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Jan Burse wrote:
> Hi
>
> Confutus wrote:
> > It's [](_ -> _} and [](_< -> _} applied to
> > the Lukasiewicz conditional that seem to have
> > gone unappreciated.
>
> What are the curly braces above ("}"), and the
> plonked less than sign ("<") ?
Pardon my typos. That should have been a parenthesis, not a curly
brace, in both cases, and the <-> is meant to be the biconditional,
that is (A <-> B) = (A -> B) & (B -> A)
> > Granted, but without a well-behaved conditional
> > this is only a fragment of a full-featured logic.
>
> BTW:
> (A->B)+ = A- v (A+ & B+) v (~A- & ~A+ & ~B-)
> (A->B)- = A+ & B-
>
> Which can easily be read off from your 3-valued
> table, thus if you had the constant u in your
> system L3 you could represent A->B by the other
> connectives. Namely by combining (A->B)+ and (A->B)-
> as I suggested in my other post.
There are a couple of ways to do it. I think
(A -> B)+ = A- v B+ v (~A- & ~A+ & ~B- & ~B+)
also works. But these both look to be rather cumbersome places to
start. Why would anyone expect these formulas to be successful, if they
didn't have some clue to start with?
> But you dont need the constant u in your system.
> You can use an arbitrary propositional variable p,
> then we have:
>
> u == ~p+ & ~p-
>
> Or when we translate it back to your "modal" operators,
> we then have:
>
> u == ~[]p & <>p
>
> further notice that ~[]~A == <>A. Thus your whole
> system could be reduced to &, v, ~ and []
Since & and v are interdefinable, as in classical logic, only one of
them is necessary. It's also possible to take -> and ~ as primitive,
and define everything else.
The question isn't whether these definitions can be made and the
consequences explored. As far as I can tell, they _could_ have been
done any time since 1920 and from more than one starting place. It's
whether anyone has actually made the definitions, explored the
consequences, and recognized their potential significance. If so, who,
when, and where is it published, and why isn't it better known already?
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