Griceian Numbers

Speranza R. Nouwen, in _http://linguistlist.org/pubs/reviews/get-review.cfm?SubID=65223_ (slightly paraphrased -- check with the original, if you are interested in the original wording). "The meaning of simple numeral expressions like 'two', 'three', 'twenty- seven' etc. has turned out to be one of the most problematic issues within" Griceianism. "Part of the problem is that there seem to be several candidates for _the_ meaning of an English cardinal." "Numerals can be used in many ways, three of which have been the focus of discussion in the pragmatic literature of the past thirty to thirty five years." "'Two'" as specifying "exact cardinality", 'two' as specifying "a lower bound" and 'two' as specifying "an upper bound." "Bultinck's book on 'Grice's numerous meanings' is an attempt at tackling theissue by comparing the most influential theoretical trend of the past three decades, the so-called Griceian programme, with the results of an extensive corpus study of numerals." Bultinck's study contains a detailed discussion of the legacy of the Oxford philosopher, born near Birmingham, H. P. Grice and his theory of conversation, with particular focus on the repercussions for the analysis of English cardinals. It is argued that the _conventional_ meaning of a numeral needs to be established by means of a corpus analysis. As Bultinck subsequently aims to show, such an analysis undermines the Griceian assumption that numerals present "a lower bound" in their coded meaning. Bultinck starts with an thorough discussion of Grice's original motives and proposals. Crucial is the distinction between conventional meanings and implicated meanings. Whereas the former are to be seen as the 'coded' or 'literal' meaning of an expression, the latter arise through inferences licensed under the assumption that the speaker observes maxims on the quantity, quality, relation and manner of what he says. Grice intends to keep the semantics ('sense') of expressions simple by showing that a single conventional meaning could give rise to more than one meaning by means of conversational implicatures. The content of the conversational principles as well as their formalisation have subsequently been much debated and Bultinck describes these developments in considerable detail. While acknowledging the general success of Grice's theory and its offspring, Bultinck argues that Grice's goal to combine a theory of conversation with the intention of preserving the logical meaning of logical expressions is misguided. Bultinck states that there is no methodological justification for taking the conventional meaning of a logical natural language expression (like 'or', 'and', 'if...then') to be exactly that of their logical counterparts. Bultinck associates what is conventional with what is familiar and therefore argues that frequency data can help determine which meaning is more conventional than others. Bultinck continues his discussion of Grice's legacy, but now focusing entirely on the literature on numerals. Most attention goes to the neo-Gricean line of theories that is labeled 'minimalism' and that is inspired by Horn's 1972 notion of 'scalar implicature', a generalisation over phenomena where a weak item on a scale implicates the negation of the stronger items. Minimalists argue that if the numerous meanings displayed by numerals are to be explained by means of conversational implicatures, then it must be the case that their coded meanings line up in an entailment scale. So, numerals are thought to form an entailment scale such that a sentence like: Two cowboys came. is entailed by the stronger Three cowboys came. (or "Mary is five years old" is entailed by "Mary is six years old"). By uttering Two cowboys came. the utterer therefore (potentially) implicates that the stronger alternative is false, thus arriving at the meaning Exactly two cowboys came. The entailments are accounted for by assuming that the conventional meaning of a numeral like 'two' is 'at least two'. Bultinck aims at showing that his corpus data falsifies this line of thinking, but in the theoretical discussion he also presents some non-empirical counterarguments, most of which are familiar from the literature. His most salient critique, however, is a repetition of the methodological critique he presented before. Bultinck argues that what Grice aimed at with his notion of conventional meaning was a standard meaning. Bultinck proposes to identify conventional meaning with ''familiar meaning''. Conventionality is thus equated with a relative high level of frequency. He argues that this implies that conventional meanings are frequent. The minimalist's choice for a conventional 'at least two' meaning, however, is not based on frequency at all. In fact, conventional meanings are solely chosen on the basis of their potential for conversational inferences. There is furthermore a short discussion of the underspecification account, where the ''logical form'' of a numeral is underdetermined and can be enriched by specifying with 'at least two', 'at most two, 'exactly two' or even 'approximately two'. Some other positions (called 'marginal' by Bultinck), like those arguing for bilateral conventional meanings or ambiguity, are discussed as well. A ''general corpus analysis'' is discussed which aims at discovering the different forms and functions of numerals. The analysis involves 1,000 occurrences of "two" from the British National Corpus. Bultinck analyses the core meaning of numerals, namely the cardinal "one". Bultinck however, focuses on a more general analysis which aside from taking the syntactic form and function into account, focuses on all possible ways of using a numeral. Apart from the core use of the specification of cardinality, these include the numeral as a label, the numeral as a temporal indicator and the numeral as a mathematical primitive. Bultinck isolates a wealth of variation in usages and discusses the underlying corpus data in great detail. Bultinck stresses that the data clearly demonstrate that it is a mistake to simply assume that the meaning of numerals can be reduced to a notion of cardinality. One clear result of the analysis, however, is a correspondence between adnominal uses and the expression of cardinality. Almost all adnominal numerals in some sense express the cardinality of a group. Bultinck tries to come to a hierarchy of numeral constructions in terms of the degree of cardinality that is involved and concludes that "the expression of cardinality is clearly the most important function of "two"" (p. 153), followed by the expression of measurement, which, as acknowledged by Bultinck, in many respects involves cardinality as well. A corpus analysis is presented that focuses on what kind of meanings cardinal uses of numerals display. It is this analysis that is supposed to contribute to the issue of the conventional meaning of 'two'. Again, Bultinck refers to the corpus method as "the methodological outcome of [Grice's] theoretical insights" (p. 168). Bultinck distinguishes four possible meanings (pp. 176,177): "at least n": necessarily n + possibly more than n "at most n": possibly n + not possible more than n; "exactly n": necessarily n + not possible more than n + not possible "less than n"; and "absolute value n": non-modal, the group of elements denoted by the NP is determined as having n elements. Crucial here is the assumption that the first three of these meanings involve modal statements about cardinality. The 'absolute value n' meaning, on the other hand, is relatively simple. In fact, Bultinck maintains that it is 'cognitively' simple, since it refers to nothing more than cardinality of a group, and that the other interpretations are therefore in some sense marked. That is, the first three meanings make what is said (understood in a non-Gricean way) about the cardinality much more prominent than the 'absolute value' interpretation does. The majority of occurrences of 'two' turn out to be either of the 'absolute value'-type or of the 'exactly n'-type. Bultinck notices that the "exactly n" readings are mostly caused by definite markers. There are no findings in the corpus of 'absolute value' uses with such markers. This observation also serves to explain the distribution of the different usages over different syntactic positions. For instance, the majority of direct objects contain numerals of the 'absolute value' type, whereas the majority of numerals in adverbial phrases are used as 'exactly n'. According to Bultinck this distribution is simply a reflex of the attested fact that direct objects are generally good candidates for introducing new topics, whereas it is less likely that material in adverbial phrases is there to (existentially) introduce a new referent. In subject position, occurrences of 'two' without definite markers are mostly 'absolute value' or 'exactly n'. But the difference between these two usages is blurred. The trend is that subject position indefinite numerals are less likely to allow for a subsequent revision of the involved cardinality than object indefinite numerals. Bultinck proposes that this is due to the fact that it is marked to use a subject for the introducing of a new referent. The focused use of the numeral hints at excluding the possibility of there being more than the 'n' elements that are expressed. This means that there is a continuum from 'absolute value' to 'exactly n' meanings. In 'pure absolute value' use there is a neutrality toward the possibility of there being more elements. This neutrality is reduced in subject position. A further finding supports this idea of a continuum. In predicative constructions (such as existential "there" sentences), most samples show the absolute value meaning of the numeral. Bultinck's idea is that such constructions hardly change the default 'absolute value' interpretation of the numeral. Although Bultinck is careful not to present it as a clear result from his corpus research, he hypothesises that the continuum from 'absolute value' to 'exactly n' is paired with a scale of syntactic constructions, ranging from existential there sentences, to objects, to subjects, to adverbials. The picture emerging from this is one where a great multitude of factors influence the 'value interpretation' of a numeral. In particular, it seems generally the case that when there is an 'exactly n' interpretation of the ''meaning complex'' that contains the numeral, this meaning can be reduced to a combination of the 'absolute value' meaning and the influence of other co-textual factors. It follows that ''[the] 'absolute value' interpretation is the starting-point for the interpretation of 'two''' (p. 225), or as Bultinck concludes in chapter six, ''the conventional meaning (the ''coded content'') of 'two' is the 'absolute value' meaning'' (p. 307). The corpus analysis shows that 'at least n' uses of numerals are highly infrequent (3,9%). This, Bultinck claims, is highly problematic for the neo-Griceans. In fact, the corpus analysis shows that the few 'at least' uses that are found are all due to the co-occurrence with a linguistic element and, in most cases, that element is 'at least'. Another finding from the corpus discredits the neo-Gricean account of numerals in another way. One of the traditional arguments for assuming the 'at least n' meaning to be conventional is that were 'exactly n' conventional, then it would be redundant to combine the numeral with 'exactly'. It is not and hence, the argument goes, 'exactly n' cannot be the coded meaning of 'n'. The corpus shows, however, some very clear facts about numeral modifiers (called 'restrictions' by Bultinck). The most common kind of modification is with 'at least' (44.8%), whereas combinations of 'two' with 'exactly' are relatively rare at 9.5%. If the neo-Gricean argument holds, exactly the reverse distribution of 'exactly' and 'at least' would be expected. The first half of the book is devoted to the discussion of the literature on (neo-)Gricean implicatures in general and the pragmatics of numerals in particular. A shorter discussion might have been more effective, since one has to wait a long time for Bultinck's main feat, the discussion of his corpus study of numerals (chapters four and five). Furthermore, the literature discussion is often overly detailed and repetitive. For instance, some of the arguments Bultinck discusses in the chapter on Gricean pragmatics are repeated in both his discussion of the literature on numerals and in the discussion of the corpus data. Nevertheless, it is certainly admirable that Bultinck so successfully weaves together discussions from linguistic pragmatics, corpus linguistics and cognitive linguistics. Although tedious at some points, the many repetitions might actually guarantee that this book is suitable for the broad audience it sets out to reach. A more serious problem is the fact that the discussion in chapters two and three is in many ways dated. Browsing the references, one finds that the most recent literature that is being discussed dates from 2001 (the book is published in 2005). Of course, many of the high points of the discussion of scalar implicatures can be traced back to the 1970s and 1980s, so it is perhaps not entirely unexpected to find mostly older literature. However, in the past few years the study of implicatures and numerals has flourished once again. Now, there is a wealth of new findings and theoretic proposals (e.g. Geurts 1998, Chierchia 2002, Recanati 2003, van Rooy and Schulz 2004). Furthermore, an increase in the interest of psycholinguists into pragmatic issues has lead to a considerable amount of empirical data challenging the traditional theoretic approaches to make more precise predictions (see, for instance, Noveck 2001, Papafragou and Musolino 2003 and, especially, Musolino 2004). Unfortunately, such recent works are completely absent from Bultinck's discussions and arguments. This may be explained by the fact that this book, as I understand it, is a published version of Bultinck's dissertation which dates from 2001. Curiously, however, this fact is not mentioned in the book. The main objective of Bultinck's corpus analysis seems to be to discredit the idea that numerals carry a conventional meaning that involves a lower bound. With this in mind, I think the three most relevant findings are: (A) the corpus is argued to display the infrequency of this alleged coded meaning; (B) the data suggest that there are 'numerous meanings' associated with English cardinals and that these are less rigidly distributed than the neo-Gricean programme would have it; and (C) the 'absolute value' meaning is the most basic one of these numerous meanings. It is not entirely clear to what extend Bultinck's 'at least n' meaning corresponds to the lower bound conventional meaning defended by the minimalists. I doubt whether the neo-Griceans really had a modal coded meaning in mind. It is certainly not the case that the lower bound meaning necessarily involves modality. It is quite easy to imagine a 'cognitively simple' lower bound analysis which simply describes the cardinality of a group as being 'greater or equal than n'. In fact, such a proposal comes very close to Bultinck's own 'absolute value' meaning. This becomes clear from Bultinck's specification of the four candidate meanings. The 'at least n' meaning is described as ''necessarily n + possibly more then n'' (p. 176). Note that in this definition, one needs to assume that the number symbol 'n' has a greater-or-equal reading itself. If the cardinality of a group is necessarily 'n', how can it at the same time be possible that this cardinality is 'more than n'? A formulation like this one presupposes once again that numerals somehow line up in entailment scales. It follows that the 'absolute value' meaning is really a lower bound meaning. Consequently, one could characterise Bultinck's proposal as minimalistic, except that the conversational implicatures have been replaced by co-textual factors that trigger modal cardinality statements. So how well does this proposal account for the data? The 'absolute value' meaning of numerals seems consistent with the data in the corpus. It is important, however, to explain in detail how the compositional meaning of numerals is defined, especially since these very meanings have turned out to be so remarkably deceptive. Unfortunately, the semantic processes Bultinck refers to are often not specified enough to assess how the sentential meanings are derived from a single core lexical meaning. Nevertheless, 'numerous meanings' contains a wealth of data and ideas that will stimulate the ongoing discussion of the semantics of simplex and complex English numerals. Anyone working on a linguistic topic that is somehow related to numeral meaning will definitely find a lot to learn in this book, especially since Bultinck's most important point, I feel, is not theoretical but methodological. The data are much more varied and complex than the neo-Gricean theories have assumed. On the basis of this, Bultinck argues convincingly that it is a mistake to search for 'the' meaning of English cardinals. REFERENCES Chierchia, G. 2004. Scalar Implicatures, Polarity Phenomena, and the Syntax/Pragmatics Interface. In Belletti, B. (ed.), Structures and Beyond: The Cartography of Syntactic Structures. Vol. 3. New York, NY: Oxford University Press. Geurts, B. 1998. Scalars. In Ludewig, P. and Geurts, B. (eds.) Lexicalische Semantik aus Cognitiver Sicht. Tuebingen: Gunter Narr. 95-117. Horn, L. 1972. On the Semantic Properties of Logical Operators in English. UCLA dissertation. Distributed by Indiana University Linguistics Club, 1976. Musolino, J. 2004. The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition 93(1): 1-41. Noveck, I. 2001. When children are more logical than adults: Experimental investigations of scalar implicature. Cognition 79: 165- 188. Papafragou, A. and Musolino, J. 2003. Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition 86(3): 253-282. Recanati, F. 2003. Embedded Implicatures, Philosophical Perspectives 17(1): 299-332. van Rooy, R. and Schulz, K. 2004. Exhaustive interpretation of complex sentences. Journal of Logic, Language and Information, 13: 491-519.