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I will use Lojban for this example, but any logical language with similar relations could be examined; this should be portable to other Loglans, at least.

{cmima} relates sets to their elements; it can be glossed as "is an element of" and represents elementhood. Note that {cmima} is a binary relation.

In contrast, {steci} also relates sets and their elements, but it is a ternary relation. {steci} relates a set, one of its elements, and an extensional property of that element. Some definitions of {steci} imply that there is an entire subset of the given set which contains only the elements which have the given property.

Now, Lojban has the feature that some components of a relation may be omitted, deleted, ignored, or removed. If we remove the first place of {steci}, which relates extensional properties, then we are left with {cmima}. In this sense, if {steci} is defined as a primitive relation, along with removal of components, then {cmima} may be defined from {steci}.

With all of that background out of the way, does {steci} imply the Axiom of Choice? Suppose that we provide {steci} with a set and an element of that set. Then, the various extensional properties of that element are choice functions which select the element (and possibly other elements) from the set. This looks like the Axiom of Choice!

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  • This is an old question within the Lojban community, and I'm curious about what the wider conlang community thinks. This can be moved to a more technical forum, I suppose, but I'm not sure which one would be more appropriate.
    – Corbin
    Jul 16, 2023 at 15:23

1 Answer 1

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My guess: It doesn't.

Sketch argumentation: {steci} allows you to talk about choice functions, but it does not guarantee that the thing you are talking about is actually a well-defined choice function, think, e.g., of the smallest element of an open interval as subset of real numbers. In other words: You can formulate sentences in Lojban that aren't true. You can lie in Lojban (as well as in any other language), and you can utter things that aren't even lies but complete nonsense.

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  • {steci} does really do what it says on the tin. I'm not asking about arbitrary Lojban utterances; I'm asking about the semantics of particular selbri. Suppose I had asked whether {mlatu} always refers to Felidae, and you replied that people can lie and say things like {mi mlatu} "I'm a cat;" that's not what I was trying to understand.
    – Corbin
    Jul 17, 2023 at 14:33
  • The fact that there is a three-valued predicate steci implemented in Lojban does not make any assumption that it can always be satisfied, as far as I can see. Therefore the pure existence of that predicate implies nothing about the Axiom of Choice (my point of view).
    – Sir Cornflakes
    Jul 17, 2023 at 15:37
  • And, by the way, as far as I understand the math behind that, you don't need to invoke the Axiom of Choice when you have an explicit Choice Function, you need AoC only in the absence of that.
    – Sir Cornflakes
    Jul 17, 2023 at 15:40

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