on that same page that on my side of the scales thereis in addition "the lingering feeling that there somehow ought to besometotalityof alltruthsorof allpropositions".
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292 ALVIN PLANTINGA AND PATRICK GRIM
But the attractionof my view here is not that it enables us to save
omniscience;omniscience sn't in any dangerin any event (whatis in
danger,as I just argued,would be at most a certainway of explicating
omniscience).Nor is the main attractiona lingering eeling that there
must somehowbe a totalityof propositions.Perhapsthere is no suchtotality (a set, a class) of propositions;sets and classes are a real
problem anyway.What has the most powerful ntuitive orce behindit,
as I see it, is rather he idea that there are universalpropositions and
properties):uchpropositions,orexample,as
(1) Everypropositions either rueor nottrue,
and(2) Thereareno genuinelyuniversal ropositions.
As I see it, (1) is an obvioustruth; hereobviously s such a proposition
as (1) andit is obviouslytrue. Therealso seemsobviously o be such a
propositionas (2) (even if, as I think, t is false), and (2) seems initially
to representyour position. "Initially", say, because (2) seems to be
self-referentiallyncoherent;it is or implies by ordinary logic, the
universalproposition
(3) for every propositionp there is a propositionq such that p is
not aboutq.
Third,(and most important): s you point out, if we proposeto reject
a premise in a Cantorianargument,we are of course committedto
rejectingany propositions hat entail that premise;amongthe proposi-
tions entailingsuch premises,you say, are "somevery basic principles
regarding otalities,truthand knowledge"; nd you add that it will be
hard to reject these.But here is my problem.Whatwillthese principles
be? In particular,won't they themselves have to be (or include) uni-
versalpropositions,and hence be such that on your view there really
aren'tany such things?This is the question I'd like to explore a bit
further.
Consider, for example, the argumentyou offer on p. 59 for theconclusionthat there are no universalpropositions.Suppose,you say,
there were such a propositionP (a propositionabout all propositions)
andconsiderP-propositions:hepropositionsP is about.Then you say
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 293
Were P genuinelyabout all propositions,of course, there would be a one-to-onemapping fromP-propositions ntopropositions impliciter..
And thenyou arguethat there can'tbe anysuchmapping.For supposetherewere;then therewouldhave to be a propositionq aboutexactly
those propositionsp which are such that f(p) is not aboutp. But then
consider the inverseimage of q underthe mappingf (call it 'r').Is qaboutr?Well, t is if andonlyif it isn't nota prettypicture.
Here we havetwopremises:
(4) For aiiy propositionp, if p is about all propositions,then
thereis a 1-1 mapping rom the propositionsp is aboutonto
propositions enerally.and
(5) For any functionf, if f is 1-1 and from propositionsonto
propositions,then there is a propositionq about exactly
thosepropositionsp suchthat (p) s not aboutp.
Now I want to make 3 points aboutthis argument.First, it initially
looks as if you are endorsing 4) and (5), or at anyrate recommending
themto me and others.You propose it willbe hardto rejectthem,that
they have considerable ntuitiveforce and considerable ntuitiveclaim
upon us. But of course on your own view (putting t my way) there
really aren'tany such propositionsas (4) and (5), since each involves
quantification ver all propositions. (4) is a universalproposition,and
both its antecedent and consequent nvolve universalpropositions,as
do the antecedentand consequentof (5).) So whatdo you proposetodo with (4) and (5)? Whatstance do you take with respect to them?Can you conscientiouslyrecommend them to me if you really thinktherearen'tanysuchpropositions,butonly, so to speak, a confusion nthe dialecticalspace I take them to occupy?Well, perhaps, n accordwith your favorite way of understandingCantorianarguments(asoutlinedon p. 71) you aren'tyourselfaccepting 4) and (5), but simply
proposingto me thatif I believe that there are any universalproposi-tions at all, then I shouldalso believe (4) and (5); this will land me inhot water;so I shouldn'tbelieve that there are any universalproposi-tions. I doubt that you can properlyrecommendthis to me, because
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294 ALVIN PLANTINGA AND PATRICK GRIM
your recommendationpresupposesthat there are universalproposi-
tions:whatyou say impliesthatit is possibleto believe(4) and (5); but
on your own view, it isn't possible, there being no such things to
believe. So perhapsyou will have to find some other way of stating
whatyoupropose.Second(andwaiving he firstdifficulty) wantto stray romthemain
topic here (the questionhow you stand relatedto (4) and (5)) and ask
parentheticallywhywe should thinkthatif I believe there areuniversal
propositions e.g.,suchalawof logicas
(6) For anypropositionp, p is notboth trueandfalse
I should also believe (4) and (5)? Why must I believe that if there issuch a propositionas (6) (one whichis about all propositions) hen (4)
and (5) are true?I don't dispute(4): I don't disputethat there is an
identitymappingon propositions,and if there is, then (4) is true. But
what about (5)? Is there really a propositionwhich is about exactly
those propositionsp such that f(p) is not about p? Take f to be the
identitymap: s therereallya propositionaboutjust those propositions
that are not aboutthemselves?Well,it certainlydoesn't look as if thereis such aproposition. f therewere, twouldpresumably avethe form
(7) For any propositionp, if p is about exactlythose proposi-
tionsnotabout hemselves,hen ...
Aboutness is a frail reed and our grasp of it a bit tenuous, but a
proposition ike (7) seems to be about all propositions,predicatingof
each propertyof being such that if it is not aboutitself, then it is ....So such a proposition sn't about only those propositions hat are not
about themselves,unless no proposition s about itself.And the same
holds for (5), the premiseof yourargument.Such a propositionwould
presumably avetheformyougive t onp. 60:
(8) For anypropositionp, if p is a P propositionand f(p) is notaboutp, then ... p ....
But a propositionof this formis not, as (5) requires,about only those
propositionsp suchthat f(p) is not aboutp (unlessallpropositionsmeetthat condition): t seems instead to be aboutevery proposition,predi-
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 295
catingof each the propertyof beingsuchthat if it is a p propositionand
f(p)is notabout t, then... p ....
So I am not at all inclined to accept(5). (6) is obvious,has a sort of
utterseethroughability, luminous ndisputability,n evidentlustre,as
Locke says.The suggestion hat it is not only not true but in fact noteven existent is a sort of affrontto the intellect - a vastly greater
affront than the rejectionof (5), which has at best a marginalplausi-
bility.Accordingly, don't think this argumentagainst he existenceof
universalpropositionss at allpowerful.
Butnow back to the maintopic:yourrelation o (4) and(5); there is
a reallyfascinatingpoint here. Let'sbriefly recapitulate. believe that
both (1) and (6) are true and also that there is an omniscientbeing(God), and I take this latter to imply that God knows, for every
propositionp, whetherp is true.You propose to make trouble or these
beliefsby wayof citingCantorianarguments.As we have seen, thereis
considerable self-referential) ifficultyn construing he relationshipn
which you standto these arguments.You suggestseveralpossibilities.
One possibilityis that you yourself accept the conclusion,but can't
state it withoutusing quotes. But then I really don't understand he
conclusion(because I don't know how the quotes are supposed to be
workinghere) and thereforecannotaccept it. A second suggestionyou
make is thatyou accept the conclusion,and the conclusion s that the
concept of a universal proposition (one about every proposition) is
incoherent,just as the concept of a round square is incoherent.But
whatis it for a concept to be incoherent? see what it is for a concept
to be necessarilyunexemplified,as is the case with the concept of a
round square. But if we say that the conclusion of the Cantorianargument s that this concept of a universalproposition s necessarily
not exemplified, hen you are again in self-referentialrouble.For this
concept is necessarilyunexemplifiedonly if it is necessarilytrue that
there aren'tany universalpropositions i.e., only if necessarily, or
every propositionp there is a propositionq such thatp is not aboutq.And of coursethatis itself (thenecessitationof) a universalproposition.
Your favoriteway of construing he argument p. 71), however, isstill different.Here the idea is thatyou don't accept or take responsi-
bilityfor the premisesor conclusionof these Cantorianarguments, ut
instead are only trying to enable me (and others) to apprehend an
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296 ALVIN PLANTINGA AND PATRICK GRIM
argumentthat shows that we have fallen into incoherence in taking
ourselves to believe (1) and (6). Now how, exactly,are we to construe
this?You don't take responsibilityor any premisesor the conclusion:
you are simply tryingto bringabout a certain effect in me. But what
effect? Well, you apparentlyhope to get me to believe that there arepropositionsI am inclined to accept from which it follows that there
aren'tanyuniversalpropositions. Here supposewe waive the problem
that I am not, in fact, at all stronglyinclined to believe (5) and the
propositionsike it to whichyoudirectme.)
But it looks as if these premises, f theyare to show thattherearen't
anyuniversalpropositions,will have to containa universalproposition
among hem.The conclusionwill be or be equivalento
(9) Every proposition s non-universal,.e.,forevery proposition
p, there s a propositionq suchthatp is not aboutq.
and to deduce this conclusion, t looks as if we shallneed at least one
universalpremise- premiseslike your (4) and (5) for example.So I
am now supposed to see that a universalpremiseI accept entails that
there areno universalpropositions. suppose we agreefurther hatif a
proposition is a universal proposition, then it is essentially a universal
proposition,couldn'thave failed to be a universalproposition.But then
thatpremise- the universalpremisethat entails that there aren'tany
universalpropositions also seemsto entail hat t doesn't tself exist!
So if you are right, I am in a dialectical situation peculiar in excelsis.
I believesomethingx from which it followsthat x isn't merelynot true,
but doesn't even exist! But then shouldn'tI stop believing x? Don't I
have a proof thatx is not true?If x entailsthat it doesn't exist, then xcan'tpossiblybe true.And the same wouldholdfor any set of premises
sufficient or a Cantorianargumentagainstuniversalpropositions: hey
can't all be true because taken togetherthey imply that one of them
does notexist.
By way of conclusion: he upshot, so I think, s that I have no reason
at all to stop believing (1), or (6), or that there is a being that knows,
for every proposition,whether it is true.For any premises that implythat there is no such proposition or being also imply that they them-
selves do not exist.If they are all true,therefore, hey do not all exist; f
theydo not all exist, they are not all true; herefore hey are not all true.
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 297
A bit more fully: suppose there is a CantorianargumentC whose
premisesentail that there are no universalpropositions.Now C may
have superfluouspremises; o note thatfor any such argumentC there
is a minimalargumentC* whosepremisesare a subset S of C's andare
such that no proper subset of S entails that there are no universalpropositions;C* will be validif andonlyif C is. C* will containat least
one universalpremise; so if the premises of C* are all true, they
don't all exist. But by hypothesisthey all exist;hence they aren'tall
true; hence there is no sound Cantorianargumentagainstuniversal
propositions!
10. GRIM TO PLANTINGA
There are importantpoints here. Let me try to addresssome of the
mainones:
1. It's true that the Cantorianproblems at issue are not first and
foremost problemsfor omniscienceper se. They seem to arise as quite
generalepistemological ndmetaphysical roblemswheneverwe tryto
bring togetherunrestrictednotions of truth, knowledge,and totality.
But thatdoesn'tmeantheyaren'tproblems or omniscience.
Omniscience s standardly lossed as being 'all-knowing' r 'knowing
everything',preciselyas its 'omni'would suggest.If there is no 'every-
thing'of the relevanttype to know, there can be no omniscience as
standardly lossed.
You suggest hat we understand mniscienceas 'a maximaldegree of
knowledge' or as 'maximalperfection with respect to knowledge'
(above, p. 73). (Isn't maximalperfection'a bit redundant?)But shouldit turnout that for any degree of knowledge there must be a greater,
it wouldappearthat there can be no 'maximalperfection'withrespect
to knowledge and thus no omniscienceas you suggestwe understand
it.
None of that means that a theist cannot continue to think of God's
knowledge as suitablydivine in kind or extent. But it may mean that
such knowledge - however divine - cannot literally qualify as omnis-cience n eitherof the sensesoutlined.
2. The most fascinatingpart of your last response is the final
argument,o the effect thatthere can be no sound Cantorianargument
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298 ALVIN PLANTINGA AND PATRICK GRIM
againstuniversalpropositions. thinkthatis a beautifulanddeep pieceof work.
Were there a sound Cantorianargumentwith the conclusionthat
there can be no universal propositions - so the argument goes - it
would requireat least one universalpropositionas a premise.But ifsound, its conclusion would be true, and thus there could be no suchproposition.If sound its premiseswould not all be true, and thus it
would not be sound. Therecan then be no sound Cantorianargument
withtheconclusion hattherecan be no universalpropositions.
Verynice.
In the end,however,I thinkthisargument implyreinforces ome of
the points we've alreadyagreed on above. We've alreadyrecognizedthat there are (self-defeating)difficultieswith the idea of a straight-forwardpositiveproposition o the effectthat therecan be no universal
propositions. t shouldthereforenot be surprising hat therewouldbe
(self-defeating)problemsfor the claimthat there was some argument,Cantorian rof anyotherkind,whichdemonstrated uchaproposition.
Here as before I think I have to turn to less direct and more
deviously dialecticalcharacterizations f what it is the argumentsat
issue reallydo. Contraryo the characterizationou give,I'mnot tryingto get you to envisageand accept an argumentwith some universalpremise and a universalconclusion to the effect that there are nouniversalpropositions. You characterizeyourself as holding certainbeliefs.I merelyhelp you to see thatyou are thereby ed to confusionandconsternation.
3. I don'tthink, hen,thatyourfinalargument howswhatyou think
itdoes.
Suppose we grantthatany Cantorianargumentwith a propositionalconclusionto the effect thatthereare no universalpropositionswouldhaveto havesome universalpropositionas a premise.By the argument
above,therecanthenbe no soundCantorian rgumento thateffect.But interestinglyenough, it doesn't seem to follow that universal
propositionsare then safe fromCantorianarguments thatyou "have
no reason at all to stop believing (1), or (6) ..." (p. 78) or otheruniversalpropositionsof yourchoice.
To see this, consideragainthe standardstructureof the Cantorianarguments hroughout.Someone proposes some set T as a set of all
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 299
truths,or claimsthat some beingB is omniscient,or offersa candidate
p as a genuinelyuniversalproposition.But then considerthe power set
of T, or of whatB knows, or of the propositionsp is about.(Although
convenient,we've seen that the use of sets here is strictly nessential.)
For each of the elementsof that particularpower set there will be atruth,or somethingB oughtto know,or somethingp oughtto be about.
But then therewill be too manyof these to put in 1-to-I correspond-
ence withthe truthsT does contain,or the thingsB does know,or the
propositions p is about: there are truths or bits of knowledge or
propositionshattheproposedcandidate eaves out.
Now the interestingthing about that core argument s that it is
written n theparticular it deals simplywith a singlecandidateset Tor beingB or propositionp. I think,moreover, hat it can in each case
be writtenpurely n the particular,withoutanyuniversalpropositionsat
all. If that is true, such a form of argumentwill continue to cause
problems or some of the thingsyou claimto believe even if it's also the
case that there is no classicialdeduction of a universalproposition
denyingthe existence of universalpropositions.At some point you will
find yourself consideringsome purportedlyuniversalpropositionor
totalityof truthsor omniscient being and we will put that particular
candidatehrough nargument f thisformand t will comeup short.
Because we can 'see' thatthis kind of trouble s boundto come up, it
is of course temptingto say that such an argumentwill hold for any
arbitrary andidate,and thus must hold for them all. It is tempting, n
other words, to read T, B, and p as variablesand then finish with a
universalgeneralization.There can be no universalpropositions.'But
that's the point at which we (or at least I) would get into trouble,announcinga positive positionwhich would also be subject o the type
of argumentat issue, and thus a positionwhich - as you point out -
could not be theconclusionof a soundargument f thistype.
What we shouldconclude,I think, is that Cantorianargumentsare
indeedverypeculiar, emptingus in some cases to try to drawuniversalconclusions that they themselvesshow us cannot be drawn. That is
something hatyour argumentpoints up magnificently. ut the fact thatthey cannot be characterizedas universal derivationsof universal
conclusions n'suchcases doesn't meanthey are somehow harmless
thatyou "haveno reason at all to stop believing 1), or (6), or that there
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300 ALVIN PLANTINGA AND PATRICK GRIM
is a being that knows, for every proposition, whether it is true."?t's
clearthatparticularandidates or universalpropositionsor fortotalities
of truth or knowledgewill still be vulnerableto particularized rgu-
mentsof theformabove.
4. If thisis right,I think,someone of yourconvictionswill be forcedback to the simplerstrategyof denyingthe diagonal n the particular
argumentswith which someone of my propensitiesassailsyou. This is
the strategyyou take withregard o (5), for example. n general,I think
it is the strategyyou shouldtake, though t will be easier in some cases
than n others.
Your approachwith respect to (5) is to deny the existence of a
proposition about precisely those propositionsp suchthat
f(p)is not
about them. The reason you give is that any such propositionwould
presumably e of theform
Vp((Pp & - A(f(p)) p .P..
readas
(8) For anypropositionp, if p is a P propositionandf(p)is not
about p, then ... p ....
But this, you insist, isn't about just those propositions.It's about all
propositions.
'Aboutness'may,as you say,be a frailreed (p. 76). But I don't think
it's that easily bent. The type of approach you outline here, as you
probablyknow, leadsdirectly o Russell'sclaimthat all propositionsare
about the universe as a whole. Our standard ntuitionswould suggest
otherwise - we'd normallythink that the last sentence is about anapproachthat leads to Russell,for example, but is not about Michael
Jackson'snose job. I agree that our grasp of 'about' may be a bit
tenuous,but itcertainlydoesn'tseemto me to be thattenuous.
It also seems to me that the strategyyou pursue with regardto (5)
can be easilycircumvented.At the point in the argumentat which you
raisethe issue above,you don't seem to have any objectionto the idea
that thereare those propositionsp suchthat f(p)is not about p. At thatstage in the argumentwe already know, moreover, that any 1-to-1
correspondencebetweenpropositionsand groupswill leave this bunch
out. Whatyou deny, as outlinedabove, is my way of gettingfrom that
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 301
bunchof propositions o a particular roposition;whatyou denyis that
therewillbe any propositionaboutthose andonlythosepropositions.
But there are lots of otherwaysof getting o a particular roposition
from here.Consideragainthat bunchof propositionsp such thatf(p)is
not aboutp. Is there not a propertywhichpreciselythese propositionsshare, and a propositionto the effect that they share precisely that
property?Are therenot variousdescriptionsunderwhichtheyfall,and
correspondingpropositionsto the effect that preciselythese proposi-
tions fall under thatdescription? s therenot a truth to the effect that
preciselythese propositionsare, say,propositions?Given any of these,
we can proceed with the argumentas before. We need not pause to
worryaboutthe vagariesof 'about'because we don't need it, here or inother Cantorianarguments.That strategyfor denying diagonals,at
least,appears o be of insufficient owerandgenerality.
As I'vesaid,I thinkthere arealso otherproblems acinga strategyof
denyingdiagonals.As indicatedat an earlierpoint, it doesn'tlook like
there can be a universalpolicy for such denials.I think the general
problemof the 'reappearing iagonal' lsoremains.
5. That is af any rate where my thinkingstandsnow. I thinkyou
might be right that we are approaching he ends of our respective
ropes, andthis may be as far as we areequipped o take the issue at the
moment.
11. PLANTINGA TO GRIM
Right; thinkwe shouldwind thisdiscussionup (or down);we'vemade
someprogress,but of coursehaven't inally ettledawhole ot.Withrespectto yourlast letter:on one point you are right: ven if, as
I suggest,we takeomniscience o be maximalperfectionwithrespectto
knowledge, it doesn't follow that your Cantorianworries don't pose
problemsfor it. For it might be thatwhat they imply is thatthere isn't
any maximaldegree of this perfection. That is surely (as you say) a
possibility.
Butof courseI am stillunconvincedhatwe havea genuineCantorian
problem orthe claim hat
(1) For every proposition p, God knows p if and only if p is
true.
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302 ALVIN PLANTINGA AND PATRICK GRIM
The problem, as you see it, attachesto the apparentquantification
over all propositions n (1); it is at thatpointthat the allegedCantorian
difficulties aisetheiruglyheads.
But I am still doubtful hat thereis a realproblemhere. We agree,I
take it, that there isn't any sound Cantorianargument or the generalconclusionthatthereare no universalpropositions propositionsabout
all propositions); ny such argument according o the argument f my
last letter)would involve at least one universalpropositionand would
thus tselffail to exist f itweresound.
You suggest, however, that there are nevertheless still Cantorian
difficulties for (1); we can instead turn to particular Cantorian argu-
ments;for any particular laim(such as (1)) thatseems to be about allpropositions, there will be a particularCantorianargumentmaking
trouble orit.As youput it,
Now the interesting hingabout this core argument s that it is written n theparticular- it deals simplywith a single candidateset T or being B or propositionp. I think,moreover,that it can in each case be written purely in the particular,withoutanyuniversalpropositionsatall.
Andyou go
on tosay
that"sucha form of argumentwill continue tocauseproblems orsome of thethingsyou claim o believe ..."
Now here I'd like to investigate briefly the claim that "thiscore
argument an be written, n each case, purely in the particular,without
any universalpropositionsat all."How would this go, for example, n
the case at hand, (1) above?The relevantCantorian rgument, resum-
ably,willbe for the conclusion hattherejust isn't any suchproposition
as(1).Howwillthatargument o?Well,perhapsas follows:
(2) If there is such a proposition as (1), then there is a 1-1
functionf from the propositions(1) is aboutonto proposi-
tions simpliciter;
and
(3) If there is such a function, then there is a propositionp
aboutexactlythosepropositionsp such thatf(p) is not aboutP.
Furthermorethe argument ontinues) he inverse imager of q underfis suchthatq is aboutrif and onlyif q isn'taboutr.
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 303
Thisis how theargumentwouldgo;andyoursuggestion,presumably,
is that (2) and (3), the central premises of the argument,are not
universalpropositions.
That'srightor at anyrate reasonable:2) as it stands sn't a universal
proposition. Its consequent,however, clearly involves quantificationoverallpropositions;heconsequentof (2) is or is equivalento
(2c) There is a 1-1 function f such that for any proposition p,
there is a propositionq suchthatp will be the value of f for
qtakenasargument.
I suppose we would agree, furthermore,hat (2) couldn't so much as
exist if its consequentdidn't;so (2) couldn'texist if (2c) didn't.But ifthere is a good Cantorianargumentagainstthe existence of (1), there
will obviouslybe an equallygood Cantorianargument one paralleling
the argument or the nonexistenceof (1)) against he existenceof (2c).But (2) can exist only if (2c) does. So if there is a good Cantorian
argument or the nonexistenceof (1), there is an equallygood one for
the nonexistenceof (2). (Obviouslythe same will go for (3); both its
antecedentand its consequent nvolve quantification ver all proposi-tions.)But (2) is an essentialpartof the Cantorian rgument gainst 1).So if this (particular)Cantorian rgument gainst 1) is sound,one if its
premisesdoesn'texist;hence it isn'tsound.
I am thereforenot inclinedto thinkthat the move to the particularwill help: true, the particularCantorianargumentagainst(1) needn't
itself have a universalpropositionas a premise, but its premiseswill
involvequantificationver allpropositions,n the sense thatif there areno propositionsthat are about all propositions, then these premiseswouldnotexist.
This has a direct bearing on a second interestingclaimyou make.You suggestthatwhenconfrontedwithone of these specificCantorian
arguments, will reject the diagonalpremise;thus in the above argu-ment I will, you think,reject(3). Right; do reject(3); it looks to me asif the proposition q proposed, the one that is about exactly those
propositionsp such that f(p) is not about p - it looks as if thatpropositionwouldhaveto be stated nsomesuchway asfollows:
for any proposition p, if....
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304 ALVIN PLANTINGA AND PATRICK GRIM
but a proposition ike that doesn'tlook to be aboutonly so and so's;it
seems to be aboutall propositions.Maybeit isn't obviousthatit really
is about all propositions;but it is certainly ar fromobviousthatit isn't.
It is thereforeat anyrate farfromclear thatthere is sucha proposition
- much essclearthatthere s such aproposition, .g.,as
Godknows, oreverypropositionpwhetherp is true.
Now here is where you make that second interestingclaim. You
suggest hatthere are otherCantorian rguments gainst 1), arguments
that don't involve the claim that there is a propositionabout exactly
those propositions hat are not aboutf(p).Well,perhapsthere are. As
you know, I don't have a perfectly general strategyfor dealing withCantorianarguments; must take them one at a time.In accordwith
thatpolicy,I'dhave to look at theseotherarguments ou saythereare,
and look at them one at a time.Thatsaid,however,I must add that it
seems likely to me that any such argumentwill contain a premise
involving,n the above sense, quantification ver all propositions.That
is, any suchargumentwill contain a premisep which is suchthat there
is a universalpropositionq so relatedto it that p can'texistunless qdoes - in whichcase,once more,theproposedargumentwillbe sound
onlyif it doesn'texist.
Thus, for example, you suggest that there is a Cantorianargument
against(1) that proceeds in terms of a propertyhad by exactly those
propositionsp such thatf(p) is not about p (rather han a proposition
aboutexactlythose propositions).Thisargument, suppose, will retain
premise 2) as it standsbutreplace 3) bysomething ike
(3*) For any functionf, if f is 1-1 and from propositionsonto
propositionssimpliciter, hen there is a property q had by
exactly hosepropositionsp suchthat (p) s not aboutp.
Andthentheargumentwouldproceed.
But of course (2) and (3*) both involve quantificationover all
propositions; hey are thereforesuch that if there is a soundCantorian
argument or the nonexistence of (1), there will be an equallysoundCantorian rgumentorthenonexistenceof them.
By wayof briefrecapitulation:argued ast timethat therearen'tany
sound Cantorianargumentsfor the conclusion that there are no
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TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS 305
universalpropositions.You agree,but point out that theremay none-
theless be particularCantorianargumentsagainst the existence of
particular niversalpropositions (1), for example;and theseneednot
necessarily nvoke as premisesany universalpropositions.Here I think
you are right.But (as we have seen) it looks as if such argumentswillnevertheless invoke premises which couldn't exist unless universal
propositions existed. I also proposed, with respect to the general
Cantorianargument or therebeingno universalpropositions, hat the
diagonalpremise.whoseconsequentaffirms he existenceof a proposi-
tion aboutjust those propositionsp such that f(p) is not about p) is
surelynot obviouslytrue and is quiteproperlyrejectable. n response
to this point, you suggest next that there may be other Cantorian
arguments gainst he existenceof (1) that do not involvethe claimthat
there is a proposition about just those propositions,but instead (for
example)endorsethe existence of a propertyhadby all andonly those
propositions).Here my strategywouldbe, whenpresentedwith one of
these arguments,o look for a premise ike (2) or (3*) - one that isn't
itself universal, but is nonetheless such that it couldn't exist if no
universalpropositionsexisted.And then the commenton that argu-
mentwould be that if it is sound,then there will be a soundargument
against heexistenceof one of itspremises:o it isn'tsound.
I thereforeremainunconvinced hat we have a realproblemhere for
(1);I suspectyou remainconvinced hat we do. No doubtthereremains
much more to be said on both sides; but perhaps for now you and I
have said about all we can usefullysay. So we haven'tcome to agree-
ment;but I havelearnedmuchfrom our discussion,andam grateful o
youforhavingraised heissue.
NOTES
Such an argument lso appears n "There s no Set of All Truths,"Analysis44 (1984)206-208 and TheIncompleteUniverse,MITPress/BradfordBooks, 1991.2 For the most part whatfollows is edited from an extendedcorrespondencebetweenthe authors.The final two sections,however,are new and were writtenwiththis piecein mind.3 I'mobliged o GaryMarforconsultation n symbolism.4 PatrickGrim,"On Sets andWorlds:A Replyto Menzel,"Analysis46 (1986), 186-191.
5 Keith Simmons actually did argue for something like this position in "On an
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306 ALVIN PLANTINGA AND PATRICK GRIM
ArgumentagainstOmniscience,"APA CentralDivisionmeetings,New Orleans,April1990.6 See G. E. M. Anscombe, "AnalysisPuzzle 10,"Analysis 17 (1957), 49-52, andSusanHaack,"MentioningExpressions,"Logiqueet Analyse17 (1974), 277-294 andPhilosophyofLogics,CambridgeUniv.Press, 1978.
ALVIN PLANTINGA PATRICK GRIM
Department fPhilosophy Department fPhilosophy
University fNotreDame StateUniversityfNew YorkNotreDame, IN 46556 atStonyBrookUSA StonyBrook,NY] ] 794-3750
USA