User:Ramcinfo/scope study

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Scope operators are things that modify predications as a whole (bridi operators) or predicates (selbri operators). They are: tenses, some (or all, depending on whom you ask) modals, {na} negation, {da}, outer quantifiers, connectives, and constructs modified by {ji'a} and {po'o}. They are called scope operators because when multiple are present in the same bridi, they modify the bridi in the order based on their order in the bridi text.

Tenses and modals

When one operator modifies the result of another, it is said for latter to be in the scope of former.[1]

 [va'o ti] [ba] culno fa lo kabri [se va'u mi]

Linking operator to selbri with {be} allows to incorporate it into the selbri. It acts on selbri only, producing new, modified selbri, so it scopes after all other operators (and even after regular sumti, in a way).

 [va'o ti] [ba] cunlo ⟨be di'a⟩ fa lo kabri [se va'u mi]

Another way to incorporate an operator into selbri is {jai}-conversion. Such a link will have scope below operators linked with {be}, if any:

 ma [va'o ti] [ba] ⟨jai gau⟩ cunlo ⟨be di'a⟩ fai lo kabri [se va'u mi]

The order of bridi operators relative to other parts of bridi (being selbri and sumti) is irrelevant. Let us move them all in front of bridi. (We're adding {ku} to {ba} to turn it into sumti tcita from selbri tcita, so it could be an independent term.)

 [va'o ti] [ba ku] [se va'u mi] (⟨di'a ku⟩ ⟨gau ma⟩ culno) fa lo kabri

The operators modify the meaning of the sumti in this way:

 (va'o ti (ba ku (se va'u mi ((di'a ku (gau ma (culno))) fa lo kabri))))

or using composition and reverse-apply operators:

 (va'o ti ∘ ba ku ∘ se va'u mi)(d'a ku ∘ gau ma ∘ culno)(fa lo kabri)
The following happens in present context:
  The following happens in the future:
    The following happens for me:
      The thing is a cup
        This is done again:
          This is done by whom, I want to know.
            The thing is full.


It should be noted what a tense stating both direction and interval is one operator; and in fact such tenses give their component in order reverse to the order of operators, e.g. {puzi} is equivalent to {ziku puku}.

Quantifiers

{na}, absent in the example above, behaves the same way, as do outer quantifiers — except the latter create {da}* variables behind the scene:

 mi na pendo ci gerku   => [na ku] [ci da poi gerku] zo'u mi pendo da
 "It is not true what each of four dogs is my pet"
 mi pendo ci gerku naku => [ci da poi gerku] [na ku] zo'u mi pendo da
 "For each of four dogs, it is not true what it is my pet"
 (e.g. no one of four dogs is my pet)

Connectives

Logical connectives could be reduced to outer quantifiers, but it easier to expand them to bridi connectives. For latter, scope of operator inside first connected bridi ends at the end of the bridi, before the connective:

  (mi [na] sipta'i) .ije (mi gleki)

"I am not sleepy. And I am happy"

Scope of any operator could be extended to the connected predicates through common prenex:

  [naku] zo'u (mi sipta'i) .ije (mi gleki)

"It is not true what (I am sleepy and I am happy)"

Other connectives are expanded to bridi connectives:

  mi [na] (sipta'i) je (gleki) => [na] zo'u (mi sipta'i) .ije (mi gleki)
  mi (sipta'i) je (gleki) naku => (mi sipta'i [naku]) .ije (mi gleki [naku])
  mi [na] (sipta'i) gi'e (gleki) => (mi [na] sipta'i) .ije (mi gleki)
  mi [naku] (sipta'i) gi'e (gleki) => [naku] zo'u (mi sipta'i) .ije (mi gleki)
  mi (sipta'i) gi'e (gleki [naku]) => (mi sipta'i) .ije (mi gleki [naku])
  mi (sipta'i) gi'e (gleki vau) [naku] => (mi sipta'i [naku]) .ije (mi gleki [naku])
  mi .e do [na] sipta'i => (mi [na] sipta'i) .ije (do [na] sipta'i)
  naku mi .e do sipta'i => [naku] zo'u (mi sipta'i) .ije (mi gleki)

Todo

  • Rewrite more clearly.
  • Review with other Lojbanists.

Credits

Study done with the help of omni___, xorxes, Ilmen, gleki and latro`a; and maybe someone I forgot to mention — sorry, then.

Followup

This study is close to la tersmu predicate logic analysis, but diverges from it in some points.

 [va'o ti] [ba] culno fa lo kabri [se va'u mi]

 kabri(c0)
 (va'o)(ti). (ba)(). (se va'u)(mi). culno(c0)
 cy no kabri
 .i [va'o ti] [ba ku] [se va'u mi] zo'u cy no culno

Fisrt, la tersmu doesn't distinguish modification of selbri with {be} from modification of bridi with sumti-tags; both threated as modification of bridi, and scoped in order of appearance:

 [va'o ti] [ba] cunlo ⟨be di'a⟩ fa lo kabri [se va'u mi]
 kabri(c0)
 (va'o)(ti). (ba)(). (di'a)() (se va'u)(mi). culno(c0)
 cy no kabri
 .i [va'o ti] [ba ku] ⟨di'a ku⟩ [se va'u mi] zo'u cy no culno

{jai}-conversion, on the other hand, scopes last, after all other modification, but still modifies the whole bridi:

 ⟨ma⟩ [va'o ti] [ba] ⟨jai gau⟩ cunlo ⟨be di'a⟩ fai lo kabri [se va'u mi]
 kabri(c0)
 ? x1. (va'o)(ti). (ba)(). (di'a)(). (se va'u)(mi). (gau)(x1). cunlo(c0)
 cy no kabri
 .i ma goi ko'a [va'o ti] [ba ku] ⟨di'a ku⟩ [se va'u mi] ⟨gau ko'a⟩ zo'u cy no culno

Compare with my analysis:

                [va'o ti] [ba ku] [se va'u mi] (⟨di'a ku⟩ ⟨gau ma⟩ culno) fa lo kabri

Second, there is slight difference in analysis of {na}: if it scopes last, la tersmu doesn't place it in prenex:

 na ku va'o ti broda
 !(va'o)(ti). broda( )
 na ku va'o ti zo'u broda

but:

va'o ti na ku broda
(va'o)(ti). !broda( )
va'o ti zo'u na ku broda

Finally, la tersmu doesn't handle scopes of connectives:

 mi [na] (sipta'i) je (gleki)
 !<sipta'i(_)){je}<gleki(_)>(mi)
 [na ku] mi (ke sipta'i ke'e) je (ke gleki ke'e)

gives same result as:

 mi (sipta'i) je (gleki) naku
 !<sipta'i(_)){je}<gleki(_)>(mi)
 [na ku] mi (ke sipta'i ke'e) je (ke gleki ke'e)