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
NON-NUMERICAL ARITHMETIC
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.
The symbols 
and 
are the simplest expressions to which more complex expressions may be simplified since they are symbols representing the primary
values or states.
THEOREM 1.
Any conceivable expression formed by a finite number of iterations of the symbol
PROOF:
Given any finite expression, however complex, we may find the innermost or deepest region.
The innermost region is either contained within a distinction
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 | |
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| and | y | |
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| So that |  |
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by Ex. 3. |
| and |  |
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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., | |
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Ex. 5. |
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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
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
px will yield x and py
This may be demonstrated as follows:
Since the procedure leading up to the placement of the symbol xor 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
Further, if any sub-region or partition of E has the value
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.
EXAMPLE:
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
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x |
x .
Thus E
by the dominance of x.
Checking by simplification,
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| by Ex. 2 | ||
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| by Ex. 3. (three times) |
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| by Ex. 3. (twice) |
Thus the indicators of the two values of our calculus remain distinct when we simplify complex expressions of distinction. This justifies the hypothesis of simplification for the arithmetic based upon the two primary equations Ex. 2 and Ex. 3. Proof of the converse, i.e., that values of the calculus remain distinct with transformations in the direction of greater complexity, is straightforward. Since any expression may be traced uniquely to one simple expression, then the new expression resulting from each step or transformation must have the same value as the original expression and the simple expression to which it reduces. Since Ex. 2 and Ex. 3 are equivalences, the direction of the transformation is reversible. Therefore, the value of any expression constructed from a given simple expression by substitutions of the reverse transformation of Ex. 2 and 3 is distinct from the value of any expression constructed from a different simple expression.
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
,
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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
The logic table on the right is used in digital circuitry and is known as the logical x-or circuit. We shall see as we develop the calculus that the use of Ex. 2 and Ex. 3, and their consequences expressed in the calculus, are considerably less difficult and more straight forward (especially in algebraic calculations) than the equivalent procedures in standard symbolic logic. The reason for this is because the calculus of distinctions more closely represents the act of making distinctions in reality. Its form and logical processes are thus simpler than those of the more arbitrary symbols of classical logic.
