It's also characteristic of this 3VL that it reduces to the two valued case when all the propositions involved are definitely true or false, and the middle value excluded. But a logic that has all the theorems of 2VL, and only those, would be equivalent to it. Not all the theorems of 2VL hold, just as not all the theorems of ordinary arithmetic hold for the integers, and not all those of real numbers apply to the complex numbers. Many theorems of 2VL hold in 3VL as well; typically those involving algebraic-type manipulations of expressions. However, some of them must be modified or restricted; typically those involving rules of inference.

As I've mentioned, direct proof via Modus ponens and transitive chains of inference doesn't work in standard Lukasiewicz logic; These rules need to be restricted to avoid dubious conditionals. Similarly, indirect proofs that rely on some form of "Reductio ad absurdum" also need to be restricted. It is not sufficient to prove P by assuming ~P and then deriving a contradiction Q and ~Q, because this isn't necessarily a contradiction in 3VL. Indirect proof is still possible, but it requires stronger contradictions of the forms "possible and impossible ("<>P & ~<>P) , or "Certainly and not necessarily []P & ~[]P", or even "Certain and impossible" (<>P & ~<>P).

There are also rules that express ideas that aren't available in 2VL. If Certainly P, then P; ([]P => P) and if P then possibly P (P => <>P) are both valid rules but their converses are not>

## Wednesday, May 25, 2005

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