[lojban] Second-order quantification has uses

[Corbin Simpson <mostawesomedude@gmail.com>]

coi

I wanted to follow up on a point of la tsani's from the thread "Reasoning by analogy". The point is raised that perhaps {bu'a} is not very useful. At least formally, though, it has enormous potential for giving categorical definitions of objects. (Here, "categorical" is used in the sense of Dedekind and Hilbert: A categorical structure is one which second-order logic can uniquely identify. The typical example is that the natural numbers, if they exist, are categorical.)

The example that I've been playing with recently is how to define {du}. Considering only the binary version of {du}, there are two definitions in jbovlaste currently. One is from la xorxes:

    lu'e x1 mintu lu'e x2 le ka ce'u sinxa makau

That is, {x1 du x2} means that x1 and x2 are references which are equivalent under the operation of looking up their referents; x1 and x2 refer to the same thing. This definition relies upon {mintu} and {sinxa}; I would hope that at least {mintu} could instead be defined in terms of {du}! (la xorxes also goes on to define {mintu} in terms of {dunli}.) Meanwhile, there's also this definition from la ilmen:

    x1 jo'u x2 simxu lo ka ro da poi selkai ce'u cu selkai ce'u .i va'i ro da se ckaji x1 .o x2

That is, {x1 du x2} means that the collection/set of {x1, x2} is self-similar/automorphic when we try to tell x1 and x2 apart by looking at the properties which characterize them; in other words, all properties apply to x1 iff they apply to x2. This is what la tsani refers to by discussing reified {ka} abstractions.

Now, consider this definition, transcribed from nonfirstorderizeable lore:

    ro bu'a zo'u x1 .o x2 bu'a

That is, {x1 du x2} means that, for any property P, P holds for x1 iff it holds for x2. No property can tell apart x1 and x2.

This is the power of second-order logic and {bu'a}, if we embrace it.

u'i .i mi'e la korvo .i co'o

u'isai ji'a Also notice that {x1 me x2} suddenly has a clean definition if we change {.o} to {na.a}, without using any plural logic whatsoever. Second-order logic knows what sets are, because it knows what properties are, and sets are merely extensions of properties.

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