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
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Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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.
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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:
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
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.
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
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.
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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".
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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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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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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
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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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
8/22/2019 Truth, Omniscience, and Cantorian Argument : An Exchange between Patrick Grim and Alvin Plantinga
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