Bergmann Elementarism

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International Phenomenological Society

ElementarismAuthor(s): Gustav BergmannSource: Philosophy and Phenomenological Research, Vol. 18, No. 1 (Sep., 1957), pp. 107-114Published by: International Phenomenological Society

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DISCUSSION

ELEMENTARISM

Cansuchwordsas 'color'and 'pitch'be replacedby circumlocutionsn

all contexts without loss of meaning?Elementarism s the affirmative

answer o this question.Or,rather,this is what I wouldsay if askedto

explainthe notion to one not familiarwith the techniquesof analytical

philosophy.The preciseexplicationuses the tool of an ideal language,L.

With this way of philosophizing assume the readerto be familiar.'

Unless L contains either the type division of Principia Mathematica

(PM)or one syntacticallyequivalentto it, the problemof elementarism

doesnot arise.In the firsttwo sectionsof thisnote I shallassumethat L isthe schema of PM, without ramification,with extensionality, supple-

mentedby a classCof undefineddescriptive onstants.C,the elementarist

claims,can be so chosenthat all its membersareof types zeroorone,i.e.,

either ndividualconstants('a','b', . . .) or first-order redicates,ncluding

relationalones ('f1', '/2', . 'rl', r2, . . .). To see the connectionbetween

this preciseformulationand the vague one with which I started, one

merelyhas to assume,quiteplausibly,that somecolorwords,say, 'green'

('gr')and 'red',are amongthe first-orderpredicatesof C. Since we say

'green is a color', 'color'would then be of the second type; thus the

elementaristwould have to insist that its transcription('Col')can be

defined n termsof his C.Thethesisof elementarismhas occupiedme for quite sometime and I

haverepeatedlyexpressed he opinion hat it is in facttrue.Thatis why I

call it a thesis,usingthe label for factualassertionsof so broada nature

that they arefelt to be "philosophical."Determinism,as I explicateit,

is another hesis.)RecentlyWeinbergand Palmierihave cometo sharemy

preoccupation.Weinberg in a difficultpapersuggests,if I understandhim correctly,that elementarismmay be true on philosophical rounds.

Palmieridevotedtwo papersto the subject.Thefirst3 is a brilliantcriti-

cismof somepointsWeinbergmade.In the second' he examines,among

other things,a definition oncegavefor'pitch'.Stimulatedby thesethree

studies,I nowpropose o do threethings.In SectionOneI shallshowthat

the only philosophicalreasonswhich one could with some plausibility

1 See The Metaphyaics fLogicalPositivism (Longmans Green&Co.,1954),hereafter

cited as MLP.2 "Concerning ndefineddescriptivepredicatesof higher evel," Mind, 53, 1954,338-

44.

3 "Higher evel descriptivepredicates,"Mind,54, 1955,544-7.4 "Second evel descriptivepredicates," his journal,16, 1956,505-11.

107








108 PHILOSOPHY AND PHENOMENOLOGICALRESEARCH

adducein favor of elementarism,or, rather,the only such reasons I canthink of, are- pecious.In Section Two I shallshow onlogicalgroundshatexceptin a finiteuniverse,where t is triviallytrue,elementarism,f true,cannot but be a factual truthof the kindI calla thesis. In Section ThreeIshallsuggesta reason orwhatI thinkis a weaknessn Palmieri'sotherwisevery penetrating econdpaper.

I

What does it mean to hold a propositionon philosophicalgrounds?Consideran example.Samehold that all undefineddescriptiveconstantsof L must referto thingswith which we aredirectlyacquainted,or,whatamountsto the same,which arewhollypresentedo us in an awarenessofthe kindcalleddirectacquaintance.5Thosesharing his belief (including,probably, Weinberg and Palmieri as well as, certainly, myself) are said to

acceptthe Principleof Acquaintance PA). To accept a propositionas aprinciples to refuseto defend t directly andto argue nstead, first, thatthe thingsit mentions areall commonsensical, nd, second,that withoutacceptingit one cannot solve all the philosophicalproblems.(This, Isubmit,is what 'principle'means n philosophy.)A propositions held on

philosophicalgrounds f it is thought to followdeductivelyfrom one orseveralprinciples, ither(oc) y themselvesor (fi) n conjunctionwithsomecommonsensicalruths about the thingswhichthey mention.

Theexample s not haphazard.Those trying to establishelementarismon philosophicalgroundsall-believe,either explicitly or implicitly,thatit follows deductively from PA in conjunction with two additional

premissesof the kind(fI).Moreover,his is the only primaface reasonfortheir belief of whichI canthink.To show that it is speciousI mustexposethe flawin the supposedargument.

Awarenesses re events.Certain tates of affairs,such as the one, call itS, referred o by 'Col(gr)', re not. Thesearethe two additionalpremisses.So far as they go, I have no quarrelwith them. 'Event', to be sure, is vague.In some contexts we call the French Revolution a single event; in some

others we don't. In the present context, though, an event is, withoutdoubt, what is whollycontained n a specious present. The fallaciousargumenthas two major parts. In the first, one concludes romthe firstpremissthat what an awareness s the awarenessof, or, as one says, itscontent,is alsoan event. In the secondpartone makesuse of the secondpremiss. If 'Col' were undefined, its referent would on some occasions be

wholly presented n a singleawarenessof the kind called directacquain-tance. So would or could therefore on other occasions S. Yet S is not what

5 Remember that, with this meaning of the phrase, we are not acquainted withanythingbut phenomenalobjects.








ELEMENTARISM 109

by this hypothesis t would have to be, namely,an event. One concludesthat 'Col'cannot be undefined. The flaw or gap is in the first part. Anawarenessand its content are two things and not one. Hence the latterneed not be an eventmerelybecause he former s one. This is the heart ofthe matter. The error s facilitatedby three subsidiaryconfusions.Thefirst stems from a certain blur surrounding wholly contained', 'whollypresented'and 'event.' That is why I italicized them when they firstoccurredand insertedat the beginningof this paragraph he qualifyingclausewhichI alsoitalicized.Thesecondsubsidiary onfusion tems fromafaulty analysis of time; the third, from an inadequateexplicationof thenotionof analyticity.I shalltake themupin this order.

1. A particular s wholly contained in a speciouspresent; a character(universal)s not. This use of 'whollycontained' s as unproblematic s isthe truth of this proposition.It follows that if 'wholly contained' and'wholly presented'weresynonymous,no universal,whateverthe type of"its" predicate, could be (partof) the content of a direct acquaintance.Onewho takesPAseriouslyand is not, thoughperhapswithoutknowing t,a nominalist must thereforeinsist on two points. First, to be whollypresented n a speciouspresent and to be whollycontained n it are twothings and not one; second, some characters are sometimes wholly

presented to a direct acquaintance even though they are not whollycontainedin "its" speciouspresent or, for that matter, in any other.These,too, I submit,arecommonsensicalruths. The crucialblurbecloudsthemby blurringhedistinctionbetweenbeingwhollypresentedandbeingwholly contained in a specious present. The haze that surrounds 'event'

derives from it.

Eventsarea kindof statesof affairs.A state of affairs s what is referredto (not "named" ) by a sentence. In our context, which "ties" events to a

specious present, an event is a state of affairs referred to by a substitution

instanceof '/(x)', say, 'gr(a)'.6But of the two constituentsof this state ofaffairs only one, the particular, is wholly contained in a specious present;

the other, which is a character, s not. In this sense no event is whollycontained n,a speciouspresent.Thatis why I spokenoncommitallyof atie. For most purposes he inaccuracydoesnot matter.In this argument,though,one may wellwonderwhetherI have not merelyshifted the blurby makingas muchas I did of eachawarenessbeing in a speciouspresent.Happily, this is not so. Closeranalysis 7 reveals that an awareness as such

isa

particular; it is an awareness of somethingy virtue of exemplifying a6 Extending this schema, as one easily could, e.g., to certain "molecular"states of

affairs,yields nothing that is relevant to my purpose.The above explication of 'state ofaffairs' s, of course,Wittgenstein'sin the Tractatus.

7 See MLP, especially essays 3 and 6, and, in greater detail, "Intentionality," inSemantica Archiviodi Filosofia, 1955),177-216.








110 PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH

character of the kind called propositional. These characters, incidentally,

have no adequate transcription in L. Thus L is not really the ideal language

Ofthis later.2. 'Green is a color' is sometimes called an eternal truth. In order to

dispel the second subsidiary confusion, I shall for the moment use 'truth'

ambiguously, once as I just did, once for the states of affairs to which true

sentences refer. Greenbeing a color is then an eternal truth. To be eternal,

if it means anything at all in analytical philosophy, is to be timeless. What,

then, does it mean for a state of affairs to be timeless? Since there is of

course no such thing as "time," it can only mean that no temporal charac-

ters (being earlier, later, simultaneous, and so on) are among its constitu-

ents. Notice that in this sense, which is the only literal sense of the term, abeing green is just as timeless as green being a color. Nor is that all. Since

the temporal characters are just characters, a being green and green being

a color are both timeless in exactly the same sense in which they are also

soundless and tasteless and in which, say, middle-c being a pitch is

colorless. The only difference is that we have no adjectives that are to

'soundless', 'tasteless', and so on, as 'eternal' is to 'timeless'.

The bearing all this has on elementarism is clear, I trust. The positions

philosophers take on the several philosophical issues tend to fall into

patterns. Quite a few patterns contain (a) PA as well as (b) the rejectionof the very idea of "eternal truths." And, if there are none, we can of

course not know any, let alone be directly acquainted with some. The

connection between (a) and (b) is, as we just saw, purely incidental or,

if you please, historical. Structurally, we also saw, its major cause is a

faulty analysis of time.

3. Considerthe two states of affairs S and S' referredto by 'Col(gr)'and

'(Ool(gr)' respectively. No character, of any type, is ever wholly presented

to uswithoutbeingexemplified.This s a partof PAwhichwehavenotyetattended o. Or,perhaps,t-is a principle f its own;thedifferencemakesnodifference. It follows that for S to be the content of a direct acquaintance

green itself must be exemplified, say, by the particular a. Similarly for S',

the particuliernvolved beingb. Thetwo particularsmay be but need not

be identical.If different, hey need not be contained n the samespeciouspresent. Hence, so a certain argument runs, if 'Col' were undefined, S and

S' could conceivably e the respectivecontents of two successivedirect

awarenesses.Thisis thoughtto militateagainstthe unquestionedanaly-

ticity of ',[Col(gr). Col(gr)]'.One concludes that 'Col' cannot beundefined.The fatal flaw of this argumentarises from an inadequate

explication of the notion of-analyticity. The analyticity of the sentence I

just wrote down depends on its being an instance of the form ', (p. p)'and on nothing else. 'Conceivably' marks the major source of the inade-

quacy. That is why I underlined it.








ELEMENTARISM 111

II

In a universe with a finite number of individuals every descriptive

predicate,of any type, can be definedby enumerating he thingswhichexemplify t. In a finite universe elementarisms therefore riviallytrue.Consider ext a universeof cardinalitygo,i.e., with a denumerablynfinitenumberof individuals.The class consistingof (1) all undefined logicalsignsof L and (2) the maximumnumberof individualconstants,one foreach individual, s then also denumerable.The class of all possibleone-term predicates,however, being equinumerouswith that of all possiblesubsetsof a denumerable et, is of cardinality2Mom);andwe knowthat,

by Cantor's amous theorem,2Mo mo.Since the classes of all orderedpairs, triples, and so on, of individuals are each also denumerable, heclassesof all possibletwo-termrelations, hree-termrelations,and so on,areeachalso of cardinality t. Sois therefore,as is easily shown, he class ofallpossibleconstantpredicatesof type one.Assumenow that L containsas of course t neednot, but this is the most favorablecase- anundefinedconstantfor eachof the Xdifferentpredicatesof type one. Call he class ofthese constants(3). This makesthe sumof classes(1), (2)and (3)of cardi-nality X. It follows readily that the class of all finite sequences,with

repetitions,of signsfrom this latter class is still of cardinality a. Everywell-formed xpressionof L not containingundefinedconstants of typeshigherthanoneis sucha sequence.It followsthat even in the most favo-rable case one can in L form at most x formallydifferentdefinitions orpredicatesof the secondtype. The numberof all possible predicatesofthis type, though, s 2tX; ndwe have, again,2x>x. This is just anotherapplicationof Cantor's heorem.Now it may of coursebe the case that allthe secondorderpredicates,definedor undefined,whichL must containin orderto be the ideallanguageare amongthose that can be defined. It

maybe the case.But since,as we saw,it is impossible o defineall possibleones- thereare,so to speak,too many for that - it neednot be the case.This is the point. I have alreadyshownthat in a denumerableuniverseelementarism,f it be true(asI believe t is), is a factualtruth, or,as I putit, a thesis. Theproof s easilyextendedto universesof highercardinality.But I shall not botherwith what interestsonly mathematiciansand ismathematicallyrather trivial at that. A philosophermay be puzzledbysomethingelse. In the introduction said that this proofrests on logicalgrounds.Yet I introducednto it thenumberof individualsn the universewhich, t is felt, is a matter of fact and not of logic. Quiteso. The ssue thisquestionraises is important indeed; I hope to return to it on an earlyoccasion.HereI canonly say that I chosemy wordsdeliberately.

To insist that elementarisms a thesis is not to deny that one can, bymeansoflogical(structural) roperties ingleoutlargeclassesof predicates








112 PHILOSOPHYNDPHENOMENOLOGICALESEARCH

of highertypes, say, the second,which are, by virtue of these properties,definablein terms of certain othersof lowertypes, say, the first. For aratherobviousexample,considera predicate F1', of type two, suchthat

(a) the emptyclass s not F1, and (fi) f /1 andf2 are differentandbothF1,then there snoindividual hat is both/1and 12.Definenow

(1) 'r1(x,y)' for '(3f) [Fl(f) *f)(y)]'

and

(2) 'F2(f)' for'(3x) (y)[/(Y)= r1(x,y)]'.

Oneproveseasily

(3) F2 F1

i.e., that 'F1'canbe defined,by (2),in termsof 'rl'. I skipthe proof,whichis trivial,andmake nsteadfour points.

1. (a)and (i) are of course he logical(structural) ropertiesn question.

Pitch and color(hue)are-two amongthe many charactershaving these

properties.If 'F1' is interpretedas "pitch,"then 'rl' namesthe relation

"equal-in-pitch."2. Instead of (3)I couldhave written(3')F2(f)= Fj(f). In otherwords,

sinceLis extensional,here s nothingproblematic boutthe '=' in (3).8. Formally,there is nothing circularabout the procedureoutlined.

Given 'F1' as an undefinedpredicate,one can, by (1), define a certain

relation.BecauseF1 has the properties a)and(fi),this relationhascertain

others.Given'rl' as an undefinedname for thisrelation,one can, by (2),define'F2'andthen,by makinguse of thoseproperties f ri, prove(3).

4. Philosophically, ne might feel thereis some sort of circularity.He

may argue, orinstance,that even though'equal-in-pitch' e a relationof

the first type, to be directlyacquaintedwith what it namesis the same

thing as to be directlyacquaintedwith what is namedby the type-two

predicate'pitch'. Or, to introduceanothercrucialword, he may insistthat one cannotknowthe meaningof 'higher-in-pitch'without knowing

that of 'pitch', and conversely.

Points 1 and 3 require no comment. Points 2 and 4 do; also, they are

involved in the difficulties with which Palmieri wrestles in his second

paper. So I shall comment on them in the next section.

III

Palmieri thinks that a definition of 'pitch' which I once suggested 8 is

circular. The crucial predicate there is 'higher-in-pitch', not, as above,

8 "Undefined descriptivepredicates,"this journal,8, 1947, 55-82. Palmieri is also

puzzledby my use of '=' in that paper.On this scoreI perhapsmisledhimby not being

as clearas I shouldhave been. Even so, I eliminatedwhateverdoubt there might have

been a year later in "Conditions or an elementaristicextensionallanguage,"Analysis,

8, 1948, 44-7 (reprinted n MLP),which Palmierihas apparentlyoverlooked.








ELEMENTARISM 113

equal-in-pitch'.But this differencemakesno difference,ust as the otherdetails of that definition do not matter. The supposedcircularity s, Ibelieve, of the kindmentioned in the last point of the preceding ection.One reason for

my believingthis is that he

explicitly accepts anotherfeature,sharedby all suchdefinitions,which some may mistakenly hinkto entail another ort of circularity.Let me explain.

Supposesomeoneproposes wo predicativecontexts, '01' and '02', asdefinitions(transcriptions) f 'duck' and 'bird' respectively.How are weto judge their adequacy?The criterion s, very roughly, hat such senten-ces as '(p)[01(9) 02(f9)]', which transcribes"all ducks are birds,"mustbe"true." I say veryroughly ortworeasons.First,there s muchmoretobe said about the matter, e.g., as to which English sentencesmust and

whichneed not satisfythe criterion.Second,whatis reallymeantby 'true'is that the sentences n question must be deductiveconsequencesof thetwo definitions n conjunctionwithsomeothersentences,containingonlyundefineddescriptivepredicates,whichwe have reason o believearetrue.9In this note I am merely nterested n the clausewhich ollows heitalicizedphrase.For it seemsto me that some think it involves somesort of circu-larity; the idea being, f I may so put it, that one merelygets out what onehas put in and thus does not really achieve anything.As I see it, theseobjectors ail to grasp he veryidea of definitional econstructionAufbau;

definitionby extensiveabstraction).On this Palmieriagreeswith me; soI shall say no more about the matter. Also, he acknowledges hat theprocedure s not formallycircular(point 3 above). I concludethat whatbothershim s the supposedcircularityof "meaning."

'Meaning' as manymeanings.Threeof them arehere relevant.In thefirstsense,which s the weakestor broadest, wo predicateshave the samemeaning if and only if they are extensionallyequivalent,as in (3'). Aslong as the languageas a whole is extensional,such expressionsare alsoidentical,at least if oneaccepts, as I do, the Leibniz-Russellxplicationofidentity. In the secondsense, whichis stricter, two predicateshave thesamemeaning f and only if the statement of extensionalequivalence sanalytic.In thethirdsense,which s the strictest,twoexpressionshave thesamemeaning f and onlyif in writing hem down we use the same tokensin the same order.In a sense,this thirdkind of meaning,called ntensionalis thus unique.The reason or this uniquenesss that differentexpressions

9 The conditions (ox) nd (fi)are merely a special case of such sentences. Their only

peculiarity s that they state nonrelationalpropertiesof a singlecharacter. n the generalcase the structural properties are relational. Concerning'structure': Let 'P' be a

(closed)sentencecontainingtwo descriptive signsof types m andn respectively (m ? n)and no other descriptive signs. Replace in 'P' the two constants by variablesof the

proper types. The resulting expression is the definiens of a logical relation of typem + 1, satisfiedby two constantsif andonly if 'P' is truefor them.








114 PHILOSOPHY AND PHENOMENOLOGICALRESEARCH

"belong" o different-concepts.10oncepts,t turns out, are a specialkind ofthe propositional haracterswhich werementioned n SectionOne.Moreimportant, the intentionalmeaningof 'meaning' annotbe accommodatedin L, which s thereforenot really the ideal language.The latter,call it L',is no longerextensional;and no two differentexpressionsare identical nL'. L is nevertheless he bulk of L'; and an even largerpartof it remainsextensional.

What stirs n Palmieriand, generally, n those who raisethe objectionofpoint 4 in the precedingsection, is, I submit, the idea of intentionalmeaning.To see that, consider hat the objectionmay be taken to assertthat, loosely speaking,"equal-in-pitch" nd "pitch" sharethe same or

nearly the same intentional meaning. I say loosely speaking, becauseintentionalmeaning s, as we saw, unique.Informally, hough,there is aconnectionof ideaswhichI can understandand appreciate. f I am rightand thisis indeedthe heart of the objection, hen t is, I believe, ll-founded.The reasonwhy I believe this is that the propercriterion or definitionalreconstructions not identityin L' but, rather,extensionalequivalenceor,whatamounts o thesame, dentityinallextensional ontextsofL'. Again,a very goodreason for this beingthe propercriterion s that identity ofdifferentexpressionsn L', which ncludes dentityof "concepts," s, as we

alsosaw,unattainable.This leads to whatmaywellbe the ultimate sourceof Palmieri'suneasiness.

Thecrucialobjectionmay also be takento assertthat since"equal-in-pitch" and "pitch" have phenomenallyhesame or nearly he samemean-ing, definition of the former in terms of the latter is an entirely emptyachievement.As farasphenomenologicalmeanings concerned, agreeandeveninsist. The aboveanalysis explicatesthe sense in whichintentionalmeaningis unique and this uniquenessreflects the sense in which thephenomenalmaterialsimply is what it is, neitheranalyzablenor recon-structible,eitherdefinitionallyor otherwise.The reasonwhy I believetheobjectionto be-nevertheless ll-foundeds that phenomenologicalescrip-tion and philosophicalanalysisare two things and not one. The latterindubitablyrestson the former.But that does not mean that it eithershouldor couldfollow it in all details.For, if it did, wouldit not betheformer?

GUSTAVBERGMANN.STATE UNIVERSITY OF IOWA.

10 I use the noncommittal 'belong' advisedly. For details, see "Intentionality,"

loc.cit., and G. Bergmannand H. Hochberg, "Concepts,"Philosophical Studies, 7,

1957, 19-27. Addition to L of the propositionalcharacters still leaves it extension-al. The only nonextensional contexts are those containing the transcription, by a

new logical primitive,of the intentionalmeaningof 'means'.

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Bergmann Elementarism