Just a couple tentative thoughts on this old thread.
Hello everybody,
I would like to ask you clarifications on the meaning of the cmavo
{zo'e}, which is defined in the CLL at
https://lojban.github.io/cll/7/7/index.html
<https://lojban.github.io/cll/7/7/index.html> as meaning “the obvious
value”, “whatever I want it to mean but haven’t bothered to figure out,
or figure out how to express”.
Let's consider the following three example sentences:
• [A] {mi tirna zo'e}
• [B] {mi tirna su'o da}
• [C] {(da'o) mi tirna ko'a} (usage of a constant {ko'a} which hasn't
been assigned a value explicitly earlier)
How does [A] semantically differ from [B] and [C]? (I suspect that the
two latters ultimately mean the same thing.)
I think you're right about [B] and [C] as full sentences -- with the {da'o} in front, there seems to be no possible world in which one is true (or false) and not the other. And if two propositions have the same truth value in all possible worlds, then they must have the same truth-conditions, and therefore the same meaning. I believe this logical equivalence is reflected in a standard rewriting/transformation rule called existential instantiation (a.k.a existential elimination) which allows you to replace an existential quantifier and its variable with a constant under certain conditions. And there is a rewriting rule that goes in the opposite direction called existential generalization (a.k.a. existential introduction).
So it's not surprising that this is a confusing issue. In a lot of cases, whether you choose an existential quantifier or a constant does not matter much.
However, assuming that {ko'a} is unquantified and presupposed, then I do not believe [B] and [C] are freely interchangeable in practice as *nested* propositions. For example, under negation:
I don't hear anything.
I don't hear [whatever it is I don't hear].
It seems to me that the {naku} causes a pragmatic difference to surface -- [B1] is a universal claim based on the dual nature of {su'o} under de Morgan's laws, while [C1] invites the listener to try to pick out a salient value. And although perhaps {ko'a} could in principle mean something like "the whole domain of discourse", that would not be very cooperative, given that {su'o da} is available.
When forming questions it seems a similar difference surfaces:
• [B2] {xu do tirna su'o da}
Do you hear anything (at all)?
• [C2] {(da'o) xu mi tirna ko'a}
Do you hear it?
We shouldn't be too surprised to discover these differences; {su'o} seems to be about plugging in totally arbitrary values, or it could be conceived as a way to generate an iterated logical-disjunction of (a possibly infinite number of) propositions, in the same way that summation with capital sigma is iterates over addition); {ko'a} is much more like an anaphor for a certain semantic value, whether we know what it is exactly or not -- whatever it is though, it's not quite arbitrary.
I think there are three possibilities for
defining {zo'e}:
1.
{zo'e} means
{da'o ko'a}, at least at first glance.
2. {zo'e} means {su'o da}.
3. {zo'e} means either depending on context.
It's not hard to find examples that pull us both directions, given usual contexts:
D {mi na djuno [zo'e]}
pulls us toward
{da'o ko'a} (option 1).
E. {mi na mlatu [zo'e]}
pulls us toward {su'o da} (option 2).
The xorlo-izing BPFK definition of {lo} makes options 2 and 3 problematic, and so ultimately, it seems to me like the choice should be option 1. The problem, however, is that it would probably necessitate debloatification and/or polymorphism for many existing predicates (entailing a change to official definitions), to avoid undesirable "na mlatu" sentences. Or maybe, teach people to say {mi mlatu no da} instead of {mi na mlatu}. Or maybe just tolerate some rather odd pragmatics. (BTW the community is the boss; I am just giving an opinion.)
How [A] should be represented in logical notation?
In the simplest cases, {zo'e} could be represented by an unbound symbol, say the letter theta, subscripted with sequential integers for each appearance. The rule of interpretation would be that theta-symbols pick out whatever works. Some people would say there is a hidden existential quantifier in there, but I say no, it's a constant-like symbol with a presupposed value determined by context.
Another device increasingly used in formal semantics is the choice operator, borrowed from Hilbert, represented by curly lowercase epsilon, which I'll skip describing for now.
{zo'e} cannot be a constant as it changes its referent(s) on each
occurrence.
Yes, ultimately, {da'o ko'a} probably isn't adequate; {zo'e} has to be some kind of function that changes meaning depending on context and even takes arguments; that's sensitive to quantifiers (allowing it to take multiple values with {ro}, as Martin Bays pointed out once on the list); and yet also impervious to the scope of {naku}. Maybe something like Skolem functions, as Guskant suggested, though how exactly to formalize this for a logical notational rewriting for Lojban (or for any other loglang) remains to be seen (I can't remember how Martin did it). Maybe something like:
ro remna cu prami zo'e
[∀x remna(x)] prami(x, zohe(x))
Where
{zohe} here is not a predicate but a contextually determined function from entities to entities.
mi'e .maik.
mu'o