George Boole invented a system of algebra which dealt with classes of objects and logic.  A few years ago, I read through his work and attempted to make sense of it, and found it difficult and cumbersome. It was greatly improved and simplifed  a few years later when logicians began using an inclusive "or" (A or B or both), rather than the exclusive sense that Boole had used and  which is more common in ordinary discourse. This took off in three directions: 1) The algebra of sets, assisted by John Venn.  2) Propositional logic; the algebra of simple, unanalyzed statements, and 3) Predicate logic, the algebra of propositional functions and quantifiers (some, and all), which is has something of an intermediate status, and which has mostly replaced the old Aristotelian syllogistic logic. 

A major development in logic was the "Principia Mathematica", which introduced an axiomatic development of logic, and attempted to base all of mathematics on logic. This wasn't entirely successful, but it had an important influence on later works.

After developing the 3-valued logic, Lukasiewicz went on to test to the propositions of Principia Mathematica to see how many of them continued to hold. Most did, many did not, and a few were contradictions in3 valued logic. I intend to follow this line and examine which ones hold up, and which do not. Lukasiewicz went on to develop a many-valued logic, which could use any number between 0 and 1 for a truth value.

At about the same time, the American logician C. I. Lewis took a different approach, and attempted to improve on propositional logic using the axiomatic approach.

Both Lukasiewicz and Lewis attemped to develop Modal logic, including notions of possibility and necessity.  These two attempts proved to be incompatible, but the reasons for the incompatibility have not been fully explored by logicians. Their works are somewhat difficult to follow, since Lukasiewicz developed a form of notation, prefix or "Polish" notation, in which an operator precedes the expressions it operates on, while Lewis used the same style of expressions used in Principia Mathematica. This has fallen out of favor and is not much used today.  Nevertheless, some comparison of the various systems is still possible.