On quantifiers and variables (Grammar update)

The purpose of this post is to document and summarize a recent change to Toaq’s variable binding syntax. This summary will only cover the new system rather than make a comparison between old and new. The old system is still documented at http://toaq.org/#quantifiers for now.

These are the quantifiers of Toaq:

Toaq Symbol Gloss
sa ∃ (existential quantifier) “some”, “there exist”, “there are”
tu ∀ (universal quantifier) “any”, “all”, “for all”
sıa ¬∃ “no”, “not any”
ke ι (iota) “the”, “those …”, “that which …”
ja λ lambda quantifier

These quantifiers make up the syntactic class SA (named after the member sa).

Here is a pseudo-grammar of the quantifiers, much simplified and omitting unrelated rules:

argument_phrase <- binding_phrase / bound_phrase

binding_phrase <- binding_phrase_1 relative_clause?
binding_phrase_1 <- quantifier t4-tone_phrase

bound_phrase <- t2-tone_phrase

This means that an argument phrase (or noun phrase) is always one of two types:

  1. binding a variable
  2. using an already bound variable

Binding a variable is done by using a quantifier before a t4 tone phrase, that is a predicate phrase carrying a falling tone. Such a phrase can optionally be followed by a relative clause, which acts as a restrictor on the domain of the quantifier.

Using an already bound variable is done by using a bare t2 tone phrase.

When binding a variable as described above, the predicate phrase following the quantifier becomes the name of the variable. It can then be re-used within the scope of its quantifier by using the same name with a t2 tone. Relative clauses are not part of the names.

Basic examples

Tỉ sa hảkō da. =
Sa hảkō bı tỉ hákō da.

[∃H : hakō(H)] tı(H).
“There are bears (here/there).”

Ke kủnē bı kảqgāı kúnē sa rảı da.
[ιK : kunē(K)] [∃R : raı(R)] kaqgāı(K,R).
“The dogs, they are seeing something.”

Ke jỉo tïjāo tu tỉeq bı dẻ jío.
[ιJ : jıo(J) ∧ [∀T : tıeq(T)] tıjāo(J,T)] de(J).
“The building that is far away from all roads, it’s beautiful.”

 

Advanced examples

It is possible to use a t2 tone phrase without the variable having been explicitly bound. In such instances, the variable is treated as having been ke-bound exophorically, i.e., by the current context, the shared knowledge or cultural background of the interlocutors, or similar.

Pronouns are by far the most common example of words that don’t need to be explicitly bound. It is usually clear who the speaker is referring to when saying “I/me” or “you”:

Mảı jí súq.
maı(J, S).
“I love you.”

Both variables (J and S in the logical translation) are bound by the context.

Variables remain bound across sentence boundaries until explicitly re-bound with a new quantifier. This is especially common with ke (“the”), but can also be useful with sa (“some”) and tu (“every”). The following examples demonstrate the resulting logic:

Chỏ jí tu rảı nïe ní kủa da. Jảq dẻ ráı da.
[∀R : raı(R) ∧ nıe(R,N)] cho(J,R). [∀R : raı(R) ∧ nıe(R,N) ∧ cho(J,R)] de(R).
“I like all the things in this room. They are so beautiful.”
“I like all the things in this room. All the things in this room that I like are beautiful.”

Kảqgāı jí sa gủosō da. Nủo gúosō da.
[∃G : guosō(G)] kaqgāı(J,G). [∀R : raı(R) ∧ guosō(R) ∧ kaqgāı(J,R)] nuo(R).
“I see some cows. They (the cows) are asleep.”
“I see some cows. All the cows that I see are asleep.”

Anaphora

Due to the way the binding syntax works, it is very easy to refer back to previously mentioned material. All one has to do is repeat the name of the variable with a t2 tone. However, sometimes the name can be a bit too long, making it impractical to repeat it verbatim. In such cases, using one of the anaphoric predicates is recommended. Here are the most important ones:

Toaq English
ho “he/she/they” (3rd person animate)
maq “it/they” (3rd person inanimate)
hoq “it/that” (3rd person abstract)
hoı …
the aforementioned …

Ke sảo ẻlū bı dủa súq chô jí .
“As for the big elephant, you know that I like him/her.”

Dẻoq sa gẻo dỉuchē déo da. Pảı hóı kỏaq jí da.
“An old scientist is talking to the children. The aforementioned adult is a friend of mine.”

fin

4 thoughts on “On quantifiers and variables (Grammar update)

  1. Thanks, I like this! I have a few questions — probably mostly just clarifications.

    > It can then be re-used within the scope of its quantifier by using the same name with a t2 tone. Relative clauses are not part of the names.

    Am I correct that this proposal doesn’t solve the problem of creating arbitrary variable names? For example, how would one say:
    [su da poi prenu ku ro de poi prenu zo’u gi ji da prami de gi de xebni da]?
    It seems like **Su poq? tu poq? bi to ru mai? poq/ poq/ ru loi? poq/ poq/** doesn’t work, because all poq refer to the second binding.

    I guess something like **Su rai? poqV da tu poq? bi …* works, but it seems like a bit of a work-around.

    > Ke jỉo tïjāo tu tỉeq bı dẻ jío.
    > [ιJ : jıo(J) ∧ [∀T : tıeq(T)] tıjāo(J,T)] de(J).

    Is tïjāo new syntax or a typo? I would have expected a relative clause, tiVjao-. Also, in that case, since there is no hoa/, I would have expected tieq to go into the 1st argument of tijao with the implicit hoa/ going in the second, not the other way around: [ιJ : jıo(J) ∧ [∀T : tıeq(T)] tıjāo(T, .)] de(J)a

    > It is possible to use a t2 tone phrase without the variable having been explicitly bound. In such instances, the variable is treated as having been ke-bound exophorically, i.e., by the current context, the shared knowledge or cultural background of the interlocutors, or similar.

    Does each such occurrence of a non-explicitly-bound variable imply a separate exophoric binding or is there just a single one? I assume it must be the former, otherwise it seems like quite a deviation from the old “default quantifier” behavior, where each non-quantified argument-tone term got its own default **ke** binding.

    Said differently, in the sentence: **mai? ji/ suq/ ru suq/**, am I necessarily loving the same person, or could the two different suq refer to different people (different bindings).

    > Variables remain bound across sentence boundaries until explicitly re-bound with a new quantifier.

    I assume this is only true for sentence-top-scope bindings, correct? In other words, in **Mai? ji/ tu poq? choV hoa/ sa ku?ne-. Gi? ku/ne-.** The second kune is nonsense, and not somehow referring to the variable bound in the scope of the relative clause, correct?

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    1. Am I correct that this proposal doesn’t solve the problem of creating arbitrary variable names? For example, how would one say: [su da poi prenu ku ro de poi prenu zo’u gi ji da prami de gi de xebni da]?
      It seems like **Su poq? tu poq? bi to ru mai? poq/ poq/ ru loi? poq/ poq/** doesn’t work, because all poq refer to the second binding.

      Correct. You need to use different names if you want different variables to be “active” at the same time. So here one could say sa pỏq tu hẻo pỏq or something similar. In English one often hears things like “if one person loves another person, then the first person has strong affection for the second person”.

      There is also room for completely arbitrary variable names (random strings of letters or a systematic range of words we could map out) as well as the option of using things like sa shỉkō pǒq na tu gủkō pǒq where shıkō and gukō don’t add any real meaning and only serve to keep the variables separate.

      Is tïjāo new syntax or a typo? I would have expected a relative clause, tiVjao-. Also, in that case, since there is no hoa/, I would have expected tieq to go into the 1st argument of tijao with the implicit hoa/ going in the second.

      Some changes to the third and seventh tones have occured. I recently discovered that there is a way to allow vowel-initial syllables into the language without introducing ambiguity, but it required some slight adjustments to the pronunciation of the third and seventh tones. The symbol/diacritic for the third tone changed as well to better reflect the new pronunciation. I will probably post a phonology update here soon.

      So tïjāo is not a typo, and starts a relative clause as you said.

      The thing about the missing hóa is another recent proposal (but we’re still experimenting with it to see how good or bad it is). Basically there are different versions of so-called auto-hoa, which has automatic place-filling of hoa in relative clauses, the details of which differ between different implementation of the proposal. Here it went in the first place automatically.

      Does each such occurrence of a non-explicitly-bound variable imply a separate exophoric binding or is there just a single one? I assume it must be the former, otherwise it seems like quite a deviation from the old “default quantifier” behavior.

      This is something that the new system changes. Previously, there was never a difference between bare :t2: and ke + :t2:, and therefore in a sentence like mảı jí súq ru súq, both súq could refer to different people, because both occurences had a quantifier (even if implicit) in front of them. In the new system this is not the case. Both occurences are bound. If you had bound them using sa or tu, you would expect them to both have the same referents. With the new system, ke now behaves exactly like the other quantifiers, so both suq mean the same no matter which quantifier binds them. If you want (potentially) different referents, all you have to do is use an explicit ke.

      I assume this is only true for sentence-top-scope bindings, correct? In other words, in **Mai? ji/ tu poq? choV hoa/ sa ku?ne-. Gi? ku/ne-.** The second kune is nonsense, and not somehow referring to the variable bound in the scope of the relative clause, correct?

      Hmm, I think not only the example you gave is probably nonsense, but any case where a variable is within the scope of another variable, even without syntactic subordination (relative clauses or content clauses), because the tu-expansion doesn’t work there (at least not straightforwardly). There may be ways to make it work in some cases, but I’m not sure yet. What’s so complicated about this is that many of these sentences (for example Dủa jí tî sa kủnē da. Nủı kúnē.) are interpretable for humans, but are hard to explain logically. I would not want to outright dismiss them. (https://en.wikipedia.org/wiki/Donkey_sentence)

      Thanks for the thoughtful response. Interestingly, many of the questions you asked also came up when the proposal was developed and discussed on Discord. I’m glad everyone is helping me make sure that nothing important gets overlooked! 🙂

      Like

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