Babel: Or, the Necessity of Violence

Babel, by R. F. Kuang, is a 2022 Alternate-History low-fantasy novel about translators who perform enchantments for the glory of the British Empire. The magic is fictional, but the translation theory is real: the Oxford translation class lectures are a legit callback to grad school. Why are translators performing magic? Because true translation is fundamentally impossible, and magic arises from the sometimes-subtle, sometimes-vast differences in meanings between attempted translations from one language to another.

Naturally, there is quite a lot of non-English representation in such a novel. Our main character, Robin, is a native speaker of Cantonese, so the first example we get is a a string of orthographic Chinese characters, which I cannot type easily to reproduce for you here--but, we immediately get diegetic transcription and translation:

'Húlún tūn zǎo,' he read slowly, taking care to enunciate every syllable. He switched to English. 'To accept without thinking.'

Note the conventional use of italics for non-English text. Here we get three parallel representations of the same bit of language, allowing the reader to understand what it actually looks like written, approximately how it sounds via romanization, and approximately what it means through Robin's translation of what he just read.

Robin is quickly introduced to the non-magical responsibilities of translation and interpretation:

This all hinged on him, Robin realized. The choice was his. Only he could determine the truth, because only he could communicate it to all parties.

The book is chock full of this kind of stuff--not just directly representing other languages, but explicitly teaching the reader about real concepts in linguistics and translation theory through the mode of having the characters learn and discuss them. Skipping ahead a bit, here is a taste of one of the theory lectures:

'The first lesson any good translator internalizes is that there is no one-to-one correlation between wrds or even concepts from one language to another. [...] If [there was], then translation would not be a highly skilled profession - we would simply sit in a class full of dewy-eyed freshers down with dictionaries and have the completed works of the Buddha on our shelves in no time. Instead, we have to learn to dance between that age-old dichotomy, helpfully elucidated by Cicero and Heironymous: verbum e verbo and sensum e sensu. Can anyone--'
'Word for word,' Letty said promptly. 'And sense for sense.'

And a bit of philosophizing later on reminded me rather strongly of the aliens from The Embedding:

We will never speak the divine language. But by amassing all the world's languages under this roof, by collecting the full range of human expressions, or as near to it as we can get, we can try.

And in fact, this is not a bad description of the project of natural language documentation and typology. 

The next instance of non-English representation makes use of footnotes to provide a non-diegetic translation for what he character already understands:

Auferre trucidare rapere falsis nominibus imperium atque ubi solitudinem faciunt pacem appellant.
Robin parsed the sentence, consulted his dictionary to check that auferre meant what he thought it did, then wrote out his translation.*
*'Robbery, butchery, and theft - they call these things empire, and where they create a desert, they call it peace.'

Although in this case, the translation does exist in the story, and so could've been included in-line, that is not so for all of the footnotes, some of which exist entirely outside of the story. For example:

for a full year Robin thought The Rape of the Lock was about fornication with an iron bolt instead of the theft of hair.*
* A reasonable error. By rape, Pope meant 'to snatch, to take by force', which is an older meaning derived from the Latin rapere.

I could continue with a detailed analysis of every sample of non-English language, as I did exhaustively for some other books earlier on in this series--but I would end up quoting from about a thrid of all pages in the book, and we'd be here all day! The range of integrative and interpretive techniques in use is actually pretty well covered by those few examples I have quoted so far. But what's really unique about the book is the extent to which it confronts the reader with concepts that you might not otherwise have to face outside of a graduate-level course in linguistics or translation, and in ways that are actually relevant to the plot. Consider:

What was a word? What was the smallest possible unit of meaning, and why was that different from a word? Was a word different from a character? In what ways was Chinese speech different from Chinese writing?

That matters for understanding the magic system and for understanding the nature of the relationships between characters. This is a masterclass in science fiction with linguistics as the underlying science... except that it's technically fantasy instead of science fiction. There's refreshingly not a single whiff of Whorfianism or UG anywhere--as there shouldn't be as those concepts would not have existed in the historical period in which this story is set!

The book also briefly addresses The Forbidden Experiment--and contributes to foreshadowing the true villainy of one of our antagonists by having him seriously entertain it as a possibility (which is unsurprising, given how he has up till then manipulated the lives of Robin and his friends).

I shall leave off with one more quote on semantic theory:

Does meaning refer to something that supercedes the words we use to describe out world? I think, intuitively, yes. Otherwise we would have no basis for critiquing a translation as accurate or inaccurate, not without some unspeakable sense of what it lacked.

 

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The Toolmaker Metaphor

Toolmaker Koan (only available from third-party sellers, which is incredibly amusing if you've read the book, but I've provided an Amazon affiliate link anyway), is a 1988 science fiction novel by John McLoughlin about first contact with two different non-human... species? (don't want to spoil too much!)... while humanity is on the cusp of an apocalyptic nuclear war. It's only a few months older than me, just barely older than the collapse of the Soviet Union, and a prime example of just how consistently late 20th century sci-fi authors failed to predict said collapse!

There is quite a range of linguistically interesting stuff going on in here. Not a large total volume of it, but a lot of small bits of different kinds of things.

We start out, upon meeting our first alien intelligence (who has been studying humanity in secret for some time, and thus conveniently speaks English), with a technical articulatory phonetic description of how to pronounce it's chosen name! ("Charon") This sort of linguistic detail doesn't show up again, but the specification that the Greeks would pronounce it with "the back voiceless stop consonant you call K" becomes relevant in the denouement for recognizing the adjective "Karonic".

A bit later, we get some Greek when discussing the significance of the alien's  name:

"The Greeks always buried their dead with a coin offering, the danake, in their mouths. Charon, I'm afraid, was a bit of a miser; with the coin the soul paid Charon's naulon, his toll."

This is straightforward diegetic translation, in which the character is using appositive definitions. I'm not really sure why the character, and thus why the author who wrote that character, bothered in this case, though; it feels very much like a forcibly inserted chance to show off cool background knowledge, which reminds me of this XKCD comic (even though the words in this case are not made up by the author!) This seems like a good time to point out that, while I am a great proponent of Interesting Linguistic Content, all of the techniques I am documenting are pointless if they are not used in support of the story! You've got to find a way to give it a function, or, as much as it pains me, cut it out.

Later, the alien Charon, by application of god-like alien technology, uh... reconstitutes a tribe of Australopithecus. Several of their vocalizations are quoted (presumably onomatopoetically), but our human protagonists obviously don't understand them (or even know if there is anything to understand in the first place!), which makes this a very clear case of Making it Irrelevant. (For reference, if the Australopithicii have language, two of their words appear to be "Skaroch!" and "Skuh!")

Eventually, Charon introduces the humans to the whileelins--or at least, their name is spelled "whileelin". How it is pronounced is anyone's guess, as their language is supposed to be whistled! (Or perhaps "sung", as it is supposed to sound similar to birdsong.) Whistling is a modality in which natural human languages actually exist, but never a primary modality; like writing, it always serves as a secondary encoding of normally-spoken language. So a non-human species that naturally whistles is a neat idea--especially since I happen to working on a primarily-whistled conlang myself right now!

Sadly, there is no indication of how information is actually encoded in the whistled signal, and no description of how the transcription system works, and thus no easy way to compare the whileelin language to human whistle languages. On the bright side, that means I am free to assume that it does not rely on absolute pitch discrimination! No indication of how data is encoded in the whistle signal. The only descriptions of the sounds or lexicon of the whileelin language that we get is that "Hwiliria"(the name of the whileelins' spaceship) sounds like a "four-toned burst of music" (a very strange description given that there are more than four letters and more than four types of letters in the name), and this bit of non-diagetic translation:

"Haijar," agreed the First.

which takes advantage of the specific semantics of the word to give you an approximate definition in the speech tag! That's not something you can get away with very often!

Prior to the humans' introduction to the whileelins, however, McLoughlin establishes some dramatic irony by shifting to the whileelins' point of view for a couple of scenes, which are used to establish a particular kind of Narrative Translation Convention, in which the use of archaic thee/thou pronouns and associated verb conjugations in English to represent the "Patriarchal mode" of the whileelin language.

Whileelins are not built like humans, physically or mentally. Thus, there are three stages to whileelin languages: They are born with an innate, genetically-programmed understanding of a basic "creche language"; upon reaching adulthood, their brains grow to unlock another innate Patriarchal/Matriarchal language. In between, there is a ten-year period of high intelligence and mental flexibility in which all the variety of arbitrary language can be learned and developed--unless a whileelin is neutered, halting their transition into full adulthood and allowing them to maintain mental flexibility indefinitely. We only ever encounter one linguistic community of whileelin in this novel, but presumably this means that whileelin languages can diverge from one another... but all possible whileelin languages would be much more similar to each other than human languages are, due to the constraints of developing from a common innate language, and needing to accommodate the integration of a second innate language--at least, as long as juveniles and castrati care about learning to understand the speech of sexually-mature adults! This is a fascinating bit of fictional linguistic science that qualifies this work as linguistic science fiction--which does not focus on the strong Sapir-Whorf hypothesis!

Returning to human languages, there are three insertions of Spanish; one short bit of code-switching right near the beginning, and two longer phrases in the last third of the book. That bit of code-switching looks like this:

"Nossir. But we're good hackers, que no?"

In this case, English syntax Makes the meaning Obvious. It's a tag question, and English is pretty liberal with what can go in the tag slot already when multiple dialects are considered (cf. "eh?", "innit?", etc.)--so when you drop some random short thing with a question mark after it (which happens in this case to be Spanish) in the tag slot, it's pretty obvious what it means just from where you put it! And while we have been told that the characters here are a multi-ethnic, multi-national group, this little bit of inserted Spanish helps to show us that--even though the rest of the dialog is English (excepting the couple of words of Greek mentioned above) for the next 212 pages!

But, on page 242, we get a reintroduction to the character who was the addressee of that tag question, who was out of the action for a god long while. And just in case you forgot who he was, his reintroduction consists of thinking "Madre de dios!" while "wiping his lap frantically" because he spilled hot coffee! The context, and exclamation mark, and implicit background knowledge that people tend to slip back into their native(or most comfortable) languages when stressed or cursing (or stressed and cursing) makes it pretty Obvious that this is an expletive, Irrelevant what the literal meaning is, and helps remind us who this character is--oh yeah, it's the guy who was addressed with Spanish!

Later on, we get this interesting passage:

--but then this Charon had claimed to be a sentient machine, one speaking like a crazy old man. Un viejo poquito loco, and it claimed to know a great deal.

Like the earlier inclusion of Greek, I cannot see an obvious purpose for this; it might just be there to remind you again, in case you forgot, that at least one character here (this one) has a native language other than English. However, regardless of purpose, the structure is fascinating. You've got the English "crazy old man" and the Spanish "Un viejo poquito loco" in direct textual juxtaposition--but, they aren't actually in the same sentence, and so not in syntactic apposition, which considerably increases the cognitive load on the reader to identify one as a (rough) translation of the other. Now, that could be a bad thing if the meaning is really important--or perhaps a good thing if you want the reader to pause and think about a particular passage. But McLoughlin side-steps the issue by simultaneously ensuring that the meaning of the Spanish is totally irrelevant. You can just delete it, and the sentence is still perfectly comprehensible, so it doesn't matter if a monolingual English reader doesn't figure it out! They still will have been shown the Spanish and given that reminder.

Finally, in the Epilogue, we get this:

    "Munirda, strangeko!" The girl glanced shyly at the Mother, back again at the Karonic. "Mensch, two, Marma, oltimaku wringlerising!"
    "And to me, my noisy descendant, you speak English!"

Now, I really hope that that's just Irrelevant, 'cause I cannot figure out what exactly it's supposed to say. (I'm pretty sure it is, 'cause the rest of the epilogue makes perfect sense.) However, several individual bits are tantalizingly familiar--which, in the context of the response, suggests that all this seeming gibberish is supposed to accomplish is to show you that this girl in the future is speaking something descended (at least partially) from English, but different from contemporary English, which helps to suggest the depth of time that has passed between the last chapter and the epilogue.

But, what does all this have to do with metaphor? Well, the linguistic content of Toolmaker Koan reminded me of the conflict between the Conduit Metaphor and the Toolmakers Paradigm, first described by Michael Reddy.

The Conduit Metaphor is a conceptual metaphor deeply embedded in the English language; it is the conception of utterances as containers into which thoughts can be placed, and sent (through the conduit of speech) to another mind. A few examples from Reddy's paper:

1. Try to get your thoughts across better.
2. None of Mary’s feelings came through to me with any clarity.
3. You still haven’t given me any idea of what you mean.
4. Whenever you have a good idea practice capturing it in words.
5. You have to put each concept into words very carefully.
6. Try to pack more thoughts into fewer words.
7. Insert those ideas elsewhere in the paragraph.
8. Don’t force your meanings into the wrong words.

Though it may feel entirely natural to speak this way, it is not a necessary conception of how to talk about language. Other frameworks are possible. And to prove this, Reddy proposed the Toolmakers Paradigm; in this alternative metaphor, we are all isolated minds living in different mental environments, and creating tools (ideas) appropriate to those environments. We can pass blueprints for tools (utterances) between our environments, but, lacking shared context outside of the blueprints themselves, there's no way to ever to tell if you actually built what someone else sent you the plans for, or if anyone else has interpreted your plans correctly. (I am aggressively summarizing here; I strongly suggest actually reading Reddy's paper.)

Any neurodivergent person who has encountered the double empathy problem, or any author who has encountered baffling analyses of their work, can easily understand the far more accurate nature of the Toolmaker Paradigm. And yet, despite being toolmakers, as Toolmaker Koan repeatedly reminds us that we are, we English speakers at least seem to really want it to not be so! It would be so nice if language actually contained thought and transmitted it accurately; and I'm sure it doesn't help that the Conduit Metaphor can be made more accurate for transmission of information between machines; but it just ain't that way for humans! In fact, I think we can even do better than the Toolmaker Paradigm as described by Reddy; language itself is a tool. No metaphor required! Even apart from acting as cognitive technology (as a I referenced in my review of Ted Chiang stories), language is a blunt tool with which we try to sculpt crude replicas of our thoughts in other people's minds. The simple fact that humans create and use languages makes us toolmakers all on its own.

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Questions & Deixis in Valaklwuuxa

I have been translating the Universal Speed Curriculum into Valaklwuuxa. This is a very simple conversational script; it's not intended to teach you a lot vocabulary, or particularly deep grammar principles- just to get you comfortable with speaking fluently in a target language and capable of asking simple questions and understanding simple answers, so that you can learn more of the target language in the target language.

As such, it starts out with sentences like "What is that?" / "That is a rock." / "Is that a rock?" Basically, you need to be able to ask content questions and polar questions, and name things by pointing (deixis), which we do in English with demonstrative pronouns. These should be easy things to handle in any language, and in fact Valaklwuuxa handles just fine... but given how subjectively weird Valaklwuuxa is, just how it manages may be non-obvious to the typical Anglophone.

If you know a little bit about Valaklwuuxa already (because you've read my previous blog posts or something), you might reasonably think "well, there aren't any normal nouns, and you don't need pronouns except the subject clitics because the verb conjugation takes care of everything else, so maybe there are extra deictic and interrogative conjugations?" And indeed, one could imagine a language that worked that way- the conjugation table would be large an unwieldy, but that never stopped a natlang! But there's a problem: if "what" and "that" are just translated by verb inflections... what gets inflected? There is, after all, no word for "is"!

Interrogatives

To resolve this, the interrogative pronouns "what" and "who" are actually translated in Valaklwuuxa by interrogative verbs, meaning roughly "to be what?" and "to be whom?" These are <k'asa> and <k'aku>, respectively. A third interrogative word, <k'axe>, is what we might be tempted to call a "pro-verb"; it most closely translates into English as "to do what?" In general, there is no morphosyntactic distinction in Valaklwuuxa between sentences like "I act" and "I am an actor- these would both translate the same way. But, Valaklwuuxa distinguishes unergative verb (with an agent-like subject) and unaccusative verbs (with a patient-like subject) in other areas of the grammar, and that is the internal distinction between <k'asa" and "k'axe>. Animate things, however, are always "things one can be" but never "things one can do", so there is only the one (unaccusative) root for "to be whom?"

Using any of these verbs as the predicate of a sentence allows asking questions like "What is it?" If you need to ask a question about an argument of some other verb (like, say "What did you eat?"), you just treat the interrogatives like any other Valaklwuuxa root, and stick them into a relativized argument phrase.

All of these interrogative roots also have corresponding answer words: <dasa> ("to be that"), <daxe> ("to do that"), and <daku> ("to be them"). These, however, are not the deictic (pointing) words that you would use in a question like "What is that?" They are more like regular pronouns (or pro-verbs)- they refer to some thing or action that has already been mentioned earlier in the discourse, which you do not wish to repeat. (And if you think that the schematicism in how answers and questions are regularly related to each other is suspiciously unnatural... well, Russian actually does exactly the same thing!)

Demonstratives

Surprisingly, the actual demonstratives turned out to work pretty much like they do in English- the exact set of them is different, and they divide up space differently, but they pretty much just look like free pronouns. Lest you think that this is not weird enough for a language with such alien-to-Anglophones morphosyntax as Valaklwuuxa... well, that's actually how natural Salish languages handle them, too.

Internally, demonstratives are considered to be pretty much the same as articles- they are things that can head argument phrases, but they can't be predicates. They just happen to be intransitive version of articles (determiners), which don't require a relative clause to follow.

The three generic, non-deictic articles, which always a require a following phrase, are as follows:

<txe> "I know which one"
<ta> "I don't know/care which one"
<kwe> "the one who/which..."

The demonstratives, which can be used with or without an explicit argument, come in pairs distinguished by animacy:

Animate/Inanimate
<tqe>/<se> "this (near me)"
<tqel>/<sel> "that (near you)"
<lel>/<lel> "yon (near it)"

Note that there is no number distinction (e.g., "this" vs." these"). Plural marking can done by attaching the clitic <=ndek> to a determiner, but is not obligatory- it is unlikely to be used, except for emphasis, if number is indicated in some other, such as by the verb conjugation or if a specific number is mentioned.

Demonstratives are also distinguished from articles in that they can also be prefixed with <we->, which is a "pointing" marker; it's not obligatory when you point at something, but can only be used if you are actually pointing at something, and can be approximated as "this/that one right here/there!"

There is also a single set (without any animacy distinction) of question/answer determiners: <k'adza>, for asking "which one?", and the answer <dadza>, used for (approximately) "the same one"/"the same thing".

Asking What Things Are

Now, we have enough to translate:

"What is that (near you)?" ~ "k'asa sel?"
"This (near me) is a rock." ~ "wonglqa se."
(Where <wonglqa> is the word for "to be a rock".)

Now you might think, why did we choose to have interrogative roots and deictic pronouns? Couldn't you just as easily do it the other way around? That would make content questions simpler, because you wouldn't have to construct a relative clause around every interrogative root. And the answer is "yes", some other language could indeed work just the same as Valaklwuuxa in every other repsect, except for flipping that one decision the other way around. But choosing to do things in this way has one really nice consequence: the structure of content questions exactly parallels the structure of their answers. If the rock is "yonder", so that both questioner and answerer use the same demonstrative, you get:

"k'asa lel?"
"wonglqa lel."

Replace the question word with its answer, and everything else stays the same. Treating interrogatives as verbs does bring up another issue, though: when using them in argument positions, which determiner do you use? Typically, you'll use <ta>, the "I don't know which one" article (because if you did know which one, why did you ask?), but any determiner is valid, and they can be used to make much more specific kinds of questions, like:

"dwu-valsk sel k'asa?" ~ "You cooked that what?" / "What's that thing that you cooked?"

A Brief Note on Polar Questions

So, that's pretty much everything you need to know about content questions- but what about polar questions, with a yes/no answer?
The simplest way to form them is simply by intonation; syntactic structure is identical to statements, but a rising-falling tone over a whole clause will turn it into a question. If you want to be more specific, though, there is an interrogative particle <k'a>, which placed immediately after whatever is in doubt. Thus, we can ask:

"dwu-valsk k'a ta wonglqa-la?" ~ "Did you cook a rock?" (as opposed to doing something else with it)
vs.
"dwu valsk ta wonglqa k'a-la? ~ "Did you cook a rock?" (as opposed to some other item)

There is of course a corresponding answer word, <da>, used to confirm the thing in doubt:

"xe-valka ta wonglqa da-la." ~ "Yes, I really did cook a rock."

And an (irregular) negative answer particle:

"xe-valka ta wonglqa pe-la." ~ "No, I did not cook a rock." (but I may have cooked something else)

Errata

In addition to the interrogatives discussed above, there are two more pairs of answer/question roots:

<skwol> / <sdwol> "how many / so many"
<k'akwo> / <dakwo> "which (ordinal) one / that (ordinal) one"

That last one is a thing for which English has no single simple question word, but many languages (like Hindi) do. If you want to elicit a response like "I am the fifth child in my family.", you can imagine a corresponding question like "Which-th child are you?" In English, that's terribly awkward, and there is just no standard way of forming that kind of question, but in Valaklwuuxa, <k'akwo> is the standard translation for "which-th", or "what number".

Now, there's one more bit of interestingness. All of the basic question roots are intransitive, but there is a generic transitivizing suffix <-(e)t>. This usually has a causative meaning ("to make something happen", "to make someone do something"), which means Valaklwuuxa doesn't need to use special verbs for "to make" or "to force" nearly as often as a language like English does, but the precise meaning of a transitivized verb is lexically specified. In the case of <kaxet>/<daxet>, the transitive versions actually mean "to do what to something?"/"to do that to something". So if you want to ask "What did he make you do?", the translation actually does use a separate word for "make" after all.

A Progressive Model of WSL Syntax & Interpretation: Part 3

Last time we saw how to introduce generalized quantifiers and arbitrary specifier positions into our model of WSL; but, in the process, we lost any recognition of the scoping effects between quantifiers. Figuring out how scoping effects work for generalized quantifiers represented by sets-of-sets can get pretty complicated and confusing, so we're gonna really slow and step-by-step.

First, let's revise and review our model of the syntax so far. The complete syntactic model that I'll be using for the rest of this post is given by the following grammar:

S  → P AP
AP → RP AP | 0
RP → QP R | QP e
QP → Q NP
NP → N NP | 0

Where an S is a Sentence, an AP is a Argument Phrase, an RP is a Role Phrase, an R is a Role, a QP is a Quantifier Phrase, a Q is a Quantifier, an NP is a Noun Phrase, and N is a Noun.

Next, let's look at some examples to get a better grasp on how scope should work. Consider the English sentence

"Everybody loves somebody."

This has several possible translations into WSL; two of them are:

1) Ka ves anz i siru jest anz jo.

and

2) Ka jest anz jo siru ves anz i.

Sentence (1) states that, for every person, there is someone whom that person loves- but not every lover necessarily loves the same lov-ee. Sentence (2), on the other hand, states that there is some single person whom everybody else loves, because the existentially-quantified patient is now outside the scope of the universally quantified agent.

Additionally, sentence (1) allows that every agent might be participating in a totally separate instance of "loving", while sentence (2) indicates that there is only one instance of "loving" going on, and every person is a simultaneous agent in it. This is because of differences in the scope of the existentially-quantified specifier ("siru", with a phonologically-null quantifier). Moving "siru" to different positions produces more subtly-different interpretations; e.g., taking (2) and moving the specifier to the end would indicate that there is one person who is separately and independently loved by everyone- possibly at different times. And adding multiple conjoined specifiers would make things even more complicated.

Now let's take a look at how Roles get assigned to QPs inside Role Phrases, as of last time:

[|RP: QP R|] = λx. {y : ∃z. z ∈ x & [|R|](z)(y)} ∈ [|QP|]

What we're doing here is constructing the set of all things that bear a particular relation to some element of the specifier set, and then asserting that that is the same as one of the potential referent sets from the Quantifier Phrase. This implicitly imposes some constraints on the identity of the specifier set as well. A Role Phrase containing a specifier works a little differently:

[|RP: QP e|] = λx.∃Y. Y ∈ [|QP|] & Y ⊆ x

This just imposes an explicit constraint on the identity of the specifier set. (Note that I have chosen here to use an upper-case Y for the referent set in the rule for specifiers, to distinguish it from the lowercase y used for an element of the referent set for normal Role Phrases.)

In order to respect quantifier scoping, we need to arrange things so that we can reconstruct all lower-scope quantifier sets and select the relevant constraints independently for every element y of the referent set that we're constructing for the current RP.

In order to do that, we first have to make the denotations of all lower-scoped RPs available during the interpretation of any given RP. That means modifying our interpretation rules for APs as follows:

[|AP|] = λx.[|RP|](x)([|AP|])

such that the denotation of next-lower-scope Argument Phrase (which contains all the remaining Role Phrases) is filtered through the current Role Phrase as a parameter. That will, of course, require updating the rules for RPs to take multiple arguments, and doing the right thing with them:

[|RP: QP R|] =
        λx.λa. {y : ∃z. z ∈ x & [|R|](z)(y) & a(x)} ∈ [|QP|]
[|RP: QP e|] =
        λx.λa. ∃Y. Y ∈ [|QP|] & Y ⊆ x & a(x)

This places the evaluation of each Argument Phrase inside the scope of the variable y bound by the next higher Argument Phrase. Although y itself is not accessible in the lower scopes (we only pass along the shared specifier set as a parameter to a), this means that lower quantifiers are re-evaluated for every y. Thus, in "Ka ves anz i siru jest anz jo.", it is possible in the evaluation of "jest anz jo" to select a different existentially-quantified referent to correspond to every member of the universally-quantified referent set of "ves anz i". This is obvious in the semantics for specifiers, where we're still cheating a little bit by using the "∃" symbol to bind the variable Y and borrowing its scoping behavior.

The interpretation for the internal structure of a QP, containing a Q and an NP, remain unchanged. If we round things out with an explicit rule for null APs, we get the following completed model for the syntax-semantics interface:

[|S|]  = [|P|]([|AP|])

[|AP: RP AP|] = λx.[|RP|](x)([|AP|])
[|AP: 0|]     = λx. true

[|RP: QP R|] =
        λx.λa. {y : ∃z. z ∈ x & [|R|](z)(y) & a(x)} ∈ [|QP|]
[|RP: QP e|] =
        λx.λa. ∃Y. Y ∈ [|QP|] & Y ⊆ x & a(x)

[|QP|] = [|Q|]([|NP|])

[|NP: N NP|] = [|N|] ∩ [|NP|]
[|NP: 0|]    = U

[|P|] = G[P]
[|R|] = G[R]
[|Q|] = G[Q]
[|N|] = G[N]

This gets us the ability to model a pretty big chunk of all WSL declarative sentences. Still to come: non-intersective Nouns, modality, alternative Projectors, subordinate clauses, and controlling semantic projection.

A Progressive Model of WSL Syntax & Interpretation: Part 2

Last time, I ended with the note that properly modelling specifier phrases would require splitting the interpretation of argument phrases in half; in particular, I had in mind the idea that variable bindings for noun phrases would need to be moved around to ensure that the specifier variable would be in-scope in the semantics for every argument phrase.

It turns out that the solution is actually much simpler. First, we will introduce a very simple change to the syntax rule for a sentence to account for sentential Projectors (a part of speech which heads independent clauses in WSL):

S  → P QP e AP

Next, we'll stop treating the specifier phrase separately, and account for it as a special case of an argument phrase:

S  → P AP
AP → A AP | 0
A  → QP R | QP e

We could also choose to treat the specifier clitic as a kind of Role, which would be slightly simpler, but this formulation better reflects my own psychological perception of what a specifier is (and thus presumably reflects the intuition of the fictional native speakers of WSL as well). Note that this allows a single clause to contain multiple specifiers, as well as putting them in arbitrary positions with respect to the other arguments; that situation is accounted for in the WSL Primer, which says that the semantics of multiple specifier phrases is the same as that of multiple conjoined specifiers, except that using multiple specifiers allows you to place them all in different quantifier scoping levels- the same as the interpretation for repeated roles.

The semantics for these bits of syntax is as follows:

[|S|]  = [|P|]([|AP|])
[|AP|] = λx.[|A|](x) & [|AP|](x)
[|A: QP R|] = λx.[|QP|](λy. [|R|](x)(y))
[|A: QP e|] = λx.[|QP|](λy. y ⊆ x)

To summarize: the denotation of a sentence is the denotation of the argument phrase chain filtered through the denotation of the Projector; the denotation of an argument phrase is the denotation of the argument given an entity variable x conjoined with the denotation of the remaining argument phrase given x; and the denotation of an argument is the denotation of a relation on the shared entity variable x and the phrase-specifier entity variable z filtered through the denotation of the quantifier phrase (which for the moment is unchanged from last time).

The case of an argument containing a QP and specifier clitic instead of a QP and a Role just contains the explicit relation that the phrasal entity z is a subset of x. Note that this requires interpreting z and x not as representing single referents, but as sets of possible referents- an idea I introduced earlier in the series on semantics for a monocategorial language.

For now, we'll only deal with a single Projector: "ka", which indicates a simple declarative sentence. It's semantics are very simple:

[|P|] = G["ka"] = λy.∃x. y(x)

This just says that some x (which now refers to a set) exists, and its identity will be constrained by y (which is bound to the denotation of the argument phrase chain). We could build this into the interpretation of an S directly, but we will have to deal with the semantics of Projectors at some point, so we might as well start here.

We now have the ability to intersperse specifiers and other arguments in any order, with the members of the specifier set constrained by the specifier phrases, and the scope of all quantifiers corresponding exactly to their surface order!

Generalizing Quantifiers

The next step in building our model of WSL semantics is to improve the handling of quantifiers. As described in this article, the denotation of a Quantifier phrase will be represented not by a proposition in predicate logic, but by a set of sets of possible referents- all those sets of referents which contain the appropriate quantity of the type of referent identified by a given Noun phrase. This helps our model conform to the intuition that a bare noun or quantifier phrase does not correspond to a logical assertion- i.e., that a given entity exists- but merely to a possible entity itself. The semantics for any individual quantifier are given by a function which takes in the denotation of a Noun phrase and uses it to construct the appropriate set of sets. The new interpretation rule for QPs is as follows:

[|QP|] = G[Q]([|N|])

And some examples of Quantifier semantics are as follows:

G["ves"] = λy. {x ⊆ U : y ⊆ x} ("every" or "all"; i.e, the set of all sets in the universe U that contain the entire set y)

G["jest"] = λy. {x ⊆ U : |y ∩ x| > 0} ("some"; i.e, the set of all sets in the universe that contain at least on element of y)

G["hiq"] = λy. {x ⊆ U : |y ∩ x| = 5} ("five"; i.e, the set of all sets in the universe that contain exactly some five elements of the set y)

This of course also requires a new formulation of the semantics for Noun phrases that produces the basis set of "referents of the right type". The new Noun rule is as follows:

[|N: n N|] = G[n] ∩ [|N|]

[|N: 0|]   = U

Rather than binding a new entity variable which we assert to satisfy a given predicate, or to be a member of a given set (given by looking up the Noun n in the lexicon), we simply directly construct the intersection of all of the sets that are the denotations of individual Nouns.

Finally, we have to update the interpretation rules for arguments (again) to handle the new kind of denotation for QPs:

[|A: QP R|] = λx. {y : ∃z. z ∈ x & [|R|](z)(y)} ∈ [|QP|]

[|A: QP e|] = λx.∃y. y ∈ [|QP|] & y ⊆ x

This says that a normal argument containing a role marker asserts that the set of all entities y which satisfy a particular relation with some element of the specifier set x is in the denotation of the QP; or, that an argument containing the specifier clitic asserts that some element y of the the denotation of the QP is a subset of the specifier set x.

Unfortunately, we've just undone our progress in allowing for the correct quantifier scopes! I constructing a compositional semantics for quantifiers in terms of sets, we have thrown out the conventions for establishing variable scopes in predicate logic- because we eliminated the variables! In order to recover the proper quantifier scopes, we're going to have to find a way to take into account the constraints imposed in lower scopes while constructing the sets for higher-scoped quantifiers.

We'll take a look at that problem in the next post in the series.

Non-intersective Nouns & Negative Scopes

This post is a follow-up to several prior discussion on formal semantics for a minimalistic monocategorial language. If you aren't familiar with them already, you may want to read part one and part two before proceeding.

Non-intersective Nouns

There is a class of adjectives called "non-intersective adjectives" because the set of referents of a noun phrase that includes them does not intersect the set of referents for the same noun phrase without.
This concept is best explained with examples: "a short basketball player" is also "a basketball player", so "short" is an intersective adjective--the set of "short things" intersects the set of "basketball players", and the meaning of the whole phrase is the intersection of those two sets.
On the other hand, "a former basketball player" is not "a basketball player", so "former" is non-intersective. The meaning of "former basketball player" is not a subset of "basketball player", and isn't formed by intersecting it with anything. It's a completely disjoint set, but one that does have a logical relationship to the set of "basketball players".

Similarly, if you "almost finish", then you do not "finish", so "almost" is a non-intersective adverb.

My formal education in formal semantics only really exposed me to one way of modelling non-intersective adjectives/adverbs: as higher-order functions that take predicates as arguments and produce new, different predicates. Thus, the predicate logic notation for a "former player" is former(player')(x) (vs. player(x)) , for some entity x.

But this model is not very compositional (i.e., if a "former player" is former(player')(x), a "former basketball player" is... what exactly?), and results in icky complicated interpretation rules. As a result, I have so far mostly avoided them in WSL, and the few times I've needed them I've punted
and just decided that they are taken care of by morphological affixes. That works as long as you just want to say something like "a former player", and leave the "basketball" (or any other additional descriptors) out of it.

Contemplating the programming language Prolog, however, provides a way out of this mess. Basic versions of Prolog do not have functions that can compute arbitrary values--just predicates that can be true or false. A Prolog system can, however, simulate functions by allowing you to query it about what the possible values are that would need to be plugged in to one or more argument positions of any given predicate in order to make it evaluate to "true". You can specify "input" values for whatever arguments you want, and for whatever arguments you leave unspecified, the Prolog system will give you a set of all possible values of those variables that satisfy the predicate as "output". Since Prolog is based on first-order logic, the exact same transformation for simulating functions works for modelling non-intersective adjectives in a predicate logic model for formal semantics.

So, non-intersective adjectives can be modeled as two-place predicates (with one input argument and one output argument) that specify the relation between the actual final referent of a noun phrase and the thing-that-it-isn't. Plus some mathematical machinery to glue it all together. That insight will allow us to add the capacity for translating non-intersective adjectives into our monocategorial language. I say "translate" because, being monocategorial, the language does not have a distinct class of "adjectives" to add non-intersective members to. Instead, it will have "non-intersective nouns" (or "non-intersective noun-jectives").

Introducing that machinery into our monocategorial language requires a little bit of updating to the existing interpretation rules. The altered syntactic rules are as follows:

[|P: w P|] = λx.λy.λz.[|w|](x, y, z, [|P|])
[|P: w ,|] = λx.λy.λz. [|w|](x, y, z, λx.λy.λz. y ⊆ z & ∃r. y ⊆ {z : ∃e. e ∈ x & r(e, z)})

Essentially, rather than immediately evaluating the denotation of the current word and the rest of the phrase with the same arguments, we pass the denotation of the rest of the phrase into the semantic function for the current word, which will allow the lexical semantics of a word to control what arguments are given to the rest of the phrase. At the end of the phrase, we pass in the "default" expressions that account for the possibility that no explicit quantifier or relation words were present. Note also that I have introduced the comma-separated argument list notation as "sugar" for repeated application of a curried function, since the large number of parameters to our lexical semantic level has started to become unwieldy.

Of course, since we've changed the lexical semantic interface in the syntactic interpretation rules, we have to also update the templates for our lexical semantic classes. The existing classes are updated as follows:

a) λx.λy.λz.λp. z ⊆ red & p(x, y, z)
b) λx.λy.λz.λp. y ⊆ {w : ∃e. e ∈ x & ag(e, w)} & p(x, y, z)
c) λx.λy.λz.λp. x ⊆ run & p(x, y, z)
d) λx.λy.λz.λp. x = y & p(x, y, z)
e) λx.λy.λz.λp. |y| > |z - y| & p(x, y, z)

In every case, we simply pass along all the original arguments directly to the semantic function for the remainder of the phrase, and logically conjoin it to the original lexical semantic expression. However, we now have the option of messing with those arguments if we so please; thus, we can introduce an additional semantic class for non-intersectives:

f) λx.λy.λz.λp.
        y ⊆ z & ∃r. y ⊆ {z : ∃e. e ∈ x & r(e, z)} &
        ∃x'.∃y'.∃z'. z ⊆ {a : ∃b. b & ∈ y' & former(a, b)} &
        p(x', y', z')

There's a lot going on here, but most of it is just book-keeping boilerplate. Let's break it down:
First, we cleanly terminate the description of the current referent; by the time we get to the end of the phrase, we won't be talking about the same thing anymore, so we have to throw in all the "just-in-case" assertions (that the referent set is some subset of the quantifier base and that it is has some relation to the sentential event) in here as well. In the next line, we assert the existence of a new quantifier base and a new referent set (z' and y', respectively) and, crucially, a new event x'--because if we're talking about a "former basketball player", then the "basketball player" which-he-isn't doesn't have any relation to this clause's event, so we have to replace it with something else. We then assert that all members of the current quantifier base have a specific relationship (in this case, "former") to some member of the new referent set. And finally, we use the rest of the phrase, denoted by p, to describe the new referent and new event.

The addition of this mechanism to the language has a few interesting long-range consequences. The most obvious is that word order actually matters. Up until now, we could've cheated on the phrase-structure rule and just said P → w w* , using the Kleene star operator for arbitrary repetition, instead of P → w P | w ,; but now, the fact that later words are in smaller phrase embedded at a lower syntactic level than previous words is very significant. Changing the ordering of words around a non-intersective noun can change which referent those words are actually describing. Similar effects are present in English; a "former blue car", for example, is not necessarily the same thing as a "blue former car". The first one may still be a car, but of a different color, while the second is definitely no longer a car, but definitely is blue.

Note that while syntax now encodes new information in word order about referent scope, the syntactic rules no longer encode information about logical connectives; the implicit conjunction of all lexical items is now a function of lexical semantics instead. We could consider this an accidental artifact of the formal framework we're using, but we might also come back to it later and exploit the lexification of logical connectives to come up with some new lexical semantic classes.

The second long-range consequence is that class-c words now have a real essential function; in the beginning of a phrase, prior to any non-intersectives, they are essentially adverbs, unconnected to the phrasal referent, which can float between phrases with no change in sentential meaning. After an intersective, however, they no longer act on the clausal event. Instead, they describe some event that the new referent is a participant in. The same applies to class-b and -d relation words.

Negative Scopes

Most non-intersectives have an implicit "not" built in to them. A "former basketball player" is not a "basketball player", a "fake Picasso" is not a "Picasso", and while an "alleged thief" might be a "thief", he also might not. And it turns out that with the interpretive machinery we've built so far, we can actually translate "not" as a non-intersective noun with the following lexical semantics:

not: λx.λy.λz.λp.
        y ⊆ z & ∃r. y ⊆ {z : ∃e. e ∈ x & r(e, z)} &
        ∃x'.∃y'.∃z'. z ∪ y' = {} &
        p(x', y', z')

I.e., "the relation between the quantifier base for the current referent set and the new referent set is that they have no elements in common".[1]

We can thus expect all non-intersectives to behave somehow similarly to negatives in the behavior they trigger in lower scopes.[2] This gives us a possible use for semantically-significant stress focus, in addition to the rising-falling intonation patterns that are used to denote phrase and sentence boundaries. We can use stress focus to indicate the specific reason that a referent "is not" something else. Referring to a previous example, the ambiguity of "former blue car" can be resolved by stating that it is either a "former blue car" or a "former blue car"--indicating that the reason the item in question is "former" is because it is no longer "blue" in the first case and no longer a "car" in the second.

More traditionally, we might want to make some special accommodations for phrases that are described by monotone-decreasing quantifiers, like "no" or "few", which are also "negative", in a slightly different way. Perhaps the "reason" for a decreasing quantifier can also be indicated by stress focus (do "few men work" or do "few men work"?) The practical impact of that level of ambiguity ("does this instance of focus refer to the quantifier or the non-intersective scope?") is likely to be minimal.

In either case, this will already be a very long blog post, so updating the syntax to recognize focused items is left as an exercise for the reader.

The Weird Ones

Non-intersective nouns, it turns out, don't have to be used only to translate what English encodes as non-intersective adjectives. They're just words that specify some arbitrary not-necessarily-intersecting relationship between the quantifier base for one referent set, and some other referent set. Y'know what else acts like that? Adnominal adpositions. E.g., prepositions that describe noun phrases. Also, genitives (for which English can use the preposition "of", but doesn't always).

Wanna say "Bob's cat climbs trees" in monocategorial form? "Cat agent of Bob, tree theme, climb event." The word "of" ends up as a non-intersective noun! Crazy! Incidentally, its semantics are as follows:

of: λx.λy.λz.λp.
        y ⊆ z & ∃r. y ⊆ {z : ∃e. e ∈ x & r(e, z)} &
        ∃x'.∃y'.∃z'. z ⊆ {a : ∃b. b & ∈ y' & ∃r. r(a, b)}
        p(x', y', z')

I.e., "all members of this quantifier base have some kind of relation with some member of the new referent set".

And there are all sorts of normal English nouns that entail a relationship to some other unspecified referent. "Father", for example. A "father" cannot exist without a child, so the concept of "father is naturally expressed in terms of a two-place predicate... which will be embedded inside the machinery for non-intersectives to allow describing both halves of the relationship, should you so desire[3]. So, "Bob's father build's houses"? In monocategorial form, it's "Agent father Bob, many house patient, build event". Note that in this case, the word order in the phrase "Agent father Bob" is critical. If we swap things around, we get the following different readings:

"father agent Bob": "Bob-who-does-something's father is somehow involved"
"Bob father agent": "Bob, who is the father of someone who does something, is somehow involved"
"Bob agent father": "Bob, who is a father, is the agent (builds houses)"

And if we want to cover both sides of the relationship at the same time: "John agent father Bob, many house patient, build event" ("Bob's father, John, builds houses.")

[1] Note that this is not the same as a translation for "no", the negative quantifier, which is addressed a little further down the page. This sort of "not" means "I'm talking a referent that is identified by not being that other thing", as opposed to saying "no referents of this description are involved in the action".
[2] While similar, this is actually not the same thing as the "negative scopes" that license negative polarity items like "anymore" in English. Those are a feature of the behavior of certain quantifiers, like the negative quantifier "no" described in [1].
[3] WSL encodes these kinds of nouns as Roles. When we get back to the semantic model for WSL in a later post, we'll see the utility of a productive morphological derivation system to turn Roles into non-intersective Nouns and vice-versa.

Generalized Quantifiers for a Monocategorial Language

At the end of yesterday's post, I briefly mentioned the concept of generalized quantifiers. Today, I want to investigate how the semantics of the minimalistic language can be extended with that concept to eliminate the need for "built-in" existential quantification.

The first step is to recognize that noun phrases do not necessarily denote single referents. In a sentence such as "Every student goes to school", for example, the phrase "every student", while grammatically singular, does not refer to just one student; rather, it denotes a set of students (in this case, all of them), and the sentence makes a statement about the properties of that set- all of it's members also belong to the set of things that go to school. Or, in other words, [|every student|] is a subset of [|goes to school|].

We can thus modify yesterday's semantics so that all entity variables actually refer to sets. In this case, the form of the syntactic interpretation rules need not change (although how we read them may be tweaked, replacing "There exists some x" with "There exists some set x"), but lexical semantics for each possible word type end up looking like this:

a) λx.λy. y ⊆ red
b) λx.λy. y ⊆ {z : ∃e. e ∈ x & ag(e, z)}
c) λx.λy. x ⊆ run
d) λx.λy. x = y

Note that predicates are defined by the set of arguments for which they are true. Thus, predicates are set-valued entities on which we can use set operators like ⊆ (subset) and ∈ (element of); the traditional predicate logic notation that we have been using so far, pred(x), is simply shorthand for the set-theoretic formula x ∈ pred.

The only major alteration introduced here is seen in the semantics for relation words, class b, which must be modified to explicitly construct the subset of entities which have a particular relation to some element of the set of events.

It is now straightforward to add a fifth semantic class of words which in some way restrict the cardinality of (or quantify) a set:

e) λx.λy. |y| = 5

(There could be an additional sixth class that restricts the cardinality of the event set, represented by the variable x, but for simplicity we will ignore that possibility for now.)

This works great for simple numerals (like 5, as shown in the example) and more vague things like "many" or "a few"; but, it causes problems for quantifiers like "most" or "one-third" which restrict the cardinality of the referent set compared to what it would have been if it were not quantified. We need some way of keeping track of that original maximal set.

In a more "normal" language, generalized quantifiers would operate at a separate syntactic level from nouns, and could take in the compositional denotation of the rest of a noun phrase all at once, and produce a new restricted set from it. In this language, however, we don't have that luxury. If we want to keep things monocategorial, we need to find some way of keeping track of the base set and restrictions on the final quantified set simultaneously as additional quantifiers and other words are added in arbitrary orders. This will require altering our lexical semantics to account for a third argument, and that will in turn require altering the syntactic interpretation rules to provide that third argument.

The altered interpretation rules look like this:

[|S|] = ∃x.[|C|](x)

[|C: P C|] = λx. ∃y. ∃z. [|P|](x)(y)(z) & [|C|](x)
[|C: P .|] = λx. ∃y. ∃z. [|P|](x)(y)(z)

[|P: w P|] = λx.λy.λz.[|w|](x)(y)(z) & [|P|](x)(y)(z)

[|P: w ,|] = λx.λy.λz. [|w|](x)(y)(z) & y ⊆ z & ∃r. y ⊆ {z : ∃e. e ∈ x & r(e, z)}

Here we have simply asserted the existence of an additional set variable, z, and added a term to define y, the set of referents for a phrase, to be a subset of z.

The new forms of the different lexical semantic classes look like this:

a) λx.λy.λz. z ⊆ red
b) λx.λy.λz. y ⊆ {w : ∃e. e ∈ x & ag(e, w)}
c) λx.λy.λz. x ⊆ run
d) λx.λy.λz. x = y
e) λx.λy.λz. |y| > |z - y|

Now, class-a noun-jective words operate on the set z, which forms the basis set for quantification. Relation words (classes b and d) act on y, which represents the actual referents of the phrase and is the result of quantification, and x, as before; and class-e quantifier words specify some relation between the set of referents y and the basis set z. The example given shows the semantics for the quantifier "most": the cardinality of the set of referents is greater than the cardinality of its difference with the basis set.

So far, we have not actually eliminated the need for logical existential quantifiers "built-in" to our semantics, but we have eliminated their semantic effect; all entities described by a sentence in the monocategorial language are no longer implicitly existentially quantified. Rather, the quantifiers in our predicate logic forms serve only to bind variables that we can use to refer to the different referent sets; and the referent sets are explicitly quantified by whatever quantifier words you feel like using. In the absence of any explicit quantifier word, sentences are evaluated as being true for some, unspecified, subset of the basis. This is exactly the same level of ambiguity present in natural languages (like Mandarin) which lack obligatory grammatical number.

Actually eliminating the existential quantifiers from our predicate logic forms would require directly constructing the relevant sets from unions and intersections, so as to eliminate the need for a common variable to use to tie the different parts together. That, in turn, requires separating the classes of quantifier and relation words from the class of noun-jectives, which cannot be done while maintaining the monocategorial analysis. It will, however, be possible to do so in WSL, which does have the necessary multiple syntactic levels.

It should also be noted that in adding quantifier words to the monocategorial language, we actually did not need to go all the way to introducing the full formalism of "generalized quantifiers"- and in fact, it looks like we can't, since the order of compositional operations which motivates the use of generalized quantifiers in the semantics of natural languages just doesn't exist here. The denotations of our monocategorial phrases are simple sets of referents, whose members have some thematic relation to the implicit event. In contrast, the denotations of natural-language noun phrases composed of nouns and generalized quantifiers are sets of possible sets of referents with the appropriately restricted cardinalities; and the correct set of referents is then extracted at the next higher level of composition, when a role is assigned to the noun phrase by a verb or adposition. Without that extra level, the monocategorial language must simply specify the final referent set directly. Again, however, WSL does have the more typical separation of nouns, quantifiers, and role-assigning words at different syntactic levels, and so we will be able to explore a more naturalistic analysis for that language.

For more thoughts on the monocategorial language, see this following post.

A Sister Language for WSL

My last post was triggered by recent discussions on CONLANG-L which inspired me to start formalizing the semantics of WSL. But after I started formalizing WSL, those discussions kept going. And as it happens, I got inspired to start on the design of a new language, similar in basic structure to WSL but also very different. So, before we get to Part II of WSL's semantic model, we're going to take a brief detour through the basic design of a new sister language.

Background: This idea came out of a discussion on how to describe the semantics of a monocategorial language whose complete syntax could supposedly be described by the following simple grammar:

S → w S | 0

Or, "A sentence consists of a list of words." That's it. Any words in the language, in any order- all of them are grammatical sentences. Which, really, is equivalent to "no syntax at all". The obvious choice for semantic rules when presented with that syntax (or lack thereof) is David Gil's polyadic association operator, but the creator was adamant that that was not a correct analysis of his language. My conclusion was that "S → w S | 0" was simply not the correct grammar as claimed, but rather that it was a two-level structure that grouped words into distinct phrases, but where phrase boundaries are maximally ambiguous. This still permits every possible linear arrangement of words as a valid grammatical sentences, since the order of words within a phrase and the order of phrases within a sentence are still completely free.

With only one class of words and completely free word order, resulting in no way to tell where phrase boundaries are and how words should be grouped together, such a language would initially seem to be fairly useless- even discounting lexical ambiguity, the number of different possible interpretations grows as the square of the number of words in a sentence- an ambiguity load that dwarfs what exists in any natural language, and would quickly swamp what you can reasonably handle with pragmatics.

In a spoken language, though, more function words or morphological words would not necessarily be required to eliminate that ambiguity- phrase and sentence boundaries could be quite adequately delimited by intonation. And intonation can in turn by encoded in text via appropriate punctuation, while still reasonably claiming that this is a monocategorial language at the lexical level (although it will have multiple types of internal syntactic nodes). I've never really played with the intonation rules for a conlang before, and especially not the effect of intonation on semantics; and I haven't seen much of that documented in other people's conlangs, either. So, this is a pretty enticing opportunity to really isolate the semantics of suprasegmental intonation.

Now, the point of WSL was to create something that very obviously does not have anything that could reasonably be called a category of "verbs" at any level, but not necessarily to be simple or minimalistic. And WSL does in fact have quite an array of different parts of speech. But for this one, the aim will be to see how far it can go before it becomes necessary to add any additional lexical classes.

The syntax of this new language ends up looking up like this:

S → C
C → P C | P .
P → w P | w ,

This reads as "A sentence consist of a clause, a clause consists of a phrase followed by another clause or a phrase followed by a period, and a phrase consists of a word followed by another phrase, or a word followed by a comma." We also specify the phonological / orthographical rule that a sequence of ", ." coalesces into a single "."

At the phonological level, the "," and "." are realized as particular intonation patterns on the preceding phrase. I'm not wedded to anything yet, but I'm thinking rising tone over the last word of a phrase for ",", and contrasting falling tone for ".". That would lead to an intonation pattern over a whole sentence that consists of a series of level tones followed by rises, and then terminated by a fall.

The extra level of rules that turns an S into a C may seem superfluous (and if we just want to describe syntactic structure by itself, they are), but the extra level makes the semantic interpretation rules much simpler.

Those basic interpretation rules look like this:

[|S|] = ∃x.[|C|](x)
"There exists some x such that the denotation of C is true for x."

[|C: P C|] = λx. ∃y. [|P|](x)(y) & [|C|](x)
[|C: P .|] = λx. ∃y. [|P|](x)(y)
"For some x, there exists some y such that the denotation of P applied to x and y is true, and
the denotation of C is true for x."


[|P: w P|] = λx.λy.[|w|](x)(y) & [|P|](x)(y)
"For some x and y, the denotation of w and the denotation of P applied to x and y are true."

[|P: w ,|] = λx.λy. [|w|](x)(y) & ∃r. r(x,y)
"For some x and y, the denotation of w applied to x and y is true and some relation r exists between x and y."

Basically, this is just a fancy mathematically formalized way of saying that a sentence describes an event which gets passed into each sub-clause, and then each phrase describes its own separate entity, and the meaning of the whole sentence is just the conjunction of the meanings of each word, applied to the whole-sentence event and the entity for that word's containing phrase, along with the assertion that the entity for a phrase has some kind of relationship to the sentence.

Every word in the language has the "semantic interface" of a two-place predicate, or a two-argument curried lambda expression, taking in an event variable and an entity variable and specifying some restriction on either or both referents and/or a relationship between them.

Some words will be simple predicates that restrict the referent of the phrase, or tell you about its properties. They will have meanings the look something like this:

a) λx.λy. red(y)

which completely discards the event and just applies some predicate (in this case, "red") to the entity variable.

Some other words will be two-place relations that tell you about the thematic role of the entity in relation to the event. They will have meanings like

b) λx.λy. ag(x, y)

which tells you that the referent of this phrase (represented by the entity variable y) is the agent of the event.

And a third class of words will tell you about the event itself. These words could come in two sub-varieties; things that look like

c) λx.λy. run(x)

which discard the entity variable and just apply a predicate to the event; and things that look like

d) λx.λy. x = y

which tells you that the entity for this phrase is, in fact, an event, and that the event is a subset or superset of (or, in this particular case, simply is) the entity described by the enclosing phrase.

Now, semantic class c has the interesting property that, since the meanings of words in that class do not depend on the entity of the phrase, they can appear in any phrase in a sentence without altering the literal meaning. That's a fairly unique behavior, and could be used to argue for recognizing them as a separate part of speech from the rest, but they don't have to be analyzed so. Their syntactic behavior is undistinguished from every other word. Even so, I'm not sure if I will want to include some in the language for "fun", or if they should be disallowed so as to avoid the argument.

Also, the boundary between classes b and d is very fuzzy, since subset, superset, and identity could just as well be modeled as binary relations between a phrasal entity and a sentential event as things like "agent" and "patient" are.

Finally, the a category, which would typically seem to correspond with nouns and adjectives, also does not have any distinguished behavior compared to classes b, c, and d. Relation words and event words can be left out, and you can have a complete sentence that consists only of class-a semantic noun-jectives, which are asserted to exist and to have some unspecified relation to some unspecified event[1]. In WSL, role markers are obligatory, but here we have the extra "& ∃r. r(x,y)" in the interpretation of phrases just to account for the case where words with the semantics of a role marker are missing.
On the other hand, you can also leave out all class-a words, and have a complete sentence that consists only of class-b relations; and the same applies to the last two classes of event words as well. Finally, there are no selection rules that cause a word of any of the four classes to disallow the use of any other particular class in the same phrase or sentence; some combinations of words may be contradictory or nonsensical, but every string of words is grammatical, and can be interpreted.

It should also be possible to represent quantifiers in this framework, as totally undistinguished words at the syntactic level which merely happen to have another different internal structure in their lexical semantics. This would allow getting rid of some of the built-in existential quantifiers, but will first require removing a few layers of abstraction from my current semantic notation in order to uncover the set-theoretic mechanics of generalized quantifiers. My efforts to that effect are detailed in this follow-up post.

Next, I'd like to figure out some useful application for stress-marked focus, which could be indicated orthographically with Capital Letters or something. That will take some thinking, since English examples often rely on the semantics of some focus-sensitive lexical item, and using it that way would provide a good argument for recognizing focus-sensitive items as a second part-of-speech. But some really simple rising/falling intonation gets us pretty dang far doing nothing but marking linear phrase boundaries!

[1]  Which means that elliptical answers to questions aren't actually elliptical at all- they're still complete grammatical sentences!

A Progressive Model of WSL Syntax & Interpretation: Part 1

Late last year, as the result of a challenge to create a language with no distinction between verbs and nouns (and more strictly, with no verb phrases at all), I began working on a new conlang called WSL. Originally, this stood for "Weird Syntax Language", but has since been backformed to an acronym for the autonym "Wjerih Sarak Lezu", whose documentation is being slowly fleshed out in the linked-to Google Doc.

In response to a follow-up challenge, I have now started the process of producing a formal description of the syntax and semantics of WSL. While relatively simple compared to most languages, WSL is still rather intimidating to make a full syntactic model of in one go. So, I'm going to take the approach of doing little pieces of it at a time, with commentary.

First, we start with the most basic requirements for predicate-argument structure:

S → a e A
A → a p A | 0

This is (almost) equivalent in expressive power to the syntax for a fully binarized neo-Davidsonian predicate logic notation that has recently been under discussion on the CONLANG-L mailing list, whose syntax is given by the single production rule

S → paaS | 0

Where p represents a binary predicate and a represents an argument variable.

Compared to that, the WSL grammar has been altered in two significant ways:

1. p follows a instead of preceding
2. Binary predicates are replaced with unary predicates on the surface, and the first (potentially Davidsonian) logical argument of each predicate is required to be identical and specified once separately (being distinguished by the e token rather than a following p).

Extracting the common first argument reduces the expressive freedom of this grammar compared to the paaS language, in that any situation that requires referring to multiple entities that both occupy first-argument positions requires multiple sequential clauses. In exchange, we get the benefit of much reduced repetition.

In a more standard predicate logic notation, like paaS, as would be variables or constants with unique referents, and we would require predicates both to indicate the argument place occupied by each variable and to restrict the referents of variables in so doing. In WSL, however, we define ps to represent two-place predicates which specify named argument positions, and as to specify one-place
predicates which each take a unique argument that is not present in the surface syntax. The e can then be a unique symbol (in fact represented by the phonologically-variable clitic <=u>), since all the information about the identity of the shared first semantic argument will be provided by the a that precedes it.

The next level of complexity looks like this:

S → N e A
A → N r A | 0
N → n N | 0

Here, I have replaced the a for argument places with n (for 'noun'), due to the insertion of an N(oun) phrase layer between the individual one-place-predicate words and the A(argument) phrase level, which contains a two-place predicate. The representation of two-place predicates has also changed, replacing p with r (for 'role'). This now allows us to use multiple logical predicates (represented by
multiple surface nouns) to describe the same argument (i.e., to take the same implicit semantic variable as their argument). This allows us to translate English phrases which, for example, use adjectives to describe nouns, or adverbs to describe verbs, except that WSL syntax does not distinguish the adjectives from the noun or the adverbs from the verb. Each shared variable can, however, still occupy only a single role. To relax this restriction, we make the following additional modification:

S → N e A
A → N R A | 0
N → n N | 0
R → r R | 0

Now, we allow multiple two-place predicates to take the same semantic arguments (where each level of A-phrase embedding introduces a new semantic variable), thus allowing for the expression of reflexives (among other things), as well as allowing multiple one-place predicates to take the same arguments. Given an appropriate range of lexical semantics selections for the predicates, this allows for the expression of arbitrarily complex semantic graphs (within the space allowed by the restriction that all two-place predicates share one common first argument) among arbitrarily-precisely described
referents.

The next significant addition to the syntax is to allow the use of explicit quantifiers ("all", "most", "some", etc.). That is done as follows:

S  → QP e A
A  → QP R A | 0
QP → Q N
N  → n N | 0
R  → r R | 0

It may at first seem like we could have avoided adding an extra rule, and just modified the production rule for S to S → Q N e A, and for A to A → Q N R A | 0; it will become important later, however, that Q and N are bound together in a Quantifier Phrase, and that Quantifier Phrases are in fact internal to Arguments

We now have enough of WSL syntax built up to describe some basic, but interesting, declarative sentences. With that foundation laid, we will introduce the interpretation rules that give meaning to the syntax.
(In actuality, all grammatical WSL sentences require an additional part of speech known as a Projector, which distinguishes, for example, declarative sentences from question. The semantics of projectors are, however, rather complex; thus, we will ignore them for now and work strictly with declarative sentences with no explicit projector). 

In the notation for interpretation, [|x|] is used to indicate the denotation the syntax x; in cases where some particular type of syntactic node may have multiple options for the kind and arrangement of daughter nodes that it contains, [|x : y...|] is used to indicate the denotation of some syntactic node x consisting of daughters y....
G[s] is used to indicate looking up the meaning of the symbol s in the lexicon, and G[x:s] is used to disambiguate homonymous symbols belonging to different syntactic categories given by x, in the case where their denotations are different.

The denotation of any null syntactic node will be assumed to be empty; there is, however, still the possibility of phonologically-null lexical items, which have contentful denotations, and occupy non-null syntactic nodes. This is the primary use-case for the G[x:s] notation- to distinguish the different kinds of null lexemes.

The interpretation rules for this subset of WSL are as follows:

[|S|]  = [|QP|]([|A|])
[|A|]  = λx.[|QP|](λy. [|R|](x)(y) & [|A|](x))
[|QP|] = λz.G[Q]([|N|])(z)
[|N|]  = λy.G[n](y) & [|N|](y)
[|R|]  = λx.(λy. G[r](x,y) & [|R|](x)(y))

And the forms of the denotations for lexical items (or phrases) in the classes of Q, n, and r are:

G[Q:_] = λz.λw. _y. w(y) → z(y); i.e., some quantifier (represented by the placeholder _) binds a variable y and provides that variable to both of its arguments, where one argument is the denotation of a noun phrase whose truth value implies the truth of the second argument.

G[n:] is always some monovalent predicate.
G[r:] is always some bivalent predicate.

Note that these definitions contain no free variables. All variables are bound by either a quantifier or a lambda expression. This allows us to freely rename variables for clarity and to ensure that we can perform valid beta reductions of lambda expressions in any order.

Temporarily ignoring the syntax and semantics of projectors, we can now fully interpret many simple sentences like

"Ka vesu jes i ajs tey mot."

The parse of this sentence (again dispensing with the projector) is

(S (QP (Q "ves") (N (n 0))) (e "=u")
   (A (QP (Q 0) (N (n "jes"))
      (R (r "i"))
      (A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
         (R (r "mot"))))))

And the first few steps of the semantic derivation are:

[|(S (QP (Q "ves") (N (n 0))) (e "=u")
     (A (QP (Q 0) (N (n "jes"))
        (R (r "i"))
        (A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
        (R (r "mot"))))))
|]
= [|(QP (Q "ves") (N (n 0)))|]([|
    (A (QP (Q 0) (N (n "jes"))
       (R (r "i"))
       (A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
       (R (r "mot")))))
  |])
= G["ves"]([|n:0|])([|
    (A (QP (Q 0) (N (n "jes")))
       (R (r "i"))
       (A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
       (R (r "mot"))))
  |])
= ∀z. [|n:0|](z) →
      [|(A (QP (Q 0) (N (n "jes")))
            (R (r "i"))
            (A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
               (R (r "mot"))))
       |](z)
= ∀z. U(z) →
      [|(A (QP (Q 0) (N (n "jes")))
           (R (r "i"))
           (A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
              (R (r "mot"))))
      |](z)

Note here that the denotation of the null noun is the universal predicate U, which always true for any argument. This, we can simplify by removing "U(z) ->" from the formula with no change in meaning.
Skipping a few steps for brevity, we get to

= ∀z. G[Q:0]([|(N (n "jes"))|])(λy. [|(R (r "i"))|](z)(y)
  & [|(A (QP (Q 0) (N (N (n "ajs")) (n "tey"))) (R (r "mot")))|](z))

= ∀z. ∃b.[|(N (n "jes"))|](b) →
  (λy. [|(R (r "i"))|](z)(y)
     & [|(A (QP (Q 0) (N (N (n "ajs")) (n "tey")))
         (R (r "mot")))|](z))(b)

Note here that the null quantifier has the semantics of an existential. After a long string of additional reductions, we end up with

= ∀z. ∃b. G[n:"jes"](b) →
    ag(z, b) & ∃d. (λy. G[n:"tey"](y) & G[n:"ajs"](y))(d) →
    G[r:"mot"](z)(y)

= ∀z. ∃b. 1sg(b) →
    ag(z, b) & ∃d. place(d) & this(d) → near(z, d)

"For all entities z, there exists some entity b such that b is me and[1] that b is the agent of z and that there exists some entity d such that d is a place and d is 'this-ish' (i.e., nearby and capable of being pointed at), which implies that z occurs near d."

Or, in normal English: "I do everything around here."

Note, however, that this is an extremely hyperbolic version of "I do everything around here." Literally, it means that there exists nothing which is not both close by some other place that's near me and of my doing.
Expressing a more typical meaning for the English sentence like "for all z such that z is near here, I do z" requires additional syntactic machinery to insert the necessary qualifications into the body of the quantifier phrase, which I may or may not ever get around to actually formalizing.
Also note that this model contains no rules for quantifier raising; thus, other possible readings of the English version, like "there is some specific place ('here') near which everything is of my doing", also cannot be expressed. It turns out that WSL does not have quantifier raising at all (not merely excluded from the subset described so far)- the scope of quantifiers in the semantics is exactly given by the order of quantifier phrases in the surface syntax. Expressing different quantifier scopes thus requires some mechanism for allowing the clause specifier (the first QP which is not part of an argument phrase, marked by the e symbol which is realized on the surface as the clitic <=u>) to move around to non-initial positions.

Formalizing the semantics for a larger subset of WSL that allows arbitrary specifier placement to control quantifier scope is rather complicated (as it require splitting the interpretation of argument phrases in half), so I shall leave that for a later installment.

[1] Alternately: "there exists some entity b such that b being me implies...."