Sunday, March 30, 2014

George Boole's Laws of Thought

Everyone who knows computing knows the Boolean operators AND, NAND, OR, XOR, and NOT.  They allow us to construct conditional statements, and programmatic branching.  Electrical engineers learn the rules of Boolean Algebra with two values OFF and ON; and with three operators,  PARALLEL, SERIES, and a simple switch for NOT.   0 and 1 are OFF and ON. The three operators  of Boolean algebra are + X and ~ corresponding to OR, AND, and NOT. They are the same as PARALLEL, SERIES, and SWITCH.  George Boole had a different view entirely.  His 1853 book, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, is a treatise on epistemology.  

We learn that correct thinking by logic is a reduction by syllogism.  All men are mortal. Socrates is a man. Therefore, Socrates is mortal.  If A =B and B = C, then A = C. The middle or the mean term is factored out. Boole granted that syllogism is one kind of valid method. However, rather than reducing statements, Boole argued by expansion for a deeper and extended understanding of seemingly simple truths.  His algebra is deceptively similar to – but different from – the forms we learned in high school; and so the learner here must pay close attention to these laws of thought. 

If x is “red” then xx means “red red” which is nonetheless still “red.”  If the essential attribute under consideration is “red” then red red things with their redness removed do not exist: xx – x = 0. Taking from Leibniz, Boole lets 0 be non-existence and 1 be the universe.   If x is our subject, then (1-x) is everything else in the universe except x. Boole rewrites xx-x as x(1-x) = 0 to give Aristotle’s Law of Contradiction: A thing cannot both have and not have the same attribute in the same way at the same time.  Boole calls this “duality.”  Also following Leibniz, Ayn Rand delivered that as  “A or non-A.”  

In 400 pages, Boole carefully applies the essential truths he discovered. He extracts more meaning – and more exacting meanings – from common statements.  However, his insistence on closely laying every step of this long journey, couched, as it was, and, indeed, of necessity, both historical and personal, must have been, in Victorian prose, renders this important work so Germanic as to be nearly unreadable.  It is well worth the effort.

“Wealth consists of things transferable, limited in supply, and either productive of pleasure or preventive of pain.” 
w = wealth
s = limited in supply
t = transferable
p = productive of pleasure
r = preventive of pain

Boole specially defines 0/0 to mean the undefined.  He includes this to stand for the indefinite class: those things not included in the present statement but which exist nonetheless.

Given that this book is 400 pages and Ayn Rand’s Introduction to the Objectivist Epistemology is 164, Boole boldly goes beyond the orbit of concept formation.  He nonetheless is in the same material space: he identifies reality; and ties concepts to objects. Boole explores the relationships between the processes of thought and perception and the expression of them through language. 

Boole even applies his method to statements about God.  His goal is not to address God’s existence but to expand and analyze statements about God asserted by Baruch (“Benedict”) Spinoza and Samuel Clarke.  (“… considered the major figure in British philosophy between John Locke and George Berkeley.” – Wikipedia here.)  From there, he examines Aristotle’s logic and Fermat’s probabilities.   The last third of the book is about the logic of probability theory.


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