Posted By Confutus

More comparison of logical theorems from Whitehead and Russel with Lukasiewicz 3-valued logic:

2.01 (P -> ~P) - >~P

This rule, one of those that is called "reductio ad absurdam", or proof by contradiction, or negation introduction in natural deduction systems, does not hold in 3VL. However, in general if a proposition holds in 2 valued logic, there is usually at least one modified or restricted version using modal concepts in 3-valued logic that does holds.   (P -> ~P) -> <>~P (If P implies not-P, then possibly not P) is valid in 3VL.  (P -> ~<>P) -> ~P (if P implies not possible P, then not P) is also valid.

2.02 Q -> (P -> Q)

Valid in 2VL and 3VL

2.03 (P -> ~Q) -> (Q -> ~P)

The contrapositive works in 3 valued logic exactly as it does in 2 valued logic; a convenience.

2.04 ((P-> (Q->R)) ->( Q -> (P -> R))

Valid in 2VL and 3VL

2.05 (Q -> R) -> ((P-> Q) ->(P -> R))

Valid in 2VL and 3VL

2.06 (P -> Q) -> ((Q -> R) -> (P - R))

Valid in 2VL and 3VL

2.07 P > (P v P)

Valid in 2VL and 3VL

2.08 P-> P

Valid in 2VL and 3VL

 
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