Subject: Re: [lojban] Not talking about imaginary worlds

From: Edward Cherlin <[email-masked]>
Date: Mon, 02 Jul 2001 13:06:27 -0700

At 09:18 AM 7/2/2001, [email-masked] wrote:
>In a message dated 7/2/2001 4:25:58 AM Central Daylight Time,
>[email-masked] 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