Semantics and the Number of English Sentences

Docimini 'Ramis . )3jurlof, AUTHOR TITLE PUB DATE NOTE - A 010 057 Thomas; Jamieson, Dale Semantics -and the Number of English .Sentences. 78 17p. . MRS PRICE DESCRIPTORS ,6 : ' 'HF-80.83.HC-S1.6i4Ius Postage. Coaprehension; Deep Structitre; *English; Grammar;* Language Patterns; *Linguistic Theory: Logic; Nominkls; *Seafinticv *Sentences;.*Sentence. Structure;_Surface Stracture;.Synta:x *Chomsky (Boas) . - , IDENTIFIERS .. . - ABSTRACT 41, as to no been said that there ate.an infinIte' It t*.is number of English sentenceo. "This is the cat- 'that caught the an English'sestenCe. So is "This is thftiCa*.that caught the rat. i that stole the cheesp...."_?.This is the cat with white paws.thaVcaugh the rat that stole the timed'', is unobjectionable as well. Since a.aear W t cutoff point cannot be'ecified, it is`temptipg to resort or "the three dots.r_This studyrproposes that the argugent from-t4e'lack of a clear cutoff,point infinity is a bad argument; the set of English. sentences may be fizzy set rather than a standard set. Furthermore, it is .argued tha !the initial question 'suppresses several quite, 1 distinct 4W4stio s: A theory constructed to acCount,for4formal, relations between 'sentences' - sight warrant the poRiting of infinite semantic-structures. 1. theory cdmisttudted to account- fOr humAn understanding probabLy would not. 'The Common claim that infinity is necessary if there are to be' novel and Areative uses'ofzlanguage is found to to be entirely without gihstante. (AnthOr/AN) . ;, .. : . - , "x 4 44, i **************7!***************4,*4*********41,*************4***O*:, * Reproductions .supplied by EDES "aief the ,best that can Ppe 'made , from the.:30riginel.,.'docupent.. '-'1 c ''.! . * * . *******************4!******************* ******4,*`*************-*********** , ., . ,, 1.." . .--I - . 40 SEMANTICS AND THE_NUM3ER OF ENGLISH SENTENCES 4. f I) CY u.AYbaPAINTAMINIT OF HEALTH. -PERMISSION TO REPRODOCE; THIS MATERIAL HAS BEEN. GRANTED BY EDUCATION a WELFARE NATIONAL INSTITUTE OP EDUCATION . P THLS DOCUMENT HAS SEEN REPRO -t DUCED EXACTLY AS RECEIVED FROM Ting PERSON OR ORGANQAT ION ta,t1wo IT POINTS Or VIEW OR OPINIONS STATED DO NOT NECESsARtLY REPerE;SENT OFFICIAL NATIONAL leesTITutg OF EDUCATION POSITION OR POLICY TO THE EDUCATIONAL RESOURCES INFORMATION SENTER (ERIC) AND I. .46 USERS OF THE ERIC SYSTEM: ". e ti - 4 ` -6 by 0 O O .17) Thomas pjurlOf Departmept of Philosophy Dale' Jamieson Unixi.pi"sity'of North Carolina ChapeltHill, NC 27514 . _Department ofePhilosophyState Univei;sity of New York pollege at Fredonia, -'Fredcnia,.!0/ . e S 14,653 . 401. " tql s 4 SemanticsAnd The Number of ..EvgliSh Sentences 1. How mapy English sentences can be understood? 4 Veryt, very 0 many. Noam Ch6msky has said and perhaps woufa'stilIIsay indefiAtely many or infinitely many (not noticing the difference, of which more later).1.. How many English .sen-' fences are there? 'Zellig Harris has said and perhaps, would still say denumerably infinitely many.2 Many or . .7 very many is not.enaugh according to Chom sky and Harris.' The dsiffetence between very large members and in-inity is .supposed to reflect interesting'" features of human languages. 2. These Views have recently, been`challenged-by,Paul Ziff.3 1 ...:there - .;;moo, important sense in which it is trlike that there are infinitely Ak many English sentences.4, Ziff would agree that there is an unimportant sense in which lgott . therE`A're'infinitely many English sentences." One van do 9.'4,00r5. arithmetic in English: But Ziff. thinks that this, possibility "Is not Elldminating with respect to English structure or '.%English grammar or anything like-it."5 Ziff has* no quarrqa. with syntactit components that generate infinitely many syntactic' structures. His qualms en with semantics. Although it is child's play to characterize powerful syntactic devices-capable of %enerating.an infinity of sentences there's no reason to believe that the semantics - of English (or of any natural language) is ota consonant or even coMparable character.° And he gores on. Inevitably one will produce nons nical sentences or sentences that are tterly incompreheil§ible to speakers of t 4 "language.) 3. Jonathan Bennett-has responded ioZiff's challenge. In his Linguistic Behaviour Bennett replies to Ziff with the following. Butthe question 'How many English sentences are there?' does not mean 'How many sentences are there which would be understood by speakers of English?' and no one would give the answer 'infinitely many' to the latter question...The notion of what has meaning iS admittedly founded upon .the notion of what is a the two are not ce-extensive. but Bennett supposas that the notion "what has moOkning" and-the notion "what is understood" have extensions.. Thus he sup- poses tHht there 3.s some-cIassose members are "what bias mpaning".and another class whose members are "what is Understood." .Bennett aIso.talks about "the class of 'English 7 sentences ".`9 But these predicates are not well-defined. Hence there is no sib .thing as : "the class of 'English sentence: ", npr tho -;cXenSipns .of the notions "what has mean. . 10'"But if we suppose that j_neo:and nwh.a7t, is lirder,ktood. goes-make sense" to talk abOli.extensions in this context, . . Bennett.is certainly corre'ct: :There are English sentences : . th.. -1. ;411 never be u nderstood- by En lish-speakers. . Some since . they will ne4T.be uttered,.some- since _they are too complex, $ some fdr ci,ther..reasoris. Bennett assumes a stronger thesis wl .7 P a 3 however. He assume -that there are English sentences that could not be uRdeAtood by English-speakers. . 4. .If Ziff's position were'that being ai,sentence (or having meaning) is co- extensive with being understood, then, Bennett uld have an easy victory and Ziff's paper could largely e ignored. Things are not thatiimpli, however. Let'us look'at an example.' dinar2y we have no diffi- culty in understanding the following sentences: (a) My uncle is a linguist. (b) My father's brother is aclinguist. (a) and (b) have (roughly) the same meaning. 11 (Notice' that-one might very well know the meaning of a entenceor that two sentences are synonymous without having understOoif the sentences. Someone might know that the sen- tence written on a piece of rper.in,his left pocket has (roughly) the sarte meaning .as the sentence written on.a piece Ad' paper in hiA right pockethim so. sentence. SozeoAle...might have told Nevertheless he might not have Understood either He might never have looked at.them.) Suppose that from tomorrow on English speakers cannot make heads or tails of subordinate groups of two Nouns if the noun?, can be re= versed. Examples Of such pairs would be 'mother's daughter', 'father's brother' and 'a circle under a triangle'. T Would - Ziff say that since (b) cannot be understood by English speakers, although : /) presents no problems, "Cb) is not an 4 an English sentence? We hope not. Failure to understand (b) need not be ettributed to the se ntics o (b), but should most likely be attributed to n urological factors. / After all, (al and (b) have (roughly) (Roan he same meaning. Jakobson discusses a type of aphasia that would ex. plain our case.12) What Ziff. would say is that failure to understand a candidate .for an English sentence may disqualify it as an English sentence if the failure can be attributed tg semantic factors.13 He does not say that a candidate for an English sentence must be disqualified if the failure can be attributed to neurological factors.''' S. One has to be careful in attributing the failure to understand a sentence to semantic factors. Some cases are fairly 114 straight forward:. The failure b3=lsomeone under some' condition to understand (c) 7.is green. could be attributed to the fact that .a number is. not +he= kind of thing that is colored; and this might be thought of as a semantic factOr. Notice that we don't claim that anyone would fail to understand an utterance of (c) under any conditions whatever, just that there are conditions under which someone/, would fail to underrstand (c) due to semantic factors. Why do we attribute this faillire of understanding to semantic factors? Because we take (c) to haver the syntactic structure MP + VP, and ence to be syntactically well-formed. 1 Thus, this failure of understanding must be due to sementic factors. Contrast (c) with 'Ball dolphin mop cathedrhl.' a string that has the Syntactic structure NP + NP + NP and thus is not syntactically well-formed. (Of course schemes have been devised that count all deviance as syntactic. Similarly schemes have been devised that count all deviance as semantic. Neieher seem particularly well-motivated.14) A FUrther complications are presented by other cases. For example _.. (d) This sentence is in English. If someone cannot understand an utterance of (d) because he cannot figure out that 'this' in '(d) refers to,(d), the fail. ure to understand (d) cannot be attributed to semantic faCtors namely, the refepence of 'this' in d) on this occasion. kather'it-can be attributed to the failure to appreciate-the ,relevant semantic factors. ."'But don't confuse the last ease with the following; -ft) This ntence- is ,falie. f One Could argue that, if conceived self-referentially, (e) could not be'understo y any English speaker. (e) is, of.' course, a version of he well-known "liar paradox." assume it to be tr e t.follows that it is-false and vice versa. No vle kno .4 7,41 at it would be like for (e)-to be true, or what it would 14 ike4for it to be false. pursue this line If we, argunient.15' We will not 6. any linguists c14 that that there are infinitely many English sentences. is an answer to. It is often unclear what question this How many English-sentences are there, of course; but as we shall see, what is asked by this question is far from clear. Here i an example: It is astonishing to find that even this truism [the implicit ability to understand indefinitely many sentences] has recently been challenged. See Dixon (1963). However, it seems that when Dixon denies that a language has infinitely maiy sentences, he is using the term "infinite" in some special and rather obscure sense. Thus on the same page (p.. 83) on which he objects to the assertion "that there are an infinite number of sentences in a language" he states that "we are clearly unable to say that there is any definite number, N, such that no sentence contains more than N clauses" (that is, he states that the' language is infinite). Either this is a blatant self-contradiction, or else he has some new sense of word "infinite" in mind.16 A - ti . But surely the mistake is Chomsky's, and Dixon is-not con* tredicting himself. . On Chomsky's view, that there is no ' 4( definite nutter of English semiencAs is equivalent to the . claim tbdc there-are infinitely many ;nglish sentences.- Why rot, not infinitely many but no definite number? In standard set theOry 17 if a set has no definite finite number of elements: 4 it. has infinitely many elements. And all finite sets have a deeinite number of eleznants. But these are standard where_ membership is we.17defined. or it is not. . Somethingw.7Ither e :Amber If we take another approach and think of the set of English sentences as a fuzzy set indefinite, does not, IR. imply infinite. In fuzzy set theory membership is graded. . An element is assigned a numbet in the inclusive interval (0,1). The number assigned to an element with respect set is the-degree to which that element beiongs It does not make sense to ask how many membe o a the set. a fuzzy set 1111. has since membership in a fuzzy set is not an either/or 'question. There is, for example, no answer to the question: . -. How many tall men are there? Some people are taller than ....,f others. S are clearly members of the (fuzzy) set of tall men. :.Some not so clearly. tall'men gre there? Some clearly not. No answer. 04t how many ° And it does not follow that there are infinitely many tall men.18 Mgybe Chomsky tholighf of this: Whis is the cat that caught the rat.' is an English sentence. So is "This it the I 1* cat that caught the rat that stole the cheese.'. 'This is the cat with white paws that caught the rat that stole the cheese.' is also an English sentence'. And it seems that one can,go on like this. There is no definite cutoff For how long? point Chomsky claims, and he is right. one can go on forever? That is, have we shown that there is unbounded redursionk English? we nee Of course not, To show this a proof of unbounded recursion and not just a proof \ that the Does it follow that is no definite cutoff point. , Chomsky does.not . provide such a proof. That there is no definite cloff point , . ,does not create any problems. Maybe we are dealing with fuzzy setS,,and. then we do not need definite `,cutoff points. I 8 7. How many'Elph sentences are there? Is the question how many English sentences car be understood? Or how many syn- tactically well-forr4e4 English sentences are there? Or hout- many syntactic structures, a. syntactic tlileory of English. shOuld generate? , . Or how many semantically non-deviant English sentences are there? Or how many English sentences a semat' tic theory of English should account for? Or how many'accept. , able English sentences are there? We will not quarrel with an ininite syntactic ent; Maybe infinity allows us to simplify-syntaotic theory. Maybe it makes it less complicated. '\ ii infinity Ziff's rejection of is based on semantics, and we will at-tend end to The question is: 8. man- , . tic structures in the following. si . , How many English_sentenc0 shoula a semantic theory of English account for? What should a semantic theory.do? But a priar question: Asvig meaning to Syntactic'. structures; - account for saneness of meani between sen'tences; . phrases,. words and Mbrphemes;'account for paraphrase'; chirac;5 terize logica and sem4ntic entailmen These are . commonly cited requirements.. If we, start with an infinite syatactic component and adopt these requirements (o'r at least some of them, and no others), then"a semantic theory should. %. account for an infinite numb:r of English sentences: - This is- .4. the picture exhibited by philosophers and log.cians such as '_ Richard Montague and Donald Davidson. A langtage is thougIt of.as an abstradt structure, notts a means of cpmmUniktiCin . 10 9 among humans But there is a piroblem. In most formal theories the syntax is constructed so that it.fits the se inc ntics, and syntactic and semantic well-formedness coLinguists have often made a distinction bqtween e. sentences which are' syntactically lcviant and those which Are semantically deviant, and thes,notions do not coincide. Sentence (c) on ,page 4 is perhaps syntactically well - formed. That is, there might be a reasonable syntactic theory of English such that (c) does not violate any rule in this theory. But it is semantically deviant. There are diffi1 culties in drawing a~ line between syntactic and semantic deviance. 20 The distInction is clear enough, however, to allow one ti6 point to sentences .Shat do not 'violate syntactic rule .but do violate semian'tic rules. -In other words: Not every syntactically well-formed sentence is semantically well-formed. This is enough to show that an _infinite syntactic component does not imply an infinite semantic- component. conclusion. We need non.syntactkedarv'm for this One such argument could be that infinity in semantics allows forsimpler semantic theories. Bdnnett argues op in-this way in Linguistic Behavior.21 . Given the primitive: state of semantic thebry this argument is AiffiCult toeeval%late, so we will remain agnostic on this .point. 9. .Something is'missing from the picture exhibited.by 411. Montague and Davidson. Sometimes more is. wanted fr/M a semantic theory than that relations. account for formal semantic Some have wanted semantic theory to lid in the ` 10 understanding of what it in to understand a sontenee (phrase, word, morpheme): This concern with understanding is a characteristic of Ziff's work in linguistics and semantics.?? It serves to distinguish Ziff's work from that of Montague .and Davidson. s Is there anything infiniti about understanding? trivial sense the answer is no: In a People's lives are finite, attention-spans are short, memories are limited. Linguists in the Chomsky-Katz mold also claim to be concerncd with understanding: We may thus regard'the development of a model of the semantic .component as taking .up the explanation of,a speaker's ability to produce and understand indefinitely many new sentences." 3ut they are unimpressed with short attention-spans and limited memories: Linguistic theory is concerned primarily with an ideal spea)er-listener,in a completely homogvnebuS speech-community, who knows its lang. uage perfectly and is unaffected by such. grammatically irrelevant conditions as memory iimitarions 1 dis-tracTions , shifTn "of`" artention and inteest( and errors (rardom or character-istic) in applying his knowledge of the language in actual performance.24 These linguists explicitly hold .that there is an infinite num.ber of English senteces. How can the concern with understand- ing be reconciled with this claim? In the following way: Ling.:istics is not about "cheap empirical facts"; it is about the behavior of ideal speaker hearers. Two lines of assault can be mountea against this position.' . One could disagree with -6-14 goals- of linguistic the° .. conceived by Chomsky and Katz. This we shall not do; .444aasssbeen well don4 elsew'here..2 5 Instead we will take another , tine of 'assault. 'Let us require of a: semantic theory that it __: -eee6count for the mploYed by speaker-hearers in employed .., 1c understanding . . sentences. What are strategies? A strategy . would not be so-called it did not aid in achieving what it is a strategy for. Strategies are like algorithms.;, both are procedures for solving problems. Howekft fall.gbrithis are like strategies ; 4.1 - strategies- are icient -proced-urs for...solving prcsblems. There are ,problems for which algorithms exist but require so 7,1\ .mUch timie.'fof their implementation.'that in practice. they can... not be , employed. (The only 'known algorithms that. are solutions to the problem of the' traveling sales tour is ;an example.26) The strategies ,employed by human speake hewers cannot be of,' this. kind. If they-Were;; 'the. rapiditylof human understands would be left. unexplained. But this argument for the efficiency. , of the strategies argument for the employed by qiuman .speaker-thearers is not an . such strategies. It oculd be the case that there are strategies like the following: finiteness of a means as means what a means means what a means %. JS- \ . That 1 12 *11. every seciuence of a's, raea,n what.:Clid initial a means: This. stiAtegy-i.rould be efficient'and would account for an ink* . Here the problem is not with under,. number of sentences. finite -- 4" standing but With-prOduction. -ftmarrspeakelphearers do notand" 17'. could produce such ,sentences Cat least past some unspecified number Of iterations). Since humanspeaker-hearers do nOt acid sudh,entences, there is no reason to believe Leould note sproducb that such "strategies are employed in'understanding. * /What 2igf'says is right: It is not pere.chance that only a finite number of sentences are uttered and understood; it could not be gtherw ise. The strategies empldyed by human speaker:-c hearers in understanding are finite, however-ideal.you want these humaris to be; and 'what-.humans opera''te on is finite as' well. lit o. An 'nfinitk'ofsentences is .a lot. One'can.go 'on and on and on and on and on and-64:*0 not,r4.Ch:infinity., HoW-many English a ' sentences are there? :Probably not as many as there,are electrons ... in the universe. . Probably not 10 79 . , But you may not be. swayed. .What about novelty? . creativity? . , What about C . What about the do-called "rationalistu'tradition in i linguistics? One bad argUMent that is "repeated in the,lingUistic. literature is that infinity is required if we are to say hew and . 1114, , . : interesting things.. But aselmostL.eveyone knows; only a finite ° number of En5;ish sentences"nae been and will be uttered. :Finnegan's Wake, 77 Dream Songs and all of'ShakesPeare and Striridberg were written from finite resources. . .; 14 - But again one ,11 7! "t might say4that none of these ;works could have been written. we . there not an infinite number of unuttered s-entences. have thought One w .N.- th.a.-t. Linguistics is an empirical science. 'Nos; we. l' . discover that it --Wj. '' h4unts the realin,of "unactualized:possib-leatc-'!-:-,',71" _ , :, ..*r% Linguists who make such claims 4.evour themselves.:2/-.., ( ,, . k -2-4,-,-' \ ''' f I'j ): // / \ 6 .r. ..6.2:. 1 1 l ; .1 a. 1 3 116 J ' NOTES, , . r of the Theory of Syntax, MIT Press, 1965, lip. 1S. 3-62.. Z. ( ris, kathematical Structures of Language, Interseience Pub - ers3 1968- pp.14--.1.1. . 3P;75.ff "The ,Number of English Sentences," Foundations of Language 1974,'pp..519-532. )Ibid. , p. 19. -N p. 521. 6Ibid., pp. 529-530. 7 p p -530 8 J. 1," Bennett; Linguistic Behaviour, Cambridge University:Press, 1976 pp. 279-280. 9 Ibid., p. 279. 10On this point see Dana Scott's excellent es ;ay, "Background to Formalization, "" in Truth., Syntax and Modality, H. LeBlanc (ed.) ,' North Holland, 1973, p. 244 -273_, `11 There is, a ,comiilication that we shall ignore: An uncle can be a -;*.zinoti-fer's%br ther or an aunts husband rather than a. fathero's brother;: 7 ti Jakobsori::-, Studies on Child Language and Aphasia, Mouton '197f- -pp 75-1'4.. fact he has said this in conversation. It is difficult to,. find this., view stated, explingitly in "The Number of English Sentences," however. 14See See forf,example: J. McCawl,ey, '''Concerning the Bas.e Component in a, Trams farmational Grammar "', Foundation of Language 1968 pp . 243 - 269;-'. and K. An.tley, "McCawley's Theory of Sel-ectional Restriction, Foundations of Language 21'97.4, pp., -257-272. 15 For this, vieur, see P. Ziff,. Semantic Analysis, Cornell' University Press, 1960, 'plb 136-138: 0 16T N.Chansky,oD.cit., footnote 9, p. 198. The inserted clause is from p. 15. The reference tg,Dixon is to his Linguistic Science and Logic, Mouton & Co., 196'3. 16 17 An elementary'introduction to set theory can be found in R. Stoll, Set Theory and Logic, W. H. Freeman and Company; 1963. 4 18 10 1 fuzzy set theory has been developed in great detail. The locus classicus is Zadeh, "Fuzzy Sets," Information and Control MS; pp.' 338-353. &See, for example, the papers by Montague sand Davidson in Semantics cof Natural Language, D. Davidson and G. Harman Cedb.:), Reidel, 1972.0 For dascussions of some of these difficulties, see: U. "ExPlorations in Semantid' Theory," in etIrrentiti,ends in Linguistics Volum* III, T. Sebeok,(ed.), Mouton,, Turnings, Cornell University Press, 19b6, pp. 119-133; G. Lakbff, "On Generative Semantics," in Semantics, D. Steirgoerg and L. Jakobovits(eds.), Cambridge University Press, 1971, pp. 232-296. 21 22 0. cit.., gip. 277-278. See, for example, Understanding Understanding, Cornell-Ufilversity Press, 1972. 2S 24 25 J. Xatz, Philosophy of Language,, aarper-&-Row, 1966, p. 151. . .- N. Chomsky, op.cit., : . . . 3. , . . See, for example, D. Hymes, "Toward .Linguistic Competence, Working Papers in Sociolinguistics 16, University of Texas at Austin,' Department of Anthropology; G. Harman, "PdYchological.Aspectsiof the Theory of .Syntax," Journal of Philosophy 1967, pp.. 75-q7; B. berwing, Transformational Grammar as a Theory. of Language Acquisition, Cambridge University Press, 1973, -pp. 259-296. ' . 26 27 . .., . ..- See H. Lewis and C; Papadimitrii.ou, " Thg Efficiency of' Algorithms ," Scientific American 1978, pp. 96-109:, earlier 1 An earlier :version of this paper was presented at a, conference on"Productivity, NoVelty, and Creativity in Language" at the University of Louisville. Ike wish to thank'JonatIlan Bennett and Paul Ziff for aluable correspondence and conversatiO t concerning these topics. r Thanks also to R. J. Haack and Richard Sharvy. '