# Lojban Wave Lessons/Quantifiers and negation

## Lesson 22: Quantifying sumti

Most other learning materials such as The Complete Lojban Language and Lojban for Beginners were written before the official adoptation "xorlo", a change in the rules about gadri definition and gadri quantification. The obsoleteness of some of the text in the older learning materials was a major cause for the motivation to write these lessons. Unfortunately for me, quantification of sumti can become a very complex topic when the implications of certain rules are discussed in detail. In order to fulfill the goal of this text being accurate enough to represent the official "gold standard" BPFK rules, this chapter was among the last ones finished and the ones most frequenty rewritten. I strongly encourage anyone who finds mistakes in this text to contact me in order for them to be corrected.

Having said that disclaimer, let's get started:

The first concept you should know about is "distributivity". In lesson fourteen I used the word "individuals" about a group of objects considered distributively. A group of objects considered distributively means that the selbri in question apply to each of the objects. This stands in contrast to non-distributivity (which masses have), in which the group has other properties than each of the individuals do. The distinction between distributivity (individual-like) and non-distributivity (mass-like) is of relevance when quantifying sumti.

Sometimes it's also mentioned how one sumti can distribute over another sumti, so I'll include this as well. What it means is that if sumti A stands in relation X to sumti B, with sumti A distributing over sumti B, then each A stands in relation X to B. Let's have an example in English:

 Three dogs bite two men.

If the dogs distribute over the men, then each of three dogs has bitten two men, meaning that between 2 and 6 different men was bitten (since one really unlucky man could have been bitten by all three dogs), whereas if the men distribute over then dogs, then two men were each bitten by tree dogs, fixing the number of men to 2, but allowing between 3 and 6 dogs.

When there can be any doubt as to which sumti distributes over which, the rule is that the first mentioned sumti always distributes over the last mentioned. This is irrespective of place structure, so if x1 and x2 are switched with se, x2, which is mentioned first, will distribute over x1.

Now, back to quantification. Let us first consider how one can quantify description sumti, which are sumti of the form GADRI BRIVLA. The number string which does the quantification can be placed before the gadri, in which case it is referred to as an outer quantifier, and it can be placed between the gadri and the brivla, in which case it's an inner quantifier. Any kind of number string can act as a quantifier.

The rules for how inner and outer quantifiers affects sumti depend on the kind of gadri which is used:

• lo and le. An inner quantifier tells us how many objects are being spoken of - how many objects are in the discourse total. If an outer quantifier is present, the sumti is distributed over that amount of objects. As stated earlier, if no outer quantifier is present, it's vague how many objects the selbri applies to (though not none), and whether it does so distributively or non-distributively. Examples are always a good idea, so here they are:

mu lo mu bakni cu se jirna - The inner quantifier of five tells us that we speak about five pieces of cattle, and the outer quantifier of five tells us that the selbri is true for each of the five. Therefore, it means "All the five cows had horns".

bakni = x1 is a cow/ox/cattle/calf etc of breed x2
jirna = x1 is the horn of x2 (metaphor: any pointed extremity)

What does the following bridi mean?

lo ru'urgubupu be li re pi ze mu cu jdima lo pa re sovda

ru'urgubupu = x1 is measures to be x2 British pounds (GBP)
jdima = x1 is the price of x2 to buyer x3 set by vendor x4
sovda = x1 is a gamete (egg/sperm) of x2

Answer: "Twelve eggs cost 2.75 British pounds" which, as the English translation, could mean both that they cost 2.75 each (distributively) or that all twelve together cost 2.75 (non-distributively)

so le ta pa pa ci'erkei cu clamau mi (notice that the ta goes before the inner quantifier)

ci'erkei = x1 plays game x2 governed by rules x3 interrelating game parts x4 {this is used to translate "play" in the sense "play a game" rather than, for instance "playing pretend" or "playing House"}
clamau = x1 is taller/longer than x2 in direction x3 my marigin x4

Answer: The inner states there are 11 players in the discourse, and the outer states that the selbri applies to nine of them distributively. Thus it means "Nine of the eleven players are taller than me"

There are a few points that needs to be raised regarding quantification of lo/le:

Using lo with an outer quantifier, "{number} lo {selbri}", works like "{number} {selbri}" in that it quantifies over individuals. Therefore, both ci gerku cu batci re nanmu and ci lo gerku cu batci re lo nanmu consider both the group of dogs and the group of men distributively. Therefore, each of the three dogs bit each of the two men, with six biting events in total.

({number} lo {selbri} is defined as {number} da poi me lo {selbri}, while {number} {selbri} is {number} da poi {selbri}. da and me will be explained later. Put simply, the lo version takes context into account more than the bare number version, which quantifies over anything and everything that fits in the selbri's x1.)

batci = x1 bites/pinches x2 at locus x3 using x4 as pinching tool.

Secondly: What if there is no outer quantifier? Does this mean that it is there, but it's elided? Nope. If there is any kind of outer quantifier, elided or not, it would force the sumti to be distributive, which would mean that neither lo nor le could be used with predicates that apply collectively. Therefore, if no outer quantifier is present, it's not only elided - it's simply not there. Sumti without an outer quantifier can be referred to as "constants".

Thirdly, it makes no sense to have an outer quantifier which is larger than the inner one. It means that the selbri holds true for more sumti than are present in the discourse - but since they appear in a bridi, they are part of the discourse. It's grammatical to do it, though.

Lastly, placing a lo or a le in front of a sumti is grammatical, if there is an inner quantifier present. "lo/le {number} {sumti}" is defined as "lo/le {number} me {sumti}".

So what would this mean? mi nelci lo mi briju kansa .i ku'i ci lo re mu ji'i ri cu lazni

briju = x1 is an office for worker x2 at location x3
kansa = x1 accompanies x2 in action/state/enterprise x3
lazni = x1 is lazy, avoiding work concerning x2

Answer: "I like my co-workers, but three out of about twenty five of them are lazy"

• la. An inner quantifier is grammatical if the name is a selbri - in this case, it's considered part of the name. An outer quantifier is used to quantify distributively over such individuals (much like lo/le) It's grammatical when placed in front of {number} {sumti}, in which case, the both the number and the sumti is considered the name.

Translate this: re la ci bargu cu jibni le mi zdani

Answer: Two "The Three Arches" are located close to my house" (Perhaps The Three Arches are a kind of restaurant?)

• loi and lei. An inner quantifier tells us how many members there are in the mass/masses in question. An outer quantifier quantifies distributively {!} over these masses

Notice here that while masses consist of a number of objects considered non-distributively, an outer quantifier always treats each of these masses as an individual.

When placed before a number string, then a sumti, loi/lei is defined as "lo gunma be lo/le {number} {sumti}" - "The mass consisting of the {number} of {sumti}".

Attempt to translate this: dei joi di'e gunma re loi bi valsi .i ca'e dei jai se jalge lo nu jetnu

gunma = x1 is a mass of the individuals x2
valsi = x1 is a word, meaning x2 in language x3
ca'e = Attitudinal: Evidential: I define
jetnu = x1 is true according to metaphysics/epistemology x2

Answer: "This very utterance, mixed together with the next one, forms a mass, consisting of two individual masses each of eight words. I define: This very utterance causes {it} to be true."

• lai. Much like la, an inner quantifier (when name is a selbri) is part of the name. An outer one quantifies distributively. Before a number+sumti, both the sumti and the number make up the name.

When a fraction is used as an outer quantifier to quantify loi, lei or lai, one speaks about only part of one mass (for instance, "half of the Johnsons" - pi mu lai .djansyn.).

• lo'i and le'i. An inner quantifier describes the amount of members of the set. An outer quantifies distributively over several of such sets. When placed before a number and a sumti, it's defined as "lo selcmi be lo/le {number} {sumti}" - "The set of {number} {sumti}".

Translate lo'i ro se cinki cu bramau la'a pa no no lo'i ro se bogykamju jutsi

cinki = x1 is an insect of species x2
la'a = Attitudinal: Discursive: Probably
bramau = x1 is bigger than x2 in dimension x3 by marigin x4
bogykamju = x1 is the spine of x2
jutsi = x1 is the species of genus x2, family x3 ... (open ended classification)

Answer: "The set of all the species of insects is probably bigger than one hundred sets of all species of vertebrates"

• la'i. As with lai

Like with the mass gadri, an outer quantifier before a set gadri enables one to speak about a fraction of a set. In front of a number and a sumti, it's defined as "lo selcmi be la {number} {sumti}" - "The set consisting of The {Number} {Sumti}" (considered a name)

• lo'e and le'e. Are for some reason not included in the currently accepted gadri proposal. If one were to extend the rules of another gadri, lo/le would probably be the best choice (since both operates with individuals rather than groups), and so one would expect the outer quantifier to force distributivity over the amount of typical/stereotypical things given by the inner quantifier.

When quantifying sumka'i representing several objects, it is useful to remember that they usually behave like lo-sumti. By definition, "{number} {sumti}" is defined as "{number} da poi ke'a me {sumti}". You will not be familiar with da until a few lessons later, so take it on faith that it means "something" in this context. Therefore, ci mi means "Two of those who belong to "us"".

Some important uses of quantification requires you to be quantify selbri or objects whose identity is unknown. This is done by "logically quantified variables". These, as well as how to quantify them will be covered in lessons twenty-seven.

Lastly, how can you quantify uncountable substances like sugar or water? One solution is to quantify it using inexact numbers. This use is vague, not only because the value of the number is vague, but also because it's not specified on what scale you're counting: The sugar could be considered a group of many crystals, counted one at a time, and the water could be quantified by the amounts of raindrops it took to make the body of water in question. While this way of counting is legitimate, it's not very exact and can easily confuse or mislead.

A way to be explicit about non-countability is to use the null operand tu'o as an inner quantifier.

tu'o = Null operand ( Ø ). Used in unary mekso.

This solution is elegant and intuitive, and also gives me an excuse to quote this horrifying, yet comical example from the original xorlo-proposal:

le nanmu cu se snuti .i ja'e bo lo tu'o gerku cu kuspe le klaji

snuti = x1 is an accident on the part of x2
ja'e = sumtcita: BAI: (from jalge): Bridi results in {sumti}
kuspe = x1 spans/extends over x2
klaji = x1 is a road/avenue/street at x2 accessing x3

What does it mean?

A second method of quantifying substances is to use the tenses ve'i, ve'a and ve'u as mentioned in lesson ten:

ti ve'i djacu - This is a small amount of water

djacu = x1 is an expanse of water/is made of water/contains water

Thirdly, of course, you could use a brivla to give an exact measurement:

le ta djacu cu ki'ogra be li re pi ki'o ki'o - "That water has a mass of 2.000 000 kilograms"

ki'ogra = x1 measures in mass x2 kilograms by standard x3

## Lesson 23: Negation

Sometimes, just saying what's the truth is not enough. Often, we want to specify what's not the truth, and we do this by using negation.

Negation in English mostly involves not, and is completely arbitrary and ambiguous. We, as Lojbanists, can't have that, of course, so Lojban contains an elegant and unambigious system for negating. What will be presented here are the official gold-standard rules. Disapproval of these "golden rules" concerning na is growing, and there is disagreement about what rule set should replace it. For now, I will stick with the official rules, and therefore, so will you, dear reader.

The first you need to know about is bridi negation, so called because it negates the bridi it's in, saying it's not true. The way to negate a bridi is to place na first in the sentence with a ku after it, or just before the selbri.

speni = x1 is married to x2 under convention x3

Thus: na ku le mi speni cu ninmu states that My spouse is not a woman. It states nothing about what my wife is, or if I even have a wife. It only states that I do not have a wife who is also a woman. This has an important implication: If the negation of a bridi is false, the bridi must be true: na ku le mi speni cu na ninmu must mean that I have both a spouse, and that she is a she.

It is possible to use bridi negations in all bridi, even the implicit bridi of descriptive sumti. lo na prenu can refer to anything non-human, whether it be a sphinx, a baseball or the property of appropriateness.

bau = sumtcita, from bangu: using the language of {sumti}
se ja'e = sumtcita, from se jalge: because of {sumti}

Often when using na, it's a problem that it negates the entire bridi. If I say mi na sutra tavla bau le glibau se ja'e le nu mi dotco, I end up negating too much, and it is not clear that I wanted to only negate that I speak fast. The sentence could suggest that I in fact speak fast because of some other reason, for instance that I speak fast in French because I'm German. To express the sentence more precisely, I need to only negate that I speak fast, and not the other things.

To only negate part of a bridi, na ku can be moved around the bridi and placed anywhere a sumti can go. It then negates any sumti, selbri and sumtcita placed after it. When placed immediately before the selbri, the ku can be elided.

Moving na ku from the left end of the bridi and rightwards effects any quantifiers in a certain way, as can be seen by this example:

There are forces within the Lojban community who, perhaps rightly, think that there is no good reason that a na placed before a selbri negates the entire bridi, whereas a na ku any other place negates only what is trailing behind it. However, in these lessons you will be taught what is still the official stance, namely that na before the selbri negates the bridi.

The use of na ku is exemplified with the following examples.

na ku ro remna cu verba It's not true that: All humans are children

su'o remna na ku cu verba For at least one human it's not true that: It's a child. See that the na ku is placed before cu, since a sumti can go only before, not after the cu. Had I only used na, it would have to go after cu - but that would have negated the entire bridi, meaning "It's not true that: At least one human is a child".

When the na ku is moved rightwards, any quantifier is inverted - that is: ro is turned into su'o. This is, of course, only if the meaning of the bridi has to be preserved. This means that when the na ku is placed at the end of the bridi, only the selbri is negated but all the sumti and sumtcita are preserved, as can be seen by these three identical bridi:

ckule = x1 is a school at location x2 teaching x3 to students x4 and operated by x5

na ku ro verba cu ve ckule fo su'o ckuleIt's not true that all children are students in a school.

su'o verba cu ve ckule na ku fo su'o ckuleSome children are students in not a single school.

su'o verba cu ve ckule fo ro ckule na kuSome children are for all schools not students in them.

The opposite of na is ja'a. This is barely ever used, since it is default in most bridi. One exception is repeated bridi (next lesson). Sometimes it's used to put more weight on the truth of the bridi, as in la .bab. ja'a melbi = "Bob is indeed beautiful".

While the mechanism of na ku resembles natural language negation, it can be difficult to keep track of exactly what is negated and how that affects the bridi. For that reason, the construct na ku is rarely seen anywhere other than the beginning of a bridi. In most cases where more specific negation is needed people resort to a different method. This method, called scalar negation, is an elegant and intuitive tool. Using it, you effect only the selbri, since the words used in scalar negation binds to the selbri much like the word se.

The name scalar negation is derived from the fact that the words which bind to the selbri can be placed along a scale from affirmation over negation and to stating that the opposite case is true:

These words are not negators in the same sense as na. They do not state that a bridi is false, but makes a positive statement that a bridi is true – the same bridi, but with a different selbri. This distinction is mostly academic, though. If, for example, I state that mi na'e se nelci "I am non-liked", I actually state that some selbri applies to me, which is also on a relevant scale with the selbri nelci. Most of the time, we assume a scale where the positions are mutually exclusive (like love-like-dislike-hate), so mi na'e se nelci implies mi na se nelci Therefore, the words no'e and to'e should only be used when the selbri is placed on some obvious scale: le mi speni cu to'e melbiMy spouse is ugly makes sense, since we immediately know what the opposite of beautiful is, while mi klama le mi to'e zdaniI go to my opposite thing of home, while grammatical, leaves the listener guessing what the speaker's opposite-home is and should be avoided. So how can you negate only the selbri without also implying that the selbri is correct at some other position on a truth-scale? Simple: Moving the na ku to the rightmost end of the bridi, as demonstrated a few lines above. This feature is very useful. Some Lojbanists prefer to prefix the rafsi nar (the rafsi of na) in front of the selbri - this has the same effect, but I advise against it, because it makes familiar brivla seem alien, and it's harder to understand when spoken casually. If a situation arises where you need to negate only the selbri, but want it to be clear before the end of the bridi, the experimental cmavo na'ei, which grammatically works like na'e, can be used na'ei: Negates the following selbri only Try to translate these sentences: My spouse is not a woman (meaning that he is a male) Answer: le mi speni cu na'e / to'e ninmu. Using scalar negation here implies that he exists, which na did not.My spouse is not really a woman Answer: le mi speni cu no'e ninmu. The scale here is presumed to be from woman to man.I don't speak fast in English because I'm German Answer: mi na'e sutra tavla bau le glibau se ja'e le nu mi dotco Also, note that whenever these words are used together with a tanru, they only affect the leftmost selbri. In order to make it bind to the whole tanru or parts of the tanru, the usual tanru-grouping words can be used. Try to say I sell something which is not yellow homes using the tanru pelxu zdani vecnu Answer: mi na'e ke pelxu zdani ke'e vecnu or mi na'e pelxu bo zdani vecnu When attempting to answer: Is the king of the USA fat?, all of these negations fail. While it's technically correct to negate it with na, since it makes no assumptions of that is true, it's mildly misleading since it could lead the listener to believe there is a king of the USA. For these scenarios, there is a metalinguistic negator, na'i.
na'i = metalinguistic negator. Something is wrong with assigning a truth value to the bridi.
Because na'i can be needed anywhere it has been given the grammar of the attitudinals, which means it can appear anywhere, and it attaches to the previous word or construct.
palci = x1 is evil by standard x2
 le na'i pu te zukte be le skami cu palciThe sought goal {mistake!} of the computer was evil, probably protests that computers can seek a goal volitionally.

Since this is a lesson on negation, I believe the word nai deserves a short mention. It is used to negate minor grammatical constructs, and can be used in combination with attitudinals, all sumtcita including tenses, vocatives and logical connectives. The rules for negating using nai depend on the construct, and so the effect of nai has been discussed when mentioning the construct themselves. The exception is sumtcita, where the rules for negation are more complex, and will not be discussed here.

Note: At the time of writing, it has been proposed to move nai to the selma'o CAI, which means the semantics of nai depend on which selma'o it follows.

## Lesson 24: brika'i/pro-bridi and ko'a

If I say that I'm called Mikhail, zo .mikail. cmene mi, and you have to say the exact same bridi, what would that be? One of the many answers is do se cmene zo .mikail.. For the bridi to be the same, you have to replace mi with do, and it doesn't matter which if you say the bridi with the se-converted selbri or not. This is because a bridi is not the words which express it – a bridi is an idea, an abstract proposition. The word mi when I say it and the word do when you do refers to the same sumti, so the two bridi are identical.

This lesson is on brika'i, the bridi equivalent of sumka'i. They are word which represent entire bridi. Here it is important to remember that a bridi consists only of sumti and the things which contain the sumti, selbri and sumtcita. Neither attitudinals, nor the semantic layer of ko or ma are part of the bridi proper, and so these are not represented by a brika'i.

There are much fewer brika'i than there are sumka'i. We will begin by going through some of the words in the most used series, called GOhA:

go'u = Repeats remote past bridi
go'a = Repeats past bridi
go'e = Repeats next-to-last bridi
go'i = Repeats last mentioned bridi
go'o = Repeats future bridi
nei = Repeats current bridi
no'a = Repeats outer bridi

Some of the GOhA-brika'i. Notice the familiar i, a, u-pattern for close in past, medium in past and distant in past.

These are very much like the sumka'i ri, ra and ru. They can only refer to main bridi of jufra, and not those contained in relative phrases or description sumti. The main bridi can contain a relative phrase, of course, but a brika'i can never be used to refer to only the relative phrase.

A GOhA acts grammatically much like a selbri, any construct which can apply to selbri can also apply to these. The place structure of a GOhA is the same as that of the bridi it represents, and the sumti are by default the same as in the bridi it represents. Filling the sumti places of a GOhA explicitly overwrites the sumti of the bridi it represents. Contrast:

A: mi citka lo plise B: go'iI eat an apple. You do. with

A: mi citka lo plise B: mi go'iI eat an apple. I do, too.

These brika'i are very useful when answering a question with xu:

A: xu do nelci le mi speni B: go'i / na go'iDo you like my wife? Yes./No.. The xu, being an attitudinal, is not copied.

When repeating bridi negated by na, that is: Bridi where na is placed in the prenex (lesson twenty-seven), in the beginning of the bridi or right before the selbri, the rules for copying over na are different from what one might expect. Any na is copied over, but any additional na in the brika'i replaces the first na. Let me show you with an example:

A: mi na citka lo plise

B: mi go'i = mi na citka lo plise

C: mi na go'i = mi na citka lo plise

D: mi na na go'i = mi citka lo plise = mi ja'a go'i

nei and no'a are not used much, except for mind-breaking purposes, which is making up bridi which are hard to parse, like dei na se du'u le no'a la'e le nei. Since nei repeats the current outer bridi, however, le nei can be used to refer to the x1 of the current outer bridi, le se nei the x2 and so on.

When using brika'i, one must always be wary of copying sentences with the personal sumka'i like mi, do, ma'a ect, and be careful not to repeat them when they are in the wrong contect, as shown in the two examples with apple eating above. Instead of replacing them one by one, though, the word ra'o anywhere in the bridi updates the personal sumka'i so that they apply for the speaker's perspective:

A: mi do prami B: mi do go'i is equivalent to A: mi do prami B: go'i ra'o

ra'o = Update all personal sumka'i so that they now fit the speaker's point of view.

The only other series of brika'i are very easy to remember:

broda = Bridi variable 1
brode = Bridi variable 2
brodi = Bridi variable 3
brodo = Bridi variable 4
brodu = Bridi variable 5
cei = Define bridi variable (not a brika'i and not in BRODA)

The first five are just five instances of the same word. They can be used as shortcuts to bridi. After saying a bridi, saying cei broda defines that bridi as broda, and broda can then be used as brika'i for that bridi in the following conversation.

While we're at it, there is an analogous series of sumka'i, which probably does not belong in this lesson, but here they are anyway:

 ko'a Sumti variable 1 fo'a Sumti variable 6 ko'e Sumti variable 2 fo'e Sumti variable 7 ko'i Sumti variable 3 fo'i Sumti variable 8 ko'o Sumti variable 4 fo'o Sumti variable 9 ko'u Sumti variable 5 fo'u Sumti variable 10

as well as the cei-equivalent for this series:

goi = Define sumti variable

These are used like the brika'i-series. Just place, for instance, goi ko'u after a sumti, and that sumti can be referred to by ko'u.

Strangely, these series are rarely used for their intended purpose. They are, however, used as arbitrary selbri and sumti in example texts, where broda and brode mean "any selbri A" and "any selbri B" and similarly for ko'a and friends:

So, is it true that the truth conditions of ko'a ko'e broda na ku are always the same as na ku ko'a ko'e broda? Nope, it isn't.