Explore the foundations of Boolean algebra and its distinction from the Algebra of Boole in this 30-minute mathematics lecture. Delve into the bifield of two elements {0,1} and its role in building the algebra of n-tuples. Discover how this abstract development influenced circuit and logic gate analysis. Examine the critical differences between Boolean algebra and the Algebra of Boole, and understand their respective relationships to set arithmetic and logic. Investigate the historical context of using mathematics to refine logic, and consider its implications for 20th-century mathematical developments. Learn about Boole's approach to syllogisms, the Square of Opposition, and Boole's proofs of Aristotle's syllogisms, including specific examples like Celarent and Camestres.