Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga

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ALVIN PLANTINGA AND PATRICK GRIM TRUTH, OMNISCIENCE, AND CANTORIAN ARGUMENTS: AN EXCHANGE (Received 6 May,1992) INTRODUCTION (GRIM) In "Logic and Limits of Knowledge and Truth" Nous 22 (1988), 341- 367) I offered a Cantorian argument against a set of all truths, against an approach to possible worlds as maximal sets of propositions, and against omniscience.' The basic argument gainst a set of all truths s as follows: Suppose there were a set T of all truths, and consider all subsets of T - all members of the power set 9 T. To each element of this power set will correspond a truth. To each set of the power set, for example, a particular ruth T1 either will or will not belong as a member. In either case we will have a truth: hat T, is a member of 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's power set theorem we know that the power set of any set will be larger han the original. There will then be more truths han there are members of T, and for any set of truths T there will be some truth eft out.There can be no set of all truths. One thing this gives us, I said, is "a short and sweet Cantorian argument against omniscience." Were there an omniscient being, what that being would know would constitute a set of all truths. But there can be no set of all truths, and so can be no omniscient being. Such s the setting or the following xchange.2 1. PLANTINGA TO GRIM My main puzzle is this: why do you thinkthe notion of omniscience, or of knowledge having an intrinsic maximum, demands hat there be a set

Transcript of Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga

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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