Predicate transformations (ZI/BI)

Particles of the ZI family (all particles starting with z-) are used to transform the immediately following predicate, before chaining or explicit bindings are performed.

To use a ZI on more than one predicate word it should be wrapped in brackets pe … pei, which itself will be prefixed with ZI.

Negations

zi allows the speaker to negate the predicate being prefixed. However, it doesn’t negate any existential variables created when performing arity mismatch resolution, other predicates in the chain, or explicit bindings. zi is called the short scope negation.

mi zi etiansa meon

\[ \begin{align} \text{mi}(c,e) &= \text{[$e$ is a speaker]} \\ \text{etiansa}(c,e,a) &= \text{[$e$ eats $a$]} \\ \text{meon}(c,e) &= \text{[$e$ is an apple]} \\ \ \\ \text{meon}_1(c,e) &= \text{meon}(c,e) \\ \ \\ \text{etiansa}_1(c,e,a) &= \color{magenta}{(\neg \text{etiansa}(c,e,a))} \wedge \text{meon}_1(c,e) \\ \text{etiansa}^w_1(c,e) &= \exists a. \text{etiansa}_1(c,e,a) \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{etiansa}^w_1(c,e) \\ \text{mi}^w_1(c) &= \exists e. \text{mi}_1(c,e) \end{align} \]

Assertion given $(c):
I and an apple exists, but I don’t eat it.
Would be false if no apple existed.

There also exists a long scope negation with particle bi, which also negates existential variables introduced by the arity mismatch resultion and other predicates in the chain or explicit bindings.

mi bi etiansa meon

\[ \begin{align} \text{etiansa}_1(c,e,a) &= \text{etiansa}(c,e,a) \wedge \text{meon}(c,e) \\ \text{etiansa}^w_1(c,e) &= \color{magenta}{\neg (\exists a. \text{etiansa}_1(c,e,a))} \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{etiansa}^w_1(c,e) \\ \text{mi}^w_1(c) &= \exists e. \text{mi}_1(c,e) \end{align} \]

Assertion given $(c):
I exist and it is false that I eat an apple.
It doesn’t imply the existence of an apple.

bi is in its own family BI as it transforms more than just the prefixed predicate.

Names

Prefixing a 1-ary predicate with za transforms it into a name: \([\text{$e$ is named with property $P$ by $a$}]\). SI can be added between za and the predicate to select which argument is used for the property. za is intransitive as it is most of the time not needed to define who named $(e) like that.

To speak about the name itself, zai must be used: \([\text{$e$ is the name corresponding to property $P$}]\).

Anaphora

ze allows the speaker to refer to the latest non-wrapped instance of the prefixed predicate instead of creating a new instance. Arguments provided to this reference predicate (ignoring the implicit context argument) are stated to match ones the instance had.

mian etiansa a mi dona ze mian

\[ \begin{align} \text{mian}(c,e) &= \text{[$e$ is a cat]} \\ \text{dona}(c,e,a) &= \text{[$e$ likes $a$]} \\ \ \\ \text{etiansa}_1(c,e,a) &= \text{etiansa}(c,e,a) \\ \text{etiansa}^w_1(c,e) &= \exists a. \text{etiansa}_1(c,e,a) \\ \ \\ \color{magenta}{\text{mian}_1(c,e_1)} &= \text{mian}(c,e_1) \wedge \text{etiansa}^w_1(c,e_1) \\ \text{mian}^w_1(c) &= \exists e. \text{mian}_1(c,e) \\ \ \\ \color{magenta}{\text{ze-mian}_1(c,e)} &= \color{magenta}{e = e_1} \\ \text{dona}_1(c,e_2,a_2) &= \text{dona}(c,e_2,a_2) \wedge \color{magenta}{\text{ze-mian}_1(c,a)} \\ \text{dona}^w_1(c,e) &= \exists a. \text{dona}_1(c,e,a) \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{dona}^w_1(c,e) \\ \text{mi}^w_1(c) &= \exists e. \text{mi}_1(c,e) \\ \end{align} \]

Assertion given $(c): A cat eats something.
Assertion given $(c): I like this cat.

When prefixing a compound, ze will refer to the lastest instance of that exact compound. To make sentences shorter, zei can be used before a predicate word to refer the latest compound containing this word.

ze eberban will refer to the latest eberban instance, while zei ban will refer to the latest compound containing ban, for exemple eberban.

Which one is the latest instance is determined by word order in the text, and using a predicate defined using this word doesn’t make it the latest again. Thus in

on gia mian blan a mian etiansa a gia dona ze mian,

ze mian refers to the mian in bold in the a sentence, and not the mian in gia’s definition.

However if the last instance of the word is indeed in a definition (or in some predicate that can be used multiple times), then ze refers to the last time it has been used. Thus in

on gia mian blan a gia dona ze mian,

ze mian refers to the mian inside gia’s definition, which is last used in the a sentence. The text can thus be translated as “A beautiful cat which likes itself”.

zeu allows to refer to the context variable used to evaluate the last instance of the predicate word, while zeiu does the same to refer to the latest compound containing this word.

TODO: Formalize how anaphora behaves when the last instance was evaluated multiple times with different values. This is related to donkey anaphora and is far from trivial to solve.

A related concept is forethought reference using KI/GI variables. A KI/GI variable is first assigned by prefixing with bo before being used in reference. Such KI variables can be used similarly to pronouns in other languages. Such GI variables are more complex to use, but are necessary to define some concepts of Eberban’s vocabulary.

Instantiation

zu/zui allows the speaker to instantiate an argument of some predicate, which is particularly useful when a predicate is shared using VI/FI or a KI variable or argument and we want to use it with some arguments. GI variables could be replaced with KI variables that are then used in chaining with zu/zui. zu instantiates the predicate with transitive behavior while zui is intransitive.

Other

zue allows the speaker to transform a predicate having an A 0-ary predicate place to a predicate having an A 1-ary predicate place which is satisfied by the E argument.

mi katmi mi jvin (I want that [I dance]) can thus be replaced by mi zue katmi jvin (I want to dance).

Note that they are not exactly equivalent, as in the first example the two mi might not refer to the same individual(s) (if there are multiple speakers), while in the second example it is necessarily the same. Aside from this slight difference both examples are here the same length and thus might seem a bit overkill, but it is more useful if the thing being repeated is longer than one syllable.