Aristotle’s Prior Analytics and Boole’s Laws of Thought: The Birth of Information Age

Symbolic logic

In 1847 George Boole published the pamphlet Mathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system, and wanted An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities to be seen as the mature statement of his views. This second book of Boole's contained Boolean algebra. Boolean logic is credited with laying the foundations for the information age. Boole maintained that:

"No general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form."

Boole was working on the main principles of Aristotle's logic and contrary to widespread belief, Boole never intended to criticise or disagree with Aristotle's logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability.

Aristotle's logic model

Aristotle’s Prior Analytics and Boole’s Laws of Thought

A 2003 article of John Corcoran: Aristotle's Prior Analytics and Boole's Laws of Thought provided a systematic comparison and critical evaluation of Aristotelian logic and Boolean logic; it also reveals the centrality of wholistic reference in Boole's philosophy of logic. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole did not dispute one point that Aristotle made, but he did “go under, over, and beyond” Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat—to assessing validity he added solving equations, and 3) expanding the range of applications it could handle—e.g. from propositions having only two terms to those having arbitrarily many.

More specifically, Boole agreed with what Aristotle said; Boole's ‘disagreements’, if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced Aristotle's four propositional forms of to one form, that of equations—by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the “perfect syllogisms”) must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce “No quadrangle that is a square is a rectangle that is a rhombus” from “No square that is a quadrangle is a rhombus that is a rectangle” or from “No rhombus that is a rectangle is a square that is a quadrangle”.

His collaboration with Alfred Tarski in the late 1970s and early 1980s led to publications on Tarski's work and to the 2007  article Notes on the Founding of Logics and Metalogic: Aristotle, Boole, and Tarski, which traces Aristotelian and Boolean ideas in Tarski's work and which confirms Tarski's status as a founding figure in logic on a par with Aristotle and Boole.

Claude Shannon: The Father of Information Theory

In 1932, Claude Shannon entered the University of Michigan, where he was introduced to the work of George Boole. He graduated in 1936 with two bachelor's degrees: one in electrical engineering and the other in mathematics.

In 1936, Shannon began his graduate studies in electrical engineering at MIT, where he worked on Vannevar Bush's differential analyzer, an early analog computer. While studying the complicated ad hoc circuits of this analyzer, Shannon designed switching circuits based on Boole's concepts. In 1937, he wrote his master's degree thesis, A Symbolic Analysis of Relay and Switching Circuits. A paper from this thesis was published in 1938. In this work, Shannon proved that his switching circuits could be used to simplify the arrangement of the electromechanical relays that were used then in telephone call routing switches. Next, he expanded this concept, proving that these circuits could solve all problems that Boolean algebra could solve. In the last chapter, he presented diagrams of several circuits, including a 4-bit full adder.

Using this property of electrical switches to implement logic is the fundamental concept that underlies all electronic digital computers. Shannon's work became the foundation of digital circuit design, as it became widely known in the electrical engineering community during and after World War II. The theoretical rigor of Shannon's work superseded the ad hoc methods that had prevailed previously. Howard Gardner called Shannon's thesis "possibly the most important, and also the most noted, master's thesis of the century."