Chisholm - Notes on the Logic of Believing

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International Phenomenological Society

Notes on the Logic of BelievingAuthor(s): Roderick M. ChisholmReviewed work(s):Source: Philosophy and Phenomenological Research, Vol. 24, No. 2 (Dec., 1963), pp. 195-201Published by: International Phenomenological SocietyStable URL: http://www.jstor.org/stable/2104460 .

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NOTES ON THE LOGIC OF BELIEVING

1. I shallset forthwhatappears o be a fruitfulmethodof investigatingcertainphilosophical onceptsand illustrate t by applying t to the con-cept of believing.The, llustrationwill show, if I am not mistaken, hatthe logic of belief-statements iffers n certainfundamental espects romthatof all the familiar

nonpsychologicalmodalities.And this fact, in turn,enablesus to formulateat least one logical mark of the psychological.2. Let us considerstatements,which are like,those of ordinaryEnglishexcept for makinguse of the apparatusof quantificationheoryand itsvariables, and let us single out two fairly simple types of such statements.The firstcomprises hose statementswhichbeginwith a universalquan-tifier,whichhave no otherquantifier,and whichare suchthat the scopeof the quantifierextendsto the end of the statement.For example:

For every x, x is material.

The second type comprisesthose statementswhich are like the first,except for havingan existentialquantifierwherethe first has a universalquantifier.Hence:

There exists an x such that x is material.

We maythinkof a modaloperator somewhatbroadly)as a phrasewhich,whenprefixedto a completestatement,yields a new completestatement.Examplesare: "It is possible,""It is permissible,"and "Jonesbelievesthat."The two types of statement ust singledout are such that a modaloperatormay be inserted either at the beginningof the statement orimmediatelyafterthe quantifier.Hencefor any modaloperator e.g., "itis possiblethat"- we may distinguish our statement orms,exemplifiedby the following:

(UC) It is possible that, for every x, x is material;(UD) For every x, it is possible that x is material;(EC) It is possible that there exists an x such that x is material;(ED) There exists an x such that it is possible that x is material.

The letters "U" and"E"refer to universaland existentialquantifications,respectively,andthe letters"C"and "D"are shortfor the medieval ermsin sensu composite and in sensu diviso.

Oneway of determininghe logicalpropertiesof anyrmodaloperator,then, is to note the relationsamongits occurrencesn these four forms.

195



196 PHILOSOPHY NDPHENOMENOLOGICALESEARCH

Thus, if we were studying he logic of "it is possiblethat," we could askwhethera statementof the formUC, in which "it is possiblethat" s the

modal operator, mplies the corresponding tatementof the UD; or, asI shall put it, "For possibility,does UC imply UD?" And we may askwhetherUC impliesEC, or ED; whetherUD impliesUC, or EC, or ED;whetherEC implies UC, or UD, or ED; and whetherED implies UC,or UD, or EC.

These twelve possibilities,for any modality,may be set forth in atable. Thus the followingthree tables indicatethe relevantpropertiesoftruth, falsehood, and moral indifference, respectively. The top line of

each table indicateswhat is being consideredas implicans,and the first

vertical column indicates what is being considered as implicate;theoccurrence of a minus-signin any particularplace indicates that theimplicationwhich thatplace representsdoes not hold;the nonoccurrenceof a minus-signn any place indicatesthat the relevant mplicationdoeshold.

UC UD EC ED UC UD EC ED UC UD EC EDUC - UC UC

UD - UD UD-

EC EC EC -ED ED ED

Truth Falsehood Indifference

The firsttable tells us, with respect to "It is truethat,"the formsUCandUD eachimplyeachof the otherthreeforms,thatthe formsEC andED are impliedby each of the otherforms,and that neitherEC nor EDimplieseitherUC or UD. For example,"Thereexists an x such that itis truethat x is material" ED) implies"It is true thatthereexists an x

such thatx is material" EC), but it does not imply "It is truethat, forevery x, x is material" UC) and it does not imply "For every x, it istrue that x material" UD).

The secondtabletells us, withrespect o "It s falsethat," hat the formsUC and ED are each impliedby each of the otherthreeforms,that UDand EC each implyeach of the otherthree forms, and that neitherUDnor EC is impliedby UC or by ED.

And the thirdtable tells us, with respectto one sense, at least, of "itis morally ndifferentwhether," hat UD impliesED, andthat,otherwise,no one of the formsimpliesany of the others.The implication romUDto ED - from "For every x, it is indifferentwhetherx smoke(s)" to"Thereexists an x such that it is indifferentwhether x smoke(s)"- isvalid on the generalassumption hat a statementbeginningwith "Forevery x" impliesthe correspondingtatementwhichbegins instead with"Thereexists an x."



NOTES ON THE LOGIC OF BELIEVING 197

3. Before considering he pattern,or believing- for "S believes.That"or "It is believedthat"- I shall make three general observationsaboutthe natureof belief. If theseobservationsare accepted,as I believe theyshould be, then the patternfor believingwill be as obvious as are thosefor truthand falsehood.

(i) We should note first that people - i.e., believers- are capable oferror: they may believe propositionswhich are false, including thosewhich are logically false; they may withhold belief from propositionswhich are true; they may have beliefs whichcontradicteach other; andthey are capableof two ratherspecial types of error, which I shall call"defective nstantiation" nd "defectivegeneralization."

A defective nstantiations to be distinguishedroma merelymistakeninstantiation.We make a mistake of the latter sort when we suppose,with respect to somethingwhich does not in fact exemplifya certainproperty, hat it does exemplify hat property.Defective instantiation, nthe other hand, is illustratedby the following:Suppose, hat materialismis true - i.e., that everything s a materialobject - and that there is a"complete: hysicist,"a man who has informedhimself with respecttoeach particularobject. He would make the mistake of defectiveinstkan-tiation if he were to believe that, in additionto these physicalobjects,there is still anotherthing- say, God. Knowingeach particular hing ashe does, he has not made the mistake of supposingany of them to benonmaterial,much less to be divine. Withoutcontradicting imself,he be-lieves that there exists an x such that x is nonmaterialand he alsobelieves, with respect to every object, that it is material.Defective in-stantiation might thus be described, paradoxically,as the mistake,ofbelieving,withrespectto somethinghat does not exist,that it does exist;or, somewhat less paradoxically,as being a mistake which is not a

mistake with respectto anythingat all."Defectivegeneralization," n the other hand, may be describedas

the mistakeof believing,with respectto some nonuniversal et of things,that those things comprise everythingthat there is. Suppose that theuniverse s exactly what our "completephysicist" houghtit to be: thereare manymaterialobjects,a, b, c ... n, andjust one otherthing, namely,God. Considernow a second "completephysicist":he, too, is well-in-formedwith respectto each and every particularphysicalobject,but hehas the mistakenbelief that they are everything hat there is, thus com-

mittingthe error of "defectivegeneralization."He then reasons:"a, b,c ... n compriseeverythinghat there is; each of these thingsis a mate-rial object; therefore everything s a materialobject."His conclusion,then, is a mistakewhichis basedupon a "defectivegeneralization." hemistake which it embodiesdoes not consist in identifyingGod with anymaterial hing,or in supposingGodto be material;ike our firstphysicist,



198 PHILOSOPHYNDPHENOMENOLOGICALESEARCH

he knows each' material thing too well to identify it with God, and. (we

may also suppose)he has no hereticalviews as to the materialnature ofGod. He has made a defective generalization a generalizationwhichisfalse, not because of the nature of the things to which it does refer,but

simply because of the nature of one of the thingsto which it does notrefer.

(ii) Having noted that believers are capable of various types of error,

we shouldalso remindourselves hatthey are capableof avoidingerror-at least, that they may occasionally have beliefs- which do not involve

any of the mistakesjust described.(iii) And, finally, I shall make the following general assumption

about the nature of belief (an assumptionwhich most philosophersnowtake for granted,but which has been denied by Meinong and others):Tobelieve with respect to any particular thing x, that x has a certain

propertyF is, in part at least, to believe that there exists an x such thatx is,F; to believe that some S are P is to believe that there exists an xsuch that x is S and x is P; and,more generally, o believe anythingat

all,, with respect to any particular hing, is to believe, with respect tosome property,that there exists somethingexemplifying hat property.

This generalassumption mpliesthat no one who is consistentcan besaid to believe,with respectto any particular hing,that that thing doesnot exist.' The consistentatheist,thoughhe may be said to believe thatGod does not exist, cannot be said to believe,with respectto God, thathe does not exist; and this is true whetherGod exists or not. That is to

say, from the suppositionhat God does exist, we would not be justifiedin inferring hat there,exists an x such that the atheist believes that xdoes not exist. The atheistbelieves, with respect to the things that do,

exist, that none of them has,the propertieswhich are peculiarto beingGod and hence that none of them is God.

Accordingto Meinong, most of us believe that all centaurs have

hooves, thus believing,with respect to certainnonexistentthings, thatthey have hooves.But it has oftenbeen pointedout thatwhat we believe,in such cases, might be expressed n one or the other of these two ways:(a) we believe, with respect to the propertyof being a centaurand the

property of having hooves, that the former includes the latter; or (b) we

believe,with respectto the thingsthat do exist, that if any of them were

1 Hence I resisted the temptation to say that, if defective instantiation is themistake of believing, with respect to something that does not exist, that it doesexist, then defective generalization is the mistake of believing, with respect to

something that does exist, that it does not exist.




a centaurit would have hooves.2 In short: from the suppositionthat,unknown o us, thereexists a centaursomewhere, t does not follow, withrespect to those of us whom Meinong describesas believing that allcentaurshave hooves, that thereexists an x such that we believe that xhas hooves.

Let us say, then, thatto believeanything,withrespectto anyparticularthing, is to have a belief with "existentialmport": t is, in part at least,to believe with respect to some property,that there exists somethinghaving that property.

We may now turnto the modal table for believing.4. To make our examplesuniform and relativelyclear, let us restrict

our universeof discourse to the people who are in a given room at acertaintime; but each examplemay be readilyrevisedso as to hold ofa universewhich is not thus restricted.Let us suppose that there areexactly three such people - Jones, Smith, and Robinson;these will bethe things which our variableswill designate (or, at least, which thevariablesof our true sentences will designate).Consider,then, the fol-lowingstatements:

(UC) S believes that, for every x, x is sitting;

(UD) For every x, S believes that x is sitting;(EC) S believesthat there exists an x such that x is sitting;(ED) Thereexists an x such that S believesthat x is sitting.

We may now turnto the columnsof our diagram.(i) UC does not imply UD. S may commit the error of "defective

generalization,"elieving hatonly Jonesand Smith are in the room,andhe may believe that each of them is sitting.If he is consistent,he willconcludethat, for every x, x is sitting (UC), thus basing a conclusion

upon his defectivegeneralization.But UD in this instance will be false;for thereexists an x, namely,Robinson- such that it is false that S be-lieves that x is sitting.

(ii) UC does not implyEC. S may believe,mistakenly, hat thereisno one in the room;hence he may also believe it to be false that thereexists an x such that x is sitting;and thereforehe will believe that, forevery x, it is false that x is sitting (UC).3But believingthat thereis no

2 Another possibility, of course, would be: we believe that there exists nothing

which is a centaur and which lacks hooves. But I think Meinong was right insaying that this is not plausible as an account of what it is to believe that allcentaurs have hooves.

3 If the reader is inclined to reject the "therefore,"he may replace the "forevery x," in its various occurrences throughout our discussion, by "it is false thatthere exists an x such that it is false that," and the inference will then be unob-jectionable.



200 PHILOSOPHYNDPHENOMENOLOGICALESEARCH

one in the room, S does not believethat there existsan x such that it isfalse that x is sitting.

(iii) UC does not imply ED. In the situation just described,S be-lieves that, for every x, it is false that x is sitting(UC); but there existsno x suchthatS believesit is false thatx is sitting.He believesthatthereis no one there, not that the people who are there are not sitting.

(iv) UD does not imply UC. S may believe that Smith, Jones, andRobinson are all sitting; hence, for every x, S believes that x is sitting(UD). But he may believe, mistakenly, hat there is a fourth person andthat this fourthpersonis standing; n which case he will not believe that,for every x, x is sitting.He will have made the mistakeof "defective

instantiation."Or he may simply be agnosticand reluctant o concludethat the threepeople, whom he believes to be sitting,compriseeveryonein the room.

(v) But UD impliesEC. If each personin the room is such that Sbelieves him to be sitting (UD), then S believes that there exists an xsuch thatx is sitting(EC). For, as we have noted, to believe withrespectto any particular hing (and hence also to believe with respect to eachparticular hing)that that thing has a certainproperty s to believe thatthere exists somethinghavingthat property.

(vi) UD impliesED. This is a consequenceof our assumption hatthere is something i.e., thatthere are people in the room. If S believessomethingwith respectto each thing, then there exists somethingwithrespectto which he believesit.

(vii) EC does not imply UC. In believingthat there exists an x suchthat x is sitting(EC), S need not believe that, for every x, x is sitting;for, as,we have noted, he need not always be in error.

(viii) EC does not imply UD. If S believes that only Robinson is

sitting,then it is not true that, for every x, S believesthat x is sitting.But it is true (EC) that there exists an x such that x is sitting.

(ix) EC does not imply ED. S may believe that some one of thethree persons is sitting and yet have no belief as to which one of thethree it happens o be; hence there will be no x suchthat S believes thatx is sitting,but S believes that there exists an x suchthat x is sitting(EC).

(x) ED does not imply UC. One of the,people, say Robinson, maybe such that S believes,him to be sitting (ED), without S also believing

that, for every x, x is sitting.(xi) Nor does ED imply UD. One of the people, again, may be such

that S believes himto be sitting (ED), without everyx being such that Sbelieves that x is,sitting.

(xii) But ED implies EC. If S believes, with respectto Robinson,that he is sitting,i.e., if there exists an x such that S believes that x is




sitting(ED), then, as we have noted, S will also believethat thereexistsan x such that x is sitting.

We have, then, completedour table for believing.UC UD EC ED The pattern has the failures of

UC both truth and falsehood; but,

UD - unlike that for moral indifference,

EC - it is such that UD implies EC,ED - and ED implies EC.

Believing5. The patterns or various"alethic,""deontic,"and causalmodalitiesdiffer, in some respect or other, from the pattern here attributedto

believing.4Thus the "alethic"and causal modalitiesdifferfrom believingin the

followingrespects,amongothers: or necessityand possibility,UC impliesEC; for impossibility, ED implies UC; and for nonnecessity, contingency,

and noncontingency, ED does not imply EC.

The "deontic"modalitiesdiffer in these respects, among others: forobligatory and not wrong ("permitted"), UC implies EC; for wrong

("forbidden"), ED implies UC; and for nonobligatory, indifference, and

nonindifference, ED does not imply EC.

The "epistemic" modalities - evidence, probability, acceptability and

the like - are more difficult to interpret.On some interpretationsheypertain o believing- to whatpeople are justified n believing- and it is

possiblethat, on some one of these interpretations,hey will conformto

the patternI have attributed o believing.But I know of no modalities which are obviously nonpsychological

andwhich conformto this pattern.It may be, therefore, hat the patterngivesus a markof intentionality.Perhapswe can say that a modalprefix

is intentional f its occurrence n our four statement-formss such that:UC does not imply UD, EC, or ED; VD impliesEC and ED, but does

not implyUC; EC does not implyUC, UD, or ED; and ED impliesEC,but does not implyUC or UD. Or, moreaphoristically: A modalprefixis intentional f it may say that some or say that all when it doesn'tsayof any, and say of each and hence that some when it doesn'tsay thatall."5

RODERICKM. CHISHOLM.BROWN UNIVERSITY.

4 I have discussed the patterns for these and for certain other psychologicalmodalities in greater detail in "On Some Psychological Concepts and the 'Logic'

of Intentionality," read at Wayne University on December 6, 1962, and to be

published by Wayne University Press in the Proceedings of the Wayne Conferenceon the Philosophy of Mind, -edited by Hector Neri Castafieda.

5 I am indebted to Robert Sleigh, Herbert Heidelberger, and Hector Neri

Castafieda.

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Chisholm - Notes on the Logic of Believing