Rings of quantifiers · Different finite sets of quantifiers are proposed and a single common algebraic structure, namely commutative rings, is shown to be applicable to all these

Rings of quantifiersA universal algebraic structure for non-numeral scalar quantifiersMichel DELARCHEUniv Paris Diderot, Sorbonne Paris Cité, CLILLAC-ARPUFR EILA, case 7002, 75205 Paris cedex [email protected]

Abstract: This paper develops an algebraic model for describing non-numeral scalar quantifiers. First, it is shown that two binary operators combined over a minimum set of quantifiers define an algebraic structure commonly known as a ring. The extendibility of this structure to new quantifiers (identified owing to their contrastive co-occurrences with the already defined ones) is then assessed. Candidates for extension are derived from the three usual transformations of quantifiers (negation, approximation and intensification) and examples in different languages (mostly English, French and Spanish) are given in support of the extensions retained. In conclusion, the impact of this model on the theory of semantic universals is discussed.

Keywords: finite commutative rings – semantic universals - quantifiers

Introduction: Semantic models for describing quantifiersEverybody has an intuitive grasp of non-numeral quantifiers such as ‘many’, ‘few’, ‘not much’, ‘some’ ‘all’, ‘almost no’ etc. They seem to constitute a really universal feature of natural languages.Starting in the 1970s, the emergence of formal mathematical models collectively known as denotational semantics and Generalized Quantifier Theory (GQT) has allowed linguists and logicians to refine their understanding of the semantics of quantifiers and describe them in terms of various abstract properties such as conservativity, monotonicity etc. [Cann & al. 2009] provide a comprehensive overview of these logico-mathematical techniques (chapters 4 and 5 of their textbook are of special relevance to the subject of this paper.) [Peters & Westerståhl 2007] provide detailed analyses of quantifiers in that framework.In the more recent field of computational semantics, another formal approach has been explored within the framework of the theory of fuzzy subsets [Glöckner 2005]. A common feature of all these modern approaches is that they describe the semantic properties of the quantifiers by considering them one by one, describing them by means of logical formulas whose variables take their values in one or more reference domains.A limitation of this formalism stems from the difficulty of objectively identifying those reference domains. This type of limitation has now been widely recognised and a conclusion reached that, for example: ‘many may be treated as underspecified for its full interpretation. To interpret an utterance of the word, therefore, the hearer has to make certain choices based on context(including world knowledge), i.e. pragmatic choices must be allowed to precede semantic interpretation if we do not want to take the easy way out and say that many is multiply ambiguous’ (Cann & al. 2009 - p114.)In parallel, the quest for semantic universals has pointed out a few dozens potential semantic primitives, including 3 non-numeral quantifiers: ‘all’, ‘many/much’ and ‘some’. Other universal primitives potentially related to quantitative measurement in a wider sense are ‘big’ and ‘small’. [Goddard & Wierzbicka 1994] ; [Goddard 2002].

Objectives and scope of this paperThe main objective of this paper is to propose a new algebraic model, potentially applicable to all natural languages, for investigating relationships among non-numeral scalar quantifiers. Different finite sets of quantifiers are proposed and a single common algebraic structure, namely commutative rings, is shown to be applicable to all these sets.For the sake of generality, ‘quantifier’ is not used here in its narrow morpho-syntactic sense (i.e. a finite set of noun modifiers expressing a number or mass count modulation of the modified noun) but in a wider semantic sense, including a variety of other modifiers used to express non-numeral quantification such as adjectives, nouns, adverbials, inflections, affixes.In the approach adopted here, quantifiers are described as a cohesive system, rather than as individual semantic entities to be represented per se as is done in denotational semantics. The model proposed doesn't rely on any topological notion nor does it require any assumption on the countability or measurability of underlying reference domains. Abstract representations of the scalar quantifiers are treated collectively as an integrated system, modelled as a single mathematical structure.In this paper, an algebraic model of quantifiers is developed incrementally, starting from the 3 universal primitive quantifiers mentioned above then combining them with other universal primitive notions such as ‘big’, ‘small’, ‘not’, ‘near’ and ‘very’ to produce derived quantifiers.Other more or less universal categorisation schemes of quantification, e.g. the singular/plural (one vs several), the non specific/specific (any vs some) the normative (too vs enough) and the proportional (minority vs majority) are not addressed by this model.

Construction of the model

Absolute versus relative quantifiersFirst we must explain the terminology we use in this paper, starting from an example based on the quantifier ‘many’ for which the relation to the reference domain is notoriously difficult to express semantically as already signalled in the introduction. In the sentence ‘Many Americans visit Paris yearly’ (source: http:// disneylandinparis.com/paris-hotels.html ) the use of the quantifier ‘many’ cannot be satisfyingly explained by formal semantic models relying on well-defined reference sets where an enumeration variable would take its values (be it here the set of all the Americans or the set of all the visitors, ‘many’ cannot be understood as a large proportion of either set.) Here, ‘many’ stands as a purely qualitative assessment of the number of Americans that come to Paris every year, not really making reference to any underlying totality; this is a case where ‘many’ is to be interpreted as an absolute (subjective) quantifier rather than as an objective (relative to a well-defined domain) one.

In such a situation ‘many’ has to be interpreted pragmatically rather than purely semantically (in this example, the pragmatic reference domain would be something like the expected size of the population of yearly American visitors to Paris).

This kind of analysis boils down to a normative pragmatic interpretation of the quantifier, based on a norm which may not be necessarily widely shared, hence the subjective qualification that I attach to it.By contrast, another sentence like: ‘Not many American visit Paris in their lifetime’ (source: www.thefalcononline.com/print.php?id=6799) can be reasonably interpreted in respect of the total population of the USA, so this occurrence of ‘many’ can be interpreted as an objective (implicitly) domain-related quantifier.

Explicitly domain-related quantifiers correspond to the “many of + <anaphora>” construct (or any semantically equivalent morpho-syntactic device used in other languages): so, for example, ‘many of the Americans who visit Paris come back every year’ is explicitly domain-related while ‘many Americans come back to Paris every year’ can be interpreted as either subjective or implicitly domain-related, depending on the wider context.The model proposed in this paper doesn’t rely on domain-based representations. In the absence of sufficient context information allowing the explicit identification of a reference domain, our model will offer the flexibility of accepting either interpretation, albeit in two slightly different frameworks developed in the next sections for absolute quantifiers and relative quantifiers respectively.

Universal classes of natural language quantifiersThe first step is to determine a basic set of candidate quantifiers.If a model representing all sorts of quantifiers, even those that can't be easily interpreted by reference to an underlying totality, is to be built, the need arise from the outset to create two different sets: a set of relative quantifiers including ‘all/every’ as one of its elements, and a smaller set of absolute quantifiers that wouldn’t include any such representation of totality.Therefore an algebraically minimal set AQ3 of absolute quantifiers based on semantic primitives is comprised of:a) AQ-n symbolizing a null quantifier ('no/not any’ in English, 'no' in Spanish and Italian, 'pas de/aucun' in French, 'kein' in German, 'нет' in Russian etc.) which would correspond to the primitive ‘not’ applied to any object.b) AQ-c symbolizing a central quantifier, representing a certain quantity, neither big nor smallIn natural languages, AQ-c can be represented by a variety of lexical or morpho-syntactical mechanisms.. Just considering a few Indo-European languages, etymologically and structurally close enough to each other, reveals 4 different ways of representing Q-c:‘I’ve bought some bread’ (specific quantifier form in English)‘He comprado Ø pan’ (empty form in Spanish)‘J’ai acheté du pain’ (definite partitive form in French)‘покупил хлеба’ (partitive genitive form in Russian.)AQ-c may also be expressed by means of words taking a quantification-related meaning (e.g. the adjective ‘certain’ followed by a plural noun in English French) or multi word expressions which are not grammatically considered as quantification determiners (e.g a certain number of’, a certain quantity of’, ‘(una) cierta cantidad de’ etc.)c) AQ-g symbolizing a great quantity (‘many/much/a lot of’, ‘mucho’, ‘molto’, ‘beaucoup de’, ‘viel’, ‘много’ etc.)Obviously, the corresponding set RQ of relative quantifiers has to contain an additional element RQ-t in order to include the notion of totality that distinguishes it from AQ.But if we want to retain the pragmatic flexibility of ascribing to either category all those occurrences of quantifiers that are not explicitly domain-related, the structures associated to AQ and RQ must be compatible, and in the rest of this paper, we will use the notation Q-x for both types of quantifiers.

Binary operators expressing the semantics of these quantifiersSince the approach adopted in this paper consists in defining the semantics of quantifiers through internal operations on finite sets thereof, operators allowing a quantifier to be expressed in terms of other quantifiers have to be introduced, under a general constraint of internal stability (that is the result of an operation on two members of a given set of quantifiers must still be a member of the same set.)

The following binary operators are proposed:a) Gap (a, b) formalizes an intuitive notion of distance between quantifiers taken as orders of magnitude (please note that this internal operator is not equivalent to the topological concept of distance which is a mapping from a pair of elements onto a positive number.)b) Min (a, b) formalizes the intuitive ordering of quantifiers through one-to-one comparisons (which is the most direct way of observing this order in natural language expressions.)These two operators can be defined by the following pair of tables for the 4 quantifiers:

Gap Q-n Q-c Q-g Q-tQ-n Q-n Q-c Q-g Q-tQ-c Q-c Q-n Q-g Q-gQ-g Q-g Q-c Q-n Q-gQ-t Q-t Q-c Q-g Q-nTable 1: Operators Gap and Min

Min Q-n Q-c Q-g Q-tQ-n Q-n Q-n Q-n Q-nQ-c Q-n Q-c Q-c Q-cQ-g Q-n Q-c Q-g Q-gQ-t Q-n Q-c Q-g Q-t

These two tables present the following properties: both operators are commutative (the above matrices are symmetrical) and have a neutral element (Q-n for Gap and Q-t for Min.) The neutral element of Gap is absorbing for Min, and the AQ subset (Q-n, Q-p, Q-g) is stable for both operators.Also, Gap is associative, that is: ∀ a, b et c ∈ Q4, Gap (a, Gap (b,c)) = Gap (Gap (a,b), c)If a = Q-n or b = Q-n or c = Q-n, associativity holds: Gap (Q-n), Gap (b, c) ) = Gap (b, c) andSince Gap (Q-n, b) = b, Gap (Gap (Q-n, b), c) = Gap (b, c)Similarly, Gap (a, (Gap (Q-n, c) = Gap (a, c) and Gap (Gap (a, Q-n) , c) = Gap (a, c)And finally, Gap (a, Gap (b, Q-n) = Gap (a, b) and Gap (Gap (a,b) Q-n) = Gap (a, b)Also, if a = b or b = c or a = c, since Gap (x, x) = Q-n, we reach the same conclusion, as both sides of the associativity equations are equal to c, a and b respectively.So we need to confirm associativity only for those triplets (a,b,c) not containing any Q-n and with the 3 elements all different. Owing to the additional property of commutativity, we can further restrict our enumeration to those triplets where Min (a, b) = a, and Min (b, c) = b, that is the upper triangle of the Gap matrix. For RQ4 the only triplet we need to evaluate is (Q-c, Q-g, Q-t) and we have: Gap (Q-c, Gap (Q-g, Q-t)) = Gap (Q-c, Q-g) = Q-g so the associativity is verified. (For the structures that are discussed in the rest of this paper, demonstrations of the associativity or non-associativity of Gap are detailed in the appendix.)We can note also that every element is its own inverse for Gap (Gap is said to be involutive.)Min is also associative, because, since we need to consider only Min-ordered triplet (a,b,c), we always have: Min (a, Min (b, c)) = Min (Min (a, b), c) = a.The reader should not believe that any apparently sensible combination of values guarantees the associativity of the operator Gap; for example, with the set (Q-n, Q-c, Q-g, Q-t) the following table for Gap could be envisaged, based on the argument that ‘many’ should be rather close to ‘all’:

Gap Q-n Q-c Q-g Q-tQ-n Q-n Q-c Q-g Q-tQ-c Q-c Q-n Q-g Q-gQ-g Q-g Q-c Q-n Q-cQ-t Q-t Q-c Q-c Q-nTable 2: Example of a non-associative table for Gap

In this alternative model, Gap is not associative because: Gap (Gap (Q-c, Q-g), Q-t) = Gap (Q-g, Q-t) = Q-c, whileGap (Q-c, Gap (Q-g, Q-t)) = Gap (Q-c, Q-c) = Q-n.

And indeed, somewhat surprisingly, the maximum distance that can be observed between ‘many’ and the totality is not ‘some’, but ‘many’ as illustrated by the following citations, easily obtained under Google by using the search key “many agreed, but many”:

Many agreed, but many disagreed as wellMany agreed, but many others had a different point of viewMany agreed but many opposed my views

So the model given as table 1 is consistent with both the associativity requirements of the ring structure and the empirical evidence which can be found on the use of ‘many’ as a relative quantifier.

Minimal and non-minimal ring structuresFor the minimum set AQ3 of 3 absolute quantifiers (Q-n, Q-c, Q-g) Min is distributive over Gap, that is: ∀ a, b, c ∈ AQ3, Min (a, Gap (b, c)) = Gap (Min (a, b), Min (b, c)).In mathematics, a set equipped with two laws featuring all those properties is named a ring. More precisely, AQ3 is a group for Gap and a commutative ring for Gap and Min. Similarly, the set RQ4 of 4 relative quantifiers (Q-n, Q-c, Q-g, Q-t) is a commutative ring for Gap and Min, with the semantics defined by Table 1. (For a general introduction to algebraic structures, the reader may refer to a textbook of algebra, e.g. [Cohn 2003])A first extension based on the list of would-be universals presented in [Goddard 2002] consist in considering the notions of ‘big’ and ‘small’ applied to quantities. The notion of a ‘big quantity’ is already captured by ‘many/much’, but the notion of a ‘small quantity’ (or ‘small part’ for relative quantifiers) is absent.So, we can introduce Q-p to represent the notion of a positive yet small quantity ('a few/a little', ‘un poco’, ‘quelques/un peu de’, ‘einige’, ‘несколько’ etc.) and the definitions of the operators Gap and Min can be extended so as to insert Q-p between Q-n and Q-r. The set (Q-n, Q-p, Q-c, Q-g, Q-t), is also a commutative ring for Gap and Min, hereafter named RQ5. (the associativity of Gap for this set is demonstrated in the appendix)Since the sub-tables obtained by striking out Q-t are stable for Gap and Min, the corresponding subset is also a commutative ring for Gap and Min that we can denote as AQ4 (in AQ4, Q-g replaces Q-t as the neutral element of Min, so it is not a sub-ring of RQ5).

Gap Q-n Q-p Q-c Q-g Q-tQ-n Q-n Q-p Q-c Q-g Q-tQ-p Q-p Q-n Q-c Q-g Q-gQ-c Q-c Q-c Q-n Q-g Q-gQ-g Q-g Q-c Q-g Q-n Q-gQ-t Q-t Q-g Q-g Q-g Q-n

Min Q-n Q-p Q-c Q-g Q-tQ-n Q-n Q-n Q-n Q-n Q-nQ-p Q-n Q-p Q-p Q-p Q-pQ-c Q-n Q-p Q-c Q-c Q-cQ-g Q-n Q-p Q-c Q-g Q-gQ-t Q-n Q-p Q-c Q-g Q-t

Table 3: Non primitive ring RQ5

New scalar quantifiers derived from primitive modifiersWe have identified so far 4 potentially universal rings of quantifiers (AQ3, RQ4, AQ4 and RQ5). In this section we investigate the feasibility of extending these structures to additional quantifiers that can be observed in different languages.However, since we no longer restrict ourselves to the minimal structures built in the preceding section, we need to define criteria for introducing the new quantifiers in our model.Our two extension criteria are:

- introducing a new value Q-x mustn’t produce a degenerate structure: Q-x must be different from all the quantifiers already identified (i.e. its lines and columns in the D and Min tables mustn’t be identical to already existing ones)

- such a new value must be identified by specific lexical and/or morpho-syntactical forms in more than one language.

Taken together, these two criteria mean that the definition of Gap and Min for a new candidate must be exhibited in several languages by means of direct or indirect contrasts with the other quantifiers already introduced; for example, most occurrences of expressions such as ‘Q-y or Q-z’ imply that Q-y and Q-z are perceived as distinct. In a less direct way, expressions such as ‘not Q-y but not Q-z’ point at a (possibly new) quantifier defined in contrast with both Q-y and Q-z.In the three next sub-sections we analyze whether the three usual modifications of basic quantifiers (negation, approximation and intensification, respectively associated to the semantic primitives ‘not’, ‘near’ and ‘very’) give birth to additional scalar quantifiers.

Negated quantifiersA strict interpretation of negation in terms of complementation over our set of quantifiers would yield the following set of equations:not (Q-t) = Q-g or Q-c or Q-p or Q-n ; not (Q-g) = Q-t or Q-c or Q-p or Q-n ;not (Q-c) = Q-t or Q-g or Q-p or Q-n ;not (Q-p) = Q-t or Q-g or Qc or Q-n ;But most of the theoretical alternative values appearing on the right hand side of these four equations require additional qualifications. For example, Q-t would never be interpreted spontaneously as Q-n, and an explicit identification of the intended meaning is required, with expressions such as: ‘not all, (even none)’.In fact, apart from Q-g, each of our four basic quantifiers can be associated with a spontaneous opposite within its complementary subset, and the use of the negation frequently adds emphasis, for example: (not Q-n) = Q-c (e.g. ‘It’s not without difficulties’ means ‘there are some difficulties’) (not Q-p) = Q-g (e.g. ‘it doesn’t cost [just] a little money’ means ‘it costs a lot of money’) (not Q-c) = Q-n (e.g. Q-n in French ‘pas de’ = ‘not’ + partitive Q-c) (not Q-t) = Q-c (e.g. ‘don’t eat all the chocolate’ means ‘eat only a certain quantity of chocolate’)However, quantifier negation is not symmetrical: Q-p cannot be said to represent (not Q-g) because it requires an additional qualification, as shown by expressions such as ‘not much but a little’, ‘pas beaucoup, mais quand même un peu’, ‘no mucho pero un poco’ etc.So, our first candidate for extending RQ5 is the so-called negative quantifier for small quantities (‘few/little’, ‘poco’, ‘peu de’, ‘wenig’, ‘мало’ etc.) which is in fact quite generally defined as ‘not Q-g’ in dictionaries.In our model, it would sit between Q-n and Q-p with respect to Min, and its internal distance to the other quantifiers can be derived from its similarity with Q-p. Renaming (not Q-g) as the minimally positive quantity Q-m, we get the following extended table which corresponds to a new ring RQ6 (since extending the table for the Min operator is a trivial task, from now on we provide a single table defining the Gap operator over the Min-ordered list of quantifiers).Gap Q-n Q-m Q-p Q-c Q-g Q-tQ-n Q-n Q-m Q-p Q-c Q-g Q-tQ-m Q-m Q-n Q-p Q-c Q-g Q-gQ-p Q-p Q-p Q-n Q-c Q-g Q-gQ-c Q-c Q-c Q-c Q-n Q-g Q-gQ-g Q-g Q-g Q-g Q-g Q-n Q-gQ-t Q-t Q-t Q-g Q-g Q-g Q-n

Table 4: the negation-extended ring RQ6

Deleting the Q-t row and column in the above table yields another ring AQ5 (which can also be seen as an extension of AQ4 to Q-m.) Since we have introduced this new element as the negation of Q-g, we can’t exclude than certain languages won’t possess a specific positive representation of Q-m, and that only this negative definition is available. However, co-occurrences of (not Q-g) and Q-p can be distinguished in those languages, it is still legitimate to expand RQ5 into RQ6.Also, certain forms may appear ambiguous with respect to the set of abstract quantifier classes proposed in this model: for example ‘немного’ in Russian (etymologically ‘not much’) may represent either Q-m or Q-p, depending on the context, but other Russian quantifiers like ‘мало’ and ‘несколько’ can be ascribed exclusively to Q-m and Q-p, respectively, thus maintaining the possibility of mapping our RQ6 and AQ5 models onto this language.

Approximated quantifiersAdverbials such as ‘almost’, ‘nearly’ or ‘practically’ create approximates of Q-n and Q-t.In any case, various natural languages contain expressions like: ‘all or almost all’, ‘not all but almost all’ and ‘many or (even) almost all’ that plead strongly in favour of introducing a distinction between Q-t and (app Q-t) while maintaining (app Q-t) distinct from Q-g, and therefore introducing (app Q-t) as a new element of our abstract modelling set.The same line of argument applies to (app Q-n) in respect of Q-m and Q-n, owing to such expressions as: ‘few or almost no’ and: ‘no or almost no’. For the sake of compactness, we denote these two new values as: Q-an and Q-at.The three following tables confirm this view on the basis of a summary exploration of three on-line corpora for English and Spanish available at http://corpus.byu.edu/ : the Corpus of Contemporary American English (aka COCA) the British National Corpus (BNC) and the Corpus De Lengua Española (CDLE), supplemented by a websearch whenever necessary.Contrastive co-occurrences of quantifiers COCA

(11/8/11)BNC

(11/8/11)CDLE

(11/8/11)Q-an / Q-nNo or almost/nearly/practically noNingun o casi ningun (+inflections)Nada o casi nada (de)

0 018

Q-an / Q-mFew/little or almost/nearly/practically noPoco o casi nada (de)Poco o casi ningun (+ inflections)

1 011

Q-m / Q-nLittle or noFew or noPoco o ningun (+inflections)Poco o nada (de)

2583 152

830 32

41

Table 5: Contrastive co-occurrences of Q-m, Q-n, Q-an

Contrastive co-occurrences of quantifiers COCA (12/8/11)

BNC (12/8/11)

CDLE (12/8/11)

Q-at / Q-tall or almost/nearly/practically allTodo o casi todo (+inflections)

43 853

Q-at / Q-gMany/much or almost/nearly/practically allMucho o casi todo (+inflections)

1 02

Q-g / Q-tMany/much or allMucho o todo (+inflections)

37 99

Table 6: Contrastive co-occurrences of Q-g, Q-t, Q-at (as noun modifiers or pronouns only)For those contrastive co-occurrences not detected in the above mentioned corpora (and also for the French language, which is not currently equipped with the same kind of online linguistic corpora) we have gleaned some representative examples by means of a websearch.

Search engines do not take punctuation into account, do not always reduce multiple references to the same piece of information, do not filter out obvious morpho-syntactic errors, and neither the search space nor the extraction algorithm are stable through time, so the associated statistical data given in the last column should be taken with a large pinch of salt.

Contrast Lang. Citations totalQ-n / Q-an EN Here's what is robbing our government: no or almost no

taxes for huge multi-billion-dollar corporations.(source: huffingtonpost.com – 2011-Mar-3)

438,000

Q-n / Q-an FR Si ce courant passe dans un fil de cuivre, il n'y a aucun, ou presque aucun effet. (If that current flows through a copper wire, there is no, or almost no effect.) (source : documents.irevues.inist.fr - article : Bistouri électrique et coagulation par plasma d'argon (APC) D. COUMAROS, P. SCHLÜTER)

12,940

Q-n / Q-an SP Ningún o casi ningún niño levanta la mano, aunque sepa la respuesta. (No or almost no child raises his hand, although he knows the answer.) (source : mundodeportivo.com - 2011-Feb-03)

14,200

Q-m / Q-an FR Dans l'armée, peu ou presque pas de citoyens.In the army, few or almost no citizens.(source : mediterranee-antique/info/fontane/rome)

44,900

Q-n / Q-m FR Grève des postes: peu ou pas de courrier ce matin encore(Post Office industrial action : still little or no mail this morning) (source : sudpresse.be - 2011-Feb-14)

6,300,000

Q-at / Q-t FR Lors d’une désintégration radioactive, un noyau conserve tous ou presque tous les protons ou les neutrons qui le constituent. (During a radioactive desintegration a nucleus keeps all or almost all the protons or neutrons which it is made of.) (source : laradioactivite.com/fr/site/pages/Fission)

467,000

Q-g / Q-at FR Il donne l'exemple, beaucoup ou presque tous devraient le suivre (he sets an example, many or almost all should follow suit) (source : lefigaro.fr – 2011-Jan-16)

250

Q-g / Q-t FR l'UNESCO, beaucoup, sinon tous ici, la connaissent bien(the UNESCO, many if not all here, know it well)Source : cidecquebec2011.org/docs/GSaoumaForero.pdf

10,800

Table 7: Contrastive co-occurrences obtained by a Google search (search keys are in bold)

Intensified quantifiersIn natural languages, all the basic quantifiers we have addressed so far can be intensified by various morpho-syntactic means:- repetition: ‘many many’, ‘mucho mucho’…- intensifying adverbials: ‘very much’, ‘very few’, ‘vraiment beaucoup’, ‘très peu’…

- intensifying prefixes: ‘re-poco’, ‘hyper-peu’, ‘ultra-much’, ‘super-beaucoup’…- intensifying suffixes: ‘muchisimo’, ‘un poquit(it)o’ etc.

It could also be argued that expressions such as ‘none at all’ or ‘absolutely all of them’ and their equivalents in other languages are intensified forms for Q-n and Q-t respectively, but that kind of intensification does not lead to the creation of new distinct values in our model, they are just a mark of rhetorical insistence and are never contrasted with the non-intensified forms.

Regarding the other 4 quantifiers introduced so far, we can try to identify new quantifiers associated with their intensified values, by adopting the same approach as in the previous section for detecting Q-an and Q-at. By ascending order with respect to Min, the first candidate is the intensified form of Q-m, which we can note as Q-im, and we can verify the existence of contrastive co-occurrences of Q-im with Q-m and Q-n, and also with the new element Q-an which has been introduced in the preceding sub-section.

Contrastive co-occurrences of quantifiers COCA (11/8/11)

BNC (11/8/11)

CDLE (11/8/11)

Q-im / Q-nVery little or noVery few or nomuy poco o ningun (+inflections)

21 3

60

2Q-im / Q-anVery little/few or almost/nearly noMuy poco o casi ningun (+inflections)

0 00

Q-im / Q-mLittle/few or very little/fewPoco o muy poco (+ inflections)

0 22

Table 8: Q-im contrast statistics in online corpora with Q-m, Q-n and Q-an

For the contrasts missing in Table 7, here are typical examples obtained through a websearch:Contrast Lang. Citations TotalQ-im / Q-n FR Encore très peu, ou pas, de quotidiens nationaux dans les

kiosques (still very few or no nation-wide newspapers in the shops) (source : franceinter.fr/chro/larevuedepresse/90833

1,240,000

Q-im / Q-an FR Ainsi, il existe à notre connaissance très peu ou presque pas de théories sur la migration forcée. (So, to our knowledge, there existe very few or almost no theories on forced migration) (source : memoireonline.com/02/10/3188/)

5,500

Q-im / Q-m FR Je crois qu'il ya peu ou très peu de chirurgiens dentistes francophones sur Shangai (I think there are few or very few French-speaking dentists in Shangai) (source : www.bonjourchine.com/.../3653-chirurgien-dentiste)

21,800

Q-im / Q-an EN Trichinosis usually gives rise to very few or almost no symptoms whatsoever in case of a minor parasitic infection(Source : primehealthchannel.com – 2011-Feb-11)

451,700

Q-im / Q-an SP Muy pocos o casi ningun cirujano se atreve a operar asi. (very few or almost no surgeon dares operate this way)(Source : forejercito.forumup.es – ingreso e formacion)

31,000

Table 9: Websearch of Q-im contrasts in French and Q-im vs Q-an in English and Spanish

Intensified forms of Q-p don’t always exist: we have ‘quite a few’ in English, but ‘un bon peu’ in French is colloquial and is only a humorous representation of Q-c or Q-g (a bit like ‘a good few’ in English) ; no intensified form is available in Spanish for explicit representations of Q-c such as ‘(una) cierta cantidad de’ (‘a certain quantity of’)Searching the COCA and BNC corpora for contrasts between ‘quite a few/little’ and ‘some/a certain number’ doesn’t yield any relevant co-occurrence.A Google websearch (conducted on August 13, 2011) produced no valid examples of contrastive co-occurrences of Q-c with the intensified forms of Q-p either (‘quite a few or some’ and ‘quite a little or some’ yielded only a handful of cases, all of them irrelevant to our investigation.)However, searching for the expression ‘quite a few or many’ yields 103,000 occurrences. Considered together, these two results indicate that the intensification of Q-p (when it exists) is in fact Q-c and not a new abstract quantifier.

Q-c cannot be intensified into any new specific quantification: other intensifying expressions such as: ‘quite a number of’ ‘quite a quantity of’ can be interpreted as representations either of Q-c itself or, more frequently, of Q-g, with which they are only marginally found in contrast, since the Google websearch conducted on August 13, 2011 produced only one occurrence of ‘quite a number or many’: ‘… many people who announced to me when they wrote in support that they opposed quite a number or many of the things that I stood for’.(source : http://www.abc.net.au/pm/stories/s714142.htm)Similarly, the same Google websearch yielded only one contrastive co-occurrence of ‘some or quite some’ (produced by a non-native speaker): ‘All you need is the open CISSP study guide plus some (or quite some) knowledge on networking (as found in MCSE)’. (source : http://agileskills2.org/blog/category/misc/page/2/)

So, although the non-existence of something can never be easily demonstrated, there is strong empirical evidence to justify the non-creation of a distinct value in our model for the intensified forms of Q-c.Regarding the intensified forms of Q-g, the three online corpora yield the following results.

Contrastive co-occurrences of quantifiers COCA (13/8/11)

BNC (13/8/11)

CDLE (13/8/11)

Q-ig / Q-gMuch/many or very-much/quite-manyA lot or quite a lotMucho o muchisimo (+inflections)

01

00

0Q-ig / Q-atQuite-many/very-much or almost/nearly allMuchisimo o casi todo (+ inflections)

0 22

Table 10: Q-ig contrasts with Q-at and Q-g in online corpora (as noun modifiers or pronouns)

However, contrastive co-occurrences of Q-g with its intensified forms can be searched on the web with expressions like ‘beaucoup ou vraiment beaucoup de’ (‘a lot or quite a lot of’ in French). Here are the results of the corresponding websearch.

Q-ig / Q-g SP Muchos o muchísimos de los que se han manifestado, y que aquí escriben tienen trabajos. (a lot or quite a lot of those who signalled themselves, and who write here have a job)

48,500

(source : eskup.elpais.com/ - 2011-Jun-20)Q-ig / Q-g FR Petite question: beaucoup ou vraiment beaucoup de monde

dans le parc ? (A little question : a lot or quite a lot of people in the park ?) (source : disneycentralplaza.com/t21175p80-disneyland-paris)

70

Q-ig / Q-at FR ‘presque/quasiment tou/t(e)/s ou/sinon/voire vraiment beaucoup’

0

Table 11: Q-ig in contrast with Q-g obtained through a websearch using Google

So the contrast between Q-at and Q-ig does exist in English and Spanish but no example was found in French. However, its presence in both the BNC and the CDLE justifies its inclusion into our list of candidates, for the sake of completeness.

Extended rings of quantifiersWe have identified in total 5 new extension candidates: Q-an, Qim, Q-m, Q-ig and Q-at, and we have already seen that the extension of the base set to Q-m gives the ring RQ6.In this section, we show that certain extensions by approximates and intensifiers also have a ring structure.

Partially extended structuresThe extension of Q6 to approximations would have the following table for Gap (the values inherited from RQ5 are italicised) and it can be named RQ8a.

Gap Q-n Q-an Q-m Q-p Q-c Q-g Q-at Q-tQ-n Q-n Q-an Q-m Q-p Q-c Q-g Q-at Q-tQ-an Q-an Q-n Q-m Q-p Q-c Q-g Q-at Q-tQ-m Q-m Q-m Q-n Q-p Q-c Q-g Q-g Q-gQ-p Q-p Q-p Q-p Q-n Q-c Q-c Q-g Q-gQ-c Q-c Q-c Q-c Q-c Q-n Q-c Q-c Q-gQ-g Q-g Q-g Q-g Q-g Q-c Q-n Q-g Q-gQ-at Q-at Q-at Q-g Q-g Q-g Q-g Q-n Q-cQ-t Q-t Q-t Q-g Q-g Q-g Q-g Q-c Q-nTable 12: approximation-extended structure RQ8a

The somewhat counter-intuitive property that Q-c is the value attained by the distance between Q-at and Q-t (rather than, say, Q-m or Q-p) can be ascertained with the two following citations:C1: Faced with the prospect of a war of all against all, humans reformulate the conflict into a war of nearly all against a few -- against the Jews or the communists or the gays or the feminists or the Mexican immigrants (citation found in the COCA)C2: Almost all icons launch Winzip; some do nothing(source: http://www.tomshardware.co.uk/forum/28864-35-almost-icons-launch-winzip-nothing )

In C1, the domain of ‘all’ (humans) can be broken into 3 sub-groups: the persecutors (‘nearly all’), the persecuted (‘a few’) and the non-persecutors/non-persecuted (a small yet unspecified quantity), while in C2 the two quantified subsets are clearly separate (their intersection is void), so it appears that ‘some’ does represent the maximum potential distance between ‘all’ and ‘almost all’.

The following examples illustrate the same phenomenon in the two other languages investigated:

L'eau est d'un prix abordable en France pour presque tous mais certains ménages démunis doivent consacrer plus de 3,5 % de leur revenu pour payer leur eau. (In France, water is at an affordable price for almost all but some impoverished households must spend more than 3.5 % of their income to pay for their water)(source : coalition-eau.org/IMG/pdf/Prix_abordable_de_l_eau_-_FR.pdf )

Es un acuerdo en el que casi todos ganan y algunos pierden.(It’s an agreement in which almost all win and some/a few lose)(source : http://elcomentario.tv/reggio/el-valor-de-un-acuerdo-de-fernando-faces-en-expansion/29/01/2011/)

This second example is doubly ambiguous: ‘algunos’ may represent either Q-p (‘a few’) or Q-c ‘some’) and the two subsets of winners and losers may not represent the whole reference set (it is conceivable that some participants in the game don’t win nor lose.)We have not found any example in Spanish that would quantify a difference between Q-at and Q-t by means of an unambiguous representation of Q-c (like ‘cierta cantidad de’ o ‘ciertos’.)The associativity of RQ8a is not verified for the triplet (Q-c, Q-at, Q-t) so RQ8a is not a ring, but RQ7a (the extension adding only Q-an) is a ring, and AQ6a is also a ring.The extension of the structure to intensified quantifiers appears impossible because the differences between Q-im and Q-m on the one hand, and between Q-g and Q-ig on the other hand, cannot be determined: the few co-occurrences found do not permit to detect any value of the difference: it seems that expressions like ‘a lot or quite a lot’, beaucoup ou vraiment beaucoup’ etc. express more a qualitative insistence than a real gap. This would mean that Gap (Q-im, Q-m) = Q-n and Gap (Q-g, Q-ig) = Q-n, but then the intensified values would become undistinguishable from the corresponding non-intensified forms in our model.

An alternative option is to consider that the gap between Q-m and Q-im is Q-m (that is the same order of magnitude that the gap between Q-an and Q-m), and that the gap between Q-g and Q-ig is similar to the gap between Q-at and Q-t, that is Q-c.

Gap Q-n Q-im Q-m Q-p Q-c Q-g Q-ig Q-tQ-n Q-n Q-im Q-m Q-p Q-c Q-g Q-ig Q-tQ-im Q-im Q-n Q-m Q-p Q-c Q-g Q-g Q-gQ-m Q-m Q-m Q-n Q-p Q-c Q-g Q-g Q-gQ-p Q-p Q-p Q-p Q-n Q-c Q-g Q-g Q-gQ-c Q-c Q-c Q-c Q-c Q-n Q-g Q-g Q-gQ-g Q-g Q-g Q-g Q-c Q-g Q-n Q-c Q-gQ-ig Q-ig Q-ig Q-ig Q-ig Q-g Q-c Q-n Q-gQ-t Q-t Q-g Q-g Q-g Q-g Q-g Q-g Q-nTable 13: intensification-extended structure RQ8i

However, like RQ8a, this structure is not a ring because the triplets (Q-c, Q-g, Q-ig) and (Q-g, Q-ig, Q-t) are not associative, but all the other triplets fulfil the condition, so RQ7i (extension to Q-im only) is a ring. AQ6i is also a ring.

Maximum ring of scalar quantifiersWe can now combine all these new quantifiers into the maximum ring structure RQ8.Gap Q-n Q-an Q-im Q-m Q-p Q-c Q-g Q-tQ-n Q-n Q-an Q-im Q-m Q-p Q-c Q-g Q-tQ-an Q-an Q-n Q-im Q-m Q-p Q-c Q-g Q-t

Q-im Q-im Q-im Q-n Q-m Q-p Q-c Q-g Q-gQ-m Q-m Q-m Q-m Q-n Q-p Q-c Q-g Q-gQ-p Q-p Q-p Q-p Q-p Q-n Q-c Q-g Q-gQ-c Q-c Q-c Q-c Q-c Q-c Q-n Q-g Q-gQ-g Q-g Q-g Q-g Q-c Q-g Q-g Q-n Q-gQ-t Q-t Q-t Q-g Q-g Q-g Q-g Q-g Q-nTable 14: Maximum ring RQ8

By suppressing the Q-t line and column of the above matrix, we can define the restriction of RQ8 to 7 absolute quantifiers, thus getting the maximum ring AQ7.

ConclusionWe have noticed the following points:

- although it is a necessary ingredient of our minimal rings, Q-n is not part of the list of semantic universals given in [Goddard 2002] ; in our opinion, starting from the universality of ‘not’, the universality of Q-n should be assessed ;

- there is an asymmetry among quantification universals : Q-g ‘represented by ‘many/much’) is considered as universal, while the small quantity Q-p isn’t ; this is reflected in our model by the derivation of Q-p from ‘small’ and ‘part’ which both belong to the would-be list of universals ; considering the ring structure RQ5, the universality of Q-p (and in particular the degree of ambiguity that may exist in certain languages between Q-p and Q-c) would need to be further investigated.

- ‘few’ is listed as a candidate for universality in [Goddard 2002]. We have introduced it in our model as a derived quantifier Q-m representing the negation of Q-g ; since ‘not’ and ‘many/much’ are both universals the universality of the ring RQ6 including Q-m appears as a strong possibility.

- the other derivations that we have explored in this paper can be described as a subset of the Cartesian product of 2 universal elements (‘near’, ‘very’) by the 6 would-be universal quantifiers that constitute RQ6 (Q-n, Q-m, Q-p, Q-c, Q-g, Q-t). We have seen that not all of the corresponding extensions could be empirically assessed (the finer the distinctions, the more difficult the search for non-ambiguously contrasting co-occurrences). All in all, Q-an, Q-im, Q-ig and Qat seem unlikely candidates for universality and the same judgement applies to those rings resulting from partial extensions. The remark made by Wierzbicka in her final review from [Goddard & Wiersbicka 1994] concerning the difficulty of ascertaining the existence of a distinction between ‘all’ and ‘almost all’ is consistent with the difficulties we have signalled in the last part of this paper.

REFERENCES

[Cann & al. 2009]Ronnie Cann, Ruth Kempson and Eleni GregoromichelskiSemantics An introduction to Meaning in LanguageCambridge Textbooks in Linguistics – Cambridge 2009

[Cohn 2003] Paul Moritz Cohn - Basic Algebra : groups, rings, and fieldsSpringer 2003

[Glöckner 2006]Glöckner, Ingo - Fuzzy Quantifiers A Computational Theory. Series: Studies in Fuzziness and Soft Computing, Vol. 193. Springer 2006

[Goddard & Wierzbicka 1994]Semantic Primitives Across Languages: A Critical Review p 445-500 ofSemantic and Lexical Universals : Theory and empirical findings Edited by Cliff Goddard and Anna WierzbickaJohn Benjamin 1994

[Goddard 2002]The search for the shared semantic core of all languages p 5-40 ofCliff Goddard and Anna Wierzbicka (eds)Meaning and Universal Grammar - Theory and Empirical Findings/. Volume I. John Benjamin 2002

[Peters & Westerståhl 2007]Stanley Peters and Dag Westerståhl Quantifiers in Language and LogicOxford 2007

APPENDIX: Assessments of associativity for the operator Gap

The associativity property is defined as: ∀ a, b et c ∈ RQ-x, Gap (a, Gap (b,c)) = Gap (Gap (a,b), c)

If a = Q-n or b = Q-n or c = Q-n, associativity holds: Gap (Q-n), Gap (b, c) ) = Gap (b, c) andSince Gap (Q-n, b) = b, Gap (Gap (Q-n, b), c) = Gap (b, c)Similarly, Gap (a, (Gap (Q-n, c) = Gap (a, c) and Gap (Gap (a, Q-n) , c) = Gap (a, c)And finally, Gap (a, Gap (b, Q-n) = Gap (a, b) and Gap (Gap (a,b) Q-n) = Gap (a, b)Also, if a = b or b = c or a = c, since Gap (x, x) = Q-n, we reach the same conclusion, as both sides of the associativity equations are equal to c, a and b respectively.

So we need to check only those triplets (a,b,c) not containing any Q-n and with the 3 elements all different. Owing to the commutativity property, we can further restrict our enumeration to those triplets where Min (a, b) = a and Min (b, c) = b, that is the upper triangle of the Gap matrix.

1°) Associativity of Gap for RQ5: Gap Q-n Q-p Q-c Q-g Q-tQ-n Q-n Q-p Q-c Q-g Q-tQ-p Q-p Q-n Q-c Q-g Q-gQ-c Q-c Q-c Q-n Q-g Q-gQ-g Q-g Q-g Q-g Q-n Q-gQ-t Q-t Q-g Q-g Q-g Q-n

We need to evaluate only the following triplets: (Q-p, Q-c, Q-g) , (Q-p, Q-c, Q-t) and (Q-p, Q-g, Q-t) since the triplet (Q-c, Q-g, Q-t) has already been checked (for the associativity of RQ4)Gap (Q-p, Gap (Q-c, Q-g)) = Gap (Q-p, Q-g) = Q-gGap (Gap (Q-p, Q-c), Q-g) = Gap (Q-c, Q-g) = Q-g

Gap (Q-p, Gap (Q-c, Q-t)) = Gap (Q-p, Q-g) = Q-gGap (Gap (Q-p, Q-c), Q-t)) = Gap (Q-c, Q-t) = Q-g

Gap (Q-p, Gap (Q-g, Q-t) = Gap (Q-p, Q-g) = Q-gGap (Gap (Q-p, Q-g), Q-t) = Gap (Q-g, Q-t) = Q-gTherefore RQ5 is associative.

2°) Associativity of Gap for RQ6:Gap Q-n Q-m Q-p Q-c Q-g Q-tQ-n Q-n Q-m Q-p Q-c Q-g Q-tQ-m Q-m Q-n Q-p Q-c Q-g Q-gQ-p Q-p Q-p Q-n Q-c Q-g Q-gQ-c Q-c Q-c Q-c Q-n Q-g Q-gQ-g Q-g Q-g Q-g Q-g Q-n Q-gQ-t Q-t Q-t Q-g Q-g Q-g Q-n

We need to evaluate only the 5 ordered triplets not already present in RQ5, which are:(Q-m, Q-p, Q-c) , (Q-m, Q-p, Q-g) , (Q-m, Q-p, Q-t) , (Q-m, Q-c, Q-g) , (Q-m, Q-c, Q-t) and (Q-m, Q-g, Q-t).Gap (Q-m, Gap (Q-p, Q-c)) = Gap (Q-m, Q-c) = Q-cGap (Gap (Q-m, Q-p), Q-c)) = Gap (Q-p, Q-c) = Q-c

Gap (Q-m, Gap (Q-p, Q-g)) = Gap (Q-m, Q-g) = Q-gGap (Gap (Q-m, Q-p), Q-g) = Gap (Q-p, Q-g) = Q-g

Gap (Q-m, Gap (Q-p, Q-t)) = Gap (Q-m, Q-g) = Q-gGap (Gap (Q-m, Q-p), Q-t) = Gap (Q-p, Q-t) = Q-g

Gap (Q-m, Gap (Q-c, Q-g)) = Gap (Q-m, Q-g) = Q-gGap (Gap (Q-m, Q-c), Q-g)) = Gap (Q-c, Q-g) = Q-g

Gap (Q-m, Gap (Q-c, Q-t)) = Gap (Q-m, Q-g) = Q-gGap (Gap (Q-m, Q-c), Q-t)) = Gap (Q-c, Q-g) = Q-g

Gap (Q-m, Gap (Q-g, Q-t)) = Gap (Q-m, Q-g) = Q-gGap (Gap (Q-m, Q-g), Q-t)) = Gap (Q-g, Q-t) = Q-g

Therefore, Gap is associative for RQ6

3°) Associativity assessment for RQ8a:Gap Q-n Q-an Q-m Q-p Q-c Q-g Q-at Q-tQ-n Q-n Q-an Q-m Q-p Q-c Q-g Q-at Q-tQ-an Q-an Q-n Q-m Q-p Q-c Q-g Q-at Q-tQ-m Q-m Q-m Q-n Q-p Q-c Q-g Q-g Q-gQ-p Q-p Q-p Q-p Q-n Q-c Q-c Q-g Q-gQ-c Q-c Q-c Q-c Q-c Q-n Q-c Q-g Q-gQ-g Q-g Q-g Q-g Q-g Q-c Q-n Q-g Q-gQ-at Q-at Q-at Q-g Q-g Q-g Q-g Q-n Q-cQ-t Q-t Q-t Q-g Q-g Q-g Q-g Q-c Q-nFor this table, we have to check only the triplets including one of the new value Q-an or Q-at (or both) since the other triplets have already been checked for RQ6.For a = Q-an

For b = Q-mFor c = Q-p: Gap (Q-an, Gap (Q-m, Q-p)) = Gap (Q-an, Q-p) = Q-p

and Gap (Gap (Q-an, Q-m), Q-p)) = Gap (Q-m, Q-p) = Q-p

For c = Q-c: Gap (Q-an, Gap (Q-m, Q-c)) = Gap (Q-an, Q-c) = Q-c and Gap (Gap (Q-an, Q-m), Q-c)) = Gap (Q-m, Q-c) = Q-c

For c = Q-g: Gap (Q-an, Gap (Q-m, Q-g)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-m), Q-g)) = Gap (Q-m, Q-g) = Q-g

For c = Q-at, Gap (Q-an, Gap (Q-m, Q-at)) = Gap (Q-an, Q-at) = Q-at and Gap (Gap (Q-an, Q-m), Q-at)) = Gap (Q-m, Q-at) = Q-at

For c = Q-t, Gap (Q-an, Gap (Q-m, Q-t)) = Gap (Q-an, Q-t) = Q-t and Gap (Gap (Q-an, Q-m), Q-t) = Gap (Q-m, Q-t) = Q-t

For b = Q-pFor c = Q-c: Gap (Q-an, Gap (Q-p, Q-c)) = Gap (Q-an, Q-c) = Q-c

and Gap (Gap (Q-an, Q-p), Q-c)) = Gap (Q-p, Q-c) = Q-c

For c = Q-g: (Q-an, Gap (Q-p, Q-g)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-p), Q-g)) = Gap (Q-p, Q-g) = Q-g

For c = Q-at: Gap (Q-an, Gap (Q-p, Q-at)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-p), Q-at)) = Gap (Q-p, Q-at) = Q-g

For c = Q-t: Gap (Q-an, Gap (Q-p, Q-t)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-p), Q-t) = Gap (Q-p, Q-t) = Q-g

For b = Q-cFor c = Q-g: (Q-an, Gap (Q-c, Q-g)) = Gap (Q-an, Q-g) = Q-g

and Gap (Gap (Q-an, Q-c), Q-g)) = Gap (Q-c, Q-g) = Q-g

For c = Q-at: Gap (Q-an, Gap (Q-c, Q-at)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-c), Q-at)) = Gap (Q-c, Q-at) = Q-g

For c = Q-t: Gap (Q-an, Gap (Q-c, Q-t)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-c), Q-t) = Gap (Q-c, Q-g) = Q-g

For b = Q-gFor c = Q-at: Gap (Q-an, Gap (Q-g, Q-at)) = Gap (Q-an, Q-g) = Q-g

and Gap (Gap (Q-an, Q-g), Q-at)) = Gap (Q-g, Q-at) = Q-g

For c = Q-t: Gap (Q-an, Gap (Q-g, Q-t)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-an, Q-g), Q-t) = Gap (Q-g, Q-t) = Q-g

For b = Q-atFor c = Q-t: Gap (Q-an, Gap (Q-at, Q-t)) = Gap (Q-an, Q-c) = Q-c

and Gap (Gap (Q-an, Q-at), Q-t)) = Gap (Q-at, Q-t) = Q-cFor a = Q-m

For b = Q-pFor c = Q-at: Gap (Q-m, Gap (Q-p, Q-at)) = Gap (Q-m, Q-at) = Q-g

and Gap (Gap (Q-m, Q-p), Q-at)) = Gap (Q-p, Q-at) = Q-gFor b = Q-c

For c = Q-at: Gap (Q-m, Gap (Q-c, Q-at)) = Gap (Q-m, Q-g) = Q-g and Gap (Gap (Q-m, Q-c), Q-at)) = Gap (Q-c, Q-at) = Q-g

For b = Q-gFor c = Q-at: Gap (Q-m, Gap (Q-g, Q-at)) = Gap (Q-m, Q-g) = Q-g

and Gap (Gap (Q-m, Q-g), Q-at)) = Gap (Q-g, Q-at) = Q-gFor b = Q-at

For c = Q-t: Gap (Q-m, Gap (Q-at, Q-t)) = Gap (Q-m, Q-t) = Q-g and Gap (Gap (Q-m, Q-at), Q-t)) = Gap (Q-g, Q-t) = Q-g

For a = Q-cFor b = Q-g

For c = Q-at: Gap (Q-c, Gap (Q-g, Q-at)) = Gap (Q-c, Q-g) = Q-g and Gap (Gap (Q-c, Q-g), Q-at)) = Gap (Q-g, Q-at) = Q-g

For b = Q-atFor c = Q-t: Gap (Q-c, Gap (Q-at, Q-t)) = Gap (Q-c, Q-c) = Q-n

and Gap (Gap (Q-c, Q-at), Q-t)) = Gap (Q-g, Q-t) = Q-gFor a = Q-g

For b = Q-atFor c = Q-t: Gap (Q-g, Gap (Q-at, Q-t)) = Gap (Q-g, Q-c) = Q-g

and Gap (Gap (Q-g, Q-at), Q-t) = Gap (Q-g, Q-t) = Q-g

So the associativity of Gap for RQ8a is not verified for the triplet (Q-c, Q-at, Q-t)

4°) Associativity assessment of RQ8i:Gap Q-n Q-im Q-m Q-p Q-c Q-g Q-ig Q-tQ-n Q-n Q-im Q-m Q-p Q-c Q-g Q-ig Q-tQ-im Q-im Q-n Q-m Q-p Q-c Q-g Q-g Q-gQ-m Q-m Q-m Q-n Q-p Q-c Q-g Q-g Q-gQ-p Q-p Q-p Q-p Q-n Q-c Q-g Q-g Q-gQ-c Q-c Q-c Q-c Q-c Q-n Q-g Q-g Q-gQ-g Q-g Q-g Q-g Q-c Q-g Q-n Q-c Q-gQ-ig Q-ig Q-ig Q-ig Q-ig Q-g Q-c Q-n Q-gQ-t Q-t Q-g Q-g Q-g Q-g Q-g Q-g Q-nFor this table, we have to check only the triplets including one of the new value Q-an or Q-at (or both) since the other triplets have already been checked for RQ6.For a = Q-im

For b = Q-mFor c = Q-p: Gap (Q-im, Gap (Q-m, Q-p)) = Gap (Q-m, Q-p) = Q-p

and Gap (Gap (Q-im, Q-m), Q-p)) = Gap (Q-m, Q-p) = Q-p

For c = Q-c: Gap (Q-im, Gap (Q-m, Q-c)) = Gap (Q-m, Q-c) = Q-c and Gap (Gap (Q-im, Q-m), Q-c)) = Gap (Q-m, Q-c) = Q-c

For c = Q-g: Gap (Q-im, Gap (Q-m, Q-g)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-m), Q-g)) = Gap (Q-m, Q-g) = Q-g

For c = Q-ig, Gap (Q-im, Gap (Q-m, Q-ig)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-m), Q-ig)) = Gap (Q-m, Q-ig) = Q-g

For c = Q-t, Gap (Q-im, Gap (Q-m, Q-t)) = Gap (Q-im, Q-t) = Q-g and Gap (Gap (Q-im, Q-m), Q-t) = Gap (Q-m, Q-t) = Q-g

For b = Q-pFor c = Q-c: Gap (Q-im, Gap (Q-p, Q-c)) = Gap (Q-im, Q-c) = Q-c

and Gap (Gap (Q-im, Q-p), Q-c)) = Gap (Q-p, Q-c) = Q-c

For c = Q-g: (Q-im, Gap (Q-p, Q-g)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-p), Q-g)) = Gap (Q-p, Q-g) = Q-g

For c = Q-ig: Gap (Q-im, Gap (Q-p, Q-ig)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-p), Q-ig)) = Gap (Q-p, Q-ig) = Q-g

For c = Q-t: Gap (Q-im, Gap (Q-p, Q-t)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-p), Q-t) = Gap (Q-p, Q-t) = Q-g

For b = Q-cFor c = Q-g: (Q-im, Gap (Q-c, Q-g)) = Gap (Q-im, Q-g) = Q-g

and Gap (Gap (Q-im, Q-c), Q-g)) = Gap (Q-c, Q-g) = Q-g

For c = Q-ig: Gap (Q-im, Gap (Q-c, Q-ig)) = Gap (Q-an, Q-g) = Q-g and Gap (Gap (Q-im, Q-c), Q-ig)) = Gap (Q-c, Q-ig) = Q-g

For c = Q-t: Gap (Q-im, Gap (Q-c, Q-t)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-c), Q-t) = Gap (Q-c, Q-g) = Q-g

For b = Q-g

For c = Q-ig: Gap (Q-im, Gap (Q-g, Q-ig)) = Gap (Q-im, Q-c) = Q-c and Gap (Gap (Q-im, Q-g), Q-ig)) = Gap (Q-g, Q-ig) = Q-c

For c = Q-t: Gap (Q-im, Gap (Q-g, Q-t)) = Gap (Q-im, Q-g) = Q-g and Gap (Gap (Q-im, Q-g), Q-t) = Gap (Q-g, Q-t) = Q-g

For b = Q-igFor c = Q-t: Gap (Q-im, Gap (Q-ig, Q-t)) = Gap (Q-im, Q-c) = Q-c

and Gap (Gap (Q-im, Q-ig), Q-t)) = Gap (Q-ig, Q-t) = Q-cFor a = Q-m

For b = Q-pFor c = Q-ig: Gap (Q-m, Gap (Q-p, Q-ig)) = Gap (Q-m, Q-ig) = Q-g

and Gap (Gap (Q-m, Q-p), Q-ig)) = Gap (Q-p, Q-ig) = Q-gFor b = Q-c

For c = Q-at: Gap (Q-m, Gap (Q-c, Q-ig)) = Gap (Q-m, Q-g) = Q-g and Gap (Gap (Q-m, Q-c), Q-ig)) = Gap (Q-c, Q-ig) = Q-g

For b = Q-gFor c = Q-ig: Gap (Q-m, Gap (Q-g, Q-ig)) = Gap (Q-m, Q-c) = Q-c

and Gap (Gap (Q-m, Q-g), Q-ig)) = Gap (Q-g, Q-ig) = Q-cFor b = Q-ig

For c = Q-t: Gap (Q-m, Gap (Q-ig, Q-t)) = Gap (Q-m, Q-t) = Q-g and Gap (Gap (Q-m, Q-ig), Q-t)) = Gap (Q-g, Q-t) = Q-g

For a = Q-cFor b = Q-g

For c = Q-ig: Gap (Q-c, Gap (Q-g, Q-ig)) = Gap (Q-c, Q-c) = Q-n and Gap (Gap (Q-c, Q-g), Q-ig)) = Gap (Q-g, Q-ig) = Q-c

For b = Q-igFor c = Q-t: Gap (Q-c, Gap (Q-ig, Q-t)) = Gap (Q-c, Q-ig) = Q-g

and Gap (Gap (Q-c, Q-at), Q-t)) = Gap (Q-ig, Q-t) = Q-gFor a = Q-g

For b = Q-igFor c = Q-t: Gap (Q-g, Gap (Q-ig, Q-t)) = Gap (Q-g, Q-g) = Q-n

and Gap (Gap (Q-g, Q-ig), Q-t) = Gap (Q-c, Q-t) = Q-g

Two triplets are not associative: (Q-c, Q-g, Q-ig) and (Q-g, Q-ig, Q-t).

5°) Associativity of Gap for RQ8:Q-n Q-n Q-an Q-im Q-m Q-p Q-c Q-g Q-tQ-an Q-an Q-n Q-im Q-m Q-p Q-c Q-g Q-tQ-im Q-im Q-im Q-n Q-m Q-p Q-c Q-g Q-gQ-m Q-m Q-m Q-m Q-n Q-p Q-c Q-g Q-gQ-p Q-p Q-p Q-p Q-p Q-p Q-c Q-g Q-gQ-c Q-c Q-c Q-c Q-c Q-c Q-n Q-g Q-gQ-g Q-g Q-g Q-g Q-c Q-g Q-g Q-g Q-gQ-t Q-t Q-t Q-g Q-g Q-g Q-g Q-g Q-gOnly the new triplets (Q-an, Q-im, Q-x) with Q-x in [Q-m .. Q-t] have to be checked. Gap (Q-an, Gap (Q-im, Q-x) = Gap (Q-im, Q-x)and Gap (Gap (Q-an, Q-im), Q-x) = Gap (Q-im, Q-x) ; so the associativity is verified.