Wednesday, August 25, 2010

George Boole and The Laws of Thought (I)

Left: George Boole: Author of the real 'Laws of Thought'

Much drivel has been written by uneducated twits about what they believe are “the laws of thought” but in this blog and the next we examine the basis of the real ones – which have formed much of the foundation of today’s digital technology-computer revolution as well as propositional logic.

George Boole (after whom Boolean Algebra is named) is the author of the genuine Laws of Thought, by which he sought to develop a symbolic system to embody logic- ultimately leading to the realization that logical and mathematical operations are, to a certain degree, interchangeable.

One caveat before we begin: Boole, even in his day, recognized his laws and symbolic logic would only be as good as the quality of the inputs. Like the financial language known as Intex (previous blog) if the wrong facts and assumptions are used, the output can only be garbage. In the case of the Laws of Thought, if incorrect facts are assumed, then false premises are created, resulting in false conclusions.

Boole, in his Chapter II of the Laws of Thought, notes:

“The elements of which all language consists are signs and symbols. Words are signs. Sometimes they are said to represent things, sometimes the operations by which the mind combines together the simple notions of things into complex conceptions…..But words, although in this and other ways they fulfill the office of signs or representative symbols, are not the only signs which we are capable of employing.”

Boole then goes on to note the existence of mathematical signs, such as X, - , + , / and so on. In this way he justifies an entire system that integrates the signs of mathematics with those of words. His critical basis is (ibid.):

It is necessary that each sign should possess, within the limits of the same discourse or process of reasoning, a fixed interpretation. The necessity of this condition is obvious and founded in the very nature of the subject”

Boole then goes on to articulate propositions by which the analysis and classification of assorted signs (words or mathematical symbols) are advanced.

For Proposition (I) we have:

“All the operations of Language, as an instrument of reasoning, may be conducted by a system of signs composed of the following elements, viz.

1st: Literal symbols as x, y etc.- representing things as subjects of our conception

2nd. Signs of operation such as +, - , x, standing for those operations of the mind by which the conceptions of things are combined or resolved as to form new conceptions involving the same elements.

3rd. The sign of identity (=).

And these symbols of Logic are, in their use, subject to definite laws, partly agreeing with and partly differing with the corresponding symbols in the science of Algebra

Clearly, from this, Boole expected anyone who would deign to use the Laws of Thought he compiled to be skilled in algebra. If one had not taken an algebra course or been exposed to it via self-education it is doubtful he’d be able to use the laws effectively, or reason rigorously. We find this, in fact, with many fundies – who are often as deficient in numerous areas of mathematics (algebra, geometry, trig, etc.) as they are in science. Boole implied in his precepts that such people would only have very crude conceptions since their lack of symbolic background wouldn’t allow them to refine what their thoughts held. (No surprise when we see the typical fundie has such limited conceptions of his god – almost like a cartoon)

Boole then shows how the algebraic –word and thought linkage occurs, e.g.

“Let it be agreed that by the combination xy – that class of things shall be represented by which the properties are simultaneously applicable. Thus, if ‘x’ alone stands for ‘white things’ and ‘y’ for sheep, then let ‘xy’ stand for ‘white sheep’. In like manner, if ‘z’ stands for ‘horned things’ then let ‘xyz’ represent ‘horned white sheep’.

Boole then considers the laws to which such symbolic combinations are subject.

First, it is evident that the order by which two symbols are written is indifferent. The expressions xy and yx equally represent that class of things to the several members of which the names or descriptions x and y are together applicable. Hence we have:

xy = yx

There may be a difference as to the order in which the conception is formed but there is none as to the individual things which are comprehended under it”

Boole is also emphatic about qualities and discerning them:

When I say ‘x’ represents ‘good’ it will be understood that it only represents ‘good’- as when a subject for that quality is supplied by another symbol, and that used alone – it will be interpreted as “good things”.

This quality property shows that dissonant qualities cannot be combined into anything acceptable in the Laws of Thought.

For example, let X represent ‘ALL Good’

Meaning the property or attribute is an ultimate one. (One could even think of ‘infinitely GOOD’) . Then if one denotes G by “God” we can say:

‘XG’ denotes an All GOOD God .

Now, let Y represent any and all evil, including anything which can be interpreted as ‘evil’ -including all things destructive. Any event, entity, abode or thing destructive cannot have associated with it the quality of ‘Good’.

Consider now the conception of ‘Hell’ – call it ‘H' and since it is destructive as conceived by certain people, then one can only frame it as ‘YH’, never as XH.

What we immediately discover here, using the Boolean laws of thought, is a condition Boole described via:

~ (G / H)

E.g. Two statements G and H, are contrary if they cannot both hold, i.e. if: ~ (XG/ YH)

Note that this is not the same as being contradictory, which implies:

two statements p, q are contradictory, if: [p -> (q) ] , [ (~p) -> ~q]

However, if many further attributes (qualities) are assigned to the preceding conceptions then it may also be possible to ascertain contradiction as well. (This would occur, as I showed in a previous blog, if one designates BOTH Hell and God as "infinite". Since infinite means all encompassing, then there cannot be two infinities- only ONE. Even more rigidly, there cannot be an infinity and anything else! (Since whatever might be "left over" would make the infinity finite- since something occupies being along with it- and hence delimits its extent.) Either there must be G, or H. Hence: [ (~G) -> ~H] , denoting a contradiction between the concepts. I did leave room to resolve this, however, by reducing G from an "infinite" to a "finite" entity - which would then "make room" for H to co-exist with it. Else it is logically impossible.)

In other words, the laws of thought reveal that an entity (GOD) designated as ‘All GOOD’ AND "infinite" is contradictory to YH or a “Hell”. Thus, if YH IS valid, then XG cannot mean or signify “All Good, infinite God” but rather YG = “Evil God”. Since then YG and YH become compatible and consistent concepts.

Such are the ways in which one can use the laws of thought to expose contradictions in words used, even when the users themselves are oblivious to their own violations of thought.

1 comment:

Parker Woody said...

We are learning about George Boole for a Civilizations class that I am currently taking. Amazing how much computer science has relied on his work. Thanks for the information.