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Jan Burse wrote:
> http://www.sapiencekb.com/logic3rev.htm
>
> Cite:
> "Negation and the non-strict conditional are defined by these axioms.
> Other connectives can be defined. The first four are standard. The
> next two have apparently not been previously defined and used in
> Lukasiewicz logic: an oversight which has prevented this logic from
> being properly appreciated for nearly 90 years. The last two have
> been previously defined but little used."
>
> can be defined & have not been previously defined
> --> logic not properly appreciated
>
> I don't believe the above claim.
I can hardly blame you, since I can scaredly believe it myself. Hasn't
*anyone* put all this together before? Where's the catch? But
disbelief isn't disproof.
> []_ and <>_ are already used elsewhere,
> namely in _+ and _- as follows:
>
> A+ == []A
> A- == ~<>A
Well, yes, and the same functions appear in Lukasiewicz original
treatments also, I do believe. But <>_ and {}_ aren't the ones I was
referring to. It's [](_ -> _} and [](_< -> _} applied to the
Lukasiewicz conditional that seem to have gone unappreciated.
> This can be used to translate L3 formulas
> over 3-valued propositional variables V
> into two boolean formulas over 2-valued
> propositional variables V+ and V-, with the
> following rules:
>
> (A v B)+ := A+ v B+
> (A & B)+ := A+ & B+
> (~A)+ := A-
> (P)+ := P+
> (A v B)- := A- & B-
> (A & B)- := A- v B-
> (~A)- := A+
> (P)- := P-
>
> Example:
>
> (~P v Q)+ = P- v Q+
> (~P v Q)- = P+ & Q-
>
> The advantage of the _+ and _- translation
> is that one, depending on its need, can
> directly read off []_ and ~<>_ without
> going through 3 valued logic.
Granted, but without a well-behaved conditional this is only a fragment
of a full-featured logic.
> One can also reconstruct the 3 valued value,
> as the following holds:
>
> A == A+ v ~(u v A-)
>
> Note the above needs the constant u,
> I am not sure whether this is part
> of your L3, or whether a formula can
> be constructed in L3 that gives u.
No, this isn't a part of L3. The truth value U would work here, but
I use the truth values only to substitute into formulas in order to
evalute them, not as constants, and there is no formula that yields
only the truth value U.
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