Yes, making quantifier expressions into sumti makes for problems, since it makes it seem they have referents rather than ranges (I note in passing that what you *say her makes it sound like sumti were things rather than expressions -- and I admit that thanks to JCB, Lojban does talk that way sometimes). If you ignore their position, however, and just look at their form (admitting again that having an implicit quantifier makes this somewhat harder) then you have an operator and a variable it binds an an occurrence of that variable. In I, you are not allowed to infer "there is a" from "not not there is a" but the ground for that is in the logic of"not", not of the quantifier. Yes, the range of second order quantifiers is predicates, but, barring the oddity of what you say earlier, this just what you *meant before.
You second point seems to play on Russell's axiom of reducibility, which says that every higher order sentence (and in Russellthat is a lot more than just second order) has an equivalent sentence in first order. The plausibility of this derives from the fact that is true of the pro sties of predicates that we immediate think of: definition, transitivity, reflexivity, symmetry, subordination, and so on. It's generally thought not true by people who still work with this system . And ignored by everyone else, who are off doing other things, either totally first (like modern set theory) or managing to do second order without this reliance (though perfectly happy to use the related sentences as shortcuts where they clarify and are independently justified).
Ro bu'a zo'u ganai ge ge da bu'a de gi roda rode rodi zo'u ganai ge da bu'a de gi de bu'a di gi da bu'a di gi roda rode zo'u ganai da bu'a de gi de bu'a da gi rada zo'u da bu'a da. ( second order but highly reduced).
The real gripe is the lack of clear rules for giving characteristic functions ("predicates" in the confused language that is common here).
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What is hackish about it is the syntax. In {ro bu'a zo'u da bu'a}, {ro bu'a} is syntactically a sumti, consisting, presumably, of all x1's of whatever referent the {bu'a} ultimately has (thinking classically; in intuitionistic logic, what I am saying is not entirely sound), since that's what it means elsewhere. And yet then the thing that is actually quantified over is a collection of predicates. It's just an arbitrary decision plopped into the CLL, changing the type of a variable abruptly. My gripe about it is irrationally amplified by my irrational mental attempts to statically type Lojban, but even putting that aside I'd say it's a valid gripe, albeit irrelevant from a practical standpoint since predicate quantification has seen no use so far as far as I know, except in my example
here.
We're also missing second order predicates, or at least things that serve as convenient
required predicates in second order discussions, in general. Saying something like "{mintu} is the opposite of {drata}", even with {me'ei} (of selma'o LE, which converts selbri into abstract predicate sumti; I do NOT think quoted words are suitable as placeholders for abstract predicates), becomes:
me'ei mintu cu dukti me'ei drata ....????
and I ultimately have no way of writing down the dukti3. (I would be pleased if you could, actually.) The only precise way of saying this (yes, I admit the English is as imprecise as a statement involving {dukti} with no dukti3, but I hope you can see the precise meaning that I want to encapsulate) that I can see is first order to its core:
ro da ro de ro di zo'u go da de di mintu gi da de di to'e drata
mu'o mi'e latros
On Tue, Jul 26, 2011 at 11:58 AM, John E Clifford
<kali9putra@yahoo.com> wrote:
I'm not sure what you take as hackish about it. Simply allows quantifiers over predicate variables, which is all that is required. To be sure, it is does not seem to allow such quantifiers any place but prenex ('ko'a (cu) suo bu'a' don't seem to compute), but the embedded quantifiers are a main source of difficulty (pace xorxes) in reconstructing the logic of Lojban, so this may not be a flaw. What then is hackish? The pattern of real Logic is followed (less a mess of sub- and superscripts that are largely irrelevant to Lojban). Yes, Lojban is based on first order, but, then, so is second order and Lojban allows that extension (and, in principle, all the other orders on up).
As I said, part of the problem is to figure just what a predicate is in Lojban.
There are several candidates (sticking to unary predicates for simplicity): the things that have the property, the set of things that have the property, the characteristic function of that set, and the property, which may or may not be what a Montagovian would call a property. Three of these have clear expressions in Lojban, but the characteristic function does not really, but is the best candidate for the predicate in what follows in second order claims. There is talk of the lambda calculus but it is unimplemented, so far as I can see (and is second order).
From: Ian Johnson <blindbravado@gmail.com>
To:
lojban@googlegroups.com
Sent: Tue, July 26, 2011 9:24:10 AM
Subject: Re: [lojban] bu'a
Erm, poor phrasing; I meant that there is no easy way to get between those three things.
.u'u .i mu'o mi'e latros
On Tue, Jul 26, 2011 at 10:22 AM, Ian Johnson
<blindbravado@gmail.com> wrote:
Quantification over predicates was implemented in a horrifyingly hackish way. This alone is a problem, in my opinion. There is also, at least not in the main body of the language, an easy way to go from predicate-as-function (selbri) to predicate-as-concrete-object (typical sumti) to predicate-as-abstract-object.
Lojban is definitely based on FOPL, though, not SOPL, and not a bizarre hybrid of the two.
mu'o mi'e latros
On Sun, Jul 24, 2011 at 2:51 PM, John E. Clifford
<kali9putra@yahoo.com> wrote:
Lojban isn't clearly of one order or the other, since it treats sets and properties and the like on a par with tree and dogs. There is no particular problem in grammar or vocabulary to treating properties of predicates and quantification over them. There are some arguments about the correct way to express a predicate as an argument, but that seems to revolve around just what a predicate is in Lojban ontology. All the answers yield grammatical and intelligible results, though sometimes different ones. None of them seem particularly stilted, but I haven't seen enough cases to get a feel for that.
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I think bu'a/bu'e/bu'i would be much much much more useful if Lojban were a second order language, because then we could talk about the existence of predicates with desired properties in a non-stilted fashion. As a first order language, though, with second order mechanisms requiring stilted language, I don't think bu'a/bu'e/bu'i are especially useful.
mu'o mi'e latros
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