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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.
[]_ and <>_ are already used elsewhere,
namely in _+ and _- as follows:
A+ == []A
A- == ~<>A
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.
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.
Bye
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