Lukasiewicz attempted to employ his 3-valued logic to use the modes of medieval logicians

It is possible that:P,

It is impossible that P

It is necessary that P,

It is contingent that P,

It is true that P

it is false that P.

For various reasons, this program had only limited success. In his prefix notation, he used Mp to indicate" It is possible that P", Np to indicate "Not P", and Cpq to symbolize "if  P then Q". 

Lewis was more successful in his developmment of modal ideas, and he used a diamond and box notation <>P to indicate "it is possible that P", and [] P to indicate "necessarily P". 

Neither of these notions corresponds exactly to the usual meanings of "possible" and necessary" in either a casual sense or a more sophisticated philosophical sense, but I hope to explore the differences later.  I prefer the Lewis box-and-diamond notation to the Lukasiewicz notation, and will be using it.

I downloaded a PDF of Russell and Whitehead's "Principia Mathematicia" so I could do more detailed comparison of the various systems. One of the the first things that stands out is the definition of the material conditional P=) Q as (~P v Q). Not p or Q.  This definition of the material conditional is standard in classical two valued logic,

The 3-valued Lukasiewicz  conditional cannot be defined this way. It allows doubtful conditionals, in which "if P then Q" is evaluated as having the middle truth value when P is true and Q has the middle truth value, or when P has the middle truth value and Q is false.  This may be explored further.