Scalar Ordering of Italian Quantifiers Scalar Ordering of Italian Quantifiers Scalar Ordering of Italian Quantifiers Scalar Ordering of Italian Quantifiers

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Scalar Ordering of Italian Quantifiers

Name: Ruggero Montalto S-number: S1591703

Adress: Meeuwerderweg 119A 9724 ER, Groningen Tel.: +31644236037

Email: r.montalto@student.rug.nl Programme: University of Groningen

Research Master Linguistics: Neurolinguistics and

Models of Grammar Supervisors: Angeliek van Hout

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1. Introduction 1. Introduction 1. Introduction 1. Introduction ... 5555 1.1 Foreword ...5 1.2 Research questions ...6 2. Quantifiers 2. Quantifiers 2. Quantifiers 2. Quantifiers ... 9999 2.1 Italian quantifiers ...9

2.2 Poco, tanto and parecchio: a morphosyntactic comparison...15

2.3 Quantifiers and numbers...20

2.3.1 Semantic aspects ...20 2.3.2 Pragmatic aspects ...26 3. Scales 3. Scales 3. Scales 3. Scales ... 292929 29 3.1 Quantifiers and scales...30

3.2 Linear-scales...35

3.3 Interval-scales...38

3.4 General Hypotheses and predictions ...41

4. Experiment 1 (Quantifier 4. Experiment 1 (Quantifier 4. Experiment 1 (Quantifier 4. Experiment 1 (Quantifier----ordering Experiment)ordering Experiment)ordering Experiment)ordering Experiment) ... 44...4444 44 4.1 Participants ...44

4.2 Design, procedure and materials ...44

4.3 Hypotheses and predictions ...45

4.4 Results ...46 4.5 Discussion...52 4.6 Conclusion...54 5. Experiment 2 (Quantifier 5. Experiment 2 (Quantifier 5. Experiment 2 (Quantifier

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5.1 Participants ...55

5.2 Materials and procedure...55

5.4 Results ...57 5.5 Discussion...61 5.6 Conclusion...63 6. Experiment 3 (Plausibility 6. Experiment 3 (Plausibility 6. Experiment 3 (Plausibility 6. Experiment 3 (Plausibility----rating Experimrating Experimrating Experimrating Experiment)ent)ent)ent) ... 64...6464 64 6.1 Participants ...64

6.5 Results and discussion ...67

6.6 Conclusion...74

7. Experiment 4 (Quantifier 7. Experiment 4 (Quantifier 7. Experiment 4 (Quantifier 7. Experiment 4 (Quantifier----comparison by Children)comparison by Children)comparison by Children)comparison by Children) ... 757575 75 7.1 Participants ...75

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Appendix A Appendix A Appendix A

Appendix A –––– Complete list of scalar Complete list of scalar Complete list of scalar----orderings from Experiment 1 Complete list of scalar orderings from Experiment 1orderings from Experiment 1orderings from Experiment 1... 999999 99 Appendix B

Appendix B Appendix B

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1. Introduction

1.1 Foreword

Each culture capable of counting has developed ways to deal with numbers linguistically. But talking in terms of exact numerosity is not always convenient. Sometimes we deal with numbers too big to be counted or with quantities that we are incapable of quantifying exactly, such as fractions. Other times it is simply more convenient or more polite not to mention a number explicitly. For these and other reasons, all human languages over time have developed terms which deal with numbers or quantities without stating explicitly a numeric value: these terms fall under the name of quantifiers. For example, even Pirahã, an Amazonian language that it is claimed not to make use of syntactic recursion (Everett, 2005), makes use of a quantification scale ranging between two degrees: hói, which means ‘one’ but also ‘a small quantity’, and aibaagi, ‘many’ (Gordon, 2004). Whereas every language in which it is possible to express bare numerals has number-words which are cross-linguistically perfectly equivalent (a quantity of 47 will be 47 whether we count in Russian, Japanese or Indonesian), the majority of languages shows very different quantification strategies, often much more articulated than in Pirahã.

In closely related languages quantifiers with a common etymological origin may not map exactly onto the same magnitude value (for instance, the Spanish mucho translates not only the Italian molto but also parecchio and tanto) and often even within one single language we are not able to find perfect agreement among speakers on the value of quantifiers; the two Italian quantifiers molto (‘many’) and tanto (also ‘many’), both referring to large numerosities, are in this case an example.

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1.2 Research questions

Quantifiers in their cardinal interpretation (e.g. as in “There are many apples” and not “Many of the apples”) and number-words (e.g. one, two, etc.) present a series of particular commonalities but also a series of deep differences. One of the differences is that scaling up (or down) number-words according to their magnitude is straightforward but doing the same with quantifiers is not, even though both lexical classes are internally ordered. The main research question concerning the present work is therefore whether it is possible for native speakers of a language to order the quantifiers of their own language on a scale. And more specifically:

(a) Are Italian native speakers able to order the quantifiers: alcuno, molto, nessuno,

parecchio, poco, qualche, tanto and tutto on a magnitude scale?

A second pair of research questions looks at the same topic but from a developmental perspective; we wonder whether children and adults order quantifiers differently or not. In other words is it possible for children who are learners of a language to order the quantifiers of their mother-tongue on a scale? And more specifically:

(b) Are Italian children able to order the quantifiers: alcuno, molto, parecchio, poco, qualche and tanto on a magnitude scale?

(c) Does their performance differ from the adults’ performance and if so, what differs?

The ideal kind of scale onto which order the Italian quantifiers should be similar, for example, to the scale of sizes used in the clothing industry. Each piece of clothing comes in different sizes which range from extra-extra-large to medium to extra-small: XXL, XL, L, M, S, XS. We could metaphorically draw correspondences between large-sizes and quantifying expressions such as many, lots of and several; the medium-size to expressions like some and finally the small-sizes to expressions such as few or very few.

All items along the clothing-size scale behave syntactically like the three adjectives from which they derive: large, medium and small. From a semantic point of view, two of these adjectives (large

and small) form together a prototypical pair of antonyms. Establishing antonymic relationships

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corresponds to a length of 35cm and S to 23cm. The sizes in our scale seem to be items that cannot establish conventional relationship with each other. That is because, in spite of their adjective-like syntactic behavior, their meanings are bidirectionally controlled by arbitrary relationship between a label (e.g. S, M, L, etc.) and a specific cardinality of, say, centimeters.

The only substantial difference between a scale of numbers and our scale of clothing-sizes is that the mapping between sizes and cardinalities is not unique, but rather a one-to-many relationship: different brands of clothes could for instance assign to the same label-size of a similar piece of clothing (e.g. a t-shirt) different sizes in centimeters. Suppose having an unfolded pile of t-shirts of different brands (so that, for instance, one brand’s L might be as big as one other brand’s M) and suppose having to fold and order them according to their sizes, what we could relatively easily do is relying on the size on labels ignoring the different measures in centimeters given by each brand.

Unlike English, that has one or two quantifiers for each label, e.g. many or lots for L, some or

several for M and few for S1_{, Italian has two or more quantifiers for each label: molti, parecchi and}

tanti for L, qualche and alcuni for M and alcuni, qualche and pochi for S. Why does Italian have so many synonymous quantifiers for the L and S labels? Is that not redundant?

In this research we offer the hypothesis that the 3 S-quantifiers and the 3 L-quantifiers are only apparently synonymous, as in fact they should refer to different magnitudes and can therefore be ordered on a scale. Their referring to different magnitudes would justify their co-existence in the language as, within each size-label, they are all three very similar, but not quite the same, because they can be ordered.

Out of a metaphor, the present research attempts to establish a scalar ordering of Italian mid-range quantifiers. In a battery of four experiments, we tried to reduce and control the role of visual and verbal context as much as possible, in the hope of better defining the conventional magnitude-values behind each quantifier (in fashion of label-sizes).

The experiments used three different methods, which looked at the research question from different perspectives. Experiment 1 (‘Quantifier-ordering Experiment’, presented in Chapter 4)

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was a magnitude-ordering task on an eight-point scale. In the experiment, the participants had to order, from less to more, eight Italian quantifiers. The quantifiers were displayed as a “cloud” of words spread across the page. Two quantifier-comparison experiments, one with adults (Experiment 2, Chapter 5) and one with children (Experiment 4, Chapter 7), used a “closed-box paradigm”. The participants had to express a meta-linguistic magnitude-judgment on two different quantificational statements, related to a visual stimulus. In the stimulus, two closed boxes of identical shape and size contained different quantities and we asked which quantity was the larger one. Experiment 3 (‘Plausibility-rating Experiment’, Chapter 6) required the participants to score the plausibility of eight quantified statements across a changing series of visual stimuli. E.g. they saw 100 dots, 45% of which were black and the rest white, and they had to score the plausibility sentences like “[quantifier] dots are black”; this last experiment was a quasi-replica of Experiment 2 from Oaksford et al. (2002). Only adults participated in Experiments 1, 2 and 3, while in Experiment 4 only children participated.

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2. Quantifiers

According to Generalized Quantifier Theory, the leading view on the interpretation of quantifier in contemporary formal semantics, quantifiers form a class of words whose meanings denote relations among sets or properties (Saeed, 2003; Nouwen, 2009). Consider Table 2.1 where the definitions of three English quantifiers (all, some and no) are reported from Saeed (2003:329). The red circle represents a set A (“apples”) and the green circle is a property B (“on the tree”).

a. b. c.

All apples are on the tree Some apples are on the tree No apples are on the tree All A are B is true if

and only if the set of things which are members of A

is a subset of B

Some A are B is true if and only if the set of things which

are members of both A and B is not empty

No A are B is true if and only if the set of things which

are members of both A and B is empty

Table 2.1 – Formal semantic definitions of all, some and no.

In the rest of this paper, we will refer to quantifiers denoting subset-relations (Table 2.1a, e.g. English all and Italian tutto/i) or empty-relations between sets (Table 2.1c, e.g. English no, Italian nessuno) as absolute quantifiers. We define them absolute because they all describe absolute states: states or situations independent of arbitrary standards of measurement.

We will refer to quantifiers denoting non-empty relations between sets (such as some, Table 2.1b) as mid-range quantifiers instead.

2.1 Italian quantifiers

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magnitude distinction relies on an obvious property of quantifiers (namely the size of the set to which a quantifier points).

On a scale of Italian quantifiers the two extremes of the scale are occupied by the quantifiers nessuno (tr. nothing, no, no one) and tutto (tr. everything, everyone). Nessuno is equivalent to zero in opposition to non-zero. Tutto is instead equivalent to the maximum amount available, opposed to non-maximum and to zero. Both nessuno and tutto are absolute quantifiers, and will always denote subset-relations or empty-relations between the members of two sets (Table 2.1a or Table 2.1c). Mid-range quantifiers will always denote not-empty relations between proportions of the members of two sets (Table 2.1b). These proportions can have either a low or a high magnitude, accordingly to the cardinality of the non-empty set of which things from both sets are members. E.g. in sentences such as “Many apples are on the tree”, the cardinality of the non-empty set denoted by many will be higher that the cardinality denoted by very few in “Very few apples are on the tree”.

The scalar space between the two Italian absolute quantifiers, nessuno and tutto, is domain of mid-range quantifiers. In the present study the focus is on six different mid-mid-range quantifiers, belonging to two magnitude groups: a low magnitude-group and a high magnitude-group. We will refer to the quantifiers in these magnitude-groups as to low-magnitude quantifiers and high-magnitude quantifiers. The low magnitude-group will include quantifiers whose meaning refers to a small proportion of set entities (typically lower than 50% in relation to the entire set), while the high magnitude-group will include the quantifiers pointing to a large proportion of the entities (typically higher than 50%). In the present study the low magnitude-group is represented by the three quantifiers: poco (‘little’, ‘few’), alcuno (‘some’, ‘a few’) and qualche (‘some’, ‘a few’). The high magnitude-group is also represented by three quantifiers and comprises molto (‘much’, ‘many’), parecchio (‘several’, ‘much’, ‘many’) and tanto (‘much’, ‘many’).

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Zamparelli (2008) writes that qualche points to a larger quantity than poco that is still limited; he extensively discusses the nature of this quantifier, in particular the fact that in different contexts it can be singular or plural in meaning, but always triggers singular verb agreement. Zamparelli argues that the properties of qualche are ambiguous. In some positions it behaves like English singular some, for example in (1), and in other positions it is a “plain existential determiner which acquires its plural meaning by scalar inferences”, (Zamparelli, 2008:293), for example in (2).

(1) Qualche mela è bacata.

Some apple is wormy.

(2) Per pranzo ho mangiato qualche mela.

For lunch (I) have eaten some apple I ate a few apples for lunch.

In this thesis we will look only at qualche used as in (2), since here its meaning has a cardinal interpretation, similar to the interpretation of the other mid-range quantifiers introduced so far.

According to both Dardano and Trifone (1997) and Zamparelli (2008), alcuno behaves as a quantifier with a cardinal interpretation only when used in its plural forms alcuni (masculine gender)

and alcune (feminine) and its meaning is equivalent to the English plural some. When qualche, alcuni

and alcune are found in predicate position (3a and 3c) they translate to English plural some (3b and 3d), whereas in subject position (4a and 4c) they can also translate as certain (4b and 4d).

(3) a. Ci sono alcune mele sul tavolo.

b. There are some/#certain apples on the table. c. C’è qualche mela sul tavolo.

d. There are some/#certain apples on the table. (4) a. Alcune mele sono sul tavolo.

b. Some/Certain apples are on the table. c. Qualche mela è sul tavolo.

d. Some/Certain apples are on the table.

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two sets we would have a qualitative-relation, e.g. “Certain apples are on the table (and other apples are not)”. Therefore in Experiments 2 and 4 the materials show the quantifier in the least-ambiguous syntactic position possible, shown in (3a) and (3c). Doing so insures that the participants’ interpretations will have focused on the set magnitude rather than on a qualitative property.

The two most informative quantifiers on the scale formed by the Italian quantifiers just introduced2 are: tutto, at the extreme left of the scale, and nessuno, at its extreme right. Nessuno and tutto are contraries, as they cannot be both true at the same time, while they can be both false instead (Levinson, 2000:64). The other six quantifiers (alcuni, molto, parecchio, poco, qualche, tanto) all occupy intermediate positions on the scale and are enclosed by the two absolute quantifiers. The low magnitude-group will cluster on the scale in direction of nessuno. The high magnitude-group will cluster instead towards tutto.

In dictionaries (Devoto-Oli, 1985; Felici-Riganti, 1987; Palazzi-Folena, 1992) quantifiers within the same magnitude-group are often cross-referenced as synonyms and some of them are cross-referenced as antonyms across the two magnitude groups. In an attempt to illustrate the synonymic properties among quantifiers of the same group, we will give a few examples for alcuni and qualche, in (6), and for pochi, in (7); let us suppose that someone who was away for a couple of days came back and, at one point in time, when back for a week already, the person utters either (6) or (7).

(6) Sono tornato qualche giorno/alcuni giorni fa.

(I) am come back a few day/a few days ago I’ve come back a few days/a few days ago.

(7) #Sono tornato pochi giorni fa.

(I) am come back a few days ago I’ve come back few days ago.

Whereas none of the sentences in (6) and (7) can plausibly refer to a lesser number of days than two, the plausibility of alcuni and qualche can stretch upwards for a number of about ten days. The plausibility of pochi is instead compromised by the context: we would hardly ever use pochi for a

2_{The two absolute and the six mid-range quantifiers introduced so far all belong to the Italian grammatical class of}

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number of days above four. So, when looking a bit closer, it appears that not all low-magnitude quantifiers seem to express exactly the same semantic relations.

Commonly used frozen sentences, such as (8) and (9) suggest that in the high magnitude-group tanto is the highest mid-range quantifier.

(8) Ti voglio tanto bene.

You (I) want much love I love you very much.

(9) Tanti auguri di buon compleanno.

Many wishes of good birthday Happy birthday.

Considering either the importance of a statement such as loving someone in (8), or the “pseudo-ritual use” of (9) during a birthday, it can be hypothesized that, in such frozen expressions, native speakers privilege the use of quantifiers pointing to the largest possible amount (as in love) or cardinality (as in wishes) by default. Molto and parecchio can appear in the same expressions, but they do so with a lower frequency in comparison to tanto.

Table 2.2 shows the frequency in the form of number of Google web-hits per query for the frozen-expression in (8) and its non-frozen variants using either molto or parecchio. “Ti voglio tanto bene” is nearly 20 times higher than the frequency count of “Ti voglio molto bene” and more than 2000 times higher than the count of “Ti voglio parecchio bene”.

SENTENCE NUMBER OF HITS

Ti voglio tanto bene 808,000

Ti voglio molto bene 41,000

Ti voglio parecchio bene 348

Table 2.2 – Number of hits for the Google query “ti+voglio+quantifier+bene” with the language filter set on Italian (8/9/2009).

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SENTENCE NUMBER OF HITS Tanti auguri di buon compleanno 390,000 Molti auguri di buon compleanno 216 Parecchi auguri di buon compleanno 2

Table 2.3 – Number of hits for the query “quantifier+auguri+di+buon+compleanno” with the language filter set on Italian (17/9/2009).

We face a chicken-or-egg problem though. Does the high frequency suggest that certain quantifiers convey more emphasis and less irony because they actually point to larger magnitudes? Or are the frozen-expressions that trigger a less emphatic (and more ironic) perception for their non-frozen variants (e.g. “Ti voglio parecchio bene”)?

Summarizing so far, the present study revolves around six mid-range quantifiers of Italian and the goal of the study is to try to order these quantifiers on a scale from tutto to nessuno. According to descriptive grammars and dictionaries and observations on everyday use of these quantifiers, it is possible to clearly distinguish the six quantifiers in two groups of three quantifiers each: a low and a high magnitude group. Combining all the pieces of information presented above, a hypothetical scale, ordered as in (10), can be outlined. The quantifiers are ordered, from left to right, from the smaller to the larger; with the border between the two magnitude groups falling between alcuno-qualche and parecchio.

(10) nessuno < poco < alcuni-qualche < parecchio < molto < tanto < tutto

The scale in (10) represents a hypothetical order based on the information presented in this chapter and, while the magnitude-group division reflects reliable information from grammars and dictionaries, the internal ordering within each magnitude-group is merely speculative.

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2.2

2.2 2.2 Poco

Poco

Poco,

Poco

, ,

, tanto

tanto

tanto and

and parecchio

and

parecchio

parecchio: a morphosyntactic comparison

: a morphosyntactic comparison

The present section focuses on specific morphosyntactic aspects of the Italian quantifiers (e.g. superlative inflections and reduplication) in order to introduce a type-distinction among the quantifiers used in the experiments. It is a theoretical distinction and it is modeled after the analyses that Kayne (2007) proposed for the English quantifiers few and many in opposition to several. If proven to be correct, this type-distinction might lead to the conclusion that there are different types of mid-range quantifiers, which cannot all be ordered as points on a scale. This is a topic that will be extensively presented and discussed in the next chapter (Section 3.3).

Certain Italian quantifiers (molto, poco and tanto among the ones used in the experiments) behave like regular adjectives and agree on gender and number with the noun of the NP to which they refer. These same quantifiers also get superlative inflections (e.g. tantissimo), can make use of syntactic reduplication (e.g. tanto tanto) and can combine with specific suffixes denoting smallness, largeness, contempt or endearment (e.g. tantino, tantuccio, tentinetto, tantinello). Italian number-words, on the other hand, do not inflect according to number or gender (the only exception is the word uno, ‘one’, which also has the feminine form una) and take neither superlative inflections nor suffixes. Other Italian quantifiers (alcuni, qualche and parecchio in the experiments) behave differently though: not many morphosyntactic possibilities available to the first group are grammatical for this second group.3

Similar observations on the distinct morphosyntactic properties of the English quantifiers few and many in comparison to several, have led Kayne (2007) to an interesting proposal on the different morphosyntactic interpretation of the two kinds of quantifiers. Kayne only made syntactic claims on the differences among these quantifiers, but we will attempt to directly relate his morphosyntactic distinctions (applied to Italian quantifiers) to the interpretation of the quantifiers.

According to Kayne (2007:835), “in all languages, modifiers with the interpretation of many and few necessarily modify NUMBER/number”. ‘Number’ written in small letters corresponds to a numeral, whereas ‘NUMBER’ written in capital letters stands for an unpronounced noun which is

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modified by an adjective such as many and few. English sentences such as (11a) and (12a) have, according to Kayne, the underlying structure in (11b) and (12b) and such an underlying structure would imply the meanings presented in (11c) and (12c).

(11) a. John has few books.

b. John has few NUMBER books. c. John has a small number of books.

(12) a. John hasn’t bought many houses this year.

b. John hasn’t bought many NUMBER houses this year. c. John hasn’t bought a large number of houses this year.

The analysis of several goes in a different direction; Kayne argues that this quantifier has a covert comparative structure such as the one in (13b), with a meaning that corresponds to the one presented in (13c).

(13) a. John has several books.

b. John has MORE THAN A several NUMBER books. c. John has more than a small number of books.

Kayne demonstrates how few, just like an adjective, can take comparative and superlative suffixes and how both few and many take degree modifiers such as too, as, so and how. The same two quantifiers, few

and many, can also be modified by certain adverbs expressing degree (e.g. unbelievably, fairly, pretty,

etc.) whereas several does not behave like an adjective and cannot be modified by any degree adverb. Proceeding from Kayne’s distinction between these two kinds of quantifiers – which from now on will be called (a) number-quantifiers when similar to few and many and (b) comparative-quantifiers when similar to several – we will distinguish the six experimental Italian mid-range quantifiers in two groups according to their morphosyntactic properties. In fact poco, molto and tanto tend to behave like few and many, while parecchio, alcuno and qualche tend to behave like several. This morphosyntactic classification crosscuts the two magnitude-groups.

Similarly to English, the ‘number-group’ allows combination with a degree adverb (14a) and superlative forms (14b):

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Really few/many/many apples

b. (Davvero) pochissime/moltissime/tantissime mele

(Really) very few/very many/very many apples

In contrast to English, not all quantifiers in this group take degree modifiers and the quantifiers that do take degree modifiers often do not take all of them, as it can be seen by comparing the examples in (15) and in (16):

(15) Incredibilmente poche/molte/tante mele

Incredibly few/many/many apples

(16) a. Troppe poche mele

Too few apples

b. *Troppe molte/tante mele

Too many/many apples

Quantifiers in the ‘comparative-group’ allow neither superlative forms, nor degree modifiers, nor degree adverbs, similarly to English several.

(17) a. *Davvero qualche mela/alcune mele

*Really some apples/some apples.

b. *(Davvero) qualchissima mela/alcunissime mele

*(Really) very some apples/some apples

Furthermore, all Italian number-quantifiers can be reduplicated, in order to express larger or smaller quantities. Superlative morphology and reduplication can combine to achieve an even higher degree of modification. The examples in (18) and (19) offer several possible combinations of these morphosyntactic strategies. None of the modifications in (18) and (19) is either grammatical or felicitous when applied to ‘comparative’ quantifiers (20), even if they are all morphosyntactically possible.

(18) a. Molte molte mele

Many many apples Veri many apples

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Many many apples Very many apples

c. Poche poche mele

Few few apples Very few apples

d. Tante tante tante tante tante tante [...] tante mele

Many many many many many many [...] many apples Very very very very very very [...] many apples

(19) a. Moltissime moltissime mele

Very many very many apples Very very many apples

b. Molte moltissime mele

Many very many aples Very very many apples

(20) a. ??Parecchie parecchie/*Alcune alcune mele

Several several/Some some apples *Very several/*Very some apples

b. *Qualche qualche mela

Some some apples *Very some apples

c. *Qualchissima mela

*Very some apples

d. *Parecchissime/*Alcunissime mele

*Very several/*Very some apples

Kayne proposed for few and many a syntactic structure in which they directly modify an unspoken

noun NUMBER, and for several a structure in which it does not directly modify a noun NUMBER,

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unreasonable – is to transpose Kayne’s morphosyntactic distinction onto a semantic level and draw a direct link to scales.

By applying Kayne’s analysis to Italian, we propose that quantifiers such as poco, molto and tanto, which use different morphosyntactic strategies to stretch the semantic range of their meanings, should position on the most external parts of the Italian scale of quantifiers. We also propose that if the analysis of several holds for the Italian quantifiers qualche, parecchio and alcuno, each ‘comparative’ quantifier should refer to a quantity that is defined by using a ‘number’ quantifier in order to establish a comparison. Parecchio, for instance, might implicitly need molto and/or tanto to be interpreted (e.g. parecchio = “less than tanto”) and ordered on a scale, similarly qualche and alcuni might need poco (e.g. alcuni/qualche = “more than poco”). For each ‘comparative’ quantifier we would always need at least one ‘numeral’ quantifier on either its left- or right-side on the scale in order to allow the formulation of comparative operations defining the meaning of the ‘comparative’ quantifiers. The existence of different quantifier-types might also make the scalar ordering of quantifiers very difficult as the quantifiers of the ‘comparative-group’ would necessarily rely on a prior scalar ordering of the ‘numeral-group’ to be ordered. But if any scalar-ordering is established by the results of our experiments we will expect to see the ‘comparative’ quantifiers clustering within the center of the scale with the ‘number’ quantifiers positioning along the external parts of the scale. The argument we provide to justify this prediction is that the magnitudes of ‘comparative’-quantifiers cannot map in between an absolute-quantifier and the lowest (or highest) ‘number’-quantifier as those positions should belong in Italian to the superlative and/or reduplicated occurrences of the ‘number’ quantifiers.4

In this section we introduced the Italian quantifiers used in the experiments, a first distinction among the quantifiers in terms of magnitude (low vs. high) and proposed a second distinction in terms of type (‘numbers’ vs. ‘comparative’). In the next section we will look at semantic and pragmatic properties of quantifiers in general and we will compare these properties with the properties of numbers.

4_{We will present this argument once more at the end of Section 3.3, after we will have introduced the notion of}

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2.3 Quantifiers and numbers

2.3.1 Semantic aspects

This section presents some semantic aspects of quantifiers and briefly explains how numbers are cognitively processed. On the basis of syntactic5_{and semantic similarities between quantifiers and} number-words (e.g one, two, etc.), it is argued that Gallistel and Gelman’s (1992) theory on number recognition and processing is relevant to the research presented here as it would license experimental methods used to compare numbers (such as ordering and magnitude-comparison tasks) as fruitful when used to compare quantifiers.

Using Saeed words, quantification can be defined as a feature common to all natural languages which allows “a proposition to be generalized over ranges of sets of individuals” (2003:298): a shared characteristic between quantifiers and number-words is that both refer to and modify properties of sets and not properties of objects in the sets.

Wiese (2003) points out that the fundamental semantic difference between quantifiers and number-words, is that numbers and number-words identify the size of sets numerically, whereas quantifiers identify the size of sets non-numerically; number-words describe the size of a set sharply and, if in case the size of the set needs to be manipulated, number-words make the relying on arithmetic combinatorics possible, while quantifiers do not.

Numeric quantification allows moving back and forth on a scale of number using arithmetic rules to establish a successor function. For instance, given: (i) a set of n apples in a basket and (ii) a rule saying that we can increase the size of our n by adding 1 apple to the set (n + 1), it will be possible to predict how many times one more apple has to be added to the basket to reach a quantity corresponding to a cardinality of nineteen apples in the basket.

Quantification through quantifiers, on the other hand, does not offer any rule allowing an exact prediction when manipulating the size of the set. There is no rule suitable to determine how many apples are some apples; neither is there an arithmetic function to predict how many apples have to be

5_{Syntactic similarities between quantifiers and number-words are, among others, their position in the NP in declarative}

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added to (or removed from) the basket so that a set of some apples will reach a cardinality of nineteen or a quantity of many apples. This happens because quantifiers simply do not follow arithmetic rules in order to define their meaning in terms of size, but rather rely on context and extra-linguistic factors such as “norm values” (Wiese, 2003).

What exactly is a norm value and how is a norm value derived and established? Norms belong to the domain of psychology and the generally accepted idea is that they are produced by aggregating a set of representations, recruited in parallel from the memory of a person, which point back to already interpreted and evaluated experiences of events (Kahneman and Miller, 1986:136). The model responsible for processing and producing norms, proposed by Kahneman and Miller, postulates that experiences related to an object, an event or a reference to a concept (such as quantifiers), when recruited from memory, evoke a set of individual elements, and each element is described by several features. In order to describe the set, all individual features are then aggregated creating the norm; Kahneman and Miller compare the norm to an “envelope of this aggregate profile” (1986:137). In the same article two types of norms are distinguished: “(a) stimulus norms, which are evoked by experiences of objects of events, and (b) category norms, which are evoked by reference to categories” (Kahneman and Miller, 1986:138). The norm values defining the specific semantic range of a quantifier belong to the group of category norms.6

Hammerton (1976) “demonstrated [with his research] that people order vague quantifiers in terms of magnitude” (Wright et al., 1994:480) and also tried to assign an approximate exact value to the vague quantifiers he used in his experiments. Goodwin, Thomas and Hartley (1977:95), in a follow-up study to Hammerton, concluded that “it is almost impossible to produce a table of the sort presented by Hammerton” mapping quantifiers onto numerical values, showing how establishing an exact semantic correspondence between a quantifier and a number is plausible but not feasible.

6_{How category norm values are drawn for quantifiers was the object of a research conducted by Wright et al. (1994),}

who investigated the mapping between numerical values and quantifiers. Wright et al.’s research uses two related hypotheses: (a) the self-information hypothesis for which “information drawn primarily from one’s own experience is an obvious candidate

to serve as a benchmark information” (1994:481) and (b) the group-norm hypothesis stating that “group information has an

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When dealing with number-words there is no need to know anything about the context or norm values; all that one needs to know is the cardinality expressed by the number to properly asses its magnitude; the two examples in (21) and (22) show how in two different contexts – “lunchbox” in (21) and “orchard” in (22) – the cardinality evoked by the quantifier many substantially changes between (21a) and (22a) while the same does not happen in between (21b) and (22b) with the number-word two.

(21) a. I have many apples in my lunchbox. b. I have two apples in my lunchbox. (22) a. I have many apples in my orchard. b. I have two apples in my orchard.

The numerosity of the cardinal quantifier many is modified in (21a) and (22a) by the speaker’s norm value. Each speaker adapts the magnitude of the quantifier context-dependently, according to his/hers personal experiences and real world knowledge. Norm values can differ across speakers to such great lengths that – to say it with Hakel’s words – “Variability is rampant. One man’s ‘rarely’ is another’s man ‘hardly ever’” (Hakel, 1968:533).

Saeed (2003) and Cao (2006) both refer to quantifiers such as many and few as being open to a double interpretation. They can in fact have either a proportional or a cardinal reading. Sometimes one or the other interpretation is triggered by the syntactic position of the quantifier, as demonstrated in (23): a proportional reading of the quantifier many is shown in (23a), where many is the syntactic head for the partitive PP “of the apples”, while in its cardinal reading the quantifier is an adjunct of “apples” (23b). In (23a) many has a partitive interpretation: the referent of the quantifier is a percentage of apples that, together with an amount of apples being somewhere else (e.g. in a basket), represents the total of all apples. In (23b) instead, the same quantifier conveys the information that the number of red apples in the orchard is high.

(23) a. Many of the apples are in the orchard. b. There are many apples in the orchard.

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one of the sentences in (24b) or (24c) disambiguates it. The reading of many will be proportional if (24a) is followed by (24b), but it will be cardinal if instead (24c) follows.

(24) a. Many apples are in the orchard… b. …and only a few are in the basket.

c. … and I am looking forward to pluck them.

A further example of cardinal vs. proportional reading, this time for the quantifier few, is given in Cao (2006). Cao argues that “few in «Few people in this conference are from Asia» may mean a small number of people, while few in «Few people in the United Kingdom are from Asia» may mean a small percentage of the population” (Cao, 2006:1).

Very often more than just one context variable plays a key-role in modifying the cardinality expressed by a quantifier. On the basis of experimental evidence, Hörmann argued that the magnitude of a quantifier “is determined […] not only by the object referred to […], but also by the spatial situation of the object referred to in the utterance. A few people before a hut are less people than a few people before a building.” (1983:229-230); in the discussion related to his experiment Hörmann remarks how “This seems, intuitively, to be correct, but there is no grammar on earth which deals with factors of this kind” (1983:230). The conclusion drawn by Hörmann is that the numerosity expressed by a cardinal quantifier “is determined not so much by the absolute size of the reference object, but by a rather holistic relation between the size of the object and the size of comparison objects also mentioned in the utterance” (1983:230).

For the most part, the experiments of the present study will look at cardinal interpretations of the Italian mid-range quantifiers. In Experiment 2 and 4 we use sentences with a syntactic structure similar to (23b). In Experiment 3 we leave both cardinal and proportional interpretations open in the verbal context, but we combine them with a visual context that strongly suggests a proportional interpretation of the quantifiers.. In Experiment 1, where we offer no visual or verbal context suggesting one or the other interpretation, the choice is entirely left to the participants.

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[assigned] to it” (2003:273). Quantifiers with a cardinal interpretation are described in the form of a function QUANT instead, which “assigns a quantity to an empirical object, just as [the function] NQ assigns a numerical quantity to an empirical object (namely, a set). In the case of ‘many’ or ‘few’, QUANT assigns set sizes above a norm value or below a norm value” (Wiese, 2003:320). This difference is relevant to our study because it implies that whenever we will ask them to order the magnitudes of quantifiers, to which they will give a cardinal interpretation, each participant will also make use of a norm value to estimate the magnitude of each quantifier.

The cardinality expressed by a number-word does not refer to a norm value in order to be interpreted and understood; but it can happen sometimes, that number-words, just like quantifiers, point to noisy quantities. The passage from (25) to (26) represents an example of how the meanings of cardinal words and quantifiers can be very similar. Similarity is often found when number-words are used either in idiomatic or in hyperbolical constructions and their meanings are not strictly lower or upper bounded.

Examples of two such expressions are given in (25). In (25a) the number “one million” can mean many as well as a huge amount of and does not necessarily represent the literal quantity of 1.000.000; similarly does the word “two” in (25b) which rather points to a smaller number, such as one or zero, or to the quantifier no.

(25) a. I’d give you one million kisses!

b. There are two copies left: the album was a success.

Not all number-words are open to such noisy, pseudo-cardinal quantifier interpretations: only numbers like two and four (mostly in idiomatic expressions) or round numbers in the order of tens, hundreds, thousands or millions, etc. are available to it (Levinson, 2000:89; Wiese, 2003:277-278) and whenever a number-word assumes this particular semantic connotation it can be replaced by a quantifier. The two sentences in (25) can in fact be rewritten as shown in (26) without a substantial change in their meaning.

(26) a. I’d give you many kisses!

b. There are (only) a few copies left: the album was a success.

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It is possible that people process quantifiers in the same way they process numbers. An important cognitive model of number processing is the one proposed by Gallistel and Gelman (1992) (based on the assumptions of Meck and Church (1983)).

Gallistel and Gelmans’ model postulates a mechanism for counting called “accumulator” common to non-verbal animals (Meck and Church, 1983) and humans (Gallistel and Gelman, 1992; id. 2000). When we count, all quantity increments are stored in the accumulator before we move the final count to memory to have it ready for further processing.

While we count, the representatives of numbers in the accumulator are mental magnitudes which retain a scalar variability. This scalar variability makes the mental representations of magnitudes noisy and the larger is the quantity the noisier will its mental representation be (Dehaene, 1997; Gallistel and Gelman, 1992; id. 2000).

In other words, our mind is not able to instantly and precisely recognize large numerosities. When we are presented a given quantity of, say, black points on a screen, we might be able to almost instantly say whether that quantity is representative of a numerosity up to 10, between 10 and 20, or larger than 20. But should the numerosity be larger than 20 though, we might still be able to approximate whether it is less or more than 50. The higher the count of numerosity is, the fuzzier is the interpretation we will have of it (Dehaene, 1997:66-77 gives of this an exhaustive presentation).

In arithmetic counting, an arbitrary verbal value is assigned to each noisy magnitude in the form of a number. Counting means to be able to partition and map magnitudes, so that each magnitude can be matched to an integer value represented by a digit.

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Summarizing, it is sometimes possible, in specific conditions, to interchange certain number-words with specific quantifiers, without altering the meaning of a sentence.

According to the same logic, we want to see with this study if the magnitude-values implicit in quantifiers (for which in the experiments we suggest a cardinal interpretation) make possible a scalar ordering of all mid-range quantifiers. If we are successful, that would lend support to the hypothesis that quantifier-magnitudes are estimated using the same cognitive mechanism which estimate the magnitudes of numbers.

This is done by using a magnitude-comparison task (Experiments 2 and 4) and a quantifier-ordering task (Experiment 1), two experimental paradigms often used in studies concerned with numbers.

2.3.2 Pragmatic aspects

Both number-words and quantifiers are sensitive to pragmatic inferences. These inferences might represent a problem when we try to establish an order among mid-range quantifiers as they can contribute to alter the interpretation given to quantifiers by altering their pragmatic boundaries.

The present section presents aspects connected to the pragmatic boundaries of quantifiers and number-words as these two lexical classes differ in how they make themselves available to pragmatic interpretation.

In accordance with the Gricean maxim of quantity, which says that the amount of information in a natural language utterance should contribute informatively to the requirements of the conversation, yet without making such a contribution more informative than required, both number-words and

quantifiers are traditionally considered by linguists like Horn, Gazdar and Levinson (Hurewitz et al., 2006:80) to be lower bounded.

In other words, when uttering sentences such as (27a) and (27b) number-words must be interpreted as it is shown in (28) with the lower bound entailed by “At leastAt leastAt least fifteen/many apples”. At least

(27) a. Fifteen apples in the box are red. b. Many apples in the box are red.

(28) At least fifteen/many apples in the box are red.

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of a language however will not embrace this logical interpretation and will rather upper-bound the statements pragmatically interpreting them as exemplified in (29).

(29) a. Exactly fifteen apples in the box are red.

b. Many apples, but not all apples, in the box are red.

Not doing so we would attribute the number-word in (29a) and the quantifier in (29b) more informativeness than they should convey. On the contrary, our doing so would assume that the statements in (27) were not adequately informative, violating therefore the above-mentioned Gricean maxim of Quantity.

Numbers present specific pragmatic properties, mostly originating from their conventional use in arithmetic operations (e.g. the result of “1+1” is always “2” and never “approximately 2” or “2 or more”). There are two theoretical positions on the pragmatic interpretation of numbers and “one position is to claim that the number-words are just ambiguous between an ‘exactly’ and an ‘at least’ interpretation, perhaps disambiguated in specific contexts. […] Another approach is to assume that the meaning of, say, three is simply indeterminate at the semantic level between ‘at least three’ and ‘exactly three’ and is then further specified […] in context as appropriate” (Levinson, 2000:88-89). Not-exact interpretation of number-words is only possible in specific cases, for instance within the numeric systems of certain languages or very specific contexts in which the number-word appears.

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In the second case, for contexts such as (25), the interpretation of a number-word is left open to quantity implicatures: in the specific case “two albums left” does not mean necessarily two, while it might be an other small numeric value larger than two as well.7

When interpreting numbers and quantifiers, context can modify the magnitude of numerosities for quantifiers but it cannot do the same for numbers (exception made for sentences where number-words do not have an exact interpretation, of course).

To summarize: in this second chapter we presented the Italian quantifiers that we will use for our experiments, we introduced two distinctions among them: one objective, in terms of magnitude (low vs. high) and one theoretical in terms of type (‘number’ vs. ‘comparative’). We also gave a description of relevant semantic and pragmatic properties of quantifiers, relating them to the properties of numbers. Last but not least, we proposed that the same cognitive model involved in number perception and estimation might be involved in the perception and estimation of quantifier-magnitudes. Most of these pieces of information will contribute to define two different types of scalar-degrees which will be introduced, together with two different scale-types in the next chapter.

7_{Hörmann (1983) experimentally demonstrated how the interpretation of the German expression} _{ein Paar,}

etymologically related to the number-word two, is reputed plausible for expressing cardinalities from 3 up to 6 by more than

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3. Scales

In order to establish a scalar ordering of Italian quantifiers, we need a scale onto which establish the order of the quantifier-magnitudes. This chapter will introduce scales in more detail. Kennedy (2001:34) gives the following definition of a scale: “generally we intend as a scale a set of objects under a total ordering, where each object represents a measure or degree”. The Merriam-Webster dictionary defines the word measurement as the act or process of measuring the magnitude of a quantity. Scales are therefore constituted by the gradual ordering of the results of two or more acts of measuring. Scales measuring the same kind of phenomena (e.g. temperature, speed, etc.) can of course differ from each other according to the combination of measurement techniques and ordering strategies applied.

Quoting Horn (1972), Hurewitz et al. (2006:80) write that both lexical classes of quantifiers and cardinal number-words refer to quantities and both have an underlying scalar structure which is internally ordered, meaning that both quantifiers and cardinal number-words can form scales from weaker to stronger elements according to the amount of information that each degree of the scale conveys (e.g. in a scale all will be more informative than some and five will be more informative than two as the meaning of five as in “at least five” entails two and not vice-versa). However, the same numerosity described with a quantifier or with a number-word is measured onto different scales and these scales will structurally differ in the same way the natures of quantifiers and number differ from each other (see Figure 3.3).

In the following pages we will look at different ways of representing magnitude-measures onto a scalar structure. The first section (Section 3.1) will introduce the reader to entailment scales (Horn scales), often used in the discussion of pragmatic aspects related to the scalar properties of quantifiers (Horn, 1989; Levinson, 2000; Blutner: 2004; Krifka, 2006), and the position of mid-range quantifiers within this kind of scales.

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3.1 Quantifiers and scales

When writing about scales and quantifiers, Horn scales (also known as entailment scales) represent a traditional starting point. Apart from their importance in the study of pragmatic implicatures, Horn scales are relevant to the present research because they clearly show how difficult it is to establish an exact ordering among the mid-range quantifiers, even theoretically.

Horn scales can be introduced as a “set of linguistic alternates” presenting specific entailment properties; more precisely, as Levinson (2000:79) defines it, a Horn scale is “an ordered n-tuple of expression alternates < x1, x2, ... , xn > such that, where S is an arbitrary simplex sentence-frame and xi > xj, S(xi) unilaterally entails S(xj)”. So for instance Horn scales are <must, may>, <necessarily, possibly>, <always, often>, <and, or>. These are all sets of words, expressing different degrees of the same information, which can all alternate within their respective syntactic positions, (1).

(1) You must/may always/often buy apples and/or oranges.

Conventionally the entailing element finds its place in a Horn scale on the left side of the comparison operator and the entailed one on its right (e.g. must > may, where must entails may and may is entailed by must). As a result of the unilateral semantic entailment in the scalar set, words placed at the leftmost extreme in the scale will always rank higher in terms of informative richness than the ones on the right.

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Figure 3.1 – A square of opposites formed by the two Horn-scales < all, some > and <none, not all>; adapted from Levinson (2000:65).

Every square of oppositions displays two Horn scales facing each other. In Figure 3.1, for instance, the two scales < all, some > and < none, not some > are represented on the two vertical sides of the square. The most informative scalar items (all and none) both collocate on the upper side of the square, whereas the less informative ones (some and not all) are at the bottom.8

Whereas the relationships between positive (all, some) and negative (none, not all) absolute quantifiers have been extensively studied and successfully understood (see for a summary Levinson, 2000:64-66), “a problem that has arisen in understanding the quantifiers is the exact relationship between midscale negative and positive quantifiers” (Levinson, 2000:83). In order to have a complete knowledge of the quantification scale it is therefore also necessary to understand the relationships between quantifiers whose position is midway along the vertical sides of the square. Such quantifiers in English are for instance many, few, several, very many, very few, etc. (whose meanings correspond roughly to the meanings of the six Italian quantifiers presented in Chapter 1.2). According to Levinson, attempting the mapping of mid-range quantifiers within a square of oppositions seems a very difficult task as there is no clear way of defining a general rule governing all the relationships among mid-range scalar items in terms of contraries, subcontraries and contradictories. To use Levinson’s words: “if we

8_{The diagram also shows how the degrees of the two Horn scales establish relationships in terms of opposition}

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are to understand quantification, we must understand the relationships between quantifiers that are midway on, or not at either end of, Horn scales” (2000:83).

Let us digress from quantifiers briefly, to see how adjectives and number-words deal with the properties of sub-opposition and contradiction. In adjectival scales relating the scalar elements is usually quite straightforward: it is normal for opposite scalar degrees to relate as antonyms (e.g. <fast, slow>), while adjacent scalar degrees may sometimes relate as synonyms (e.g. <…, cool, cold>); scales of number-words are very clear in terms of entailment, since arithmetic counting accounts for a sure way to define numerical magnitudes (e.g. <…, three, two, one, zero>), but it is impossible to establish antonymic or synonymic relationships. In both arithmetic and semantic terms, eight can be neither a synonym nor a contrary for nine, or for any other number. Similarly, expressing an exact judgment on the extent to which many is a better synonym for very many than lots of, is hard; thinking in terms of synonymic and antonymic relations is then probably not the best way to order mid-range quantifiers on a scale.9_This should not be surprising if we keep in mind how cardinal quantifiers can replace number-words (e.g. “I’d give you one millionone millionone million/manyone million/many/many/many kisses!”). The example suggested already a clear and strong conceptual link between the lexical categories of cardinal quantifiers and number-words.

Levinson (2000:85-86) explains how Larry Horn (1989:237), in spite of the difficulty of the task, attempted to establish a system of correspondences on an arithmetic base between positive and negative mid-range quantifiers of English (a distinction roughly corresponding to the one made in Chapter 1 between high and low magnitude Italian quantifiers). First of all, Horn assigned each corner of the square of oppositions (Figure 3.1) a value. The upper-left corner got a value of 1.0, the lower-left and lower right a value of 0.0 and the upper-right a value of -1.0; then two half-way marks were placed on the median point of both vertical sides of the square. The left median point was scored 0.5 and the right -0.5. All items above the positive median point (e.g. all, every, most, a majority) are said by Horn

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to have negative items as contraries, while all positive items falling under the median point (e.g. very many, many, several, quite a few, etc.) have negative items as subcontraries (Figure 3.2).

Figure 3.2 – The arithmetic scale for English quantifying phrases, adapted from Horn (1989:237); in Levinson (2000:85).

In order to establish which items fall above the median point(s) and which under (e.g. many being placed above several and not vice versa), Horn used “suspender” tests. These tests consisted in contrasting the felicitousness of sentences such as exemplified in (2) and (3).

(2) a. Many if not most of them came. b. ??Most if not many of them came. (3) a. Several if many of them came.

b. ??Many if not several of them came.

Horn’s “suspender” tests, joint to other tests for tolerant versus intolerant logical operators (cf. Löbner, 1987:62-66; Levinson, 2000:86), make it possible to order any mid-range quantifier on either the positive or the negative scale on the square of oppositions.

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have, which we think is demonstrated by the fact that no synonymic or antonymic relations are possible among numbers, which do have magnitudes indeed.

Since our experiments (presented in Chapters from 4 to 7) attempt to expose the magnitudes implicit in the meaning of quantifiers, without focusing on any pragmatic aspect at all, we restrict our hypotheses on the scalar ordering of Italian quantifiers (Section 3.4) only to linear and interval-scales (which we are going to introduce in Sections 3.2 and 3.3).

Even though it has already been established that no correspondence exists between quantifiers and numbers (Moxey and Sanford, 2000), there are ranges of values in a numeric scale onto which a given quantifier will most probably project its semantic range. The quantifier-scale in Figure 3.3 represents, from top to bottom, an interval-scale of quantifiers, a linear-scale of number-words and a scale of physical referents. The physical referents at the bottom are eleven sets of apples, whose magnitudes range from 0, on the extreme left, to 10, on the extreme right. The intermediate scale comprises the number-words from zero to ten. The different distances between each number-point represent the perception-noise which increases together with the number of apples. Finally the uppermost scale uses five different quantifiers: two absolute (no, all) and three mid-range (few, several, many). The ranges of the five quantifiers are represented by areas of different color and – for reader’s information – their ranges are here intuitively delimited.

Figure 3.3 – Parallel scales of apples, number-words and quantifiers.

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number-word scale will be shifting onto a higher degree, one in this case. For each apple added to the physical set, the scale of number-words will always have an exact corresponding value available to maintain the symmetry. The quantifier-scale does not reproduce such symmetry in its reference neither to number-words, nor to physical objects. In the example, where we have a finite set of ten apples, the only exceptions are the two absolute-quantifiers no and all, which correspond respectively to 0/zero apples and 10/ten apples. If the set was not finite there would be only one quantifier which would have found a symmetric correspondence and that would be no.

In the next sections we will discuss in greater detail first linear-scales (Section 3.2) and then interval-scales (Section 3.3), to finally formulate our general hypotheses and predictions about the scalar ordering of Italian quantifiers (Section 3.4).

3.2 Linear

3.2 Linear----scales

scales

Linear-scales have a linear structure, the simplest kind of scalar structure possible. In a linear-scale a degree either precedes and/or follows other degrees on the scale. This is very clear when looking at the scale of number-words in Figure 3.3. The same scale of number-words also shows how easy it is to move back and forth along the scale when the size of a set is quantified numerically. That is because in such cases it is possible to use arithmetic rules to determine a successor function. Successor functions give us the faculty of predicting increments (and decrements) in the magnitude of the scalar degrees at a given point of the scale. Figure 3.2 shows for instance that, given a set of n apples equal to 0 and a rule that increases the size of n by 1 apple (n + 1), it is predictable when in the set there will be exactly 1 apple, then 2 apples, then 3 apples and so on. The rule n+1 is the successor function for both the physical and number-word scales in Figure 3.3. Summarizing, linear-scales are ideal for ordering anything discretely measurable (e.g. integer numbers, such as 1, 2, 3, etc.): each degree on the scale will correspond to a point in a linear function predicting all possible increments and decrements of magnitude.

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Kaufman et al., 1949; McCloskey, 1992; Moyer and Landauer, 1967; Pinel et al., 2001; Starkey and Cooper, 1981; Starkey, Spelke and Gelman, 1993).

Applying a comparison task paradigm to magnitudes which align along a linear-scale always results in the elicitation of a cognitive constant reported for the first time by Moyer and Landauer (1967) and called distance effect. Whenever “humans are asked to judge which of two physical magnitudes [...] is greater” (Gallistel and Gelman, 1992:59), the distance effect manifests itself with slower reaction times in the responses and more precision errors in the answers whenever the values of the magnitudes are close to each other; when they are further apart faster reaction times and more precise answers are witnessed. The distance effect can be represented as a continuously decreasing concave-upward curve relating reaction times to scalar distance (Gallistel and Gelman, 1992) (see Figure 3.4).

Figure 3.4 – Diagram of the distance-effect function: “Each black dot shows the average response time to a given number. Responses become increasingly slow as the target

numeral gets closer to 65. Data from Dehaene et al. 1990)”; Dehaene (1997:75).

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In natural languages, all that is arithmetically quantifiable in terms of continuous values, has a tight connection to certain pairs of antonymic adjectives.Sometimes adjectival scales introduce intermediate degrees; of course, those scales are not bipolar but they comprise one or more mid-range degrees, as shown in (5). The kind of scale in (5) is similar to the one already seen in Chapter 1, used to label the size of clothes. (4) a. < long, short > b. < high, low > c. < heavy, light > d. < fast, slow > e. < full, empty > f. < bright, dark > (5) a. < black, gray, white>

b. < hot, warm, lukewarm, cool, cold >

None of the adjectival scales seen in (4) and (5) is properly linear though. We saw that linear-scales offer the advantage of establishing exact orderings among their degrees by using predictor functions. The big drawback of linear-scales, when applied to natural language items, though, consists in the fact that there are no predictor functions for adjectives or cardinal quantifiers.

It may be hard, for instance, deciding when exactly the temperature of a glass water ceases to be cold and becomes cool. When represented on a linear-scale, natural language terms will always display a range of semantic fuzziness where two or more quantifiers or adjectives overlap (e.g. cool and cold, , many and lots, parecchio, molto and tanto).

Such a representation would go against the nature of linear-scales though, as on a linear-scale degrees can only precede or follow each other, with no overlapping allowed. Moreover, the linear-scale incapability of handling phenomena of overlapping makes impossible for linear-scales to deal with positive and negated forms of words at the same time.

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(9) ??My room is not eighteen square meters wide.

Negated forms are instead plausible scalar degrees for both adjectival (10) and quantifier-scales (11).

(10) <happy, not happy, not unhappy, unhappy> (11) <many, not many, not few, few>

Nevertheless, the presence of just one negated form in a linear-scale interferes with the continuity drawn by the other scalar degrees, because the directionality of the negated scalar degrees is opposite to the one of their respective non-negated forms. This also makes comparison operations among the scalar degrees and their negated counterparts difficult, as they would require reversing the scale in our mind prior to the compare-operation. Moreover, it is necessary to deal with the overlap of semantic ranges such as the ones of many and not few or few and not many.

The following section will introduce interval-scales and present them as a possible solution to the problems generated by the overlapping of one or more semantic ranges in the linear scalar structure.

3.3 Interval

3.3 Interval----scales

scales

Interval-scales retain a fundamentally linear shape, but overcome the limits of a simple linear structure by introducing the idea of scalar degrees which configure as intervals. Kennedy (2001) proposed a new ontology of scalar degrees, suggesting that scalar degrees that can intertwine antonymic relations and form comparative constructions with one another. Kennedy writes that scalar-degrees “must be formalized as intervals on a scale […] rather than as points on a scale” (2001:36).

He also makes a structural distinction between “two sorts of degrees: positive and negative degrees. […] positive degrees are intervals that range from the lower end of a scale to some point, and negative degrees are intervals that range from some point to the upper end of a scale” (Kennedy, 2001:52).10_So for instance the positive interval-degree for the scale <tall, short> will range from 0 to a given point and will move upwards, as shown in (12) while the negative interval-degree will move downwards towards the zero and stop at a given point, as we see in (13).