Re: [lojban] Not talking about imaginary worlds

[Edward Cherlin <edward.cherlin.sy.67@aya.yale.edu>]

At 09:18 AM 7/2/2001, pycyn@aol.com wrote:
> In a message dated 7/2/2001 4:25:58 AM Central Daylight Time,
> edward.cherlin.sy.67@aya.yale.edu writes:
...
> <We can carry on such discussions either with a good deal of hand-waving, or
> by proposing a modal logical theory in which to carry on our discourse.
> We're not too keen on hand-waving here, and we certainly don't agree on a
> Lojbanic theory of modal logic sentence modifiers. This results in the
> typical infinite regress so familiar from the Tortoise and Achilles, where
> one of us says, "It's obvious!" and the other says, "No it isn't, it's
> impossible, and even if it were possible, I still wouldn't believe it.">
>
> Which is why I am suggesting turning to a "uses of language" approach, rather
> than trying to work this out in terms of other worlds, taking (almost) all
> language as descriptive.

OK. We agree in principle, and we can discuss the details.

> <The usual case is that we wish to suspend the operation of reductio ad
> absurdum (and excluded middle along with it) and use a somewhat limited
> form of positive, even constructive logic. "Let us suppose X" says the
> mathematician, physicist, or science fiction writer, "then ignoring the
> obvious contradictions, what happens?">
>
> Classically one or the other version of relevance logic or some sort of
> paraconsistency, but I repeat that that is missing a useful alternative in
> favor of a nearly useless formalism in descriptive language.

References, please. This sounds promising.

I think that we should have the *option* of specifying the logic we are 
using, just as some gismu have options for an ontology. At this point, I 
don't know what I might want to specify, but I want some rather general 
method for specifying it. For now, I will think about explicit methods of 
specification using existing grammar. Perhaps by the end of the grammar 
freeze, we will have something worth adding.

...
> <Building non-standard arithemetic and analysis requires that we work in two
> different logics simultaneously. Technically they are called first-order
> and second-order logic. We don't have a good way of describing this
> situation either in natural languages or in Lojban. If we did, I think it
> would go a long way toward clarifying the grammar puzzles that are
> exercising us today.>
>
> Well, Robinson's non-standard artihmetic does not involve second-order logic
> explicitly (or, any more than ordinary arithmetic does).  It is more a matter
> of object language and metalanguage:  The formulae look normal but what they
> mean is something else (Goedel's proof shows this more clearly, since we get
> interesting metalanguage readings of apparently uninteresting object language
> formulae.  Well, you get that in Robinson, too, but the metalnaguage readings
> are a lot less clear).

The object language is first-order, and the metalanguage must be at least 
second-order (although there must also be a non-formal language somewhere 
up the chain). First-order theories can talk about sets, and second-order 
theories can talk about sets of sets. Although this is rarely made 
explicit, it is necessary to change points of view constantly in developing 
non-standard arithmetic and analysis. Ordinary mathematical discourse 
discusses sets in a form that encompasses all levels of membership. This is 
not a well-defined concept, since it turns out that there are models of set 
theory with non-standard levels of nesting.

> I don't think this has a lot to do with the present problem, though.

Possibly. But "all progress depends on the unreasonable man", so I'll keep 
at it.

> <We will have to do what the mathematicians do--Work out how to express
> ourselves clumsily in the current language, and then invent a better one
> when we have a better idea of what we are doing.>
> Amen.

Selah.



Edward Cherlin
Generalist
"A knot! Oh, do let me help to undo it."
Alice in Wonderland