Unit 8. Numerals II
Cantor illustrated the concept of infinity for his students by telling them that there was once a man who had a hotel with an infinite number of rooms, and the hotel was fully occupied. Then one more guest arrived. So the owner moved the guest in room number one into room number two; the guest in room number two into number three; the guest in three into room four, and so on. In that way room number one became vacant for the new guest. What delights me about this story is that everyone involved, the guests and the owner, accept it as perfectly in order to carry out an infinite number of operations so that one guest can have peace and quiet in a room of his own. That is a great tribute to solitude.
Translating English numerals
As we have seen in the previous unit, the cardinal numerals (‘one, two’), as well as indefinite and weighting numerals used in the same way, work like adjectives. They form brackets or coordinations, including – especially the weighting numerals, as might be expected – partitive ones.
|roomÌ trÌy. ⇔ trÌ roomÌy.||rooms, (which are) three individuals||three rooms|
|room-acc1 three-acc-acc2. ⇔ three-acc1 room-acc-acc2.|
|trÌ roomÌyn.||three individuals from the set of rooms||three of the rooms|
|jnÌ roomÌyn.||everything from the set of rooms; everything of the type ‘room’||all rooms|
The latter phrase is an example of a partitive construction that describes all items from its set, a possibility left open by the partitive cases’ definition. The corresponding cumulative bracket would mean ‘all things, which are rooms’ or ‘everything, which is a room’, i.e. there are no other things than rooms.
The multiplicative numerals (‘once, twice’) work like adverbally used adjectives: they form factive brackets.
|sneezeà dwÌa.||two individuals of sneezing||She sneezes twice.|
As an individual need not necessarily be coherent, two individuals of a continuous action such as ‘eat’ or ‘sleep’ do not necessarily mean eating or sleeping twice in the sense of two temporally separate actions. But then, even in English we speak of sleeping twice if we are thinking of two nights, even if we wake up in the middle of the first night and then go to sleep again. It’s not a question of continuity but of what is perceived as a unit (an individual).
Distributive numerals (‘one each, in pairs’) are yet another case for the indefinite numeral eachÌ., this time with a cumulative bracket. To express ‘each of something’, we also need the familiar partitive bracket. An inversion gets the sentence into the desired shape.
|eachÌ hostìy dwÌi.||Each [pair] is two guests.||pairs of guests, two guests each|
|each-acc1 host-dat-acc2 two-acc-dat3.|
|eachÌ hostiÌn giveìy roomÌy rÌy.||Each of the guests is [one] given / the recipient of giving one room.|
|each-acc1 host-dat-partacc2 give-dat-acc2 room-acc-acc3 one-acc-acc4.|
|⇔ giveà eachÌi hostiýn roomÌy rÌy.||Each of the guests gets one room. The guests get one room each.|
|give-fact1 each-acc-dat2 host-dat-partacc3 room-acc-acc2 one-acc-acc3.|
The inverted sentence has the advantage that eachÌ. is left with a single object – the partitive –, so we can form a compound. (This is as if we had derived the reflexive expression speakà eachÌi vèyn. ‘They are talking to themselves’ from the ‘uninverted’ form eachÌ speakeÌn ìy.)
|giveà hostieachiì roomÌy rÌy.||The guests get one room each.|
|give-fact1 host-dat-each-dat-dat2 room-acc-acc2 one-acc-acc3.|
Adjectives of possession containing a numeral (‘two-windowed, four-legged’) need similar constructions, as roomÌ windowÙy twoÌy. would mean ‘one or several rooms with two windows (altogether)’. The same is true of related nouns and such.
|roomyRÌ windowÙy dwÌy.||Each of the rooms has two windows.||two-windowed room(s)|
|room-acc-each-acc1 window-ben-acc2 two-acc-acc3.|
|RÌ dnÌy gwÌnqu.||Each [of the individuals] has four legs (tool noun).||quadruped(s)|
|each-acc1 walk-acc-acc2 four-partacc-ins3.|
|groupyRÌ qÌfi.||Each of the groups is made of seven individuals.||group(s) of seven|
|workà dayilRÌoR.||Each of the days is an episode for working.||days’ work|
‘day’ defines a certain quantity and is therefore multiplied with ‘each’ by an epenthetic consecutive; see Measuring in unit 12 and Units of measurement in the appendix for more on this.
Recall that pnà. ‘make five individuals’ implies making them one after the other. Consequently, ‘the fifth’ is the (temporal or spatial) end point of making five: either the egressive (iR, closing time, temporal end point) or the illative (ir, spatial end point) case is appropriate here.
|pnìR.||the (temporal) end point of making five individuals||the fifth|
|pnìr.||the (spatial) end point of making five individuals|
|mìR rynjÌ pnÌny.||(mà.-desorption)||the twenty-first|
|make-egr1 16-partacc-acc2 five-partacc-acc2.|
‘the fifth room’ is the end point of making five rooms. Looking at the cardinal numerals, we see that the rooms are the accusative object of ‘five’. An egressive or illative bracket has a different meaning.
|pnìr roomÌy.||the fifth room|
|pnìr roomÌyn.||the fifth of the rooms|
|pnìr roomÌir.||the fifth thing, a room|
|dwìR meetày.||I met her for the second time.|
|mlìR meetày.||I met her again.|
|rìR crystalàiR iceÌi.||At first, ice crystals form.|
|one-egr1 crystal-fact-egr2 ice-acc-dat3.|
Repetitive ordinal numbers (‘every sixth’) are constructed similarly to distributives and fractions.
|eachÌ rìRyn. ⇒ riReachìR swÌhyn.||each of the first [individuals] per six; the first of six each||every sixth|
|each-acc1 one-egr-partacc2. ⇒ one-egr-each-egr1 six-acc-partacc2.|
Of course we could also translate this as ‘the second of six each’ or ‘the sixth of six each’, but the difference is only relevant for rows with a defined starting point, i.e. to distinguish ‘the 6th, 12th, 18th, …’ from ‘the 1st, 7th, 13th, …’ and so on.
To translate numerals describing composition (‘twofold/double/binary, threefold/triple/ternary’), a cardinal numeral (a bracket or coordination) will often suffice.
|wallÌ dwÌy.||a twofold/double wall, two walls|
|sneezeà dwÌa.||a twofold/double sneeze. She sneezes twice.|
To highlight the integritive (twofold, i.e. one thing in two parts) or the composite aspect (double/binary, i.e. two parts forming a whole), we use the grouping numerals ensembleÌ. or groupÌ.. The difference between integritive and composite is essentially in the placement of the dative. The difference between uncompounded and compounded phrasings is in the importance of the numeral. (Compare Grammatical number below.)
|wallÌ ensembleÌy dwÌi. ⇔ ⇒ wallÌ ensembleidwÌy.||a wall, an ensemble of two||a wall of two components ⇒ a twofold wall|
|wall-acc1 ensemble-acc-acc2 two-acc-dat3. ⇔ ⇒ wall-acc1 ensemble-dat-two-acc-acc2.|
|wallÌ dwÌy ensembleìy. ⇒ wallÌ ensembleidwìy.||walls, two forming an ensemble||two walls in an ensemble ⇒ a double wall|
|wall-acc1 two-acc-acc2 ensemble-dat-acc3. ⇒ wall-acc1 ensemble-dat-two-dat-acc2.|
Technically, the inversion and compoundings above are achieved by first removing ‘wall’ and reintroducing it afterwards; this is possible with brackets.
‘two/three sorts/kinds/types of’ is translated with the grouping numeral sortÌ.. The things which are sorted into a number of kinds of course want the dative. Again, the difference between uncompounded and compounded phrasings is in the importance of the numeral.
|sortÌ swyhÌ snÌwi. ⇔ ⇒ sortyswÌh snÌwi.||six sorts of snow|
|sort-acc1 six-acc-acc2 snow-acc-dat2. ⇔ ⇒ sort-acc-six-acc1 snow-acc-dat2.|
In counting, we use the inner factive.
|rà. dwà. trà. —||it has become one individual, they have become two individuals, they have become three individuals, …||one, two, three, …|
|one-fact1. two-fact1. three-fact1. ….|
Room number eight is the room having an ‘8’, the one marked with an ‘8’; so what we need is an adjective of possession. To distinguish this from a room having eight things, we enclose the numeral in quotes: <8>.
|roomÌ ‹8›Ùy.||the room with an ‘8’||room (number) eight|
For consecutively numbered rooms, ordinal numerals (‘the eighth room’) are an alternative.
Digit sequences such as numbers with many positions after the decimal point (including recurring ones) – or, say, credit card numbers – are also read as quoted digits.
|‹—035266A15A,1Ì›.||1.A51A662530…hex = 1.6449340668…|
|‹— rÌ. fiveÌ. tenÌ. pointÌ. rÌ.›|
|“— one-acc1. five-acc1. ten-acc1. point-acc1. one-acc1.”|
Grammatical number and collective nouns
We can get close to the idea of grammatical number (singular, plural, dual and others) by demoting a numeral to a compound modifier. Some examples of compounding have already been shown in the chapters above; to grasp the difference to uncompounded constructions, it may help to think of them as weaker forms of numerals, just as ‘a room’ is weaker than ‘one room’ with respect to the number ‘one’.
Don’t be tempted to translate grammatical number like this on principle; in most situations, the number of things is irrelevant.
Similar compounds, this time with the grouping numeral ensembleÌ., form collective nouns.
|noteyensembleÌ.||notes forming an ensemble||melody|
|ensembleinoteÌ.||an ensemble of notes|
The first possibility is parallel to the translation of grammatical number and thus highlights the notes at the expense of the ensemble. Its disadvantage is that objects of the compound refer to the notes, not the melody, so adding an adjective does not yield ‘a beautiful melody’ but ‘a melody of beautiful notes’. The second possibility makes it clear that there is more than notes; the ensemble is highlighted. The fact that the notes’ inner case is lost is irrelevant here, but can cause problems with, say, an ensemble of gifts. If nothing else helps, we can always resort to an uncompounded form.
Other collective nouns include ‘team, flock, mankind, lexicon (vocabulary)’.
Strengthening, weakening and opposition
Adjectival verbs generally denote living beings or things that are on the ‘positive’ side of some scale; for example, wiseÌ. is on the positive side of a wisdom scale, and beautifulÌ. on the positive side of a beauty scale. Compounds with the weighting numerals 5/8Ì. 3/4Ì. 7/8Ì. can be used to convey a more specific positive meaning, and 3/8Ì. 1/4Ì. 1/8Ì. indicate the less positive meanings ‘of little wisdom or beauty’, ‘slow’, and the like. The negator nÌ. marks the zero point, and kÌ. is on the negative side: wiseilnÌ. ‘unwise, someone without wisdom’ is definitely more impolite than wiseil1/4Ì., and wiseilkÌ. is downright rude. fillilnì. means ‘not filled, empty’, while fastilnÌ. does not mean ‘slow’ but ‘not proceeding’. amountÌ. marks an arbitrary point on the scale, which we will need in unit 11.
The epenthetic consecutive is used because the numerals weight the abstract nouns ‘wisdom, speed’, as explained in the chapter on negators – wisey1/4Ì., by contrast, means ‘a few wise ones’ (or, more accurately, ‘a bit, which is a wise one or are wise ones’). Again, recall the negative numbers from the previous unit, which are also formed with inner consecutives.
Uncompounded adjectives are another issue handled by pragmatics: literally, fastÌ. just describes someone who is moving, speaking or otherwise acting, however slowly, but is usually taken to mean someone acting at a speed relevant to the context; greenÌ. is usually understood as something that is completely or predominantly green, and not just greenish. This issue can also be understood as leaving out the weighting numeral if the weight is clear from context.
There is no generally applicable definition of the ‘positive’ side of an adjective; it is part of its lexical meaning. For example, there is no compelling reason why rightÌ. ‘right’ is expressed by an uncompounded word and rightilkÌ. ‘left’ is its opposition, and not the other way round. (See unit 12, Temporal and spatial verbs.)
Adjectives are not the only words that can be strengthened or weakened. Weighted actions often, but not necessarily, have an epenthetic factive.
|jnà làxty. ⇒ laxtjnà.||want at all costs|
|1/1-fact1 want-fact-acc2. ⇒ want-fact-1/1-fact1.|
|crà speakày. ⇒ speakacrà.||speak a bit|
|1/4-fact1 speak-fact-acc2. ⇒ speak-fact-1/4-fact1.|
|càwb liveày. ⇒ liveacèwb.||hardly alive|
|1/8-fact1 live-fact-acc2. ⇒ live-fact-1/8-nom1.|
|nà liveày. ⇒ liveanè.||dead|
|not-fact1 live-fact-acc2. ⇒ live-fact-not-nom1.|
|àv jnÌy. ⇔ jnà Ìvy. ⇒ yvjnà.||eat all||eat up|
|eat-fact1 1/1-acc-acc2. ⇔ 1/1-fact1 eat-acc-acc2. ⇒ eat-acc-1/1-fact1.|
By the way: ‘He eats a lot (= a large amount)’ has the weighting numeral in the accusative, ‘He eats a lot (= He does a lot of eating)’ in the factive.
A grammatical form expressing a stronger action is generally called intensive aspect, one expressing a weaker action is called diminutive aspect. ‘eat up’, expressing a completed action, is the Lemizh equivalent of the completive aspect.
The sentence ‘She doesn’t eat it because of the vitamins’, with stress on the vitamins, is usually understood as ‘She eats it, but not because of the vitamins’. The seemingly obvious translation eatanà vitaminÌOl. negates the whole sentence and consequently does not claim anything about the eating. Rather, it says ‘It is not true that she eats it because of the vitamins’, effectively meaning ‘She might eat it, but if so, it’s not because of the vitamins’. (While nà1 eataÌ2 vitaminÌOl2., with the vitamins as an object of the negator, means ‘She doesn’t eat it, because of the vitamins’.) Instead, we need to negate the vitamins, yielding nà vitaminÌy. ⇒ vitaminynÌ. ‘something that is not a vitamin; non-vitamins’. Again, an epenthetic consecutive would be an alternative, but this time the concrete accusative seems more appropriate.
|àv vitaminynÌOl.||She eats it because of non-vitamins.||She doesn’t eat it because of the vitamins.|
|àv vitaminyÒl tasteynÌOl.||(coordination)||She eats it because of the vitamins, not the taste.|
|eat-fact1 vitamin-acc-psu2 taste-acc-not-acc-psu2.|
|Other modifiers and epenthetic cases also occur:|
|àv vitaminaRRÌbvOl.||The times of vitamin-making are 3⁄8, i.e. she eats it because of something that is sometimes vitamins.||She sometimes eats it because of the vitamins.|
She eats it, sometimes because of the vitamins.
The same type of construction, but with outer cases other than the persuasive, is found in the following sentences:
- ‘He didn’t give it to his brother [but to someone else].’ (compound with nà., dat)
- ‘I won’t come today [but tomorrow].’ (nà., temp)
- ‘She came without the mechanic.’ (nà., com)
- ‘I’m not paying you to ask questions.’ (nà., fin)
- ‘I can manage with just the tiniest bit of help.’ (càwb. ‘1⁄8’, ins)
- ‘Not many arrows hit the target.’ (crà. ‘1⁄4’, acc – don’t use a negation of dmà. ‘3⁄4’)
Likewise, ‘someone/something else’ and ‘different’ are negations of the definite pronoun, not of the whole sentence.
|giveà tynÌi.||He gave it to someone else.|
|drinkà glassÌer tynÌy.||She is drinking from a different glass.|
|drink-fact1 glass-acc-ela2 this-acc-not-acc-acc3.|
Objects can also be modified with modal verbs and adverbs (‘She should eat it because of the vitamins’, ‘Maybe she eats it because of the vitamins’, unit 13) or ‘that’-clauses (‘They say that she eats it because of the vitamins’, unit 15).
Modified objects cannot express that she eats it partly because of the vitamins, as this is the domain of the inner partitive. We can use the partitive alone or combine it with weighting numerals for specification (‘mostly, marginally’). These constructions incidentally illustrate the subtleties of cumulative vs. partitive brackets with weighting numerals.
|àv vitaminynÒl tasteÌnOl.||She partly eats it because of the vitamins, partly because of the taste.|
She eats it because of the vitamins and the taste.
|eat-fact1 vitamin-partacc-psu2 taste-partacc-psu2.|
|àv 3/4ÌnOl vitaminÌy.||Much of the reason for eating is vitamins.||She mostly eats it because of the vitamins.|
She eats it, mostly because of the vitamins.
|eat-fact1 3/4-partacc-psu2 vitamin-acc-acc3.|
The second example might look like it would mean ‘She eats it because of many vitamins, among other things’. However, the inner partitive indicates that 3/4Ì. ‘much’ is taken from the set of reasons for eating, and the cumulative bracket with the vitamins equates ‘much from the reason for eating’ with the vitamins.
Other examples of partitive modifications include:
- ‘I wasn’t speaking about you for the most part’ (1/4à., acc)
- ‘He is mostly eating fish’ (3/4à., acc)
As you know, an object in an outer partitive case indicates a part of something in the sense of one or some of a set of things. But what about a fraction of a whole, as in ‘I see part of the hotel’? While what I see is not taken from a set of hotels, it is taken from a set of things (walls, windows, roofs, chimneys) that, taken together, form a hotel. So we have to use groupì hotelÌy. ‘things grouped to form a hotel’, the outer accusative equating the hotel with the group (as opposed to the grouped things). We can also compound: inversion yields hotelÌ groupÌy. ‘a hotel, which is a group’ ⇒ groupyhotelì. (demoting the hotel to a modifier and losing its inner case, which, as with collective nouns, can be ambiguous). Both ways, we need an outer partitive to specify that we are only talking about some of these grouped things.
|dmàt qmìen hotelÌy.||I see some of the things grouped to form a hotel.||I see part of the hotel.|
|see-fact1 group-dat-partnom2 hotel-acc-acc3.|
|dmàt qmyhotelìen.||I see a hotel-part.|