 Lewis thought that there are an unlimited number of possible systems of logic. One example is the extensional propositional calculus of Principia according to which there are two truth values, true and false; other examples are the various systems of many valued logic that Lewis surveyed in SL, and, of course, Lewis’ own various modal systems S1–S5. Lewis thought that each of these systems is valid so long as it is internally consistent. The principles of the various alternatives simply define the meaning of the logical concepts and operators such as negation, truth/falsity, disjunction, implication, and thus they are all true (Lewis 1932, in Lewis 1970, 401). Bivalent systems simply have a different notion of truth and falsity from non-bivalent ones. Nonetheless, some systems may accord better than others with notions of truth or implication or deduction that are implicit in our everyday reasoning. Logics can thus be assessed pragmatically by their sufficiency for the guidance and testing of our usual deductions, systematic simplicity and convenience, and accord with our psychological limitations and mental habits. However, Lewis denied that he was claiming that principles of logic could be true without being necessarily true, or necessarily true without being necessarily necessary. A logic in which \(\Box p \fishhook \Box \Box p\) holds simply operates with a different notion of necessity from one in which it doesn’t.