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Date: Sun, 25 Jun 2000 05:05:47 EDT
Subject: Re: [lojban] RE:Trivalent Logics
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In a message dated 00-06-24 20:09:46 EDT, xorxes writes:
<< >Ooops! For functional completeness the system needs min(x,y), too and
that
>seems harder to get. Once it is gotten, however, it alone generates all of
>the connectives (binary, unary, more-ary), or rather the Sheffer function,
>min(x,y)+1, does.
Is there a simple :) way to see that this is true? >>
I think the short answer is "No," but, as you note, min is a natural "and"
and, it turns out, +1 is _a _ natural negation, so that we have a kind of
NAND here. If you feel comfy generating all 2-value connectives from NAND (I
have met people who do -- I am not one of them), then this goes just the same
(but it takes three negations to get back to where you started, of course).
<>
Yes. I have not looked at cases that are generated directly to see if there
are any really significant things missing. Everything I could think of (how
many can that be in three-valued logic?) -- various generalizations of binary
conditionals, disjunction, conjunctions, negations, equivalences, fixed-value
functions and the like are easy to get to, though, with the basic formula
(and usually several dozen different ways, given the different ways each of
the unary functions can be presented).
<>
I suspect we have more than enough to work with. As I noted, there seem to
be several system we could use -- evidentials, confidentials (I forget the
regular name, but you get the idea), probability, necessity-possibility,
concern (there is one unary that seems to mean, "I don't know and I don't
give a damn"). Whether these systems are separated and each complete in
Aymara, I haven't worked out yet (and probably can't given the state of the
paper) but we can fiddle a bit in Lojban. There is not, I think, unary
Sheffer function nor much info about minimal conditions for a complete set
(bivalent systems regularly got to a binary connective for (1,1) and (0,0)
and I think even combinatorics does).
I am not sure that I agree with your assignment of values above, but then I
am not sure I understand what many of these connectives (or these "truth"
values) are meant to mean either. I would, for example, have taken {pe'i} to
be (1,1,-1)-true if true or in doubt, false if false. (0,0,0) is what
Guzman assigns to the "I don't care" marker, maybe comparable to answers to
"Have you stopped beating your wife?"