Syntactic sugar, part 2

A thing I forgot to mention last time was the existence of two modifiers grammatically equivalent to 〈(〉 and 〈)〉, but more restrictive. I'll call them 〈{〉 and 〈}〉. 〈{〉 is analogous to 〈(〉, but hides the following selbri to every tag cluster except the one directly to its right. 〈}〉 does the same, but reflected to the opposite direction. I'll be using them in this post.

The main focus of this post is on making new selbri from old ones. This sets it apart from the last post, which was full of “moving around” words. Were this a manual for basic Vim, the last post would document f, F, h, l, % et cetera, whereas this would document y, yy, d, p, P et cetera.

I'll start with something I introduced before: 〈vei ... ve'o〉. This takes a selbri with type \(\text{fa}:a; \: \text{fe}:b; \: \text{fi}:a\), which appears after the 〈vei〉, and any number of tag clusters of type \(b\) connected to a 〈ju'e〉 or the 〈ve'o〉. If nothing is connected to the 〈ve'o〉, the selbri should have a defined default for 〈fi〉, which is implicitly connected to the 〈ve'o〉. The structure is evaluated via a right fold of all the 〈ju'e〉s (and the 〈ve'o〉) in order.

Taking the example from before (well, slightly modified; I got it wrong), we get this reduction chain:

1. vei ce'o 2 faju'e 3 faju'e 4 faju'e 7 faju'e ve'o
2. kefa { 2 fafe ce'o fifa vei ce'o 3 faju'e 4 faju'e 7 faju'e ve'o ke'e
3. kefa { 2 fafe ce'o fifa kefa { 3 fafe ce'o fifa vei ce'o 4 faju'e 7 faju'e ve'o ke'e ke'e
4. kefa { 2 fafe ce'o fifa kefa { 3 fafe ce'o fifa kefa { 4 fafe ce'o fifa vei ce'o 7 faju'e ve'o ke'e ke'e ke'e
5. kefa { 2 fafe ce'o fifa kefa { 3 fafe ce'o fifa kefa { 4 fafe ce'o fifa kefa { 7 fafe ce'o fifa vei ce'o ve'o ke'e ke'e ke'e ke'e
6. kefa { 2 fafe ce'o fifa kefa { 3 fafe ce'o fifa kefa { 4 fafe ce'o fifa kefa { 7 fafe ce'o fifa kunti ke'e ke'e ke'e ke'e

〈kunti〉, by the way, comes from the Lojban word for “empty”, so represents the empty list (or set in other contexts; it's polymorphic). But the upshot is that we get what we expected – a grammatically atomic list – which can easily be translated into a more general form which uses simpler grammar. Even 〈ke... ke'e〉 can be rephrased, but it involves some messy pointer work.

This phrasing can also be used for things like sums. Taking 〈sumji〉 – “fa is the sum of fe and fi”, 〈vei sumji 2 faju'e 3 faju'e 4 faju'e 7 fave'o fafa 16〉 holds. Notice that, this time, it's okay to connect the last element straight to the 〈ve'o〉, since 4 + 7 works, whereas 4 :: 7 didn't. It has to be 4 :: 7 :: []. (‘::’ stands for the append-to-list function, type \(a\rightarrow [a]\rightarrow [a]\)).

I was going to do a section on logical connection, using these brackets to extend from binary connection to multiary connection. But I still need to work on that. Instead, I'll go over something that looks like a J conjunction. To pay homage to J, I'll write it using the J symbol ‘&’.

〈&〉 is called “appose”. It's difficult to define directly, but the rule is 〈kefi { broda fafa & fefa brode } foke'e〉 =_meaning 〈kefe brode fife broda fifi brode feke'e〉, where 〈broda fa〉 and 〈brode fa〉 are functions relating their respective 〈fe〉 and 〈fi〉 terms. This will make more sense when I start using semantic tags, rather than trying to repurpose Lojban's positional tags. But, in short, if we wanted to say “Alice and Bill have the same height”, we'd say 〈.alis. fafi { du fafa & fefa cisni } fofa .bil.〉, where 〈du〉 means “fe = fi” and 〈cisni〉 means “fe has measurement fi (in dimension/aspect fo)”. This alleviates the need for places like Lojban's dunli3, and generalises well.

Because 〈{ ... fafa & fefa ... }〉 is quite heavy on words, there should be an alternative. I'll call it 〈&:〉 for now. So, the example translates as 〈.alis. fafa du &: cisni fefa .bil.〉.

A final thing I want to introduce is the existence of a math mode, delimited by 〈li ... lo'o〉. In this, the tag clusters are worked out automatically, using attributes of the selbri present. For instance, 〈li 2 sumji 3 lo'o〉 gets expanded to 〈kefa { 2 fafero sumji firofa 3 } ke'e〉. There could, potentially, be many different versions of math mode, and I haven't decided on a default one yet. If one sets out the conventions clearly and has all the words, one can use math mode to encode many other unambiguous languages.

CCy cmavo in Lojban

This is a post exclusively about Lojban. If you don't know Lojban, there's no point reading it. In this post, I argue that cmavo made up of an initial consonant pair followed by the shwa (and a pause) could be incorporated into Lojban. I propose that they be used as BY, for things like diacritics and non-Lojban letters. We would gain 48 such possible cmavo.

General CCV cmavo are not allowed because they'd connect to any following brivla. In that situation, they are indistinguishable from rafsi, and are assumed to be rafsi. However, CCy cmavo, like Cy cmavo, are followed by a pause. A pause causes an unconditional word boundary, so CCy cmavo have no chance of combining with a following word.

What about combining with the previous word? Well, were any type of CCV cmavo to combine with the previous word, the only type of word it could form would be a brivla. However, brivla never end with ‘y’, so we don't have this problem.

That's wrapped up, so let's be a bit more ambitious: CGy cmavo, where G ∈ {i, u}. That would yield cmavo like *{kuy.}, which seems like a good fit for ‘Q’. The previous arguments work for this form: ‘.’ can't appear in the middle of a word and ‘y’ can't appear at the end of a brivla. These work for any form which is able to hold together as a single word and ends with ‘y’. We could have CV'y and more.

What we do need to worry about is which CG pairs are valid initials. No native Lojban words, except attitudinals, have on glides at all, and I can't even think of any fu'ivla that contain a CGV cluster.

Filtration as implication

\(
\newcommand{\d}{\mathbin\cdot}
\newcommand{\abs}{\text{abs}}
\)In Marianesque mathematical notation, all mathematical statements are functions acting upon an implicit argument. This argument is taken to be the domain of the statement, and is hence a set. Implicit mapping does its magic, and everything is how we'd expect it. But often we want to make claims about part of the domain. In conventional mathematics, we'd say things like \(n>3\implies {}_\d^2n>9\). Indeed, we can still do that, but there are alternatives.

In this article, Jakub Marian defines square brackets to denote the filter function. It is defined thus for sets:\[
[p]\ xs = \{x:x\in xs\wedge p\ x\}
\]How do we use it? Well, Jakub translates classical:\[
{}_\d^2x+{}_\d^2y=1\implies\abs\ (x\d y)\in\left[0,\frac 1 2\right]
\]into:\[
\abs\ (x\d y)\ [{}_\d^2x+{}_\d^2y=1]\subseteq\left[0,\frac 1 2\right]
\]where \(x\) and \(y\) are essentially the fst and snd functions, and the implicit argument is the set of all pairs of real numbers. I use \(\abs\) rather than vertical bars for two reasons: vertical bars don't work well and I plan to have other uses for them in conjunction with brackets (to make syntax that looks similar to Haskell's quasi-quoters). Most notably, I need a notation for zip lists.

Getting back on topic, I think that this notation can be taken further. Notice how the \(\abs\ (x\d y)\) and \(\left[0,\frac 1 2\right]\) have become split up. This is not ideal. So, we can rephrase the whole thing as this:\[
\left(\abs\ (x\d y)\in\left[0,\frac 1 2\right]\right)\ [{}_\d^2x+{}_\d^2y=1]
\]I'd like to say that this has some new and interesting properties, but I can't find any right now. It does, though, give a different way of thinking about implication. It's more visual than the pure logical version, given that it uses sets (which can act like spaces).

I'd like to make some notes about relations. To make things work well, usual equality – along with the greater-than and less-than relations – are not defined for functors. This is because equality between functions makes more sense when the domain is considered. We should be able to say \((\Gamma\ \text{succ}=\text{fac})\ \mathbb{N}\). But if equality also worked on sets, our claim would become \(\Gamma\ \text{succ}\ \mathbb{N}=\text{fac}\ \mathbb{N}\), which is a weaker claim than we wanted. Hence, we have to map from the inside out, mapping over the functions, then the set on the outside. \(\equiv\) is provided for equality between anything other than functions.

An alternative that always uses \(\equiv\) is to cast sets to zip lists before forking. \((\Gamma\ \text{succ}\equiv\text{fac})\ \text{zipList}\ \mathbb{N}\) is still the stricter claim. This is because lists maintain their order, so elements can be compared pairwise. However, I'm not sure whether it's realistic to expect people to remember to cast sets to lists just to allow forking. I think, though, that this is preferable, since the result is \(\top\) or \(\bot\), not \(\{\top\}\), \(\{\top,\bot\}\) or \(\{\bot\}\) as it would be with the set solution. Maybe we should default to zip lists, even for things like \(\mathbb{C}\), then cast them to sets (or just non-zip lists) as necessary.

Syntactic sugar

My aim with my language (which really needs a name!) is for everything to be expressible using a small subset of the grammar, just as Haskell has its underlying Core language. As I currently imagine it, my core language contains rules about tokenisation, understanding tag clusters and using jumps to change which selbri a tag points to. But to make the language bearable, we need a bit more. I've introduced a few features already, particularly in the last post.

I'll start with some features that change what selbri tags refer to. I've already introduced 〈)〉, but I'll start with jumps. Jumps, which I'll write 〈<〉 and 〈>〉, move the choice for the next selbri either to the left or right. This setting lasts until the end of the tag cluster. For example, we can say 〈gerku plipe <fafa〉 and 〈>fafa gerku plipe〉, which both mean the same as 〈gerku fafa plipe〉. Possibly a more realistic example is 〈2 fafe<fi + fafa 4〉, where both the 〈fe〉 and the 〈fi〉 refer to 〈+〉. Additionally, there are 〈<|〉 and 〈>|〉, with macro expansions 〈<|tag〉 → 〈<tag>〉 and 〈>|tag〉 → 〈>tag<〉. These can be called “temporary jumps”. To make 〈<|<|〉 work as expected, the grammar contains “tag ::= jump tag”.

Jumps are fairly low-level, and could get complicated if used a lot. They're a bit like GOTOs, but they can't do anything crazy like reaching inside function brackets. A pair of alternatives, which I've partially introduced, is 〈(〉 and 〈)〉. 〈(〉 goes before a selbri, and hides it from all preceding tags. Similarly, 〈)〉 goes after a selbri, and hides it from all following tags. These are useful for creating sentences with more than one “noun” to one side of the “main verb”. These have less power than jumps, but are usually enough. They leave us with as much power as Lojban has!

A final necessary feature for tag clusters is 〈ku〉. It's used to separate tag clusters, and can be used to join multiple pairs of places from adjacent selbri, without joining the pairs to each other. I use it in the example from the next paragraph, and I think that what it does is obvious.

Another thing that can give text structure is use of function brackets. I've shown the use of 〈ke ... ke'e〉 where the expression's 〈fa〉, 〈fe〉 and 〈fi〉 places (function, first argument and second argument, respectively) are filled, but other places can be filled, too. In reality, my language is to have semantic tags (like BAI, or cases from natural languages), rather than positional tags (FA), so there are many tags to choose from. What tags the expression shows on the outside are defined inside using the selbri 〈cei〉. This makes it easy to define arbitrary selbri on-the-fly. For example, {batkyci'a} (“fa writes fe on medium fi using keyboard fo”) can be replicated by the expression 〈ke ciska fafa ku fefe ku fifi ku fofofa cei gunma fefa batka ke'e fa〉. Lojban requires paraphrasing to achieve the same kind of definition.

I have some more features to explain (I haven't even got to 〈vei ... ve'o〉 yet), but I've decided to split this post into two. By the way, any ideas about what I can call this language? I'm thinking of translating “join language” or “link language”, but I haven't even decided how to make compound words yet. I could also go for something idealistic, like “symmetry language”, but then I'd be worried about the possibility of there being a language which is more symmetrical.

Functions in speech

Functional programming languages, and also modern imperative/multi-paradigm programming languages, often allow us to use functions without declaring it remotely. These are either called “anonymous functions” or “lambda expressions”. The only difference is that anonymous functions tend to be made of statements (see JavaScript), whereas lambda expressions contain, um, an expression (see Haskell). There is a third way (as seen in J and stack-based languages) in which anonymous functions are constructed using combinators. This is also seen in Haskell-like languages, in which it is called “points-free style”.

In a spoken language, the idea of side effects doesn't really apply. Hence, we don't need anonymous functions made up of statements. What look like statements in spoken languages can be seen as expressions that return a boolean value (true, hopefully). Other values can be derived by describing them in terms of predicates they satisfy. So we just need to find a way to create boolean-yielding functions.

Fortunately, our languages are full of boolean-yielding functions. We call them “selbri” or “verbs”. But we need more. In Lojban, there are 4 basic ways to make new selbri from old ones: tanru, lujvo, {be} phrases and abstractions. For my language, I consider tanru grammatically identical to {be} phrases, with some tags of the two component selbri linked together in normal tag clusters. There is no need for a word {be}, since links between selbri are always made explicitly.

That just leaves lujvo and abstractions. I'll start with abstractions. Lojban abstractions do a variety of things:

• {du'u} refers to the fact of whether the contained bridi is true.
• {nu} refers to events at which the contained bridi is true.
• {ka} makes a function that takes a sumti as a parameter and describes how that sumti fits with the rest of the contained bridi.
• {ni} refers to the extent to which the contained bridi is true, presumably as a number between 0 and 1.

There are some others, too. Of those, {ka} is the one that stands out. The rest just need a bridi, but {ka} needs a bridi containing 1 or more {ce'u}s, where {ce'u} is a cmavo confusingly glossed as “lambda”.

Property abstractions in my language are fairly similar to Lojban's {ka} abstractions, but aren't necessarily bound to a contiguous place in the sentence (though that does usually make things easier). 〈ka〉 is like any other selbri, but has a tag 〈ce'u〉 described as the “test set”. I call it this because 〈ka fa〉 acts like a function that tests its argument for membership in the set specified at 〈ce'u〉.

That's all we need, so here are the obligatory jumping dogs. I'll use the Lojban word {kakne} – “fa can do fe”. (I drop “under conditions x3” because any single dog is in exactly 1 condition. This follows from what I said about tenses. Noöne uses that place, anyway.) The result is 〈gerku fafa kakne fefa ka ce'ufa plipe〉, meaning “at least 1 dog can leap”.

{ka} is also used in Lojban to make functions with more arguments. This is done by adding more {ce'u}s, where the order of occurrences of the {ce'u}s determines which argument they get (alternatively, it's possible to name them in a {ce'ai} prenex to give them a different order). In my language, using a place again just adds to the claims of what's in that set. Because sets are unordered, this can't create ordered parameters. I could add more 〈ce'u〉-like places, but I don't have the vocabulary for that right now. Instead, I'll introduce a syntactic alternative which works well for binary functions.

As I said earlier, my language doesn't have tanru, so I can repurpose {ke} and {ke'e}. I'll use these for “function brackets”. Any tags immediately to the right of 〈ke〉 is connected to the first parameter, and any tags immediately to the left of 〈ke'e〉 are connected to the second parameter. A special rule says that both tags refer inwards by default. From the outside, the whole expression is grammatically atomic, and behaves like a brivla. 〈ke ... ke'e fa〉 is the test function on two arguments.

Examples of binary {ka} in Lojban are difficult to find. One I can think of is the x2 place of {porsi}. Let 〈porsi〉 have the definition “list fa is sorted by comparison fe from set fi”. 〈porsi fe〉 is then a function like “<”, which would be used to sort numbers into ascending order. To avoid introducing too much grammar, I'll stick to sorting a set of numbers. I do, though, have to introduce some grammar to deal with constructing lists and sets. In short, constructors are contained within a 〈vei〉...〈ve'o〉 pair. 〈vei〉 is followed by a selbri specifying the type of connection, then each item is specified by 〈ju'e〉s in order, with tags connected to each (tags adjacent to a 〈ju'e〉 refer away from it, by default). The last element's tag can connect to 〈ve'o〉, or the identity element can be taken from the selbri.

Sets use the selbri 〈ce〉 (“fa is the set made of element fe connected to set fi”), and lists use 〈ce'o〉 (“fa is the list made by appending element fe to the left of list fi”, or, in Haskell “fa = fe:fi”). The pattern also allows other monoids, and even non-monoids, to have similar constructors, but I digress. We now have enough for my planned example: 〈vei ce 2 faju'e 7 faju'e 4 faju'e 3 fave'o fafi porsi fefa kefa mleca feke'e ) fifa vei ce'o 2 faju'e 3 faju'e 4 faju'e 7 fave'o〉. In real speech, the 〈ke〉...〈ke'e〉 structure would be redundant; we'd extract a function directly from 〈mleca〉 (“fa < fe”). But if we had a more complex relationship to satisfy, the function brackets would become necessary.

After that example, I really want to describe quicksort! We need some more grammar and vocabulary, though.

Tense

In this post, I'm going back to some more specific points from my new language. This can be seen as an application of both the referential transparency ideas and the quantification ideas. I will use Lojban as the “other language” because it's simple and formalised, but most points will also apply to English and other languages.

In Lojban, every concrete sumti looks like a smear across space-time. This immediately looks as if it breaks referential transparency. I'm not who I used to be, for instance. But that sentence gives us a clue as to why referential transparency can still hold. “am” is not the same verb as “used to be”. Verbs essentially choose some piece of space or time, and then act upon a few relevant slices of those irregular hypersmears we call “nouns”.

“I'm not who I used to be” is quite an awkward thing to translate into Lojban. I get {mi na du lo pu du be mi} – “I'm not equal to a thing that was equal to me”. But I'm not sure whether this is actually correct. Following the above logic, the main {du} is acting upon here-and-now slices of its arguments: {mi} and {lo pu du be mi}. But what is the here-and-now slice of {lo pu du be mi}? Surely, {lo pu du be mi} satisfies {da} in {da pu du be mi}. So, we're taking a past slice of {da} and equating it to a past slice of {mi}. But {mi} (uniquely) satisfies this {da}. Hence, that sentence wasn't correct, because it reduces to {mi na du mi}. Incidentally, the English sentence is probably also false by the same logic, unless we arbitrarily fix tenses of nouns that are being used with certain verbs. But that's messy and non-general.

All of this prohibits us from relating nouns from different tenses unless we can give nouns tenses directly. So, why don't we do that? I don't see why not. If our nouns, rather than our verbs, have tense, all of this stuff becomes trivial. But there are a few things to remember. In order to do this, we get rid of the smear concept, and with it formally lose the notion of continuity. In maths, we've rederived such things anyway, so this is not as big a problem as it may sound. It does, though, throw up some counterintuitive interpretations (counterintuitive according to those used to smear languages).

The number of past-dogs that jump is much larger than the number of dogs that have jumped. Why? In non-smear languages, there is no intrinsic connection between a dog and that dog in the past. Every time a dog jumps, it becomes a different dog. Therefore, the number of past-dogs that jump is equal to the number of times any dog has ever jumped.

Of course, some way to get back the meaning of “the number of dogs that have jumped” is necessary. What we have to do is invent a predicate similar to “fa is a successor/predecessor to fe in tense fi by standard fo”. When put through this predicate, some of the past-dogs end up having the same successor in the present tense, and are reduced to a single present-dog. Those present-dogs can be considered “dogs who have jumped”.

The “by standard” place is important. The “fa” and “fe” places need not and do not have a formal connection. We can make the predicate as strict or loose as we like. For example, in translating the claim “Germany have won the World Cup 4 times”, we can specify that we consider West Germany to be a predecessor of present-day Germany, so their titles do in fact count for Germany. A literal translation of an overly-precise statement would go something like “4 predecessors/successors of Germany from the past by standard that the predecessors/successors largely occupy predecessors/successors of the land of Germany win the World Cup”. But the standard is nearly always unnecessary (and I'm not sure whether I've picked a good standard in the example).

This scheme also makes it possible to make truly tenseless claims. In Lojban, every bridi is cryptotyped with tense information about space and time. That applies even to {li re sumji li pa li pa}, in which all of that information is meaningless and weakens the claim, technically. Time and space have no bearing on whether 2 is the sum of 1 and 1, so it makes little sense to mention them. In a non-smear language, 1, 2 and the sum predicate can be taken to be universal constants, referring to things that don't happen to exist in space-time.

Referential transparency

This is another language post, since referential transparency is a given in mathematics. It is more general than the previous ones, and doesn't mention my new language or Lojban at all. This serves as a consolidation of my opinions from a conversation with la .az. about Quine's Quantifiers and Propositional Attitudes. I use Death Note for examples throughout, but I only reference events up to episode/chapter 2. No spoilers!

The essence of referential transparency in logic is that if two entities are identical to each other, there are no functions which can be applied to them that give different results. All languages, I believe, can achieve referential transparency under the right interpretation. Spoken languages and non-transparent programming languages will often rely on implicit information to tell apart predicates or arguments, but once all of this information is added, things will work out.

To see how referential transparency can work, consider “I believe that Kira has killed people” and “I believe that Light hasn't killed anyone”. Naïve interpretations relying on referential transparency would claim that, since Kira and Light are the same person, these beliefs are contradictory, and the speaker is being irrational. However, that's not the case. The speaker doesn't know that they are the same, so can legitimately hold both views.

At this point, Quine gives up on referential transparency, inventing the idea of “opaque contexts”, in which values can't be substituted freely. But I think that this is a bit extreme. If Yagami Souichirou is speaking, he can substitute “Light” for “my son”. Matsuda can also substitute “Light” for “Yagami's son”. The key problem in their reasoning is that they don't know that Light and Kira are the same. Words like “believe” do induce a context, but not an opaque one.

So, we are forced to concede that “Light” and “Kira” refer to different entities, since there exist functions to tell their referents apart. But they are, in some sense, the same. To the reader/viewer's logic, and to Light's own logic, they are indistinguishable. To formalise this, I use the notions of “referrers” and “phenomena”. The words “Light” and “Kira” refer to distinct referrers. We make all of our claims on the referrer level, since this is as much as we can know. Referrers contain information about claims people have made about them. Some of these claims are that they are identical to another referrer. If a person believes that two referrers refer to the same phenomenon (i.e, believes that they are “the same”), they can unify their knowledge about both referrers. Light knows that the referrers “Light” and “Kira” both refer to the same phenomenon, but Souichirou doesn't. Hence, Light knows that he's killed people, but his dad doesn't.

Furthermore, early on, Light believes that Lind L Tailor is L. According to him, both of those referrers have the same referent, so he believes that killing Lind L Tailor will kill L. When he realises that they are, in fact, different, he is able to separate all of his claims about L from his claims about Lind L Tailor. The obvious one is “Lind L Tailor is dead. L isn't.”.

Naturally, unification of knowledge about different referrers is done via fuzzy logic. In this post, I've used clear-cut examples to illustrate the points. There are many cases of this in real life, like aggregating experiences with friends. Each time, you're pretty sure that they're who they were before. But L, thinking that Light might be Kira, would be rash to completely unify all of his knowledge from those two referrers, but would be overly cautious to not do it at all.