Explicit binding (VI/FI/SI/PE/KI/GI/BA)

Left atom place selection

When the speaker wants to interact with an argument that normal chaining doesn’t select (due to transitivity, or because they want to interact with O or U arguments), it is possible to attach additional bindings using the VI family of particles after a predicate, followed by another (chain of) predicate(s), which is closed by the vei particle to return to the previously seen chaining behavior.

vei is the only member of family VEI, and not a member of VI.

ve, va, vo, vu allow binding atom (or generic) arguments :

mi duna [vo mo vei] meon

\[ \begin{align} \text{mi}(c,e) &= \text{[$e$ is a speaker]} \\ \text{duna}(c,e,a,o) &= \text{[$e$ gives $a$ to $o$]} \\ \text{mo}(c,e) &= \text{[$e$ is a listener]} \\ \text{meon}(c,e) &= \text{[$e$ is an apple]} \\ \ \\ \text{mo}_1(c,e) &= \text{mo}(c,e) \\ \text{meon}_1(c,e) &= \text{meon}(c,e) \\ \ \\ \text{duna}_1(c,e,a,o) &= \text{duna}(c,e,a,o) \color{magenta}{\wedge \text{mo}_1(c,o)} \wedge \text{meon}_1(c,a) \\ \text{duna}^w_1(c,e) &= \exists a \exists o. \text{duna}_1(c,e,a,o) \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{duna}^w_1(c,e) \\ \end{align} \]

Given $(c), $(e):
$(e) is a speaker who gives an apple to a listener.

To bind more than one argument of the same predicate we have to use the FI family, which interacts with the same predicate as the last non-closed VI.

The FI family follows the same pattern of vowels as VI. The above sentence can thus be rewritten as duna [ve mi, fo mo vei] meon or even as duna [ve mi, fo mo, fa meon vei]. FI have additional members: feu to bind the same place as the previous FI or VI in the chain, while fau binds the next place (this allows bind places of a predicate having more than 4 slots, which should however be rare).

In the last exemple it is possible to omit the final vei. It is, however, not possible to do so in the previous exemples as we would get the chain mo meon which is not what we want.

Arguments list

Between VI/FI and the inner predicate an arguments list can be provided by having zero or more KI/GI/BA terminated with be, in which case the bindings will occur with those arguments instead of the arguments of the predicate after be.

KI (all particles starting with k-) represent an atom or generic argument $(x) and have meaning \(ki(c,e) = [\text{$e$ is variable $x$}]\).

GI (all particles starting with g-), however, represents a predicate whose arity and type will be inferred from its usage in the sentence. All GI starting with gi- have intransitive sharing behavior, while the others starting with ge/ga/go/gu- have transitive behavior. For transitive GI, chaining to a GI ending with -i will be done by equivalence, otherwise by sharing (like the final -i rule for roots).

Any BA used in the argument list allow skipping this argument if it is not used in the inner predicate. Which BA member is used doesn’t matter.

mi dona [va ke be: mian bure ke]

\[ \begin{align} \text{mian}(c,e) &= \text{[$e$ is a cat]} \\ \text{bure}(c,e,a) &= \text{[$e$ eats $a$]} \\ \ \\ \text{bure}_1(c,e,a) &= \text{bure}(c,e,a) \wedge \color{magenta}{\text{ke}_1(c,a)} \\ \text{bure}^w_1(c,e) &= \exists a. \text{bure}_1(c,e,a) \\ \ \\ \text{mian}_1(c,e) &= \text{mian}(c,e) \wedge \text{bure}^w_1(c,e) \\ \text{mian}^w_1(c) &= \exists e. \text{mian}_1(c,e) \\ \ \\ \text{va}_1(c,e) &= \color{magenta}{\text{ke}_1(c,e)} \wedge \text{mian}^w_1(c) \\ \ \\ \text{dona}_1(c,e,a) &= \text{dona}(c,e,a) \wedge \text{va}_1(c,a) \\ \text{dona}^w_1(c,e) &= \exists a. \text{dona}_1(c,e,a) \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{dona}^w_1(c,e) \\ \end{align} \]

Given $(c), $(e):
$(e) is a speaker who likes [something eaten by a cat].

BA can also be used outside of the argument list, which adds an argument at the end of the list and uses it directly. ba declares an atom/generic argument (like a KI), bahi a transitivite predicate argument (like a gi-initial GI), and bahe an intransitive predicate argument (like a non gi-initial GI). bai, baihi and baihe on the other hand always adds an argument at the end of the sentence arguments list, which is mostly useful when asking questions.

The above example can thus be shortened as mi dona [va be mian bure ba].

Right place and chaining selection

When it is only needed to bind one or two places of a predicate, using VI/FI and multiple lists of multiple arguments quickly becomes verbose. For that reason, predicates can be prefixed with particles of family SI (all particles starting with s-), which override the chaining behavior.

se, sa, so, su select the place corresponding to its vowel both for the argument bound with a predicate on its right, and for the single argument that is exposed in the combined predicate. The right place is bound by sharing, while adding a final -i makes it bound by equivalence.

mian se bure blan

\[ \begin{align} \text{blan}(c,e) &= \text{[$e$ is beautiful]} \\ \ \\ \text{blan}_1(c,e) &= \text{blan}(c,e) \\ \ \\ \text{bure}_1(c,e,a) &= \text{bure}(c,e,a) \wedge \text{blan}_1(c,\color{magenta}{e}) \\ \text{bure}^w_1(c,\color{magenta}{e}) &= \exists a. \text{bure}_1(c,e,a) \\ \ \\ \text{mian}_1(c,e) &= \text{mian}(c,e) \wedge \text{bure}^w_1(c,e) \\ \end{align} \]

Given $(c), $(e):
$(e) is a cat that eats something, and $(e) is beautiful.

SI particles with two vowels (except i) allow to select both, the two vowels representing the two exposed place, and the second vowel corresponding to the slot bound with a predicate on its right. As with single vowel the right place is bound by sharing, while adding a final -i makes it bound by equivalence.

meon sae bure mian

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

Given $(c), $(e):
$(e) is an apple which is eaten by a cat.

SI particles follow a more general pattern to support more slots and usages:

  • si followed by a single vowel makes the predicate “transparent” by re-exposing all the places of the predicate on its right, which is bound to the slot corresponding to the vowel used. This is mostly useful with predicates having a proposition (0-ary) place, such that the predicates before and after can be bound together “across” the predicate with the proposition place.
  • Otherwise, SI particles are composed of an s followed either by one or many e/a/o/u or a single i, which can optionally be followed by h followed by a single e/a/o/u. The one or many e/a/o/u lists the slots of the predicate to expose, while a single i exposes none. The h followed by a single e/a/o/u selects the slot bound with a predicate on the right. When absent, the last e/a/o/u is used instead. Final -i encodes if the place is bound with sharing or equivalence.

Exemples :

  • sia: Transparent A
  • sea: Expose E and A, chain to A with sharing
  • seho: Expose E only, chain to O with sharing
  • saeoi: Expose A, E and O (in this order), chain to O with equivalence.

Left predicate place selection

Using ve, va, vo, vu on a predicate argument will not provide its definition but instead share it like an atom argument with the following predicate. The bound place must have the same predicate argument type however. To provide a definition of the predicate argument the particles vie, via, vio, viu (and FI equivalents) must be used. FI have again additional members: fei to bind the same place as the previous FI or VI in the chain, while fai binds the next place.

If we take the example tce mian from the previous chapter it is equivalent to tce via mian. Sharing with ve, va, … can be used like so:

mi katmi [va sae tuli mo]

\[ \begin{align} \text{katmi}(c,e,A) &= \text{[$e$ wants $A$ (0-ary) to be true]} \\ \text{tuli}(c,e,A) &= \text{[$e$ needs $A$ (0-ary) to be true]} \\ \text{mo}(c,e) &= \text{[$e$ is a listener]} \\ \ \\ \text{mo}_1(c,e) &= \text{mo}(c,e) \\ \ \\ \text{tuli}_1(c,e,A) &= \text{tuli}(c,e,A) \wedge \text{mo}_1(c,e) \\ \text{tuli}^w_1(c,A) &= \exists e. \text{tuli}_1(c,e,A) \\ \ \\ \text{katmi}_1(c,e,A) &= \text{katmi}(c,e,A) \color{magenta}{\wedge \text{tuli}^w_1(c,A)} \\ \text{katmi}^w_1(c,e) &= \exists A. \text{katmi}_1(c,e,A) \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{katmi}^w_1(c,e) \\ \end{align} \]

Given $(c), $(e):
$(e) is a speaker who wants some proposition (which is needed to be true by a listener) is true.

While using via, it has a different meaning :

mi katmi [via sae tuli mo]

\[ \begin{align} \text{mo}_1(c,e) &= \text{mo}(c,e) \\ \ \\ \text{tuli}_1(c,e,A) &= \text{tuli}(c,e,A) \wedge \text{mo}_1(c,e) \\ \text{tuli}^w_1(c) &= \exists e. \exists A. \text{tuli}_1(c,e,A) \\ \ \\ \text{katmi}_1(c,e,A) &= \text{katmi}(c,e,A) \color{magenta}{\wedge A \Leftrightarrow \text{tuli}^w_1} \\ \text{katmi}^w_1(c,e) &= \exists A. \text{katmi}_1(c,e,A) \\ \ \\ \text{mi}_1(c,e) &= \text{mi}(c,e) \wedge \text{katmi}^w_1(c,e) \\ \end{align} \]

Given $(c), $(e):
$(e) is a speaker who wants that [some truth is needed by a listener] is true.

Brackets

pe and pei are like spoken brackets that wrap a predicate or chain of predicates to define a new one, and have higher priority than chaining : A B pe C D pei E F will be chained in order A (B ([C D] (E F))) instead of A (B (C (D (E F)))). pe can also be followed by an argument list to define the arguments of this new predicate.

Using VI/FI/SI is often preferred and more simple than using pe and pei, but in some cases it is not possible. Those cases will be presented in later chapters.

Multi-places VI

VI family extends to particle having multiple EAOU vowels, each optionaly preceeded by a I. Each non-final EAOU will bind a single predicate from the following chain, while the last vowel binds the rest.

  • mi dona vao: mo, meon: I give you an apple.
  • mi kelo vaio: za ubob, sae coriu vihon: I’m thanksfull to Bob that I own a car.
  • drie veao: mi, meon, e uiuro sfia jo ta: I buy apple(s) for 0.2 EUR (total).