Boolean Many-Valued Logic

Contrary to the popular thought, classical logic, in the deductive sense, is not tied to bivalence (i.e., 2-valuedness) in its semantics, for any many-element Boolean algebra, e.g., 4-element, 8-element, 16-element, \(\ldots \) , or infinite-element Boolean algebra, can be used for the semantics equally well. Classical logic with a Boolean many-valued semantics is called Boolean many-valued logic. In philosophy we encounter situations in which we want to say that there is no fact of the matter about whether certain statements are true or false, such as the situations described in the Fission Problem for personal identity, the Sorites Paradox, and the Liar Paradox. In such situations, it is reasonable to give the relevant sentences neither Truth (or 1) nor Falsity (or 0) but intermediate truth values between 0 and 1. In the Boolean many-valued approach advocated in this book, we assign intermediate Boolean values to those sentences. This way, we can have not just two but many truth values at our disposal but retain classical logic as our base logic at the same time. As a preliminary to the philosophical discussions given in the subsequent chapters, Chap. 1 offers short but rigorous accounts of classical logic, many-element Boolean algebras, and the Boolean many-valued semantics of classical logic. It is maintained, among other things, that classical logic is not just a set of deduction rules but rather a set of rules that include also metadeduction rules; and that the notion of logical consequence to be employed in the Boolean many-valued semantics ought to be not that of truth preservation but that of so-called bottomline preservation.