Thursday, August 26, 2010

George Boole and the Laws of Thought (II)

We continue now examining the basis of the Boolean Laws of Thought. Boole, continuing in his Chapter II, now makes some extraordinary associations and conclusions which are important to follow. This section again deals with the use of letter symbols (e.g. x, y, z etc.) and their meaning. As he notes the following is a “law of thought and not a law of things”:

He writes:

“We are permitted, therefore, to employ the symbols x, y, z etc. in the place of the substantives, adjectives and descriptive phrases subject to the rule of interpretation, that any expression in which several of these symbols are written together shall represent all the objects or individuals to which their several meanings are together applicable, and to the law that the order in which the symbols succeed each other is indifferent”

In the previous blog we saw the symbolic example (given x = white and y = sheep) that:

xy denotes white sheep, but also yx denotes white sheep, since the order of the symbol doesn’t alter the meaning of the conception (combination). Thus: xy = yx

Boole is careful to emphasize that while this illustrates the commutative law of algebra, it is not in the sense of actual algebraic multiplication but rather a process of “logical combination” by which xy means a definite conception.

More abstractly, he allows that (since the logic is commutative for the conception) we need not use either xy or yx but simply ‘x’.

“As the combination of two literal symbols xy expresses the whole of that class of objects to which the names or qualities represented by x and y are together applicable, it follows that the two symbols have exactly the same signification, their combination expresses no more than either of the symbols taken alone would do. In such case we should therefore have xy = x

As y, however, is supposed to have the same meaning as x we may replace it in the above equation with x and we thus get:

xx = x

Now, in common algebra the combination xx is more briefly represented by x^2. Let us adopt the same mode of notation here; for the mode of expressing a particular succession of mental operations is a thing in itself quite as arbitrary as the mode of expressing a single idea or operation. In accordance with this then, the above equation assumes the form:

x^2 = x”

Again, I have to interject here that we are not seeing any violations of known mathematics, but rather the outcome for a logical process of combination. Boole is simply compacting and simplifying notation for logical combinatorial operations, much like the inventors of tensor calculus did with their notation.

So what is Boole getting at? Is he really and truly saying we can let ‘x’ denote ‘white sheep’ or ‘good people’ instead of writing xy?

Yes, and he confirms it:

The reader must bear in mind that although the symbols x and y in the examples previously formed received significations different from each other, nothing prevents us from attributing to them precisely the same signification. It is evident that the more nearly their actual significations approach to each other, the more nearly does the class of things denoted by the combination xy approach to identity with the class denoted by x, as well as with the class denoted by y. The case supposed in the demonstration of the equation: xx= x is that of absolute identity of meaning”

Boole, obviously not sure if his readers have grasped this, goes on to clarify using examples:

“The law which it (equation) expresses is practically exemplified in language. To say ‘good, good’ in relation to any subject, though a cumbrous and useless pleonasm, is the same as to say ‘good’. Thus, ‘good good men’ is equivalent to ‘good men’”

Applying this to the example at the end of Part I, we saw that we denoted an “all Good God” by XG and an “all evil Hell” by YH.

Using Boole’s compactified notation we can write either G or H to mean the same thing, with no loss of generality. Thus: XG = GX and we may let X=G so GG = G. Another way of stating or saying this is that the very word “GOD” elicits GOOD – if used in terms of the prior conception, so no adjectival X is required. When one uses the word “GOD” then ALL Good is understood.

In a similar fashion, we can write: YH = HY and we may let (based on prior conceptions) Y = H therefore: HH = H, or Hell is an ‘All Evil’ state of Being or abode,

We will further process this in terms of Boole’s next section on numerative classes.

Boole’s next law is prefaced:

“Signs of those mental operations where we collect parts into a whole, or separate a whole into parts”

He illustrates this using the example of letting x = women and y = men

So all (adult) people living on Earth is:

x + y = y + x

In this case, the signification is not only by quality (sex) but numerical as well, since the total of men plus women (x + y) must equal the total of adults living on Earth. Less generalized operations can also apply, for example – let ‘z’ denote European, then:

z(x + y) = zx + zy

and we have the juxtaposition of two literal symbols to represent their algebraic product.

Now, take the earlier symbols we used for God (G) and Hell (H).

Can we manipulate them in a similar fashion to the preceding? This allows an excellent illustration of the power of the laws of thought in action.

First, we identify – for the sake of clarity, G with ‘B’ or 'Being'.

Then does: G + H = B?

Only if both G and H are finite, and G is not 'all good'. But if G= oo, and G = XG = GG

G + 0 = G, or G + (H+ (-H)) = G

(since oo +1 = oo)

Hell is negated by the presence of XG, the "All Good" or an infinity GOOD- which eliminates any finite evil.

In other words, only GOD exists! What has happened? How can this possibly be?

A clue is provided in Boole’s own discussion. Take two finite groups of men, x and y, where x denotes the main class (‘men’) and y denotes “Asiatic men”. Then when one writes “All men except Asiatics” it implies: x – y

But bear in mind this is for finite classes. However, once one assigns G= oo, one no longer has a finite class for God as Being. One has exhausted ALL being. (The very definition of infinite).

Thus, one cannot write: “All Being except Hell” or B – H

The only option is: GG = G implying H = 0

Once more we see that the conceptions G and H are mutually exclusive if G = oo. One can incorporate both only if G and H represent finite classes, then:

B = G + H or B – G = H

Now, a side problem for readers: Let X denote “GOOD” and apply it to the above such that the operation is distributive, i.e.: z(x+y) = zx + zy. How do you interpret this?

Moving on, we go to Boole’s next class of law under the header:

Signs by which relation is expressed, and by which we form propositions

And the law follows via axioms:

"1st. If equal things are added to equal things, the whole are equal.
2nd. If equal things are taken from equal things, the remainders are equal.

Hence, we may add or subtract equations, and employ the rule of transposition as in common algebra.”

Let’s apply this to the previous example: B = G + H or, using transposition: B – G = H

How do the significations hold up against the new axioms for Boole’s law of thought?

In Boole’s formulation (Proposition IV), “nothing” and “Being” are the two limits of class extension. He allows that numerical values can be applied here: 0 for nothing, and 1 for Being.

His statement of Prop IV is: “That axiom of metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for any being to possess a quality, and at the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x^2 = x”

Or as we have seen before: GG = G.

Now, Boole rewrites his equation for the fundamental law of thought (using transposition) in the form: x^2 – x = 0 or x (1 – x) = 0

Where x denotes ‘being’ and (1- x) ‘not being’.

But if we take: GG = G and rewrite: G^2 – G = 0 or G(1-G) = 0

Then we have G denotes “God” and (1- G) denotes “not God’ or presumably “Hell”.

But what if G= oo?

Then: G^2 = (oo)(oo)

But then: G^2 – G = oo(1 – oo) = oo = G

In other words, (1 – G) doesn’t apply or factor in. This conforms, as Boole notes, to Aristotle’s “fundamental axiom of all philosophy”:

It is impossible that the same quality can belong and not belong to the same thing…this is the most certain of all principles.”

In other words, we have shown – again – using the Boolean fundamental law of thought, that an infinite God cannot coexist with any type of Hell. It would mean the same quality (evil) would have to apply to the same thing, or to put it another way, Hell would have to be part of the Divine being.

In a future blog we will consider the application of Boole’s laws to logic gates, as used in digital components.