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Jan Burse wrote:
> Hi
>
> Confutus wrote:
> >>But I doubt that three value logic is useful for reasoning.
> >>Because we can get away without, and still have undefined.
> >>For example the empty theory neither entails p nor ~p,
> >>so this might be interpreted as undefined.
> >
> > The ability to deal consistently with that odious and uncomfortable
> > never-never land of the forbidden middle and the "unknown" and
> > "undefined" is precisely why it is useful in reasoning and has such
> > potentially broad application.
>
> FOL or propositional logic also deals consistently with
> the case that neither p nor ~p are entailed. You don't
> need to switch to 3-valued logic.
>
> 2-valued logic does the job. It does the job because
> consistency is not affected by undefined. Completness
> is affected by undefined, but consistency is not.
> Consistency is affected by over definition.
>
> For example if you have in a theory p and ~p, you get
> inconsisteny. But I said the theory contains neither
> p nor ~p.
In the main stream of mathematics, 2VL does the job, because
mathematicians work very hard to eliminate uncertainty. But
uncertainty repeatedly comes up anyway, for instance in working with
expressions whose numeric values are undefined, discussions of
infinity, discussions of provability, and discussions of completeness.
And, in the larger world outside mathematics, uncertainty is
commonplace.
> In your 3-valued logic you cannot define over definition,
> or lets call it error. So arguing similar you have argued
> in favor of 3-valued logic, I could argue in favor of
> 4-valued logic, because 3-valued logic is not able to
> represent error.
I've never said that L3 doesn't have its own limitations, because it
most certainly does. But it's not what you can do without it, nor what
you can't do with it, that determines whether a theory is useful. It's
what you can do with it and can't do without it (or can do more
easily with it than without it) , that determines whether it's useful.
> There was a thread about para consistent logic in
> sci.logic. But lets stick to undefined, and not confuse
> it with issues of consistency.
>
> > The modal operators of L3 have the same effect of
> > limiting uncertainty and containing it within the more
> > tractable 2-valued approach. That's partly why I don't
> > use the constant U, because then uncertainty would
> > tend to proliferate.
>
> I showed you in another post that a constant u is not
> necessary. So it is present, whether you want it or not,
> as soon as you have the operator [], you can talk about
> undefined.
I didn't quite follow the whole discussion of
A == A+ v ~(u v A-)
because still don't have an equivalent for the constant u. I can use
?A or one of its equivalent formulas, ( ~A- & ~A+, in your notation,
is one of them), to state that A has the value of U, but this is a
different assertion than A itself and has a different truth-value.
> Further I don't understand your slang, things like "limit
> uncertainy" or "tendency of uncertainty proliferation"
> need to be specified. Up til now, you didn't talk about
> these issues.
>
> Could you please elaborate.
One of the things I can do with this logic that people without it
can't is to discuss ideas from classical, modal, intiutionistic, and
paraconsistent logic in one formal language instead of four. Those are
just the areas I know about. I'm repeatedly and forcibly reminded of
the Blind Men and the Elephant.
For instance, one of the theorems of the logic can be stated,
informally, that if P and ~P are both possible, then P is equivocal,
(meaning that it has the truth value U). Another can be stated that a
contradiction (P and ~P) means that either P is impossible or P is
equivocal. If contradictions are impossible, you have the classical
two-valued case. If contradictions may be equivocal, then
paraconsistent logic may apply. These are intuitively agreeable,
common-sensical notions which are difficult to formalized in 2VL.
Yes, I can discuss uncertainty using <> and []. The third group of
theorems does exactly that. I speak of limiting uncertainty in the
sense that, for instance, in natural language, it's easy enough to
remain noncommital about arbitrary statements, e.g. "I don't whether
or not to allow P to have a third value". L3 requires a decision.
This is analogous to the "excluded middle" in classical 2VL. There are
many ways to restate this limitation, and there are also ways to escape
it. But the ways around this limitation are outside the scope of L3.
Even the fact of requiring formulas to have values of T in all cases
before they can be called theorems limits uncertainty.
By "proliferation of uncertainy" I mean that if I used U as a
constant, it would have the effect of introducing new functions, (for
instance D:= ( []P & U) would be a kind of denial of P. But
allowing just one of these would, in turn, allow me to introduce all
27 possible unary operators and any of the 3^9 binary connectives. This
is also (but in a different way) outside the scope of L3.
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