# Lojban Wave Lessons/22

 Lojban Wave Lessons: Foreword | ← Lesson 21 | Lesson 22 | Lesson 23 →

## 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?