Lojban Wave Lessons/30

Lesson 30: Semantics of functions

Functions are a group of two-three types of abstractions. The term's not official, but I'll use it here anyway.

The definition of functions is closely related to the neat little word ce'u. ce'u is a sumka'i which fills one sumti place. It's only found usage inside abstractions which are also functions. All functions can have at least one ce'u somewhere in the abstraction - that's what makes them functions. The ce'u can be elided, in which case it's most often assumed to fill the first elided sumti place of the function, unless context provides a more reasonable alternative.

What does it actually do? Let's have a look at its definition:

ce'u = Pseudo-quantifier binding a variable within an abstraction that represents an open place.

Well, that wasn't very helpful, so let me try explaining it with another approach:

Putting ce'u in a sumti place leaves the sumti place empty. The place is not erased, like if you fill it with zi'o, but the place is not filled with anything - not a specific thing, not a zu'i, not a zo'e, nothing. In that manner, the empty sumti places are reminiscent of the x1, x2, and x3's we put in the sumti places of English definitions of brivla - marking "This is where something else can be put".

Thus mi citka lo ti badna is "I eat this banana", but mi citka ce'u is "I eat X".

Of course, "I eat X" is meaningless unless that X is filled by something, and indeed the sentence mi citka ce'u is senseless in Lojban as well.

In order to put it to use, we need a function abstraction. We'll begin with the most often-used: The selbri abstraction ka. Let's see its official gloss:

ka = Property/quality abstractor (-ness); x₁ is quality/property exhibited by BRIDI.

Under the understanding which I will teach, this gloss is mildly misleading. Instead, ka should probably be glossed such:

ka = Predicate/selbri abstractor: x₁ is the predicate/selbri of BRIDI (needs at least one open variable i.e. a "ce'u")

Using a selbri abstraction, "I eat X" can make sense, as in the following example:

ckaji = x₁ is characterized by selbri x₂

lo ti badna cu ckaji lo ka mi citka ce'u - "This banana is characterized by the selbri: "I eat X"", which may be rephrased as "This banana fits the selbri: "Being eaten by me"", which is of course equivalent to mi citka lo ti badna - "I eat this banana".

For the statement to make sense, the sumti place held open by ce'u usually, but not always, must be filled by something. The main selbri of the statement, in this case ckaji, gives us a clue how to fill the open sumti place. Such selbri almost always fill it with a sumti from the main selbri. How ce'u is given a non-zero value has been a subject of minor debate in Lojbanistan, but the issue is more or less settled: ce'u keeps a sumti place open, and the main selbri then fills it with something, and what fills the place depends on the selbri in question.

Though it often is, the ce'u place need not always be filled by the selbri in order for the abstraction to make sense: On its own, lo ka ce'u te vecnu lo finpe means: "buying a fish", or "to buy a fish". This can be used in a sentence without the selbri filling the ‘’ce’u’’ in:

lo se lisri cu srana lo ka ce'u te vecnu lo finpe - "The plot is about buying a fish". Here, srana does not apply anything to the ce'u-place, and the abstraction is instead seen as the selbri on its own.

An alternative way of explaining ce'u is by regarding the word as representing variables in a lambda function. For instance, consider the sentence:

la .alis. cu djica lo ka ce'u te vecnu lo finpe - "Alice wants to buy a fish"

Here, the first argument of djica is the one who wants something, namely Alice. The second argument is the selbri that Alice wants to fulfill: Buying a fish.

We can view ce'u as a free variable, which then becomes bound by a lambda abstraction, namely ka. Now, ka ce'u terve'u lo finpe can be seen as a lambda function:

\ x -> te vecnu(x,lo finpe,zo'e,zo'e),

and in this case djica supplies the lambda function with Alice.

Lambdas can be stored, allowing them to be passed around and use them in various situations:

ca'e ko'a ka ce'u dansu .i mi ko'a ckaji .i do ko'a djica .i ma'a ko'a kakne - It is dancing. I am doing it. You want it. Everyone can do it."

Now, using ka, you can correctly phrase "I can run on my arms". How?

Answer: mi kakne lo ka {ce'u} bajra fi lo mi birka

A lot of often-used gismu take selbri as one of their sumti, which means lo ka is used quite often. A few notable examples are troci, kakne, djica, zukte, snada and fraxu:

lo xasli na’e kakne lo ka silcu la'e la'oi X-files - "The donkey cannot whistle the X-files song"

.e'o ko lo jai se zgike cu fraxu lo ka darxi lo damri ca lo nu do sipna - "Please forgive the musician for striking the drum when you were sleeping!"

At least one selbri can fill two ce'u within a ka-abstraction, namely ‘’simxu’’. What does the following jufra mean?

mi lo pampe'o cu simxu lo ka {ce'u ce'u} gletu

Answer: Me and my lover have sex with each other mutually"

Of course, the ce'u need not be placed in the beginning of the ka-abstraction, though it is by default. One could very well speak of:

lo ka la .bab. melbi ce'u - "The selbri of: "Bob is beautiful according to X"", or in other words: "Thinking that Bob is beautiful".

Indeed, moving the ce'u around in an function creates very different meanings:

lo ka ce'u panzi la .maik. - "The selbri: "X is a child of Mike"" = "Being Mike's child", versus

lo ka la .maik. panzi ce'u - "The selbri: "Mike is a child of X"" = "Being the parent of Mike".

One could even imagine a statement in where the ce'u is placed in a very unconventional place, that nonetheless is quite intuitive:

mi .e nai do ckaji lo ka lo bruna cu jbocre, wherein the ce'u is elided, but most probably hiding in lo bruna be ce'u, therefore meaning "I and not you is characterized by the selbri: "The brother of X is good at Lojban"", which is the same as "I have a brother who's good at Lojban, but you don't".

One can make a function, like a "ka"-abstraction, and fill all sumti places, leaving no place for a ce'u. The resulting bridi are weird:

mi kakne lo ka mi merko lo mi bangu - "I can my language is American". This is clearly a type error. Some people regard functions without any ce'u to be equivalent to bridi abstractions, so that:

mi krici lo ka mi vrude la cevni is the same as mi krici lo du'u mi vrude la cevni - "I believe that I am good in the eyes of God", and is just as good a sentence in Lojban as its translation is in English. In my opinion, one should refrain from using any of the function abstractors if one doesn't want to use a function. If you mean du'u, use du'u.

The other abstractor which clearly can provide a function is ni. Like ka, a ce'u can be placed in a ni abstraction, but unlike with ‘’ka’’, using a ‘’ce’u’’ with ‘’ni’’ is not mandatory. Thus, if no ce'u is placed in a ni-abstraction, one cannot assume that it's elided - it might simply not be there. If the main selbri is not one which clearly tells us how to fill a ce'u-place, such as zmadu or mleca, there's probably no ce'u at all.

In all other aspects, the way ce'u works within the abstraction is just like ka, so the difference is purely semantical. Whereas ka creates a selbri, ni creates an amount. Here's the definition of the word:

ni = Amount abstraction: x₁ is the amount of BRIDI on scale x₂

Being familiar with ka, the usage of ni should be straightforward:

mi zmadu do lo ni {ce'u} xekri - "I exceed you in amount: "X is black"", or: "I'm blacker than you." As stated in lesson twenty-eight, all agree that this makes total sense because the brightness of one's skin could be measured by a camera. However, some people will not accept the unmeasurable:

mi zmadu do lo ni mi pendo la .maik. - "I am more of a friend of Mike than you are". I think using amounts to quantify the unmeasurable is fine, but that is an issue I swept under the carpet two lessons ago, and I'm not gonna take it on here.

It's absolutely clear, however, that it's wrong to use ni as a way to enumerate how many objects fit a selbri - it's always about to which extent certain sumti fit a selbri. Thus:

do mleca mi lo ni panzi ce'u means "You are less of a parent than I am", and not "You have fewer children than me".

In case you're curious (I was), the jufra zo'e panzi ce'u in the previous example actually refers to two distinct bridi, because the selbri fills the open ce'u-place twice, once for do, and once for mi, making the two sub-bridi: zo'e panzi do and zo'e panzi mi. Since these two bridi are considered different, the zo'e need not refer to the same object.

What does it mean if you don't use a ce'u inside a ni-abstraction? Well, then the main selbri can't fill any of the sumti in the abstraction, so when using selbri like zmadu and mleca, there's a good chance it won't make any sense. However, if ni itself is the main selbri, it's totally fine to avoid using any ce'u at all:

li du'e ni do nelci lo vanju You like wine too much.

The last of the abstractors we treat in this lesson is si'o, the concept abstractor. si'o may be considered a function, or it may not be considered a function. A si'o-abstraction certainly contains a ce'u - in fact, under the understanding which I am teaching, a si'o-abstraction always contains nothing but ce'us! These ce'us, unlike those of ka or ni, remain open and cannot be filled by any selbri. In other words, the function cannot be applied to anything, which is what makes it a maybe-function.

si'o = Concept abstractor: x₁ is x₂'s concept of BRIDI

Let's have a few examples:

lo si'o xebni, which, because all the sumti places are filled with ce'u is equivalent to:

lo si'o ce'u xebni ce'u - "The concept of: "X hates Y"" = "The concept of hate" = Hate

The mythical creatures Balrog from Lord of the Rings are described as being "shadow and flame", the poesy of which appears much stronger in Lojban: la balrog cu si'o fagri joi manku is asserting not only that it’s made out of shadow and flame, but also suggesting that it’s the prototypical Shadow and Flame, from which all other shadow and flame derives.

For good measure, it should be stated that etymologically, "si'o" derives from "sidbo", "idea", but in current usage an idea is considered a text and not a concept.

The difference between the three abstractors ka, ni and si'o can be illustrated with a few more examples for comparison:

lo ka crino cu pluka mi Being green pleases me

lo ni crino cu pluka mi How much {zo'e} is green pleases me (no ce'u!)

lo si'o crino cu pluka mi Greenness pleases me

mi nitcu lo ka sipna ku lo ka kanro I need sleep in order to be healthy

mi nitcu lo si'o sipna lo ka tavla fi lo sipna I need the concept of sleep in order to speak about sleeping things

And I was tempted to write mi nitcu lo ni sipna ku lo ka vreji ri - "I need the amount of how much {zo'e} sleeps", but that doesn't seem to make a lot of sense.