THE CALCULUS OF DISTINCTIONS
TABLE OF CONTENTS

Calculus of Distinctions
Non-Numerical Arithmetic
The Algebra of Distinction
Types of Distinctions
Logical Interpretations of the Calculus

Taking Ex. 2 as the form of repetition (or condensation and expansion) and Ex. 3 as order (or creation and cancellation), we may, through formal consideration of patterns arising from these primary expressions, develop a non-numerical arithmetic. Once identified and verified by rigorous proof, these patterns may be called theorems.

THEOREM 1.

Any conceivable expression formed by a finite number of iterations of the    symbol  can be simplified to one of the two primary values or states.

 

PROOF:

Given any finite expression, however complex, we may find the innermost or deepest region.

The innermost region is either contained within a distinction ,  or it is not. If it is not, then the expression is already simple. If it is contained, then it is either void, or it will, with the containing distinction, reduce to the void state by Ex. 3. We may continue this procedure until, after a finite number of applications of Ex. 3 we either arrive at a simple form or a form which will exhibit the form of repetition (Ex. 2), which may be reduced to  or the form of cancellation (Ex. 3), which reduces to  .  Therefore, in any case, the original finite expression may be reduced to either  or  .

 

While the value to which a given expression simplifies may be unique, the sequence of calculations by which we obtain it clearly is not. So the question arises whether or not the simplification of an expression by one sequence of steps will always result in the same simple value as that of any other sequence of steps that reduces the expression to a simple value. While Theorem 1 establishes the universality of the simplification of finite expressions, it does not guarantee uniqueness of the result of simplification.

It is obvious that

Ex. 4.

Since is void and may be dropped without loss of meaning,  may be called the dominant value. There are only two values in our arithmetic; they are  and  . It is clear that there are some expressions that may be simplified to one unique value. Let x stand for any expression that simplifies uniquely to the value  by some finite sequence of steps, and let y stand for any expression that is uniquely reduced to   by some finite sequence of steps. Then xy  x  by substitution in Ex. 4.

By definition	x
and	y
So that				by Ex. 3.
and

We have defined  and   as the only simple expressions. Complex expressions may be built up by utilizing the transformations of Ex. 2 and 3 by the principle of equivalence,

i.e.,				Ex. 5.
and				Ex. 6.

A complex expression may define distinct regions exhibiting iteration and order or depth. In terms of the universe defined by the expression, each iterative part of the expression may be said to represent a sub-region exhibiting order or depth.

For example, the expression below exhibits two primary distinctions, each with a depth of three distinctions representing two primary regions, each with two sub-regions. The innermost region of the left-hand primary distinction is divided into two partitions. Note that the partitions occur at the same depth.

Figure 1

A complex expression may exhibit any number of distinctions with any number of sub-regions and/or partitions. A partition may also have sub-regions and/or sub-partitions within it.

THEOREM 2.

Any given finite expression may be simplified to one and only one simple expression. That is, the result of the simplification of any finite expression is unique.

PROOF OF THEOREM 2. (Proof due to G. Spencer Brown)

Let  E be any complex expression. First, place the symbol x outside of each innermost distinction in  E. This will not change the value of  E since

 

x and

			x

Next, place either x or y outside the next distinction, depending upon whether the region encompassed by the next distinction contains a distinction or is void:  If it contains the value , place a y outside. If it is void, place an x outside. As demonstrated above, this procedure will not change the value of  E. Continue this procedure until you reach the outermost distinction of all partitions. The symbol, x or y, placed outside the last distinction of a partition will indicate the value of the partition since none of the steps, up to and including placing the final symbol, can change the value of the partition and evaluation of the new expression,

px will yield x and py  y.

This may be demonstrated as follows:

Since the procedure leading up to the placement of the symbol x or y outside the outermost distinction cannot change the value of the expression, then if  p represents the first sub-region, and if the outermost symbol is x, the only value of  p that will remain unchanged in  px   is  p .

Conversely, if the value of  p is , then the outermost symbol cannot be x, since the value of  px which is a change from the value of  p.

Further, if any sub-region or partition of  E has the value , i.e., the outermost symbol is x, then the value of  E is . Therefore, only if all the sub-regions and partitions in  E have the value (outermost placed symbol y) can the value of  E be .

We have developed a procedure by which any sub-region or partition of any expression,   E, may be evaluated without changing the value of  E. Since the values of the sub-regions and partitions uniquely determine the value of  E by Ex. 2, and  E may be any expression, we have proved Theorem 2.

Let  E be the expression

.

1.) Place x outside the deepest distinctions in  E:

.

2.) Place x or y outside the next deepest distinctions as follows:

a.) x if the distinction is empty (i.e., void), and

b.) y if the distinction defines a region

containing a distinction.

Continuing:

and

x

and  x .  Thus,  E by the dominance of  x.

Checking by simplification,

		by Ex. 2
		by Ex. 3. (three times)
		by Ex. 3. (twice)

Because of the consistency of our calculus demonstrated in the proof of Theorem 2 and its converse, we conclude that the values   and  , along with any and all expressions developed from them, will obey the basic rules of symbolic logic. The truth of this statement may be demonstrated by the development of standard logic tables from the primary expressions of the calculus of distinctions.

Using the definitions of the primary expressions, the forms of Ex. 2 and 3,

	and ,

and applying the rule of dominance to map all possible combinations of and .

we get

If we replace with the logical truth value of true and  with false, indicated by T and F, and relate them to standard logical syntax, we obtain the standard logic tables

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