elements. Assigning Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra because the universal quantifier maps to the infimum operation, and the existential quantifier maps to the supremum; this is called a Boolean-valued model. All finite Boolean algebras are complete. Suszko's thesis In order to justify his claim that true and false are the only logical values, Roman Suszko (1977) observes that every structural Tarskian many-valued propositional logic can be provided with a bivalent semantics. Criticisms Future contingents A famous example is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9: Imagine P refers to the statement "There will be a sea battle tomorrow." The principle of bivalence here asserts: Either it is true that there will be a sea battle tomorrow, or it is false that there will be a sea battle tomorrow. Aristotle denies to embrace bivalence for such future contingents; Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both the philosophy of time and the philosophy of logic. One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet-undetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer; see Łukasiewicz logic. Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be." (Which is true if "tomorrow" eventually occurs.) Vagueness Such puzzles as the Sorites paradox and the related continuum fallacy have raised doubt as to the applicability of classical logic and the principle of bivalence to concepts that may be vague in their application. Fuzzy logic and some other multi-valued logics have been proposed as alternatives that handle vague concepts better. Truth (and falsity) in fuzzy logic, for example, comes in varying degrees. Consider the following statement in the circumstance of sorting apples on a moving belt: This apple is red. Upon observation, the apple is an undetermined color between yellow and red, or it is mottled both colors. Thus the color falls into neither category " red " nor " yellow ", but these are the only categories available to us as