Boolean Thinking

Pure logic and mathematics began to come together in the mid to late 1800s in Europe. The publication in 1853 of a book titled An Investigation of the Laws of Thought, by the remarkable mathematician George Boole, established a foundation rapidly accepted and extended by others through the remainder of the century and beyond.[1]

Boole developed a kind of algebra in which the variables x, y, z… take on only the values zero and one. A curious equality, which he introduces early and to which he returns repeatedly in the book, is the following relation:

(1)                                      x = x2

The equation (1) is easily verified by substituting zero or one for x. Of course, (1) departs from our customary algebra of continuous quantities. Some, not all, of the usual algebraic rules were carefully transferred by Boole into his new system. Simple rearrangement of (1) gives

(2)                                     x (1-x) = 0

and setting either factor equal to zero, we return to

(3)                                     x = 0  or  x = 1

Boole recognized that such a variable x could stand for a logical proposition with a value of false (zero) or true (one). The dual statement (3) above then means “x is either false or true, but not both” and this can also be incorporated in a single quadratic equation such as (1) or (2). Boole named this the Law of Duality.

Letting various letters such as x, y, and z stand for propositions, Boole the master teacher summarized the process as follows:

…any system of propositions may be expressed by equations involving symbols x, y, z, which, whenever interpretation is possible, are subject to laws identical in form with the laws of a system of quantitative symbols, susceptible only of the values 0 and 1. But as the formal processes of reasoning depend only upon the laws of the symbols, and not upon the nature of their interpretation, we are permitted to treat the above symbols, x, y, z, as if they were quantitative symbols of the kind above described. We may in fact lay aside the logical interpretation of the symbols in the given equation; convert them into quantitative symbols, susceptible only to the values 0 and 1; perform upon them as such all the requisite processes of solution; and finally restore to them their logical interpretation. [Boole’s italics]

In a stunning synthesis of math, logic, and practical application, Boole then developed additional meanings for the variables x, y, z and the values 0 and 1. He identified x with the class of objects having a property signified by x. Thus x = 1 if an individual object belongs to the class, and x = 0 if an individual object does not belong to the class. Similarly y, z, or any other variable you might wish to name can identify objects according to their membership in a class. The value 1 serves a further purpose. He recognized 1 as standing for the universe of objects under consideration, the “class in which are found all the individuals that exist in any class.”[2] The value 0 correspondingly comprises the empty class or “Nothing.”

Here is the way Boole symbolically represents negation –

…let x represent the class men, and let us express, according to the last Proposition, the Universe by 1; now if from the conception of the Universe, as consisting of “men” and “not-men,” we exclude the conception of “men,” the resulting conception is that of the contrary class, “not-men.” Hence the class “not-men” will be represented by 1 − x. And, in general, whatever class of objects is represented by the symbol x, the contrary class will be expressed by 1 − x. [chapter III, p. 34]

In the new system, algebraic multiplication is transformed into successively more refined definitions of classes of objects. In his words,

…if x represents opaque substances, y polished substances, z stones, we shall have,

xyz = opaque polished stones;

xy(1 − z) = opaque polished substances which are not stones;

x(1 − y)(1 − z) = opaque substances which are not polished, and are not stones;

and so on for any other combination. [chapter IV, p. 39]

George_Boole_color
George Boole, circa 1860

We do not have space or time to develop Boole’s algebra step by step. His book is a marvel of clear writing, and I highly recommend it. I do think that summarizing an example which he gives in Chapter 8 can be helpful. Boole wants to show how his new algebra of logic can clarify an argument presented in Dr. Samuel Clarke’s book titled “Demonstration of the Being and Attributes of God.” There was no incongruity in this example, because Boole thought deeply about religion.[3]

In this particular train of reasoning, Clarke seeks to demonstrate that “Something has existed from eternity.” Boole distills Clarke’s written paragraph into a series of propositions, each of which may be found either true or false, as follows:

x =  Something is.
y =  Something always was.
z =  The things which now are have risen from nothing.
p =  It exists in the necessity of its own nature.
            (i.e., the something spoken of above)
q =  It exists by the will of another Being.

Among these propositions, y is the conclusion sought, or to be more specific, y = 1, which will be interpreted as “It is true that something always was.”

Clarke’s reasoning began with the following 5 premises, which I show first in words and then in Boolean symbols:

1st.  “Something is.” translates to     x = 1

2nd.  “If something is, either something always was, or the things that now are have risen out of nothing.”
translates to     x = v (1–x) + (1–y) ]

3rd.  “If something is, either it exists in the necessity of its own nature, or it exists by the will of another being.”
translates to    x = (1–q) + (1–p) ]

4th.  “If it exists in the necessity of its own nature, then something always was.”
translates to    p = vy

5th.  “If it exists by the will of another being, then the hypothesis, that the things which now are have risen out of nothing, is false.”
translates to   q = v (1–z)

Thus the premises have been transformed into these 5 equations:[4]

x = 1
x = v [ y (1–x) + x (1–y) ]
x = v [ p (1–q) + q (1–p) ]
p = vy
q = v (1–z)

Using the rules of his own algebra, Boole solves these equations in 12 steps, of which I show only (11) and (12) below:

(11)                 1 – y = 0, or,  y = 1.

(12)                 x = 0.

Finally, the interpretation of (11) is that it is true that something always was, and the interpretation of (12) is that it is false that the things which now are have risen from nothing.[5]

Boole’s new algebra could be used to solve either practical or philosophical problems. Its application to “the Being and Attributes of God” was, somewhat surprisingly, not the end or goal of the book. He went on to show how his new algebra could answer questions of probability, which we now recognize as belonging to the field of statistics. From the historic distribution of rainfall on the fields, Boolean algebra could be used to calculate the probability of a good harvest.

But I doubt that even George Boole would have predicted that his new algebra would underpin the binary logic of digital computers.

Think about your cell phone, listen to digitally recorded music, check out the spreadsheet data behind a chart of progress in your field, whatever it is. As you do, remember the words of George Boole…

…any system of propositions may be expressed by equations involving symbols x, y, z, which, whenever interpretation is possible, are subject to laws identical in form with the laws of a system of quantitative symbols, susceptible only of the values 0 and 1….

Beautiful!

 

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Images:  Long Hall, University College Cork, Cork, Ireland, by Bjørn Christian Tørrissen – http://bjornfree.com/galleries.html, CC BY-SA 3.0, from Wikimedia Commons. Color portrait of George Boole, Wikimedia Commons, public domain.

[1] Boole, George. An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. 1853, is available online through Project Gutenberg, accessed at http://www.gutenberg.org/ebooks/15114 on June 27, 2014.

[2] Ibid, chapter III, p. 34.

[3] George Boole was born to a working class family in Lincolnshire, England. He cared about and conversed with common folk, and he engaged repeatedly in efforts at social reform. See this link on the website of University College Cork.

[4] The astute reader may notice in the equations that form the premises an extra notation not previously defined, that is, v. This is a special symbol that Boole found necessary to incorporate the meaning of “some” or “sometimes”as in “either none, some, or all,” or “either never, sometimes, or always.”

[5] Perhaps your own assessment will not judge kindly the conclusions brought forward by Clarke and Boole. If that is the case, don’t blame the logic. Instead think hard about the 5 premises with which Clarke began his deductive process. Today one person or another might choose different premises. A positivist, for example, might begin with the premise “What is true is only that on which rational observers can agree.” A pragmatist might hold the premise “Unless it makes a difference in somebody’s disposition to act, then it makes no difference.” Elsewhere we examined the premises of personal viewpoint – “Every sentence is first person” and “The overarching viewpoint is not allowed.” Boolean analysis has this virtue – It brings clarity to the deductive consequences of our first principles. It may tell us, then, that the 4 premises I have just listed produce a contradiction, that 3 of them may be sufficient to negate the 4th. What do you think?

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