It has become a truism that every statement is either true or false. It might be supposed that this principle must be disproved before one can write a serious work on many-valued logic. This is by no means the case. In fact, the present volume will not constitute a disproof of such a principle. However, in the following chapters, systems of many-valued logic through the level of the first order predicate calculus will be constructed in such a manner that they are both consistent and complete. The tools of construction will include the logical procedures of ordinary English and those of some formalized systems of two-valued logic. Thus, in effect, we shall use the truism in constructing many-valued logics. It does not follow from this that ordinary two-valued logic is necessary for the construction of many-valued logic, but it does follow that it is sufficient for such constructions. The ability to establish such a sufficiency is certainly no more mysterious than the fact that a Harvard graduate can learn Sanskrit using his native English.
At this point, however, a word of warning is in order, for the treatment of many-valued logic which follows is concerned with the behavior of many-valued statements and not with their meaning. This indifference toward the meaning of many-valued statements indicates that we have no prejudices regarding the possible interpretations of our systems of many-valued logic. As far as our treatment is concerned, the meaning of a many-valued statement could be a linguistic entity such as a many-valued proposition or a physical entity such as one of many positional contacts. For that matter, the meaning of a many-valued statement might be quite different from either a proposition or a positional contact. In any case, regardless of the possible interpretations of many-valued logical systems, it is our opinion that . . .