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From: scope845hlang343jbo@icebubble.org
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Subject: [lojban] Multiple-variable abstractions (WAS: Re: Reasoning by analogy)
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Date: Wed, 06 Jan 2021 19:13:03 +0000
In-Reply-To: <103d554a-d7f5-918e-bb82-6e025e41131b@mail.jerrington.me> (Jacob
	Thomas Errington's message of "Sat, 02 Jan 2021 16:14:16 GMT")
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Jacob Thomas Errington <jake@mail.jerrington.me> writes:

> On 2021-01-01 19:56, scope845hlang343jbo@icebubble.org wrote:
>
>> What about something like this:
>>
>>    la lojban. bangu mi'o  <-->
>>
>>    la lojban. ce'o mi'o ckaji loka ce'u bangu ce'u
>
> The problem there is using a binary relation {lo ka ce'u ce'u bangu}
> where ckaji2 should be a unary relation. If we allow this, then we
> can't unambiguously interpret {ko'a ce'o ko'e ckaji lo ka broda}. Is
> it unpacking the tuple or not?

An example of a sentence exhibiting that ambiguity might be {ko'a ce'o
ko'e ckaji loka porsi}.  Would that mean {ko'a ce'o ko'e porsi} or {ko'a
porsi ko'e}?  It would be ambiguous.

A way to resolve such ambiguity might be to treat {lo ka broda} as a
solitary sumti (not a sequence) when the {ka} phrase doesn't contain
{ce'u}, but as a sequence when it contains multiple {ce'u}.  Such a rule
could become unwieldy when the property contains many arguments, i.e.
{lo ka ce'u broda ce'u ce'u ce'u ...}.  So, I might propose the
following rule:

  (1) The property is interpreted as a sequence when two or more {ce'u}
      are explicity expressed within it.

  (2) Otherwise (when there are zero or one {ce'u}), it is interpreted
      as a solitary sumti (as opposed to a sequence).

  (3) Any unexpressed places following the second {ce'u}, if any, are
      assumed to be {ce'u}.

This rule would introduce two questions: (A) How would one speak about a
sequence with just one element?, and (B) How would one differentiate
between properties with different numbers of arguments (arities)?

If I say {ko'a ce'o zi'o ckaji lo ka ce'u porsi}, I'm speaking about a
sequence containing one element, {ko'a}, but I'm not saying that {ko'a
porsi}.  On the other hand, if I say {ko'a ckaji loka ce'u ce'u porsi},
then it'd be true that {ko'a porsi}; it'd be the sequence {lo porsi ce'o
lo se porsi be ri}.  The problem of specifying a sequence with just one
element (A) is a challenged posed by Lojban's infix syntax, not by
abstractors like {ka}.

Issue (B), however, does not suggest any easy solutions.  Would {lo ka
ce'u broda ce'u} be a property with two arguments, or a property with
more than two arguments, such as {lo ka ce'u broda ce'u ce'u} or {lo ka
ce'u broda ce'u ce'u ce'u}, with the trailing {ce'u} unexpressed?  It
might be possible to quantify the number of arguments.

A ternary property, {lo ka ce'u broda ce'u ce'u}, could be expressed as
something like {lo ci ka broda} or {lo cimei ka broda}.  However, the
tanru form would introduce the semantic ambiguity inherent in tanru.
And the form with an inner quantifier might run afoul of Lojban's normal
interpertation of quantifiers.

This could be solved, however, by treating quantified abstractions as
"magic", kind of like the way quantified variables of selma'o GOhA are
treated magically in the terms of a prenex (preceeding {zo'u}).  I.e.,
{ro bu'a zo'u broda} doesn't mean the same thing as {roda zo'u broda},
because {ro bu'a zo'u} is magic.  So, if we consider {lo su'o ka} magic,
could we use this mechanism to speak about multi-variable properties?

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