Posted By Confutus

Continuing on the subject of 3-valued Lukasiewicz logic, Lukasiewicz initially proposed an interpretation of "possible" for his third truth value, but its behavior doesn't really sustain such an intepretation, and he channged it later.

In common with classical logic, one of the basic functions is negation. In common with classical logic, The negation of a true statement is false, and the negation of a false statement is true. The negation of a statment with the third truth value has the third truth value.  This makes sense; if I don't know whether "Today is Friday" is true or false, I also don't know whether "Today is not Friday" is true or false, either.

There are two other possibilities for negation.  A strong negation of a statement with the third truth value would be false. A weak negation of a statement with the third truth value would be true.  I don't expect to make much use of the strong or the weak negation, because these can be expressed in other ways.

 

There is a larger problem with his notion of equivalence. He uses logical equivalence and defines it as a biconditional; if A is true, then B is true, and if B is true, then A is true. However, One of the elementary concepts of matthematical structure is an equivalence relation, which is defined as a relation that is reflexive (A = A), symmetric (if A = B, then B = A), and transitive (if A = B and B = C, then A = C). The relation Lukasiewicz defined fails the third property. His logical equivalence is not a mathematical equivalence. Most of the other three-valued logics that have been devised also fail this test.  In contrast, the classical two-valued biconditional is a mathematical equivalence.

 

This deficiency can be repaired using concepts Lukasiewicz defines later, but this an apparenly unrecognized and unremarked  major reason three-valued logic hasn't been as useful as classical two-valued logic.


 
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