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.

ALSO ON NECESSARY FACTS

Knowledge Maps

Visualizing Complex Data

Gregory M. Browne's Necessary Factual Truth

Understanding Objectivism by Peikoff and Berliner

Browne on Kuhn

Gregory M. Browne is the author of Necessary Factual Truth (University Press of America, 2001), the inspiration for the title of this blog.  He offers these comments on my recent reviews here and here of Thomas Kuhn’s Structure of Scientific Revolutions.

Kuhn makes some good points, most of which are a reaction to the philosophy of science of the Logical Positivists.  But both the Logical Positivists and Kuhn share faulty assumptions that lead them to misunderstand what I call “Deep Kinds”. 

In short, they are unaware of the existence of Deep Kinds, or at least are unaware that some kinds have deep essences—that is, deep sets of necessary attributes, more than what one may be aware of any given time, more than what one learns when one learns the meaning of the term referring to them, more than what one may put in a short definition or concept.  Such are most kinds in chemistry and biology, such as gold and horses.   These have attributes that it took as much investigation to discover—the atomic structure of gold and the DNA of horses—which nonetheless were as necessary to the kinds as the known attributes we had put in our dictionary definitions.  Aristotle understood this, but this was forgotten in the reaction to Scholasticism and not rediscovered, by most philosophers, until the work of Hilary Putnam and Saul Kripke in the 1970s.  Ayn Rand realized this truth independently, and, through Leonard Peikoff, influenced me to elaborate the idea and gives this type of kinds a name.

In the interlude, the doctrine that only defining attributes are necessary, and that definitions are just the short “Nominal Definitions” that we find in dictionaries, whose function is merely to express meaning and not summarize knowledge, prevailed by the early 20^th century reign of the Logical Positivists.  So “All triangles have 3 sides” was considered necessary (but non-factual, because it expressed a conventional definition), but “Gold has 79 protons” was not considered necessary (though admittedly factual).

But by the 1930s it was seen that this simple model was inadequate, even by many disciples of the Logical Positivists.   The Ordinary Language philosophers came up with “cluster concepts” and “family resemblance concepts”, and in general “open textured” concepts, W. V. Quine came up with a holistic view of knowledge, and others extended this to a holistic theory of meaning.  Meaning and knowledge came to be widely seen as relative to a conceptual scheme.  And Kuhn concluded that scientists with different conceptual schemes or different “paradigms” could sometimes not communicate, because what was said in one scheme or paradigm was incommensurable with what was said another, and that they were not talking about the same things, because the meanings and references of their terms differed.

However, the later Putnam and Kripke rejected this, as he said that the scientists in different paradigms were talking about the same things, that their terms had the same referents, the “natural kinds” (Deep Kinds) even though their definitions might differ.  So scientists from ancient Greece on down who talked of gold were talking of the same kind of thing, the same natural kind with the same essence.

 

And Rand avoided the mistakes too, since she did not consider that what was excluded from the definition was not necessary, and insisted that new discoveries need not invalidate old theories but rather can add to our knowledge.

Nonetheless, Kuhn deserves credit for exposing some of the weaknesses in Logical Positivist philosophy of science.

Dr. Browne and I met at Eastern Michigan University.  I was walking the halls waiting for a criminology class to begin and I saw “Ayn Rand” written on the blackboard at the end of a couple of columns of philosophers starting with Thales.  Over the semester, I stood outside and eventually took a seat to watch the lectures.  He now teaches online for Yorktown University in Denver.