Finite-valued logic

In logic, a finite-valued logic (also finitely many-valued logic) is a propositional calculus in which truth values are discrete. Traditionally, in Aristotle's logic, the bivalent logic, also known as binary logic was the norm, as the law of the excluded middle precluded more than two possible values (i.e., "true" and "false") for any proposition. Modern three-valued logic (ternary logic) allows for an additional possible truth value (i.e. "undecided"). The term finitely many-valued logic is typically used to describe many-valued logic having three or more, but not infinite, truth values. The term finite-valued logic encompasses both finitely many-valued logic and bivalent logic. Fuzzy logics, which allow for degrees of values between "true" and "false"), are typically not considered forms of finite-valued logic. However, finite-valued logic can be applied in Boolean-valued modeling, description logics, and defuzzification of fuzzy logic. A finite-valued logic is decidable (sure to determine outcomes of the logic when it is applied to propositions) if and only if it has a computational semantics. Aristotle's collected works regarding logic, known as the Organon, describe bivalent logic primarily, though Aristotle's views may have allowed for propositions that are not actually true or false. The Organon influenced philosophers and mathematicians throughout the Enlightenment. George Boole developed an algebraic structure and an algorithmic probability theory based on bivalent logic in the 19th century. Jan Łukasiewicz developed a system of three-valued logic in 1920. Emil Leon Post introduced further truth degrees in 1921. Stephen Cole Kleene and Ulrich Blau expanded the three-valued logic system of Łukasiewicz, for computer applications and for natural language analyses, respectively. Nuel Belnap and J. Michael Dunn developed a four-valued logic for computer applications in 1977. Since the mid-1970s, various procedures for providing arbitrary finite-valued logics have been developed.