No, trivalent (and other multivalent) logic does not deal with possible worlds, it is entirely cisworld, not transworld. You could combine the two, allowing multivalence in each possible world and perhaps even get some explanations of one logic in terms of the other, e.g, that a sentence gets the value 1 in a world just in case it got the value 2 in some world alternate to that one. But multivalent logics do not depend on such notions, nor reflect them very well at all (the metatheorem just cited doesn't go over to a theorem very well).
"Possible worlds" covers a multitude of different systems, not all of which can be conveniently brought into a single scheme, the Lewis series, say, or the slightly more generous Kripke one, or the Prior tense-based sets. Some things
are common to all (possible worlds, I suppose, for one) but techniques appropriate in one area, temporal modalities, say, do not apply in deontic or epistemic modalities nor in the various subjunctive realms: fiction, representation, contrary-to-fact conditionals, etc. For one major example, the role of the "real world" (the world of primary evaluation) in very different in these, critical in some cases, irrelevant in others, one world among many in still others. Similarly, the metrics which may be laid upon the the world-connecting vectors, have a variety of different rules, ranging from highly complex to non-existent. The genesis of the possible worlds is important in some cases, irrelevant (indeed, unaskable) in others. Lojban provides AT MOST a way of starting to talk in some of these ways, but nothing like the a fully functioning language for any of them.
I am not sure whether Lojban has even a fully
functional trivalent language, though I suspect that xorxes six unary functions and an appropriate understanding of the given connectives would permit us to build up something, a la Guzman. But it would be, at best, a very poor substitute for any possible world system or even the small practical parts, like the various subjunctives.
From: Gleki Arxokuna <gleki.is.my.name@gmail.com>
To: lojban@googlegroups.com
Sent: Sunday, August 5, 2012 12:16 PM
Subject: [lojban] Revising mu'ei and CAhA once again. Possible worlds.
Continuation of http://www.lojban.org/tiki/mu'ei
Note:This topic should be analysed from the Trivalent logic point of view as the latter also deals with Possible worlds.
But let's get started with more simple stuff.
mu'ei has always been a problem for me. Although the wiki was simple in describing it I felt something incomplete or illogical there.
Luckily, Lojbanistan has some authority and one can always ask how others solve the same problem.
Here is the log.
<gleki> Do you use mu'ei in real life? Do you have any thoughts of making a more generalised abstraction that will include both mu'ei and ba'oi?<robin>I did for a bit and then stopped; I just use {da'i} tricks now.<gleki>!!! just da'i or pada'i, su'oda'i, roda'i? how can you distinguish between ba'oi and mu'ei then?<robin>I don't find ba'oi useful at all. Just da'i.<gleki>but how can we distinguish two meanings? i just wanna some examples how we can use da'i for each case. If i can't use conditionals then i cant speak this language. Conditionals are the basics. What are your solutions for su'omu'ei, romu'ei, mu'ei. I can clearly see differences in their meaning important when speaking. Regardless the theory of alternate realities behind MUhEI I need words with such semantics. ko sidju mi<robin>So use mu'ei ? There's nothing wrong with them. su'o mu'ei is clearly ka'e. I have no idea what use ro mu'ei has; it looks totally pointless to me. Erm, as a bridi tag; as a sumti tag it's fine. Looking at http://www.lojban.org/tiki/mu'ei , for "If the train breaks down I'll be late" is {da'i mi lerci ri'a lo nu le trene cu spofu} "If the train breaks down I might be late" is not a structure I usually have to produce, but if I did I would just use cumki ; {lo nu mi lerci cu cumki lo nu le trene cu spofu}
So having this absolution granted from lojbo nolraitru I started revising mu'ei.
Here is what I came up with.
(if you can't see the image look here).
We have two layers. One describes alternate (possible worlds). And it's {ka'e}.If you have balls of one color only there are no alternate worlds. i.e. only bag in the middle has more than one output at M-level.
Therefore I opine that mu'ei is not a good cmavo as it's trying to express two levels and therefore two meaning at once. But cmavo should express one meaning each (being more close to semantic prims).
Strangely enough {pu'i} was out of consideration on mu'ei pages on lojban.org wiki. That's why mu'ei scheme is not complete and comprehensive.
Other issues including unsettled.
romu'ei is absurd.
bi'ai is described as naka'ena which in my scheme is equal to {ca'a}. But actually in the examples from the wiki bi'ai is used more like {pu'i}. In any case it's meaning is covered by the existing
cmavo.
ba'oi has extra meaning of alternate world identical to This World up to the present. This meaning is yet to be defined using new cmavo if my criticism of mu'ei is accepted.
da'i and va'o look like non-logical conditionals. Their meaning is out of my understanding. But I'm gonna use da'i more like Robin in those cases when I'm not sure what alternate-world-cmavo to use or in order to reach ambiguity.
ka'e is used more like an abbreviation of kakne. If the latter meaning of ka'e is fixed we need to find another cmavo for that purpose (for A-level).
naka'e has no cmavo for the output at M-level. Luckily naka'e is short enough to be used on it's own.
Conclusion.
mu'ei is not needed. If you wanna describe potential i.e. alternate worlds at A-level use naka'e, ka'e or naka'ena=ca'a.
If in possible worlds some balls are black and some are white then it's ka'e
that can result either in nu'o or in pu'i.
You can use all those cmavo as sumtcita as well which staisfies the need in most conditional sentences
(conditionals are sentences like "If I hadn't swum I would have been healthy" or similar).
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