possible worlds and mu'ei
Very brief explanation of what modal logic gives us, and what 'possible worlds' means. Then at the bottom, why Jordan DeLong now think mu'ei (and the related cmavo ba'oi and such) is in fact a good thing, despite having initially completely misunderstood it.
Hopefully this may be of use to anyone who lacks formal logic training, like myself, in understanding exactly what mu'ei is all about. (to those of you where leni do certu loi logji cu zmadu tu'a mi, please correct any mistakes here, or feel free to expand stuff, etc.)
First let's say what a 'world' is, in the context of logic. A world is defined by the set of things which exist in that world, and the truth value of every proposition which can be made in the world. In modal logic, two new concepts are added to propositional logic which allow making claims across worlds, but it's really only one concept since either can be written in terms of the other and negation. Specifically,
- [] - necessary ([]p -> ~<>~p (not possible that not p))
- <> - possible (<>p -> ~[]~p (not necessary that not p))
'Necessary' means that something must be true across all the worlds which can be 'accessed' from the current world, and 'possible' means it is true in at least one of those worlds. But what's it mean to 'access' worlds? Accessibility of a world from a different world is defined by a function R which takes the two worlds as arguments. The precise meaning of what it is for something to be 'necessary' or 'possible' is based on what R returns. For example, if we are dealing with logical necessity, for a given pair of worlds w1 and w2, R(w1, w2) returns true iff the logical rules of inference in both w1 and w2
yield results consistent with the truth values in the world - in other words you can only access worlds which conform to the rules of the logical system under which it is necessary. Different types of functions for R can be used to serve different purposes - allowing discussion of different types of necessity (for example epistemological necessity or moral necessity); essentially any definition for R can give us a different modal system (and this appears to be how mu'ei lets us do a certain kind of if).
The term 'possible worlds', then, refers to the set of worlds which are accessible from the current world. I.e. the set of worlds w for which a true []p (in *this* world) makes a true p in w (and note for pedantry that there's nothing to prevent that set from including this world also, and many definitions of R (such as the example logic-system based R) will be true for R(w, w) where w is the current world).
The necessary and possible operators [] and <> are in this way somewhat analogous to the universal and existential quantifiers of predicate logic (which, btw, makes it incidently nice that the mu'ei equivalents also involve ro and su'o).
romu'ei is the same as the [] operator and su'omu'ei means the <> operator. This means that
naku romu'eiku naku == su'omu'eiku
naku su'omu'eiku naku == romu'eiku
Because mu'ei is in ROI and generalized to simply take a PA, we can make statements about things like so'emu'ei (I guess it's 'mostly necessary'), though I don't think this feature is particularly compelling compared to giving us the [] and <> operators. The portion of the mu'ei cmavo (the tagged sumti) which is intended to allow rendering of counterfactual correlation essentially allows a way to specify some information about the R function: what is stated is that R(w, w') (where w is this world) will return true if the proposition which is contained in the tagged sumti is a true one in w' (though it may be true or false anyway - it is just a hint to what R() means).
Discussion
May I dumb it down even further? A 'world' is a hypothetical or real situation. If you have world w1 and world w2 - hypothetical situation A and hypothetical situation B - one is accessible from the other if both of them 'make sense', and follow the same rules of logic.
So if A has me blond and B has me a redhead, then A is accessible from B - both make sense, neither ruptures the time-space continuum or whatever. But if A has me blond and B has me be my own father, then B is not accessible from A: it violates rules of logic which presumably hold in A.
- pc:
- The rules of logic are the same from world to world (for this purpose, anyhow), what is decisive here are certain facts in each world, including generalizations like "'father of' is irreflexive."
If A is our real world, then B is any hypothetical situation that still follows the rules of logic. B can include you coming to work 10 minutes later than you did. It cannot include you killing your own grandfather before you were born. That scenario does not follow the rules of logic. You can tell, because it's a paradox. 'Possible world' really means just 'non-paradoxical scenario'.
- pc:
- Yes, I think it is useful to view possible worlds as less than complete worlds (strictly, I suppose, sets of complete worlds which coincide in the scenario. Notice, though, that totally different worlds may be accessible from a world - that is worlds with no overlapping scenario. I take 'non-paradoxical' to mean 'not self-contradictory', but I am unsure whether - for counterfactuals, e.g. - we need to insist on that.
So in modal logic, we reason over the totality of hypothetical situations. If there's no way something could happen without violating the rules of logic, that's the same as saying there is no possible world in which it could happen. Any hypothetical scenario in which I kill my grandfather before I am born is not a possible world; it's an impossible world.
- pc:
- relative to rules about procreation (and so to those worlds in which the meaning of 'grandfather' does not undergo major changes). Also I could, of course, kill the person in an alternate world who is my grandfather in this world without contradiction; modal logics make scope problems big time.
The reason 'worlds' are invoked is because hypothetical scenarios don't really otherwise fit into logic. Logic is all about propositions (bridi) about the world: they are either true or false. If you say "but it might have turned out different", propositional logic barks "whaddaya mean? It either happened or it didn't happen." Hypothetical scenarios have to be countable and orderable somewhere, if a logic is to be able to deal with them. (And there isn't anything that logically outlandish about "but it might have turned out different", after all.) So modal logic calls these scenarios 'worlds', because it has to count them somehow. The possible world constraint is there so you don't use this as an excuse to say logically outlandish things. And it is, as it turns out, the standard logical way of dealing with the hypothetical.
- nitcion:
- Sorry to dumb it down yet further, Jordan, but I will likely write a textbook entry on this one of these days...
- pne:
- Don't be sorry - I think I understood your explanation better than Jordan's.
- la xod:
- Only silly stereotyping and deservedly maligned "common sense" differentiate the worlds that are possible from those impossible. We have no license to discuss worlds that conflict with our corpus of observations: we can discuss the possible world wherein Hillary is elected president in 2008, but not any where she was elected in 2000.
- pc:
- Why not? Surely the pluperfect subjunctive (or some such thing) is a valuable as any other. For "what if" fiction, if nothing else.
- And Rosta:
- Natural language if is normally taken to involve not possible worlds in general, but rather, contextually relevant possible worlds. So, ro mu'ei should be understood as meaning "in all contextually relevant possible worlds". I think this covers Nick's point & xod's objection. Needless to say, some possible worlds are irrelevantly surreal in some contexts, but entirely pertinent in, say, sci-fi contexts.
- maikl:
- It seems to be that this whole notion depends upon making a distinction where none can be made. As if the outcome of a coin toss were actually exactly-50/50 (a computer can, given the initial forces, predict this outcome, however: it is itself a convention to assume they are equal)--whereas the outcome of a political election would be qualitatively different; & a world in which "ears discover fire" more different still. What it comes down to, is a judgment upon the intellectually-separated subevent X, that it seems to have been either largely certain or largely contingent: & this is itself dependent upon your understanding (does the "butterfly effect" control your coin toss? depends on how much you've read, & how much you believe of what you've read) of what CONTEXT means. But for all this, i suppose it is useful to have a way of talking about these things bau la .lojban.. But i would prefer to have it arduous (using cei broda) rather than simple: there are too many assumptions built into this usage, & Lojban does well to make us examine them.
- And Rosta:
- All this stuff about possible worlds etc. is just an attempt to model the meaning of natural language conditionals, the sort of everyday logic that all people use quotidianly. I don't understand what you would prefer for us to say. I'd be interested to know, but it should maybe go under the if page?
- Skorgu:
- I'm not sure if this is the right place to pose this, but here it is anyway. From the point of view of surrealist literature how would one go about proposing an impossible world and comparing it to our current system? Clearly you can't do this and have logic remain constant, but is there/could there be some method of signaling that logic might not hold for literary purposes? In other words, how would you discuss the paradoxical situation where I kill my own grandfather?
- You would discuss it the same way as in English. You can say, for example:
- mi na ka'e zvati ti su'omu'ei le du'u mi pu catra le mi roryrorci pu le nu ri rorci le rorci be mi
- I couldn't be here had I killed my grandparent before he/she engendered my parent.
- But you can also say:
- mi pu ca'a catra le mi roryrorci pu le nu ri rorci le rorci be mi ije ku'i ue mi za'o zvati ti
- I did actually kill my grandparent before he/she engendered my parent, but (surprise!) I'm still here.
- pc:
- Note, as above, that this is not about logic, but about premises -- semantic axioms, say; what does "x is grandfather of y" entail.